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Among other merits, Transformers are witnessed as capable of learning long-range dependen- +cies and spatial correlations, which is a clear advantage over convolutional neural networks (CNNs), which have been the de +facto standard in Computer Vision problems so far. Thus, Transformers have become an integral part of modern medical image +analysis. In this review, we provide an encyclopedic review of the applications of Transformers in medical imaging. Specifi- +cally, we present a systematic and thorough review of relevant recent Transformer literature for different medical image analysis +tasks, including classification, segmentation, detection, registration, synthesis, and clinical report generation. For each of these +applications, we investigate the novelty, strengths and weaknesses of the different proposed strategies and develop taxonomies +highlighting key properties and contributions. Further, if applicable, we outline current benchmarks on different datasets. Finally, +we summarize key challenges and discuss different future research directions. In addition, we have provided cited papers with +their corresponding implementations in https://github.com/mindflow-institue/Awesome-Transformer. +Keywords: Transformers, Medical Image Analysis, Vision Transformers, Deep Neural Networks +1. Introduction +Convolutional neural networks (CNNs) have been an inte- +gral part of research in the field of medical image analysis for +many years. By virtue of convolutional filters whose primary +function is to learn and extract necessary features from medi- +cal images, a wealth of research has been dedicated to CNNs +ranging from tumor detection and classification [1], detection +of skin lesions [2, 3, 4] to segmentation of intervertebral discs +[5, 6], brain tumor segmentation [7, 8], to name only a few. +CNNs have also contributed significantly to the analysis of dif- +ferent imaging modalities in clinical medicine, including X-ray +radiography, computed tomography (CT), magnetic resonance +imaging (MRI), ultrasound (US), and digital pathology. De- +spite their outstanding performance, CNNs suffer from concep- +tual limitations and are innately unable to model explicit long- +distance dependencies due to the limited receptive field of con- +volution kernels. Moreover, the convolutional operator suffers +from the fact that at inference time, it applies fixed weights re- +gardless of any changes to the visual input. To mitigate the +aforementioned problems, there have been great research ef- +forts to integrate attention mechanisms, which can be regarded +as a dynamic weight adjustment process based on input fea- +tures to the seminal CNN-based structures to improve the non- +local modeling capability [9, 10, 11]. To this end, Wang et al. +[12] designed a non-local flexible building block, which can be +plugged into multiple intermediate convolution layers. SENet +[13] suggested a channel attention squeeze-and-excitation (SE) +block, which collects global information in order to recalibrate +each channel accordingly, in order to create a more robust rep- +resentation [14]. Inspired by this line of research, there has +been an overwhelming influx of models with attention variants +proposed in the medical imaging field [15, 16, 17, 18, 19]. Al- +though these attention mechanisms allow the modeling of full +image contextual information, as the computational complex- +ity of these approaches typically grows quadratically with re- +spect to spatial size, they imply an intensive computational bur- +den, thus making them inefficient in the case of medical im- +ages that are dense in pixel resolution [20]. Moreover, despite +the fact that the combination of the attention mechanism with +the convolutional operation leads to systematic performance +gains, these models inevitably suffer from constraints in learn- +ing long-range interactions. Transformers [21] have demon- +strated exemplary performance on a broad range of natural lan- +guage processing (NLP) tasks, e.g., machine translation, text +†Corresponding author: Dorit Merhof, Tel.: +49 (941) 943-68509, E-mail: dorit.merhof@ur.de. +January 10, 2023 +arXiv:2301.03505v1 [cs.CV] 9 Jan 2023 + + Transformers + Classification + Pure Transformers + Hybrid Models + Segmentation + Pure Transformers + Hybrid Models + Other Architectures + Reconstruction + Low Dose Enhancement + Sparse-View Reconstruction + Undersampled Reconstruction + Super Resolution Reconstruction + Report Generation + Reinforcement Learning-based Systems + Graph-based Systems + Memory-based Systems + Other Systems + Registration + Deformable Registration + Affine Registration + Rigid Registration + Detection + Backbone + Neck + Head + Synthesizing + Intra-Modality + Inter-Modality +Figure 1: Overview of the applications covered in this review. +classification, and question answering. Inspired by the emi- +nent success of Transformer architectures in the field of NLP, +they have become a widely applied technique in modern Com- +puter Vision (CV) models. Since the establishment of Vision- +Transformers (ViTs) [22], Transformers proved to be valid al- +ternatives to CNNs in diverse tasks ranging from image recog- +nition [22], object detection [23], image segmentation [24] to +video understanding [25] and image super-resolution [26]. As +a central piece of the Transformer, the self-attention mecha- +nism comes with the ability to model relationships between el- +ements of a sequence, thereby learning long-range interactions. +Moreover, Transformers allow for large-scale pre-training for +specific downstream tasks and applications and are capable of +dealing with variable-length inputs. The immense interest in +Transformers has also spurred research into medical imaging +applications (see Figure 1). Being dominant in reputable top- +tier medical imaging conferences and journals, it is extremely +challenging for researchers and practitioners to keep up with the +rate of innovation. The rapid adoption of Transformers in the +medical imaging field necessitates a comprehensive summary +and outlook, which is the main scope of this review. Specif- +ically, this review provides a holistic overview of the Trans- +former models developed for medical imaging and image anal- +ysis applications. We provide a taxonomy of the network de- +sign, highlight the major strengths and deficiencies of the exist- +ing approaches and introduce the current benchmarks in each +task. We inspect several key technologies that arise from the +various medical imaging applications, including medical im- +age segmentation, medical image registration, medical image +reconstruction, and medical image classification. So far, review +papers related to Transformers do not concentrate on applica- +tions of Transformers in the medical imaging and image analy- +sis domain [27]. The few literature reviews that do focus on the +medical domain [28, 29], despite being very comprehensive, +do not necessarily discuss the drawbacks and merits of each +method. In our work, we explicitly cover this aspect and also +provide a taxonomy that comprises the imaging modality, organ +of interest, and type of training procedure each paper has se- +lected. More specifically, in Section 3 (Medical Image Classi- +fication), we comprehensively elaborate on the most promising +networks along with their key ideas, limitations, the number of +parameters, and the specific classification task they are address- +ing. In Section 4 (Medical Image Segmentation), we analyze +network architectures in terms of their design choice and pro- +pose a detailed taxonomy to categorize each network to provide +insight for the reader to understand the current limitations and +progress in segmentation networks based on the Transformer +architecture. +In Section 5 (Medical Image Reconstruction), +we take a different perspective to categorize networks based +on their network structure and the imaging modality they are +built upon. We categorize the synthesis methods in Section 6 +based on their objective (intra-modality or inter-modality) and +then provide detailed information regarding the network archi- +tecture, parameters, motivations, and highlights. In the sections +related to detection (Section 7), registration (Section 8), and re- +port generation (Section 9) we briefly summarize the state-of- +the-art (SOTA) networks and provide detailed information re- +garding the network architectures, advantages, and drawbacks. +Moreover, due to the swift development of the field, we believe +that the community requires a more recent overview of the lit- +erature. +We hope this work will point out new research options and +provide a guideline for researchers and initiate further inter- +est in the vision community to leverage the potential of Trans- +former models in the medical domain. Our major contributions +are as follows: +• We systematically and comprehensively review the ap- +plications of Transformers in the medical imaging do- +main and provide a comparison and analysis of SOTA ap- +proaches for each task. Specifically, more than 200 papers +2 + +Linear Projection of Flattened Patches +Patch + Position +Embedding +Extra learnable +[class] embedding +Embedded +Patches +LN +Multi-Head +Self-Attention +LN +MLP +Class +Benign +Malignant +Transformer Encoder +MLP Head +Figure 2: Architecture of the Vision Transformer as proposed in [22] and the detailed structure of the Vision Transformer encoder block. In the Vision Transformer, +sequential image patches are used as the input and processed using a Transformer encoder to produce the final classification output. +are covered in a hierarchical and structured manner. +• Our work provides a taxonomized (Figure 1), in-depth +analysis (e.g. task-specific research progress and limita- +tions), as well as a discussion of various aspects. +• Finally, We discuss challenges and open issues and also +identify new trends, raise open questions and identify fu- +ture directions. +Paper Organizations. The remaining sections of the paper +are organized as follows. In Section 2, we provide an overview +of the key components of the well-established Transformer ar- +chitecture. Moreover, this section clarifies the categorization +of neural network variants in terms of the position where the +Transformer is located. +Section 3 to Section 9 comprehen- +sively review the applications of Transformers in diverse med- +ical imaging tasks as depicted in Figure 1. For each task, we +propose a taxonomy to characterize technical innovations and +major use cases. Section 10 presents open challenges and future +perspectives of the field as a whole, while finally, Section 11 +concludes this work +2. Background +In this section, we first provide an overview of the over- +all architecture of the Transformer module and the key ideas +behind its feasible design. Then, we outline a general taxon- +omy of Transformer-based models, characterized by their core +techniques of using Transformers, i.e., whether they are purely +Transformer-based, or whether the Transformer module is ei- +ther used in the encoder, decoder, bottleneck, or skip connec- +tion, respectively. +2.1. Transformers +The original Transformer [21] was first applied to the task for +machine translation as a new attention-driven building block. +The vanilla Transformer consists of an encoder and a decoder, +each of which is a stack of L tandem of consecutive identi- +cal blocks. The Transformer module is convolutional-free and +solely based on the self-attention mechanism or attention mech- +anism in short. Specifically, these attention blocks are neu- +ral network layers that relate different positions of a single se- +quence to compute the sequence’s representation. Since the es- +tablishment of Transformer models, they have attained remark- +able performance in diverse natural language processing tasks +[30]. Inspired by this, Dosovitskiy et al. proposed the Vision +Transformer (ViT) [22] model as illustrated in Figure 2. When +trained on large datasets, for instance, JFT-300M, ViT outper- +forms the then state-of-the-art, namely ResNet-based models +like BiT [31]. In their approach, an image is turned into fixed- +sized patches before being flattened into vectors. These vectors +are then passed through a trainable linear projection layer that +maps them into N vectors with the dimensionality of D × N is +the number of patches. The outputs of this stage are referred +to as patch embeddings. To preserve the positional information +present within each patch, they add positional embeddings to +the patch embeddings. In addition to this, a trainable class em- +bedding is also appended to the patch embeddings before going +through the Transformer encoder. The Transformer encoder +is comprised of multiple Transformer encoder blocks. There +are one multi-head self-attention (MSA) block and an MLP +block in each Transformer encoder block. The activations are +first normalized using LayerNorm (LN) before going into these +blocks in the Transformer encoder block. Furthermore, there +3 + +are skip connections before the LN that add a copy of these ac- +tivations to the corresponding MSA or MLP block outputs. In +the end, there is an MLP block used as a classification head that +maps the output to class predictions. The self-attention mech- +anism is a key defining characteristic of Transformer models. +Hence, we start by introducing the core principle of the atten- +tion mechanism. +2.1.1. Self-Attention +In a self-attention layer (Figure 3a), the input vector is firstly +transformed into three separate vectors, i.e., the query vector +q, the key vector k, and the value vector v with a fixed dimen- +sion. These vectors are then packed together into three different +weight matrices, namely WQ, WK, and WV. A common form +of Q, K, and V can be formulated as Equation (1) for an input +X +K = WKX, Q = WQX, V = WVX, +(1) +where WK, WQ, and WV refers to the learnable parameters. The +scaled dot-product attention mechanism is then formulated as +Attention(Q, K, V) = Softmax +�QKT +√dk +� +V, +(2) +where √dk is a scaling factor, and a softmax operation is ap- +plied to the generated attention weights to translate them into a +normalized distribution. +2.1.2. Multi-Head Self-Attention +The multi-head self-attention (MHSA) mechanism (Fig- +ure 3b) has been proposed [21] to model the complex relation- +ships of token entities from different aspects. Specifically, the +MHSA block helps the model to jointly attend to information +from multiple representation sub-spaces, as the modeling ca- +pability of the single-head attention block is quite coarse The +process of MHSA can be formulated as +MultiHead(Q, K, V) = [Concat (head1, . . . , headh)]WO, +(3) +where headi = Attention +� +QWQ +i , KWK +i , VWV +i +� +, and WO indi- +cates a linear mapping function to combine multi-head repre- +sentation. Note that h is a hyper-parameter set to h = 8 in the +original paper. +2.2. Transformer modes +While the Transformer was originally introduced with an +encoder-decoder pipeline, many modern architectures gener- +ally exploit the Transformer architecture in different fashions, +which generally depend on the target application. The usage of +Transformers in vision tasks can broadly be classified into pure +and hybrid designs. +2.2.1. Pure Transformers +Due to the deficiency of CNN-based architectures in learning +global and long-range semantic information interactions, which +stems from the locality of convolution operation, a cohort study +has investigated the purely Transformer-based models without +any convolution layer. These models usually consist of encoder, +MatMul +Scale +SoftMax +MatMul +(a) Self-Attention +Linear +Linear +Scaled Dot-Product +Attention +Linear +Concat +Linear +(b) Multi-Head Self-Attention +Figure 3: (a) The process of self-attention. (b) Multi-head attention. The MSA +consists of multiple SA blocks (heads) concatenated together channel-wise as +proposed in [21]. +bottleneck, decoder, and skip connections directly built upon +the ViT or its variants. In this criteria, there are usually multiple +multi-head self-attention modules in both encoding and decod- +ing sections that allow the decoder to utilize information from +the encoder. Examples of such methods are the Swin-Unet [32] +and the TransDeepLab [33] networks which, as their name sug- +gests, try to model the seminal U-Net [34], and DeepLab [35] +architectures. +2.2.2. Transformer: Hybrid +The hybrid Transformer models usually modify the base +CNN structure by replacing the encoder or decoder modules. +Encoder: Encoder-only models such as the seminal BERT +[36] are designed to make a single prediction per input or a sin- +gle prediction for an entire input sequence. In the computer +vision era, these models are applicable for classification tasks. +Moreover, as utilizing a pure Transformer can result in lim- +ited localization capacity stemming from inadequate low-level +features, many cohort studies try to combine CNN and Trans- +former in the encoding section [24]. Such a design can enhance +finer details by recovering localized spatial information. +Decoder: Transformers can also be used in a decoding fash- +ion. Such a causal model is typically used for generation tasks +such as language modeling. Besides that, the modification can +apply to the skip connections of the decoder module. Skip con- +nection is a widely-used technique to improve the performance +and the convergence of deep neural networks. It can also serve +as a modulating mechanism between the encoder and the de- +coder. To effectively provide low-level spatial information for +the decoding path, the idea of exploiting Transformers in de- +signing skip connections has emerged. This notable idea can +lead to finer feature fusion and recalibration while guaranteeing +the aggregation scheme of using both high-level and low-level +features [24, 37]. +4 + + Medical Image + Classification + Pure + Hybrid + Original ViT Structure + 11. TransMed + Other ViTs + 15. LAT + 16. DT-MIL + Original ViT Structure + 7. Covid-Transformer + 6. ViT-BUS + 5. ViT-vs-CNN + 4. FESTA + 3. XViTCOS + 2. MIL-ViT + 1. COVID-VIT + Other ViTs + 8. POC-Former + 9. RadioTransformer + 10. COVID-VOLO + 12. 3DMET + 13. Femur-ViT + 14. Hybrid-Covid-ViT + 18. HATNet + 17. TransMIL + 19. GTP +Figure 4: Taxonomy of ViT-based approaches in medical image classification. Methods are categorized based on their proposed architecture into pure and hybrid +methods, in which they adopt the vanilla ViT or present a new type of vision Transformer in medical image classification. Notably, we utilize the prefix numbers +in the paper’s name in ascending order and denote the reference for each study as follows: 1. [38], 2. [39], 3. [40], 4. [41], 5. [42], 6. [43], 7. [44], 8. [45], 9. +[46], 10. [47], 11. [48], 12. [49], 13. [50], 14. [51], 15. [52], 16. [53], 17. [54], 18. [55], 19. [56]. +3. Medical Image Classification +Image classification is still one of the challenging problems +in computer vision, which aids in segregating extensive quan- +tities of data into meaningful categories. Vision Transformers +(ViT) have recently demonstrated outstanding results in vari- +ous image classification tasks and offer significant advantages +over conventional CNNs [57, 58, 59, 60, 61, 62]. These advan- +tages include long-range relationships, adaptive modeling, and +attention maps that yield intuition on what the model deems +more important inside an image [42]. Due to these alluring ad- +vantages, there is rising interest in building Transformer-based +models for medical image classification. Therefore, highly pre- +cise classification is becoming increasingly vital for facilitating +clinical care. +In this section, we exhaustively examine ViTs in medical im- +age classification. As illustrated in Figure 4, we have broadly +classified these methods based on the role ViT plays in their +architecture. These categories include pure Transformers and +Hybrid Models. Generally, a vision Transformer-based clas- +sification architecture consists of three modules: (1) a back- +bone for capturing input features, (2) an encoder for model- +ing the information, and (3) a classification head for generat- +ing output based on the specified task. Therefore, the Trans- +former can be adopted in each module. However, some works, +including Lesion Aware Transformer (LAT) [52] and De- +formable Transformer for Multi-Instance Learning (DT-MIL) +[53], take a different approach and utilize encoder-decoder +structures. LAT proposes a unified encoder-decoder system for +Diabetic Retinopathy (DR) grading, and DT-MIL introduces +a Transformer-based encoder-decoder architecture for classi- +fying histopathological images, where the deformable Trans- +former was embraced for the encoder part. In the following, we +will go into great depth on both hybrid and pure models. +3.1. Pure Transformers +Since the emergence of Transformers, there has been a grow- +ing debate regarding whether it is time to entirely switch from +CNNs to Transformers. Matsoukas et al. [42] conduct a se- +ries of experiments to answer this critical question. They take +ResNet50 [63] and the DeiT-S [64] models to represent CNN +and ViT models, respectively. They train each of these two +models in 3 different fashions: a) randomly-initialized weights, +b) pre-trained on ImageNet (transfer learning), and c) pre- +training on the target dataset in a self-supervised scheme using +DINO [65]. Their findings show that when utilizing random +initialization, ViTs are inferior to CNNs. In the case of transfer +learning, the results are similar for both models, with ViT being +superior for two out of three datasets. Additionally, ViT per- +forms better when self-supervision on the target data is applied. +They conclude that Vision Transformers, indeed, are suitable +replacements for CNNs. +Transformers have had a profound effect on medical devel- +opment. Researchers have thoroughly investigated adopting the +ViT in medical image classification tasks since its introduction. +However, the limited number of medical images has hindered +Transformers from replicating their success in medical image +classification. ViT-BUS [43] studies the use of ViTs in medi- +cal ultrasound (US) image classification for the first time. They +propose to transfer pre-trained ViT models based on the breast +US dataset to compensate for the data-hunger of ViTs. Eval- +uated results on B [66], BUSI [67], and B+BUSI datasets in- +dicate the predominance of attention-based ViT models over +CNNs on US datasets. Likewise, COVID-Transformer [44] +utilizes ViT-L/16 to detect COVID from Non-COVID based on +CXR images. Due to the limitation of sufficient data, they intro- +duce a balanced dataset containing 30K chest X-ray images for +multi-class classification and 20K images for binary classifica- +tion. The published dataset is created by merging datasets [68], +[69], and [70]. They fine-tune the model on the dataset with +5 + +a custom MLP block on top of ViT to classify chest x-ray +(CXR) images. Moreover, COVID-Transformer exploits the +GradCAM Map [71] to visualize affected lung areas that are +significant for disease prediction and progression to display the +model interpretability. Similarly, Mondal et al. [40] present +xViTCOS for detecting COVID-19 in CTs and CXRs. xViT- +COS employs a model that has been pre-trained on ImageNet- +21k [72]. Nevertheless, the training data capacity might over- +shadow the generalization performance of the pre-trained ViT +to transfer the knowledge from the learned domain to the tar- +get domain. +By training the model on the COVIDx-CT-2A +dataset [73], a moderately-sized dataset, xViTCOS overcomes +this problem. However, due to the shortage of the insufficient +amount of CXR images, the pre-trained ViT model is fine- +tuned using the CheXpert dataset [74]. In addition, xViTCOS +leverages the Gradient Attention Rollout algorithm [75] to vi- +sually demonstrate the model’s prediction on the input image +for clinically interpretable and explainable visualization. +In +experiments using COVID CT-2A and their custom-collected +Chest X-ray dataset, xViTCOS significantly outperforms con- +ventional COVID-19 detection approaches. MIL-VT [39] sim- +ilarly suggests pre-training the Transformer on a fundus image +large dataset beforehand, initialized by the pre-trained weight +of ImageNet, then fine-tuning it on the downstream retinal dis- +ease classification task in order to encourage the model to learn +global information and achieve generalization. Unlike previ- +ous approaches, they apply some modifications to the vanilla +ViT structure. In the classic ViT, embedded features are ne- +glected for classification; instead, only the class token, which +retains the summarization of embedded features’ information, +is used. Yu et al. [39] propose a novel multiple-instance learn- +ing (MIL)-head module to exploit those embedded features to +complement the class token. This head comprises three sub- +modules that attach to the ViT in a plug-and-play manner: 1) +the MIL embedding submodule that maps the feature embed- +dings to a low-dimensional embedding vector, 2) the attention +aggregation submodule that outputs a spatial weight matrix for +the low-dimensional patch embeddings; this weight matrix is +then applied to the low-dimensional embeddings to ascertain +each instance’s importance, 3) the MIL classifier submodule +that determines the probability of each class through aggre- +gated features. In the downstream task, both MLP and MIL +heads use the weighted cross-entropy loss function for train- +ing. The outputs of both heads are then weight-averaged for +the inference time. Results indicate the effectiveness of the +proposed training strategy and the MIL-head module by dra- +matically boosting the performance over APTOS2019 [76] and +RFMiD2020 [77] datasets when compared to CNN-based base- +lines. In contrast to the previous 2D-based methods that employ +transfer learning, COVID-ViT [38] proposes training ViT to +classify COVID and non-COVID cases using 3D CT lung im- +ages. Given that a COVID volume may contain non-COVID 2D +slices, COVID-ViT applies a slice voting mechanism after the +ViT classification result in which the subject is categorized as +having COVID if more than a certain percentage of slices (e.g., +25%) are predicted to be COVID. The findings reported for the +MIA-COVID19 competition [78] confirm that ViT outperforms +CNN-based approaches such as DenseNet [79] in identifying +COVID from CT images. +Besides the remarkable accuracy of Transformers compared +to CNNs, one of their major drawbacks is their high computa- +tional cost, thereby making them less effective for real-world +applications, such as detecting COVID-19 in real-time. In light +of the prevalence of COVID-19, the rapid diagnosis will be ben- +eficial for starting the proper course of medical treatment. CXR +and lung CT scans are the most common imaging techniques +employed. However, CT imaging is a time-consuming process, +and using CXR images is unreliable in identifying COVID-19 +in the early stage. In addition, vision Transformers are compu- +tationally expensive to deploy on mobile devices for real-time +COVID-19 classification. Therefore, Perera et al. [45] present a +lightweight Point-of-Care Transformer (POCFormer). The +compactness of POCFormer allows for real-time diagnosis of +COVID-19 utilizing commercially accessible POC ultrasound +devices. POCFormer reduces the complexity of the vanilla ViT +self-attention mechanism from quadratic to linear using Lin- +former [80]. The results display the superiority of POCFormer +in the real-time detection of COVID-19 over the CNN-based +SOTAs on the POCUS dataset [81]. +In addition, despite the great potential shown by ViTs in Im- +ageNet classification, their performance is still lower than the +latest SOTA CNNs without additional data. These Transform- +ers mainly focus on a coarse level by adopting a self-attention +mechanism to establish global dependency between input to- +kens. +However, relying only on a coarse level restricts the +Transformer’s ability to achieve higher performance. Thus, Liu +et al. [47] leverage a pre-trained version of VOLO for an X-ray +COVID-19 classification. VOLO [82] first encodes fine-level +information into the token representations through proposed +outlook attention, alleviating the limitations of Transformers +that require a large amount of data for training, and second ag- +gregates the global features via self-attention at the coarse level. +Through the outlook attention mechanism, VOLO dynamically +combines fine-level features by treating each spatial coordinate +(i, j) as the center of a K × K local window and calculating its +similarity with all its neighbors. The findings indicate that fine- +tuning VOLO on Dataset-1 [83] leads to 99.67% top1 accuracy +on Dataset-1 test cases and 98.98% top1 accuracy on unseen +Dataset-2 [84], which demonstrates the generality of the ap- +proach. +Furthermore, accessible labeled images have considerably +influenced research on the use of Transformers to diagnose +COVID-19. Considering the shortage of labeled data, data shar- +ing between hospitals is needed so as to create a viable central- +ized dataset. However, such collaboration is challenging due +to privacy concerns and patient permission. Motivated by Fed- +erated Learning (FL) and Split Learning (SL), Park et al. [41] +present a Federated Split Task-Agnostic (FESTA) framework +that uses ViT for multi-task learning of classification, detection, +and segmentation of COVID-19 CXR images. FESTA benefits +from the decomposable modular design of ViT to train heads +and tails via clients and share the server-side Transformer body +across clients to aggregate extracted features and process each +task. The embedded features from the body Transformer are +6 + +then passed to their task-specific tail on the client side to pro- +duce the final prediction (Figure 5(a)). Figure 5(b) illustrates +the single-task learning scheme and (c) the multi-task learning +scheme. In multi-task learning, heads, tails, and a task-agnostic +Transformer body are first jointly trained for 6000 rounds (see +Figure 5(c)). Then, heads and tails are fine-tuned according to +the desired specific task while freezing the weights of the Trans- +former body. FESTA merits from 220000 decentralized CXR +images and attains competitive results compared to the data- +centralized training approaches. The experimental results also +demonstrate the stable generalization performance of FESTA, +where multi-task learning enhances the performance of the in- +dividual tasks through their mutual effect during training. +Figure 5: Overview of the FESTA framework [41], which utilizes ViT for +multi-task learning of COVID-19 CXR classification, detection, and segmenta- +tion. (a) FESTA leverages ViT’s decomposable modular design to train heads +(H) and tails (T ) via clients while sharing the server-side Transformer body +(B) between clients to integrate retrieved features. Final predictions are then +derived by feeding embedded features to their task-specific tails on the client +side. (b) illustrates the single-task learning scheme, and (c) two steps multi-task +learning scheme. The former is trained for 12000 rounds, while the latter un- +dergoes two training steps. First, the whole parts train in 6000 rounds. Then by +freezing the weights of the Transformer body, the heads and tails are fine-tuned +for 6000 steps based on the desired specific task. +Most attention-based networks utilized for detection and +classification rely on the neural network to learn the neces- +sary regions of interest. Bhattacharya et al. [46] in Radio- +Transformer argue that in certain applications, utilizing ex- +perts’ opinions can prove beneficial. Specifically, they apply +this notion to leverage radiologists’ gaze patterns while di- +agnosing different diseases on medical images; then, using a +teacher-student architecture, they teach a model to pay attention +to regions of an image that a specialist is most likely to examine. +The teacher and the student networks consist of two main com- +ponents: global and focal. The global component learns coarse +representation while the focal module works on low-level fea- +tures, and both these segments are comprised of Transformer +blocks with shifting windows. In addition, the global and focal +components are interconnected using two-way lateral connec- +tions to form the global-focal module; this is to address the +inherent attention gap between the two. The teacher network is +first directly pre-trained on human visual attention maps. Then, +the entire model is trained for different downstream tasks, e.g., +object detection and classification. Furthermore, the authors +propose a self-supervised Visual Attention Loss (VAL) that in- +corporates both GIoU and MSE loss. The student network is +trained to predict probability values for different classes and at- +tention regions. These attention regions are then compared to +those obtained from the teacher model, and the weights are op- +timized using VAL. +Visual Attention Loss +TEACHER +STUDENT +Global-Focal +Global-Focal +Human Visual Attention +Training +Input Image +Visual Attention +Predicted Attention +TEACHER +Global-Focal +HVAT +Disease Classification +Eye gaze points +Figure 6: Overview of RadioTransformer [46]. Human Visual Attention Train- +ing (HVAT) block first uses radiologists’ visual observations of chest radio- +graphs to train a global-focal teacher network. The pre-trained teacher network +is then utilized to distill the teacher’s knowledge to a global-focal student net- +work through visual attention loss, enabling the student to learn visual infor- +mation. Following the teacher-student strategy and incorporating radiologist +visual examinations leads to an improvement in the classification of disease on +chest radiographs. +3.2. Hybrid Models +In spite of the vision Transformers’ ability to model global +contextual representations, the self-attention mechanism un- +dermines the representation of low-level details. +CNN- +Transformer hybrid approaches have been proposed to ame- +liorate the problem above by encoding both global and local +features using the locality of CNNs and the long-range depen- +dency of Transformers. +TransMed [48] proposes a hybrid CNN-Transformer net- +work that leverages the locality of CNNs and the long-range +dependency character of Transformers for parotid gland tumor +and knee injury classification. Multimodal medical images pri- +marily have long-range interdependencies, and improving per- +formance requires an effective fusion strategy. TransMed pro- +poses a novel image fusion strategy. Firstly, three neighboring +2D slices of a multimodal image are overlaid to create three- +channel images. Then, each image is partitioned into K × K +patches. This fusion approach allows the following network +to learn mutual information from images of different modali- +ties. Patch tokens are fed into a CNN network to capture their +low-level features and generate patch embeddings. The classic +ViT is then used to determine the relationship between patch se- +quences. TransMed’s final results verify the effectiveness of hy- +brid models in classifying multimodal medical images by out- +performing all its counterparts by a large margin. TransMed-S +enhances average accuracy on the PGT dataset by about 10.1% +over its nearest counterpart, BoTNet [86], while requiring fewer +parameters and FLOP count. Comparably, Tanzi et al. [50] de- +velop a new CAD system (Femur-ViT) based on Vision Trans- +7 + +(a) +(b) Single-task learning scheme (e.g. classification) +Classification +Single body +Client 1 +Hcls +Training for 12,000 rounds +cls +6 +6 +Client 1 +: +Client 6 +Task-agnostic +Hcls +Transformer +cls +... +Body +Client 6 +B +Segmentation +Hcls +6 +Client 7 +Task-agnostic +seg +Transformer +(c) Multi-task learning scheme +Body +Client 8. +Hseg + Step 2: Training for 6,oo0 rounds (body fixed) +Step 1: Training for 6,oo0 rounds (body learnable) +B +6 +6 +6 +Client 1 +Client 1 +Detection +-cls +-cls +Task-agnostic +Task-agnostic +Client 9 +Tap +Transformer +Transformer +... +.. +.. +Body +Tdet +Body +Client 10 +Client 10 +B +B +apn +Hdet +Client 10... +apH +2Image +Bags +Words +n +B1 +I +W0 +m +W1 +3 +W1 +2 +W1 +1 +n +CNN +n +m +d +B1 +cnn +Word-to-word +attention +Bw2w +Word-to-bag +Attention +Word-to-bag +attention +n +d +n +d +Bw2b +�Bw2b +Bag-to-bag +attention +Bb2b +Bag-to-image +attention +Ib2i ∈ Rd +Classifier +Benign +Atypia +DCIS +Invasive +Bi +cnn +Multi-head +attention +Feed forward +network (FFN) +Bi +w2w +Word-to-word attention +�Bw2b +Bw2b +Multi-head +attention +Multi-head +attention +Feed forward +network (FFN) +Bb2b +�Bb2b +Bag-to-bag attention +Bi +w2w +Function +Ψ +Linear +Softmax +Dot-product +B +i +w2b +Word-to-bag attention +(a) +(d) +(c) +(b) +Figure 7: The overall architecture of [85]. HATNet hierarchically divides an input image into n × m words, which are then fed into the CNN encoder to provide +word-level representations for each bag. Then by performing a bottom-up decoding strategy and applying a linear classifier, breast biopsy classification results +are obtained. Notably, bag-to-image attention has the same procedure as word-to-bag attention, shown in (c). +formers for diagnosing femoral fractures. First, YOLOv3 [87] +is utilized to detect and crop the left and right femur regions. +Afterward, a CNN (InceptionV3 [88]) and a hierarchical CNN +(different InceptionV3 networks in cascade) [89] are applied to +the dataset, and the results serve as baselines for the classifi- +cation. Then, they use a modified ViT to classify seven differ- +ent fracture types. Finally, a clustering approach is proposed +as an evaluation technique for the ViT encoder. +This study +highlights the power of using ViT models for medical image +classification and the ability of the proposed CAD system to +significantly increase clinicians’ diagnostic accuracy. 3DMeT +[49] proposes applying a 3D medical image Transformer for +assessing knee cartilage defects in three grades: grade 0 (no de- +fect), grade 1 (mild defect), and grade 2 (severe defect). Primar- +ily, using medical 3D volumes as an input to the Transformer +is computationally expensive, thereby making it impractical. +3DMeT resolves the high computational cost problem by re- +placing conventional linear embedding with 3D convolutional +layers. The weights of convolutional layers are adopted using +the teacher-student training strategy. 3DMeT takes an exponen- +tial moving average from the first one/few-layer(s) of the CNN +teacher’s weights and uses it as convolutional layers’ weights. +This method enables Transformers to be compatible with small +medical datasets and to benefit from CNNs’ spatial inductive +biases. Lastly, the Transformer and CNN teacher’s outputs are +combined in order to derive the classification results. +Operating Transformers over Whole Slide Images (WSIs) +is computationally challenging since WSI is a gigapixel im- +age that retains the original structure of the tissue. MIL and +CNN backbones have demonstrated practical tools for acting +on WSI. MIL is a weakly supervised learning approach that +enables deep learning methods to train high-resolution images +like WSI. Since annotating such images at the pixel level is +impractical, MIL proposes to divide an input WSI into a bag +of instances and assign a single label to the bag of each im- +age based on pathology diagnosis. The bag has a positive label +if it contains at least one positive instance, and it is consid- +ered negative if all the instances in the bag are negative. Then +CNN backbones are employed to down-sample and extract the +features of each instance and allow Transformers to operate +according to the generated feature maps and currently avail- +able hardware. Therefore, DT-MIL [53] proposes to compress +WSIs into compact feature images by embedding each patch of +the original WSI into a super-pixel at its corresponding posi- +tion using EfficientNet-B0 [90]. The resulting thumbnail image +feed into a 1 × 1 Conv for feature reduction, followed by a de- +formable Transformer encoder that aggregates instance repre- +sentations globally. A similar approach is adopted by Holistic +ATtention Network (HATNet) [55], where they first divide an +input image into n non-overlapping bags, each broken down +into m non-overlapping words (or patches). n × m words are +fed into the CNN encoder to obtain word-level representations +for each bag. HATNet aims to develop a computer-aided di- +agnosis system to help pathologists in reducing breast cancer +detection errors. According to the World Health Organization +(WHO), breast cancer is the most frequent non-skin cancer in +women, accounting for one out of every four new female can- +cers annually [91]. As illustrated in Figure 7, HATNet follows +a bottom-up decoding strategy such that it first performs multi- +head attention to words in a word-to-word attention block, then +considers the relationship between words and bags in word-to- +bag attention, followed by bag-to-bag attention to attain inter- +bag representations. The acquired bag features are then ag- +gregated in bag-to-image attention to build image-level repre- +sentations. A linear classifier is ultimately applied to achieve +the final results. Furthermore, unlike most MIL methods that +take all the instances in each bag independent and identically +distributed [92, 93, 94], TransMIL [54] suggests that it is es- +sential to consider the correlation between different instances +and explore both morphological and spatial information. Two +Transformer layers address the morphological information, and +a conditional position encoding layer named Pyramid Position +Encoding Generator (PPEG) addresses the spatial information. +The proposed PPEG module has two merits: 1) It handles posi- +tional encoding of sequences with a variant number of instances +by using group convolution over the 2D reshaped patch tokens, +8 + +and 2) It enriches the features of tokens by capturing more con- +text information through convolutions. In contrast to conven- +tional iid-based MIL methods requiring many epochs to con- +verge, TransMIL converges two to three times faster by using +morphological and spatial information. TransMIL also outper- +forms all the latest MIL methods [95, 96, 97, 98, 92] in terms of +accuracy and AUC by a significant margin in binary and multi- +ple classification tasks and exhibits the superiority of taking the +correlation between different instances into account and con- +sidering both morphological and spatial information. +Previous methods mainly rely on weakly supervised learn- +ing or dividing WSIs into image patches and using supervised +learning to assess the overall disease grade. Nevertheless, these +approaches overlook WSI contextual information. Thus, Zheng +et al. [56] propose a Graph-based Vision Transformer (GTP) +framework for predicting disease grade using both morphologi- +cal and spatial information at the WSIs. The graph term allows +for the representation of the entire WSI, and the Transformer +term allows for computationally efficient WSI-level analysis. +The input WSI is first divided into patches, and those that con- +tain more than 50% of the background are eliminated and not +considered for further processing. +Selected patches are fed +forward through a contrastive learning-based patch embedding +module for feature extraction. A graph is then built via a graph +construction module utilizing patch embeddings as nodes of the +graph. In the graph Transformer section, a graph convolution +layer followed by a mincut pooling layer [99] is applied first to +learn and enrich node embeddings and then lessen the number +of Transformer input tokens. Since the graph adjacency matrix +contains spatial information of nodes, by adding an adjacency +matrix to node features, GTP obviates the need for adding ex- +tra learnable positional embeddings to nodes. The final Trans- +former layer predicts the WSI-level class label for three lung tu- +mor classes: Normal, LUAD, and LSCC. GTP also introduces a +graph-based class activation mapping (GraphCAM) technique +that highlights the class-specific regions. GraphCAM exploits +attention maps from multi-head self-attention (MHSA) blocks +in the Transformer layer and maps them to the graph space +to create a heatmap for the predicted class. The experiments +show that GTP performs as a superior interpretable and effi- +cient framework for classifying WSI images while considering +morphological and spatial information. +Diabetic Retinopathy (DR) is an eye disorder that can cause +impaired vision and sightlessness by damaging blood vessels +in the retina. Most deep-learning approaches view lesion dis- +covery and DR grading as independent tasks that may produce +suboptimal results. In contrast to conventional methods, LAT +[52] proposes a unified encoder-decoder structure that com- +prises a pixel relation-based encoder to capture the image con- +text information and a lesion filter-based decoder to discover le- +sion locations, which the whole network jointly optimized and +complemented during training. The encoder is particularly in +charge of modeling the pixel correlations, and the Transformer- +based decoder part is formulated as a weakly supervised lo- +calization problem to detect lesion regions and categories with +only DR severity level labels. In addition, LAT proposes two +novel mechanisms to improve the effectiveness of lesion-aware +filters: 1) Lesion region importance mechanism, g(·|Φ), to de- +termine the contribution of each lesion-aware feature, and 2) +Lesion region diversity mechanism to diversify and compact +lesion-aware features. The former is a linear layer followed by +a sigmoid activation function that generates importance weights +for lesion-aware features, and the latter adopts a triplet loss +[100] to encourage lesion filters to find diverse lesion regions. +In the DR grading branch, LAT presents a DR grading classi- +fication module that calculates a global consistency loss based +on the lesion-aware features, indicated as h(·|σ). Eventually, +the final DR grading prediction is achieved by calculating the +cross-entropy loss between the predicted labels obtained from +the fusion of g(·|Φ) and h(·|σ) and the ground truth. The total +loss is the aggregation of cross-entropy loss, global consistency +loss, and triplet loss. Visual results of LAT regarding the lesion +discovery are depicted in Figure 8. +Ground-truth +LAT +CAM +Figure 8: LAT [52] vs. CAM [101] visual comparison. The ground truth con- +sists of microaneurysms, hemorrhages, soft exudates, and hard exudates, which +are colored as green, yellow, green, and blue dots, respectively. +3.3. Discussion and Conclusion +Section 3 thoroughly outlines 19 distinctive Transformer- +based models in medical image classification. We have cat- +egorized the introduced models based on their architectures +into hybrid and pure. These approaches differ according to +whether they adhere to the original structure of the vanilla +ViT or provide a new variant of the vision Transformer that +can be applied to medical applications. In addition, we have +presented details on the studied classification methods re- +garding their architecture type, modality, organ, pre-trained +strategy, datasets, metrics, and the year of publication in +Table 1. Additional descriptions of the methods, including +their model size, contributions, and highlights, are described +in Table 2. +As is evident in the storyline of this section, we have dis- +cussed methods in each paragraph regarding the underlying +problems in medical image classification and introduced so- +lutions and how they address such issues. However, the need +for more research on these problems is crucial to making +such approaches widely applicable. +9 + +Table 1: An overview of the reviewed Transformer-based medical image classification models. +Method +Modality +Organ +Type +Pre-trained Module: Type +Datasets +Metrics +Year +Pure +ViT-vs-CNN +[42] +Dermoscopy +Mammograms +Multi-organ +2D +ViT: Self-supervised & Supervised +1APTOS-2019 [76] +2ISIC-2019 [102] +3CBIS-DDSM [103] +Kappa +Recall +ROC-AUC +2021 +ViT-BUS +[43] +Ultrasound +Breast +2D +ViT: Supervised +1B [66] +2BUSI [67] +ACC +AUC +2021 +POCFormer +[45] +Ultrasound +Chest +2D + +POCUS [81] +Recall, F1 +SP, SE, ACC +2021 +MIL-VT +[39] +Fundus +Eye +2D +ViT: Supervised +1Private Dataset +2APTOS-2019 [76] +3RFMiD-2020 [77] +Recall, F1 +ACC, AUC +Precision +2021 +COVID-VIT +[38] +CT +Chest +3D + +MIA-COV19 [78] +ACC, F1 +2021 +xViTCOS +[40] +X-ray +CT +Chest +2D +ViT: Supervised +1COVID CT-2A [73] +2CheXpert [74] +Recall, F1 +Precision +SP, NPV +2021 +FESTA +[41] +X-ray +Chest +2D +ViT: Supervised +1Four Private Datasets +2CheXpert [74], 3BIMCV [104] +4Brixia [105], 5NIH [106] +6SIIM-ACR [107], 7RSNA [108] +Recall, F1 +SP, SE, AUC +2021 +COVID-Transformer +[44] +X-ray +Chest +2D +ViT: Supervised +1[68], 2[70] +3[69] +Recall, F1 +ACC, AUC +Precision +2021 +COVID-VOLO +[47] +X-ray +Chest +2D +ViT: Supervised +1[83] +2[84] +ACC +2021 +RadioTransformer +[46] +X-ray +Chest +2D +ViT: Supervised +1RSNA [109], 2Cell Pneumonia [110] +3COVID-19 Radiography [83, 111] +4NIH [106], 5VinBigData [112] +6SIIM-FISABIO-RSNA [113] +7RSNA-MIDRC [114, 115] +8TCIA-SBU COVID-19 [116, 117] +Recall, F1 +ACC, AUC +Precision +2022 +Hybrid +TransMIL +[54] +Microscopy +Multi-organ +2D +CNN: Supervised +1Camelyon16 [118] +2TCGA-NSCLC [119, 120] +3TCGA-RCC[121] +ACC +AUC +2021 +LAT +[52] +Fundus +Eye +2D +CNN: Supervised +1Messidor-1[122] +2Messidor-2[123] +3EyePACKS[124] +AUC & Kappa +2021 +TransMed +[48] +MRI +Ear +Knee +3D +ViT: Supervised +CNN: Supervised +1PGT [48] +2MRNET [125] +Precision +ACC +2021 +3DMeT +[49] +MRI +Knee +3D +CNN: Supervised +Private dataset +ACC, Recall, F1 +2021 +Hybrid-COVID-ViT +[51] +X-ray +Chest +2D +CNN: Supervised +CheXpert [74] +AUC, ACC +SP, SE +2021 +Femur-ViT +[50] +X-ray +Femur +2D +ViT: Supervised +CNN: Unsupervised +Private dataset +Recall, F1 +Precision, ACC +2022 +DT-MIL +[53] +Microscopy +Lung +Breast +2D +CNN: Supervised +1CPTAC-LUAD [116] +2BREAST-LNM [53] +Recall, F1 +AUC, precision +2021 +GTP +[56] +Microscopy +Lung +2D +CNN: Self-supervised +1NLST [126], 2CPTAC [127] +3TCGA [128] +Precision, Recall +SP, SE, ACC, AUC +2022 +HATNet +[55] +Microscopy +Breast +2D +CNN: Supervised +Breast Biopsy WSI Dataset [129] +ACC, ROC-AUC +F1, SP, SE +2022 +Data availability in the medical domain is one of the most +challenging aspects of developing Transformer-based mod- +els since Transformer models are known for being data- +hungry to generalize. Reasons for data scarcity in the med- +ical field can be referred to as privacy concerns of patients, +the time-consuming and costly process of annotation, and +the need for expert staff. To this end, the use of genera- +tive models [130, 131, 132] and their integration with Trans- +former models can become prominent since they are capable +of creating synthetic data that is comparable to genuine data. +In addition, another way to attack this problem is by utiliz- +ing federated learning, such as [41]. Nevertheless, there is +still room for improvement when it comes to privacy con- +cerns since, in federated learning, communication between +the client and server is required. +Despite their SOTA performance, Transformer-based net- +10 + +Table 2: A brief description of the reviewed Transformer-based medical image classification models. The unreported number of parameters indicates that the +value was not mentioned in the paper, and the code was unavailable. +Method +# Params +Contributions +Highlights +Pure +ViT-vs-CNN +[42] +22M +• They investigate three different weight initialization approaches on three medical datasets: (1) random +initialization, (2) transfer learning using supervised ImageNet pre-trained weights, and (3) self-supervised +pretraining on the target dataset using DINO [65]. Their final verdict is that ViTs can replace CNNs. +• Utilize three different training schemes on three different datasets to conclude whether ViT can replace CNN. +• They repeat their processes five times to be certain of the outcome. +• Comparing only two models, DeiT-S and Resnet-50, on only three datasets cannot generalize the conclusion of the superiority of each of the +Transformer and CNN. +ViT-BUS +[43] +ViT-Ti/16: 5.53M +ViT-S/32: 22.87M +ViT-B/32: 87.44M +• Proposes the use of ViT for the classification of breast ultrasound images for the first time. +• Transferring pre-trained ViT models based on small ultrasound datasets yields much higher accuracy than CNN models. +POCFormer +[45] +Binary CLS: 2.8M +Multiclass CLS: 6.9M +• Proposes a lightweight Transformer architecture that uses lung ultrasound images for real-time detection +of COVID-19. +• POCFormer can perform in real-time and be deployed to portable devices. +• POCFormer can be used for rapid mass testing of COVID-19 due to its compactness. +MIL-VT +[39] +22.12M +• Proposes to first pre-train the Vision Transformer on a fundus image large dataset and then fine-tune it on +the downstream task of the retinal disease classification. +• Introduces the MIL-VT framework with a novel Multiple Instance Learning(MIL)-head to effectively +utilize embedding features to improve the ViT performance. +• The MIL head can significantly enhance the performance by easily attaching to the Transformer in a plug-and-play manner. +• MIL-VT efficiently exploits the embedding features overlooked in the final prediction of the ViT. +COVID-VIT +[38] +52.81M +• Offers utilizing ViT to classify COVID and non-COVID patients using 3D CT lung images. +• COVID-ViT performs better in classifying COVID from Non-COVID compared to DenseNet. +• The reported result is not enough to conclude. They only compare ViT with DenseNet. +xViTCOS +[40] +85.99 +• Proposes and explores using ViT for detecting COVID-19 from CXR and CT images. +•2 xViTCOS makes use of the Gradient Attention Rollout algorithm [75] for visualization and clinical +interpretability of the output. +• Uses a heatmap plot to demonstrate the model’s explainability. +FESTA +[41] +Body: 66.37M +(CLS) Head: 13.31M, Tail: 2k +(SEG) Head: 15.04M, Tail: 7.39M +(DET) Head: 25.09M, Tail: 19.77M +• Proposes a Federated Split Task-Agnostic (FESTA) framework that leverages ViT to merit from federated +learning and split learning. +• They use multi-task learning to classify, detect, and segment COVID-19 CXR images. +• The proposed FESTA Transformer improved individual task performance when combined with multi-task learning. +• Experimental results demonstrate stable generalization and SOTA performance of FESTA in the external test dataset even under non- +independent and non-identically distributed (non-IID) settings. +• FESTA eliminates the need for data exchange between health centers while maintaining data privacy. +• Using FESTA in the industry may not be safe because it may encounter external attacks on the server that may lose the parameters of the +entire network. In addition, using this method may jeopardize patient information through privacy attacks. +• Authors did not evaluate the robustness of their approach to difficulties through communication, stragglers, and fault tolerance. +COVID-Transformer +[44] +ViT-L/16: 307M +• COVID-Transformer investigates using ViT for detecting COVID-19 from CXR images. +• COVID-Transformer introduces a new balanced chest X-ray dataset containing 30K images for multi- +class classification and 20K for binary classification. +• Uses a heatmap plot to demonstrate the model’s explainability. +COVID-VOLO +[47] +86.3M +• Proposes fine-tuning the pre-trained VOLO [82] model for COVID-19 diagnosis. +• Using VOLO enables capturing both fine-level and coarse-level features resulting in higher performance in COVID-19 binary classification. +RadioTransformer +[46] +3.93M +• Presents a novel global-focal RadioTransformer architecture, including Transformer blocks with shifting +windows, to improve diagnosis accuracy by leveraging the knowledge of experts. +• Introduces an innovative technique for training student networks by utilizing visual attention regions +generated by teacher networks. +• Outperform counterpart backbones on multiple datasets. +• Model’s explainability +Hybrid +TransMIL +[54] +2.67M +• Presents a Transformer-based Multiple Instance Learning (MIL) approach that uses both morphological +and spatial information for weakly supervised WSI classification. +• Proposes to consider the correlation between different instances of WSI instead of assuming them inde- +pendently and identically distributed +• Proposes a CNN-based PPEG module for conditional position encoding, which is adaptive to the number +of tokens in the corresponding sequence +• Converges two to three times faster than SOTA MIL methods. +• The proposed method can be employed for unbalanced/balanced and binary/multiple classification with great visualization and interpretability. +• TransMIL is adaptive for positional encoding as token numbers in the sequences changes. +• It needs further improvement to handle higher magnification than ×20 of WSIs - Higher magnification means longer sequences, which in turn +require more memory and computational costs to process. +LAT +[52] +- +• Proposes a unified Transformer-based encoder-decoder structure capable of DR grading and lesion detec- +tion simultaneously. +• Proposes a Transformer-based decoder to formulate lesion discovery as a weakly supervised lesion local- +ization problem. +• Proposes lesion region importance mechanism to determine the importance of lesion-aware features. +• Proposes lesion region diversity mechanism to diversify and compact lesion-aware features. +• Unlike most approaches that confront lesion discovery and diabetic retinopathy grading tasks independently, which may generate suboptimal +results, the proposed encoder-decoder structure is jointly optimized for both tasks. +• Despite existing methods that only perform well for discovering explicit lesion regions, LAT can also detect less dense lesion areas. +• The proposed LAT is capable of identifying Grades 0 and 1, which are hard to distinguish. +TransMed +[48] +TransMed-T: 17M +TransMed-S: 43M +TransMed-B: 110M +TransMed-L: 145M +• Proposes a hybrid CNN-Transformer network for multimodal medical image classification. +• Proposes a novel image fusion strategy for 3D MRI data. +• TransMed achieves much higher accuracy in classifying parotid tumors and knee injuries than CNN models. +• Requiring fewer computational resources compared to SOTA CNNs. +3DMeT +[49] +- +• Replaces conventional linear embedding with 3D convolution layers to reduce the computational cost of +using 3D volumes as the Transformer’s inputs. +• Obtains weights for 3D convolution layers by using a teacher-student training strategy. +• The proposed method makes the Transformers capable of using 3D medical images as input. +• 3DMeT Uses significantly fewer computational resources. +• Adopting CNN as a teacher assists in inheriting CNN’s spatial inductive biases. +Hybrid-COVID-ViT +[51] +- +• Proposes a Vision Transformer that embeds features for high-level COVID-19 diagnosis classification +using a backbone trained to spot low-level abnormalities in CXR images. +• Different from SOTA models, the proposed model does not use the ImageNet pre-trained weights while archiving significantly better results. +• They examine the interpretability of the proposed model. +Femur-ViT +[50] +- +• Investigates using ViT for classifying femur fractures. +• proposes using unsupervised learning to evaluate the ViT results. +• Achieves SOTA results compared to the CNN models. +GTP +[56] +- +• Proposes a graph-based vision Transformer (GTP) framework for predicting disease grade using both +morphological and spatial information at the WSIs. +• Proposes a graph-based class activation mapping (GraphCAM) method that captures regional and contex- +tual information and highlights the class-specific regions. +• They use a self-supervised contrastive learning approach to extract more robust and richer patch features. +• They exploit a mincut pooling layer [99] before the vision Transformer layer to lessen the number of +Transformer input tokens and reduce the model complexity. +• In contrast to SOTA approaches, the proposed GTP can operate on the entire WSI by taking advantage of graph representations. In addition, +GTP can efficiently classify disease gade by leveraging a vision Transformer. +• The proposed GTP is interpretable so that it can identify salient WSI areas associated with the Transformer output class. +• GTP obviates the need for adding extra learnable positional embeddings to nodes by using the graph adjacency matrix. It enables diminishing +the complexity of the model. +• The proposed GTP takes both morphological and spatial information into account. +DT-MIL +[53] +10.88M +• Presents a novel embedded-space MIL approach incorporated with an encoder-decoder Transformer for +histopathological image analysis. Encoding is done with a deformable Transformer, and decoding with a +classic ViT. +• An efficient method to render a huge WSI is proposed, which encodes the WSI into a position-encoded +feature image. +• The proposed method selects the most discriminative instances simultaneously by utilizing associated +attention weights and calibrating instance features using the deformable self-attention. +• The proposed method efficiently embeds instances’ position relationships and context information into bag embedding. +An extensive analysis of four different bag-embedding modules is presented on two datasets. +HATNet +[55] +(w/ MobileNetv2): 5.59M +(w/ ESPNetv2): 5.58M +(w/ MNASNet): 5.47M +• Presents a novel end-to-end hybrid method for classifying histopathological images. +• HATNet surpasses the bag-of-words models by following a bottom-up strategy and taking into account inter-word, word-to-bag, inter-bag, +and bag-to-image representations, respectively. (word → bag → image) +works still face challenges in deploying their models in the +real world due to computational limitations. As shown in +Table 2, most approaches have a high number of parame- +ters which provokes a serious problem. Different novel ap- +proaches have been introduced to reduce the quadratic com- +plexity of self-attention, which can be leveraged in the med- +ical domain. +Furthermore, though ViTs have shown im- +pressive capabilities in ImageNet classification, their per- +formance is still lower than the latest SOTA CNNs with- +out additional data [47]. Hence, existing methods mostly +follow pre-training strategies on the ImageNet dataset to +build the pre-trained weights for the subsequent downstream +tasks. However, despite the enhancement, the domain of +natural images is significantly different from medical data, +thereby may restrict the performance of further improve- +ment. Therefore, we believe efficient Transformers will con- +siderably influence the future research of Transformer-based +models. +4. Medical Image Segmentation +Medical segmentation is a significant sub-field of image seg- +mentation in digital image processing [133]. +It aims to ex- +tract features from a set of regions partitioned from the en- +tire image and segment the key organs simultaneously, which +can assist physicians in making an accurate diagnosis in prac- +tice. X-ray, positron emission tomography (PET), computed +11 + +tomography (CT), magnetic resonance imaging (MRI), and ul- +trasound are common imaging modalities used to collect data. +The CNN-based U-Net [34, 133] has been the main choice +in this field due to its effective performance and high accu- +racy. Nevertheless, it cannot extract long-range dependencies +in high-dimensional and high-resolution medical images [134]. +Therefore, the flexible combination of the U-Net structure with +Transformers become a prevalent solution to the segmentation +problem at present. Take the multi-organ segmentation task as +an example: some networks can achieve state-of-the-art multi- +organ segmentation performance on the Synapse dataset (as +shown in Figure 9) for abdominal images. +Figure 9: Transformer-based models can perform image segmentation on med- +ical image datasets. Figure a and c illustrate two 2D slices of raw images with +the labels from Synapse dataset [135]. Figure b and d show the 3D visualiza- +tion of the labeled organs from different angles. These images were generated +with MITK Workbench [136]. +In this section, we present the application of ViTs in seg- +mentation tasks. First, we divide the approaches into two cate- +gories: pure Transformers and hybrid Transformers, where the +pure Transformer denotes the use of the multiple multi-head +self-attention modules in both the encoder and decoder. Hy- +brid architecture-based approaches fuse the ViTs with convolu- +tion modules as the encoder, bottleneck, decoder, or skip con- +nection part to leverage information about the global context +and local details. Furthermore, we review some methods with +other architectures that propose several novel manners for self- +supervised learning. Figure 10 demonstrates the different di- +rections of the methods employing Transformers in the U-Net +architecture. +4.1. Pure Transformers +In this section, we review several networks referred to as +pure Transformers, which employ Transformer blocks in both +the encoding and the decoding paths. Despite the great suc- +cess of CNN-based approaches in medical segmentation tasks, +these models still have limitations in learning long-range se- +mantic information of medical images. The authors proposed +Swin-Unet, a symmetric Encoder-Decoder architecture moti- +vated by the hierarchical Swin Transformer [57], to improve +segmentation accuracy and robust generalization capability. In +contrast to the closest approaches [142, 141, 140, 149] using +integrations of CNN with Transformer, Swin-Unet explores the +possibility of pure Transformer applied to medical image seg- +mentation. +As shown in Figure 11, Swin-Unet consists of encoder, bot- +tleneck, decoder, and skip connections utilizing the Swin Trans- +former block with shifted windows as the basic unit. For the +encoder, the sequence of embeddings transformed from im- +age patches is fed into multiple Swin Transformer blocks and +patch merging layers, with Swin Transformer blocks perform- +ing feature learning, and patch merging layers downsampling +the feature resolution and unifying the feature dimension. The +designed bottleneck comprises two consecutive Swin Trans- +former blocks to learn the hierarchical representation from the +encoder with feature resolution and dimension unchanged. +Swin Transformer blocks and patch-expanding layers con- +struct the symmetric Transformer-based decoder. In contrast to +the patch merging layers in the encoder, each patch expanding +layer is responsible for upsampling the feature maps into double +resolutions and halving the corresponding feature dimension. +The final reshaped feature maps pass through a linear projec- +tion to produce the pixel-wise segmentation outputs. Inspired +by the U-Net, the framework also employs skip connections +to combine multi-scale features with the upsampled features at +various resolution levels to reduce the loss of fine-grained con- +textual information caused by down-sampling. +In contrast to the CNN-based methods showing over- +segmentation issues, the proposed U-shape pure Transformer +presents better segmentation performance resulting from learn- +ing both local and long-range dependencies. Compared to the +previous methods [150, 24], the HD evaluation metric of Swin- +Unet shows an improvement in accuracy for better edge predic- +tion. The experiments on the Synapse multi-organ CT dataset +and ACDC dataset from MRI scanners also demonstrate the ro- +bustness and generalization ability of the method. +Compared to Swin-Unet and DS-TransUNet, nnFormer +[137] proposed by Zhou et al. preserves the superior perfor- +mance of convolution layers for local detail extraction and em- +ploys a hierarchical structure to model multi-scale features. It +utilizes the volume-based multi-head self-attention (V-MSA) +and the shifted version (SV-MSA) in the Transformer blocks +instead of processing 2D slices of the volume. The overall ar- +chitecture of nnFormer is composed of an encoder and a de- +coder. Each stage in the encoder and decoder consists of a +Transformer block applying V-MSA and SV-MSA and a suc- +cessive upsampling or downsampling block built upon convo- +lution layers, which is referred to as the interleaved architec- +ture. V-MSA conducts self-attention within 3D local volumes +instead of 2D local windows to reduce the computational com- +plexity by approximately 98% and 99.5% on the Synapse and +ACDC datasets, respectively. +nnFormer is first pre-trained on the ImageNet dataset and uti- +lizes symmetrical initialization to reuse the pre-trained weights +of the encoder in the decoder. The results of experiments that +compare nnFormer with prior Transformer-based [24, 32] and +CNN-based arts [151] illustrate nnFormer makes significant +progress on the segmentation task. +12 + +a +b +c +d Medical Image + Segmentation + Pure + 1. Swin-Unet + 2. nnFormer + 3. MISSFormer + 4. TransDeepLab + Hybrid + Decoder + 5. SegTran + Encoder + 6. Trans-UNet + 7. TransBTS + 8. TransFuse + 9. MedT + 10. UNETR + 11. Swin UNETR + Skip Connection + 12. CoTr + 13. HiFormer + Other Architectures + 14. T-AutoML + 15. Cross Teaching + 16. Self-pretraining with MAE +Figure 10: An overview of ViTs in medical image segmentation. Methods are classified into the pure Transformer, hybrid Transformer, and other architectures +according to the positions of the Transformers in the entire architecture. The prefix numbers of the methods denote 1. [32], 2. [137], 3. [138], 4. [33], 5. [139], +6. [24], 7. [140], 8. [141], 9. [142], 10. [143], 11. [144], 12. [145], 13. [37], 14. [146], 15. [147], 16. [148]. +. +Figure 11: The architecture of the Swin-Unet [32] which follows the U-Shape +structure. It contains the encoder, the bottleneck and the decoder part which are +built based on the Swin Transformer block. The encoder and the decoder are +connected with skip connections. +Although the recent Transformer-based methods improve the +problem that CNN methods cannot capture long-range depen- +dencies, they show the limitation of the capability of modeling +local details. Some methods directly embedded the convolu- +tion layers between fully-connected layers in the feed-forward +network. Such structure supplements the low-level information +but limits the discrimination of features. Huang et al. propose +MISSFormer [138], a hierarchical encoder-decoder network, +which employs the Transformer block named Enhanced Trans- +former Block and equips the Enhanced Transformer Context +Bridge. +The Enhanced Transformer Block utilizes a novel efficient +self-attention module that illustrates the effectiveness of spa- +tial reduction for better usage of the high-resolution map. The +original multi-head self-attention can be formulated as follows: +Attention(Q, K, V) = S oftmax( QKT +√dhead +)V, +(4) +where Q, K, and V refer to query, key and value respectively +and have the same shape of N × C, dhead denotes the number +of heads. The computational complexity is O(N2). In efficient +self-attention, the K and V are reshaped by a spatial reduction +ratio R. Take K for example: +new K = Reshape(N +R ,C · R)W(C · R,C) +(5) +K is first resized from N × C to N +R × (C · R) and then pro- +jected linearly to restore the channel depth from C · R to C. +The computational cost reduces to O( N2 +R ) accordingly. Further- +more, the structure of the Enhanced Mix Feed-forward network +(Mix-FFN) extended from [152] introduces recursive skip con- +nections to make the model more expressive and consistent with +each recursive step. +The U-shaped architecture of the MISSFormer contains the +encoder and decoder built on the Enhanced Transformer blocks +connected with an enhanced Transformer context bridge. +Multi-scale features produced from the encoder are flattened +and concatenated together and passed through the Enhanced +Transformer Context Bridge. +The pipeline of the Enhanced +Transformer Context Bridge is based on the Enhanced Trans- +former Block to fuse the hierarchical features. The output of the +bridge is split and recovered to each original spatial dimension +to pass through the corresponding stage in the decoder. The +results of experiments show a robust capacity of the method to +capture more discriminative details in medical image segmen- +tation. It is worth mentioning that MISSFormer trained from +scratch even outperforms state-of-the-art methods pre-trained +on ImageNet. +The results in Figure 12 show that the performance of +MISSFormer for prediction and segmentation of edges in pure +13 + +W +H +-× 48 +Patch Partition +Linear Projection +X +W x H xClass +4 +4 +Linear Embedding +Patch Expanding +W ×H ×C(4x) +Skip Connection +Swin Transformer +1/4 +Swin Transformer +W +H +x +4 +4 +Block x2 +Block x2 +4. +4 +Patch Merging +Patch Expanding +Skip Connection +1/8 +W +H +Swin Transformer +Swin Transformer +W +H +-×2C +×2C +8 +8 +Block x2 +Block x2 +8 +8 +Patch Merging +Patch Expanding +Skip Connection +1/16 +W +H +Swin Transformer +Swin Transformer +W +H +×4C +×4C +16＇16 +Block x2 +Block x2 +16＇16 +4 +Patch Merging +Patch Expanding +Encoder +Decoder +W..H +Swin Transformer +Swin Transformer +×8C +32＇～32 +Block x1 +Block x1 +BottleneckTransformer network structures is more accurate compared +to TransUNet and Swin-Unet. Comparing MISSFormer and +MISSFormer-S (MISSFormer without bridge), MISSFormer +has fewer segmentation errors because the bridge is effective +for integrating multi-scale information. +Figure 12: A visual comparison with the state-of-the-art approaches on Synapse +dataset. Above the red line shows the successful cases of segmentation, and +below the red line are the failed cases with relatively large errors [138] +. +Inspired by the notable DeepLabv3 [153] which utilizes the +Atrous Spatial Pyramid Pooling (ASPP) to learn multi-scale +feature representations and depth-wise separable convolution +to reduce the computational burden, the authors propose Trans- +DeepLab [33] to combine the DeepLab network with the Swin- +Transformer blocks. Applying the Swin-Transformer module +with windows of multiple sizes enables the fusion of multi- +scale information with a lightweight model. +TransDeepLab is a pure Transformer-based DeepLabv3+ ar- +chitecture, as shown in Figure 13. +The model builds a hi- +erarchical architecture based on the Swin-Transformer mod- +ules. TransDeepLab first employs N stacked Swin-Transformer +blocks to model the input embedded images into a deep-level +feature space. 2D medical images are first to split into non- +overlapping patches of dimension C and size 4 × 4. The ensu- +ing Swin-Transformer block learns local semantic information +and global contextual dependencies of the sequence of patches. +Then, the authors introduce windows with different sizes to pro- +cess the output of the Swin-Transformer and fuse the resulting +multi-scale feature layers, which are then passed through Cross +Contextual attention. This design, referred to as Spatial Pyra- +mid Pooling (SSPP) block, replaces the original Atrous Spa- +tial Pyramid Pooling (ASPP) module exploited in DeepLabV3. +A cross-contextual attention mechanism is utilized to explore +the multi-scale representation after fusion. This attention mod- +ule applies channel attention and spatial attention to the out- +put from windows of each size (from each layer of the spatial +pyramid). Finally, in the decoder part, the low-level features +from the encoder are concatenated with the multi-scale features +extracted by Cross Contextual Attention after bilinear upsam- +pling. The last two Swin-Transformer blocks and the patch ex- +panding module generate the final prediction masks. +Stacked N blocks +… +… +Cross Contextual +Attention +Multi-scale Representation +Encoder +Low-level +Features +Concat +Upsample +Decoder +Idea: Pure transformer model to model DeepLab3 model with additional attention mechanism +1. +Using swim transformer strategy to reduce time complexity +2. +Transformer structure to better model long-range contextual dependency +3. +Novel structure using transformer +Swin Transformer +Block × 𝟐 +Patch Merging +Patch Partition +Linear Embedding +2 × 2 Window +7 × 7 Window +𝐼𝑚𝑎𝑔𝑒 𝑃𝑜𝑜𝑙 +Swin +Transformer +Block × 𝟐 +Patch +Expanding +… +… +Figure 13: The overview architecture of TransDeepLab, which comprises +encoder and decoder built on Swin-Transformer blocks. +It is the pure +Transformer-based extension of DeepLabv3++ [33]. +4.2. Hybrid Models +Hybrid Transformers concatenate Transformer blocks with +convolution layers to extract local details and long-range de- +pendencies. We further classify this category into Transformer: +Encoder, Transformer: Decoder and Transformer: skip con- +nection according to the position of the combined module in +the U-Net architecture. +4.2.1. Transformer: Encoder +Starting with TransUNet [24], multiple methods in the med- +ical image segmentation field adopt the self-attention mecha- +nism in the encoder. +Transformers have developed as an alternative architecture +for modeling global context that exclusively relies on attention +mechanisms instead of convolution operators. However, its in- +ner global self-attention mechanism induces missing low-level +details. Direct upsampling cannot retrieve the local informa- +tion, which results in inaccurate segmentation results. The au- +thors propose the TransUNet architecture, a hybrid approach +that integrates CNN-Transformer hybrid as the encoder and +cascaded upsampler as the decoder, combining the advantages +of Transformer and U-Net to boost the segmentation perfor- +mance by recovering localized spatial information. +The framework of the TransUNet is illustrated in Figure 14. +The proposed encoder initially employs CNN as a feature ex- +tractor to build a feature map for the Transformer input layer, +rather than the Transformer directly projecting the raw tok- +enized picture patches to latent embedding space. In this way, +the intermediate CNN feature maps of different resolutions can +be saved and utilized in the following process. +For the decoder, the Cascaded Upsampler (CUP) is proposed +to replace naive bilinear upsampling, applying several upsam- +pling blocks to decode the hidden feature and output the final +segmentation result. Finally, the hybrid encoder and the CUP +constitute the overall architecture with skip connections to fa- +cilitate feature aggregation at different resolution levels. This +strategy can compensate for the loss of local fine-grained de- +tails caused by the Transformer encoder and merge the encoded +14 + +(b) MISSFormer +(a) GroundTruth +(c) MISSFormer_S +(d) Swin-Unet +(e) TransUnetglobal information with the local information contained in in- +termediate CNN feature maps. +The experiments show that TransUNet significantly outper- +forms the model consisting of pure Transformer encoder and +naive upsampling, as well as the ViT-hybrid model without +skip connections [24]. Comparisons with prior work [34, 150] +also demonstrate the superiority of TransUNet over competing +CNN-based approaches in terms of both qualitative visualiza- +tion and the quantitative evaluation criteria (i.e.average DSC +and HD). TransUNet integrates the benefits of both high-level +global contextual information and low-level details as an alter- +native approach for medical image segmentation. +reshape +1/4 +1/8 +1/2 +Conv3x3, ReLU +Upsample +Segmentation head +(n_patch, D) +(D, H/16, W/16) +(512, H/16, W/16) +(256, H/8, W/8) +(128, H/4, W/4) +(64, H/2, W/2) +(16, H, W) +Transformer Layer +… +(n = 12) +Hidden Feature +Linear Projection +CNN +Hidden Feature +Downsample +Feature Concatenation +Transformer Layer +Embedded Sequence +𝒙𝒑𝟏, 𝒙𝒑𝟐, … , 𝒙𝒑𝑵 +Layer +Norm +MSA +Layer +Norm +MLP ++ ++ +𝒛𝟏 +(a) +(b) +Figure 14: The overview architecture of the TransUNet [24]. The Transformer +layers are employed in the encoder part. The schematic of the Transformer is +shown on the left. +Wang et al. [140] propose the encoder-decoder architecture, +TransBTS, which leverages Transformer on learning global +contextual information and merits the 3D CNN for modeling +local details. In contrast to the concurrent Transformer-based +model [24], which analyzes 3D medical volumetric data in a +slice-by-slice manner, TransBTS also explores the local fea- +tures along the depth dimension by processing all the image +slices at once. +The network encoder initially employs a 3D CNN to capture +volumetric spatial features, simultaneously downsampling the +input 3D images, yielding compact volumetric feature maps. +Each feature map is projected into a token and fed into the +Transformer encoder to investigate the global relationships. +The full-resolution segmentation maps are generated by the +3D CNN decoder after the progressive upsampling while us- +ing the feature embedding from the Transformer. For the en- +coder part, TransBTS first utilizes the 3 × 3 × 3 convolution +blocks with downsampling to process the 3D input medical im- +age data, which boosts the effective embedding of rich local +3D features a cross spatial and depth dimensions into the low- +resolution feature representation F. They apply a linear projec- +tion to the feature representation F to obtain the sequence f, +which is then integrated with position embeddings, as the input +for the Transformer encoder. The Transformer encoder con- +sists of multiple Transformer layers, each of which comprises +a Multi-Head Attention(MHA) block and a Feed-Forward Net- +work(FFN). The output sequence of the Transformer encoder +passes through the feature mapping module to be reshaped to a +4D feature map Z of the same dimension as F. The approach +employs cascaded upsampling and convolution blocks to pro- +gressively restore the segmentation predictions at the original +resolution. Furthermore, skip connections combine the fine- +grained details of local information with the decoder modules, +resulting in more accurate segmentation masks. +The authors conduct comparisons between the proposed +TransBTS and the closest method TransUNet [24]. TransUNet +essentially processes 3D medical images slice by slice, while +TransBTS is a 3D model that explores the continuous interac- +tion through the depth dimension by processing a 3D medical +image in a single pass. In contrast to TransUNet, which adopts +pre-trained ViT models on other large-scale datasets, TransBTS +is trained on the dataset for the specified task without relying on +pre-trained weights. +The framework is evaluated on the Brain Tumor Segmenta- +tion (BraTS) 2019 challenge and 2020 challenge. Compared +to the 3D U-Net baseline, TransBTS achieves a significant en- +hancement in segmentation. The prediction results indicate the +improved accuracy and the superiority of modeling long-range +dependencies. +Previous approaches [141] primarily focus on replacing +convolution operation with Transformer layers or consecu- +tively stacking the two together to address the inherent lack +of pure Transformer-based models to learn local information. +In this study, the authors propose a new strategy-TransFuse +which consists of the CNN-based encoder branch and the +Transformer-based branch in parallel fused with the proposed +BiFusion module, thus further exploring the benefits of CNNs +and Transformers. The construction of the Transformer branch +is designed in the typical encoder-decoder manner. The input +images are first split into non-overlapped patches. The linear +embedding layer then projects the flattened patches into the raw +embedding sequence which is added to the learnable position +embedding of the same dimension. The obtained embeddings +are fed into the Transformer encoder, which comprises L layers +of MSA and MLP. The output of the last layer of the Trans- +former encoder is passed through layer normalization to obtain +the encoded sequence. +The decoder part utilizes the same progressive upsampling +(PUP) approach as SETR [154]. The encoded sequence is first +reshaped back to a sequence of 2D feature maps. Then they +employ two stacked upsampling-convolution layers to restore +the feature scales. The feature maps with different spatial res- +olutions generated by each upsampling-convolution layer are +retained for the subsequent fusion operation. +For the CNN +branch, the approach discards the last layer of the traditional +CNNs architecture and combines the information extracted +from the CNNs with the global contextual features obtained +from the Transformer branch. A shallower model is yielded as +a result of this design, avoiding the requirement for extremely +deep models that exhaust resources to get long-range depen- +dencies. For instance, there are five blocks in a typical ResNet- +based network where only the outputs of the 4th, 3rd, and 2nd +layers are saved for the following fusion with the feature maps +from the Transformer branch. +The BiFusion module is proposed to fuse the features ex- +tracted from the two branches mentioned above to predict the +segmentation results of medical images. The global features +15 + +from the Transformer branch are boosted by the channel atten- +tion proposed in SE-Block [13]. Meanwhile, the feature maps +from the CNN branch are filtered by the spatial attention which +is adopted in CBAM [155] block to suppress the irrelevant and +noisy part and highlight local interaction. Then the Hadamard +product is applied to the features from the two branches to learn +the interaction between them. They concatenate the interaction +feature bi with attended features ˆti and ˆgi and feed the results +through a residual block to produce the feature f i, which suc- +cessfully models both the global and local features at the orig- +inal resolution. Finally, the segmentation prediction is gener- +ated by integrating the f i from different BiFusion modules via +the attention-gated (AG) [156] skip connection. +They evaluate the performance of three variants of Trans- +Fuse on four segmentation tasks with different imaging modal- +ities and target sizes. +TransFuse-S is constructed with +ResNet-34 (R34) and 8-layer DeiT-Small (DeiT-S) [64]. Be- +sides, TransFuse-L is composed of Res2Net-50 and 10-layer +DeiT-Base (DeiT-B). TransFuse-L* is implemented based +on ResNetV2-50 and ViT-B [22]. +For polyp segmenta- +tion, Transfuse-S/L outperforms significantly the CNN base- +line models with fewer parameters and faster running time. +TransFuse-L* also achieves the best performance among the +previous SOTA Transformer-based methods with a faster speed +for inference. It runs at 45.3 FPS and about 12% faster than +TransUNet. The experiments for other segmentation tasks also +show the superiority of the segmentation performance. +Despite the powerful results of applying Transformers to seg- +mentation tasks [157, 154], the dilemma is that properly train- +ing existing Transformer-based models requires large-scale +datasets, whereas the number of images and labels available +for medical image segmentation is relatively limited. To over- +come the difficulty, MedT [142] proposes a gated position- +sensitive axial attention mechanism where the introduced gates +are learnable parameters to enable the model to be applied to a +dataset of arbitrary size. Furthermore, they suggested a Local- +Global(LoGo) training strategy to improve the segmentation +performance by operating on both the original image and the +local patches. +The main architecture of MedT, as shown in Figure 15 (a), +is composed of 2 branches: a shallow global branch that works +on the original resolution of the entire image, and a deep local +branch that acts on the image patches. Two encoder blocks and +two decoder blocks comprise the global branch, which is suf- +ficient to model long-range dependencies. In the local branch, +the original image is partitioned into 16 patches and each patch +is feed-forwarded through the network. +The output feature +maps are re-sampled based on their locations to obtain the out- +put feature maps of the branch. Then the results generated from +both branches are added and fed into a 1 × 1 convolution layer +to produce the output segmentation mask. The LoGo training +strategy enables the global branch to concentrate on high-level +information and allows the local branch to learn the finer inter- +actions between pixels within the patch, resulting in improved +segmentation performance. +Figure 15 (b) and (c) illustrates the gated axial Transformer +layer, which is used as the main building block in MedT, and +Global Branch +Local Branch +Encoder +Block +Decoder +Block +Image +Segmentation +Mask +1x1 +Conv +Add +Conv +1x1 +Norm +Gated +Multi- +Head +Attn +Height +Gated +Multi- +Head +Attn +Width +Conv +1x1 ++ +Norm +Input +Encoder - Gated Axial Transformer Layer +(a) +(b) +Conv +Block +X +WV +WK +WQ +rQ +rK +GQ +GK +rV +GV1 +GV2 +softmax +yY +Gates +Positional +Embeddings +Weights +Matrix +Multiplication +Addition +(c) +Gated Axial Attention Layer +Resample +Patches +Patches +Figure 15: Overview of the MedT [142] architecture. The network uses the +LoGo strategy for training. The upper global branch utilizes the first fewer +blocks of the Transformer layers to encode the long-range dependency of the +original image. In the local branch, the images are converted into small patches +and then fed into the network to model the local details within each patch. The +output of the local branch is re-sampled relying on the location information. +Finally, a 1 × 1 convolution layer fuses the output feature maps from the two +branches are to generate the final segmentation mask. +the feed-forward structure in it. They introduced four learnable +gates GV1,GV2,GQ,GK ∈ R that control the amount of informa- +tion the positional embeddings supply to key, query, and value. +Based on whether a relative positional encoding is learned ac- +curately or not, the gate parameters will be assigned weights +either converging to 1 or some lower value. The gated mecha- +nism can control the impact of relative positional encodings on +the encoding of non-local context and allows the model to work +well on any dataset regardless of size. +Unlike the fully-attended baseline [157], MedT trained on +even smaller datasets outperforms the convolutional baseline +and other Transformer-based methods. In addition, improve- +ments in medical segmentation are also observed since the pro- +posed method takes into account pixel-level dependencies. +In contrast to multiple proposed methods [154, 24, 142, 141] +that investigate the task of 2D medical image segmentation, +UNETR [143] proposes a novel Transformer-based architec- +ture for 3D segmentation which employs the Transformer as +the encoder to learn global contextual information from the +volumetric data. In addition, unlike the previous frameworks +proposed for 3D medical image segmentation [145, 140], the +encoded feature from the Transformer of this proposed model +is directly connected to a CNN-based decoder via skip connec- +tions at different resolution levels. The U-shaped UNETR com- +prises a stack of Transformers as the encoder and a decoder +coupling with it by skip connections. They begin by generating +the 1D sequence of patches by splitting the 3D input volume +in a non-overlapping manner. The flattened input patches are +then passed through a linear projection layer to yield K dimen- +sional patch embeddings. They attach a 1D learnable positional +embedding to each patch embedding taking into account the +spatial information of the extracted patches. After the embed- +ding layer, the global multi-scale representation is captured us- +ing Transformer blocks composed of multi-head self-attention +16 + +modules and multilayer perceptron layers. +They resize and +project the sequence representation extracted from the Trans- +former at different resolutions for use in the decoder in order to +retrieve spatial information of the low-level details. +In the expanding pattern of the framework, the proposed +CNN-based decoder combines the output feature of different +resolutions from the Transformer with upsampled feature maps +to properly predict the voxel-wise segmentation mask at the +original input resolution. +The paper claims UNETR achieves new state-of-the-art per- +formance on all organs compared against CNN [158, 35, 159, +160] and competing for Transformer-based [145, 24, 154] base- +lines on BTCV dataset, with significant improvement in perfor- +mance on small organs in particular. In addition, it outperforms +the closest methodologies on brain tumor and spleen segmen- +tation tasks in MSD dataset. UNETR shows the superiority of +learning both global dependencies and fine-grained local rela- +tionships in medical images. +Figure 16 presents qualitative segmentation comparisons for +brain tumor segmentation on the MSD dataset between UNETR +[143], TransBTS [140], CoTr [145] and U-Net [34]. It can be +seen that the details of the brain tumor are captured well by +UNETR [143]. +Ground Truth +0.86 +UNet +TransBTS +CoTr +UNETR +0.83 +Figure 16: Comparison of visualization of brain tumor segmentation on the +MSD dataset. The whole tumor (WT) includes a combination of red, blue, and +green regions. The union of red and blue regions demonstrates the tumor core +(TC). The green regions indicate the enhanced tumor core (ET) [143]. +As opposed to other methods which attempted to utilize the +Transformer module as an additional block beside the CNN- +based components in the architectures, UNETR [143] lever- +ages the Transformer as the encoder instead of the CNN-based +encoder. The Swin Transformer [57] is a hierarchical visual +Transformer featuring an efficient shift-window partitioning +scheme for computing self-attention. Inspired by these two ap- +proaches, a novel model termed Swin Unet Transformer (Swin +UNETR) [144] is proposed for brain tumor segmentation in +this work. +The proposed framework applies a U-shape architecture with +the Swin Transformers as the encoder and a CNN-based mod- +ule as the decoder connected to the encoder via skip connec- +tions at different resolution levels. The model initially converts +3D MRI images with four channels to non-overlapping patches +and creates windows of a specific size with a patch partition +layer. +The Swin UNETR encoder is composed of 4 stages. Each +stage comprises 2 Transformer blocks and a patch merging +layer. In the Transformer blocks, the self-attention is computed +with a shifted windowing mechanism. Swin UNETR employs +the windows of size M × M × M to partition the 3D token with +resolution of H′ × W′ × D′ into regions of ⌈ H′ +M × W′ +M × D′ +M ⌉ at +layer l. The partitioned window regions are then shifted by +(⌊ M +2 ⌋, ⌊ M +2 ⌋, ⌊ M +2 ⌋) voxels at the following l + 1 layer. The patch +merging layer after the Transformer components reduces the +resolution of feature maps by a factor of two and concatenates +them to form a feature embedding with the doubled dimension- +ality of the embedding space. +For the decoder of the architecture, the output feature repre- +sentations of the bottleneck are reshaped and passed through the +residual block containing two convolutional layers. The subse- +quent deconvolutional layer increases the resolution of feature +maps by a factor of 2. The outputs are then concatenated with +the outputs of the previous stage and fed into another residual +block. After the resolutions of the feature maps are restored to +the original H′ ×W′ × D′, a head is utilized to generate the final +segmentation predictions. +The authors conduct the experiments to compare Swin UN- +ETR against the previous methodologies SegResNet [161], nn- +UNet [151]and TransBTS [140] in this work. +The results +demonstrate that the proposed model has prominence as one +of the top-ranking approaches in the BraTS 2021 challenge. +It is due to the better capability of learning multi-scale con- +textual information and modeling long-range dependencies by +Swin Transformers in comparison to regular Transformers with +a fixed resolution of windows. +4.2.2. Transformer: Decoder +Another direction is to modify the decoder of the U-shape +structure to aggregate the Transformer-CNN-based modules. +In the Segtran framework [139], Squeeze-and-Expansion +Transformer is proposed to ”squeeze” the attention matrix and +aggregate multiple sets of contextualized features from the out- +put. A novel Learnable Sinusoidal Position Encoding is also +employed to impose the continuity inductive bias for images. +The Segtran consists of five components: a CNN backbone +to extract image features, 2) input/output feature pyramids to +do upsampling, 3) the Learnable Sinusoidal Positional Encod- +ing, 4) Squeeze-and-Expansion Transformer layers to contex- +tualize features, and 5) a segmentation head. The pretrained +CNN backbone is first utilized to learn feature maps from the +input medical images. Since the input features to Transformers +are of a low spatial resolution, the authors increase their spa- +tial resolutions with an input Feature Pyramid Network (FPN) +[162] to upsample the feature maps by bilinear interpolation. +Then the proposed Learnable Sinusoidal Positional Encoding +is added to the visual features to inject spatial information. In +contrast to the previous two mainstream PE schemes [163, 22], +the new positional embedding vector, a combination of sine and +cosine functions of linear transformations of (x, y), brings in the +continuity bias with adaptability. The equation of the encoding +strategy varies gradually with pixel coordinates. Thus, close +17 + +units receive similar positional encodings, increasing the atten- +tion weights between them towards higher values. The encod- +ing vectors generated from the addition of positional encodings +and visual features are then fed into the Transformer. +The novel Transformer architecture combines Squeezed At- +tention Block (SAB) [164] with an Expanded Attention Block. +Here this method employs the Induced Set Attention Block +(ISAB) proposed by [164] as a squeezed attention block. The +Squeezed Attention Block computes attention between the in- +put and inducing points and compresses the attention matrices +to lower rank matrices, reducing noises and overfitting. The +Expanded Attention Block (EAB), a mixture-of-experts model, +outputs Nm sets of complete contextualized features from Nm +modes. Each mode is an individual single-head Transformer +and shares the same feature space with each other. That is as +opposed to multi-head attention in which each head outputs an +exclusive feature subset. All features are then aggregated into +one set using dynamic mode attention. The dynamic mode at- +tention can be obtained by doing a linear transformation of each +mode feature and taking softmax over all the modes. +Compared with representative existing methods in the exper- +iments, Segtran consistently achieved the highest segmentation +accuracy and exhibited good cross-domain generalization capa- +bilities. +skip connection +conv + max pool 2x2 +up-conv 2x2 +Downsampling +path +Upsampling +path +Figure 17: Segtran network extracts image features using a CNN backbone and +combines the features with the position encoding of pixels flattened into a series +of local feature vectors. Multiple squeezed and extended transform layers are +stacked to process the local feature vectors. Finally, an output FPN after the +Transformer upsamples the features to generate the final prediction [139]. +4.2.3. Transformer: Skip Connection +In this section, Transformer blocks are incorporated into the +skip connections to facilitate the transmission of detailed infor- +mation from the encoder to the decoder. +Although Transformer-based methods overcome the limi- +tation of capturing long-range dependency, they present ex- +treme computational and spatial complexity in analyzing high- +resolution volumetric image data. Some studies [163, 24] em- +ploy hybrid structures, fusing CNN with Transformer in an at- +tempt to reduce the training requirement on huge datasets. The +recent approach, TransUNet [24], shows good performance. +However, it is difficult to optimize the model due to the inner +self-attention mechanism of the vanilla Transformer. First, it +takes a long time to train the attention, which is caused by ini- +tially distributing attention uniformly to each pixel within the +salient regions [21]. Second, a vanilla Transformer struggles to +handle multi-scale and high-resolution feature maps due to its +high computational cost. +Motivated by this, [145] proposes a novel encoder-decoder +framework, CoTr, which bridges CNN and Transformer. The +architecture exploits CNN to learn feature representations. An +efficient deformable self-attention mechanism in the Trans- +former is designed to model the global context from the ex- +tracted feature maps, which reduces the computational com- +plexity and enables the model to process high-resolution fea- +tures. And the final segmentation results are generated by the +decoder. +As shown in Figure 18, the DeTrans-encoder consists of +an input-to-sequence layer and multiple DeTrans Layers. The +input-to-sequence layer first flattens the feature maps at differ- +ent resolutions extracted from CNN-encoder into 1D sequences +{fl}L +l=1. +Then the corresponding 3D positional encoding se- +quence pl is supplemented with each of the flattened sequences +fl to complement the spatial information. The combined se- +quence is fed as the input into the DeTrans Layers. Each of +the DeTrans Layers is a composition of an MS-DMSA and a +Feed-Forward Network (FFN). In contrast to the self-attention +mechanism which casts attention to all the possible locations, +the proposed MS-DMSA layer only attends to a small set of +key sampling locations around a reference location. As a re- +sult, it can achieve faster convergence and lower computational +complexity. The skip connection is utilized after each DeTrans +Layer for preserving the low-level details of local information. +The output of the DeTrans-encoder is successively upsampled +by the pure CNN encoder to restore the original resolution. Be- +sides, they apply skip connections and a deep supervision strat- +egy to add fine-grained details and auxiliary losses to the pre- +diction outputs. +The results of experiments indicate CoTr with the hybrid ar- +chitecture has superiority of performance over the models with +pure CNN encoder or pure Transformer encoder. It also outper- +forms other hybrid methods like TransUNet [24] in processing +multi-scale 3D medical images with reduced parameters and +complexity. +HiFormer [37] is proposed to aggregate a fusion module +in the skip connections to learn richer representations. Fig- +ure 19 demonstrates the end-to-end network structure of the +strategy that incorporates the global dependencies learned with +the Swin Transformer and the detailed local features extracted +by the CNN modules. The encoder is composed of two hierar- +chical CNN, Swin Transformer modules and the novel Double- +Level Fusion module (DLF module). First, medical images +are fed into a CNN module to obtain a local fine-grained se- +mantic representation. After the CNN layer catches the shal- +low feature layers, HiFormer introduces the Swin Transformer +modules to complement the global feature information. The +Swin Transformer module employs windows of different sizes +to learn the dependencies between multiple scales. To reuse +18 + +DeTrans Layer +DeTrans Layer +... +F +F +F +R +R +R +DeTrans Layer +Positional encoding +Upsampling +Flatten +Reshape +CNN-encoder +DeTrans-encoder +Decoder +MS-DMSA +Feed Forward +Reference point +Layer Norm +Layer Norm +Deformable Transformer Layer +Figure 18: Overview of the CoTr [145] architecture. It is composed of a CNN- +encoder, a DeTrans-encoder and a decoder. The CNN-encoder models the lo- +cal information of the input images and provides the outputs at each stage. +The outputs of different resolutions are flattened, fused and passed through the +Deformable Transformer Layers along with positional encoding. The decoder +reshapes the processed sequences from the DeTrans-encoder and produces the +final predictions. +the shallow and deep multi-scale feature information in the en- +coder, HiFormer designs a novel skip connection module, the +DLF module. The deep-level semantic and shallow-level lo- +calization information are fed into the DLF module and fused +by the cross-attention mechanism. Finally, both generated fea- +ture maps are passed into the decoder to produce the final +segmentation prediction results. The experiments conducted +on the Synapse dataset [135], SegPC [165], and ISIC 2017 +dataset [166] demonstrate the superiority of the learning abil- +ity of HiFormer. Moreover, the lightweight model with fewer +parameters also exceeds CNN-based methods and previous +Transformer-based approaches with lower computational com- +plexity. +𝐶𝑜𝑛𝑣 1 × 1 +Patch merging +Patch merging +DLF Module +Conv Block +Segmentation +Head +Transformer +Encoder × 𝑺 +GAP +Cross +Attention +Transformer +Encoder × 𝑳 +GAP +𝐶𝑜𝑛𝑣 1 × 1 +𝐶𝑜𝑛𝑣 1 × 1 +𝐇/𝟒 × 𝐖/𝟒, 𝐃 +𝐇/𝟖 × 𝐖/𝟖, 𝟐𝐃 +𝐇/𝟏𝟔 × 𝐖/𝟏𝟔, 𝟒𝐃 +𝐇/𝟒 × 𝐖/𝟒, 𝐃′ +𝐇/𝟖 × 𝐖/𝟖, 𝟐𝐃′ +𝐇/𝟏𝟔 × 𝐖/𝟏𝟔, 𝟒𝑫′ +𝑯 × 𝑾 × 3 +ConvUp +Ps +Pl +× 6 +× 2 +× 2 +ConvUp +Figure 19: HiFormer comprises the CNN-Transformer encoder, the CNN-based +decoder and the Double-Level Fusion Module (DLF).The feature layers of the +shallowest level pl and of the deepest level ps are fed into the DLF module for +the fusion of hierarchical information. Blue blocks and orange blocks refer to +Swin Transformer and CNN modules, respectively [37]. +4.3. Other Architectures +Most ViT-based models rely on pre-training of large natu- +ral image datasets to obtain pre-weights and then solve down- +stream tasks by transfer learning. Several works explore train- +ing in a self-supervised or semi-supervised manner to effi- +ciently utilize medical image datasets of limited size or datasets +without manual labels. Furthermore, some approaches apply +Transformers to seek the design of architectures that implement +medical image segmentation, instead of using the Transformers +to act directly on the input image. +Unlike the previously proposed methods that employ the +Transformers to act directly on the medical image for feature +extraction, this method [146] adopts the AutoML for automat- +ically designing the network architecture without much human +heuristics or assumptions, where the Transformer is applied to +encode the embedding vector regarding the architecture config- +urations. The approach reduces the workload of algorithm de- +sign by automatically estimating ”almost” all the components +of the framework instead of manually designing for the network +and training strategies. That improves the model performance +of segmentation simultaneously. +The proposed Transformer-based T-AutoML inspired by +SpineNet [167] leverages neural architecture search (NAS) with +a larger search space to optimize the selection of the network +connections. This framework can connect the feature maps at +different spatial levels of the network with another one arbitrar- +ily, compared with the previous methods that only search for the +encoder-decoder U-shape networks [168, 169, 170]. The can- +didates of different blocks in the network consist of 3D residual +blocks, 3D bottleneck blocks, and 3D axial-attention blocks. +The residual blocks and bottleneck blocks are effective in alle- +viating the vanishing gradient. The axial-attention blocks are +applied to model the long-range dependency in the 2D medi- +cal images. Another upsampling layer (linear interpolation) is +utilized at the end of the architecture to produce the results of +feature maps at the original volume size. +To search for the optimal architecture and training configu- +ration, the authors first encode the necessary components in the +search space to form a one-dimensional vector v. The search +space contains candidates of different configurations with re- +gard to data augmentation, learning rates, learning rate sched- +ulers, loss function, the optimizer, the number and spatial reso- +lution of blocks, and block types. +After the obtainment of the encoding vector v, the proposed +new predictor predicts the binary relation of validation accuracy +values between vi and v j. The predictor employs the Trans- +former encoder to encode the vector v of varying lengths into +feature maps of a fixed resolution. Then the feature maps are +passed through the Multiple FC layers to generate the binary +relation predictions denoted as GTvi,vj. Since the predictor is +designed for ranking the vectors with respect to the accuracy +values and estimating the relations, the actual values of the pre- +dicted accuracy are not necessary to be calculated for each vec- +tor. Thus, the new predictor requires less overall training time. +The experiments indicate that the proposed method can +achieve the state of the art(SOTA) in lesion segmentation tasks +and shows superiority in transferring to different datasets. +Despite the promising results achieved by the CNNs and +Transformer-based methods with large-scale images, these ap- +proaches require expert labeling at the pixel/voxel level. Expen- +sive time costs and manual annotation limit the size of the med- +ical image dataset. Due to this dilemma, the proposed semi- +supervised segmentation [147] provides a low-cost and practi- +cal scheme, called Cross Teaching between CNN and Trans- +former, to train effective models using a little amount of cor- +19 + +rectly labeled data and a large amount of unlabeled or coarsely +labeled data. +Inspired by the existing works [171, 172, 173] for semi- +supervised learning which introduce perturbation at different +levels and encourage prediction to be consistent during the +training stage, the designed cross teaching introduces the per- +turbation in both learning paradigm-level and output-level. As +shown in Figure 20, each image within the training set con- +taining labeled and unlabeled images is fed into two different +learning paradigms: a CNN and a Transformer respectively. +For the unlabeled dataset with raw images, the cross teaching +scheme allows the cross supervision between a CNN ( f c +φ(.)) and +a Transformer(f t +φ(.)), which aims at integrating the properties +of the Transformer modeling the long-range dependency and +CNN t0 learn local information in the output level. +The unlabeled data initially passes through a CNN and a +Transformer respectively to generate predictions pc +i and pt +i. +pc +i = f c +φ(xi); pt +i = f t +φ(xi); +(6) +Then the pseudo labels plc +i and plt +i are produced in this manner: +plc +i = argmax(pt +i); plt +i = argmax(pc +i ); +(7) +The pseudo label plc +i used for the CNN training is generated by +the Transformer. Similarly, the CNN model provides pseudo +labels for Transformer training. The cross-teaching loss for the +unlabeled data is defined as follows: +Lctl = +LDice(pc +i , plc +i ) +������������������������ +supervision for CNNs ++ +LDice(pt +i, plt +i) +���������������������� +supervision for Transformers +(8) +which is a bidirectional loss function. One direction of the data +stream is from the CNN to the Transformer, and the other di- +rection is from the Transformer to the CNN. For the labeled +data, the CNN and Transformer are supervised by the ground +truth. The commonly-used supervised loss functions, i.e. the +cross-entropy loss and Dice loss, are employed to update model +parameters. +Lsup = Lce(pi, yi) + LDice(pi, yi) +(9) +where pi , yi represent the prediction and the label of image xi. +And the overall objective combining the cross-teaching branch +and supervised branch is defined as: +Ltotal = Lsup + λLctl +(10) +where λ is a weight factor, which is defined by time- +dependent Gaussian warming up function [174, 175]: +λ(t) = 0.1 · e +� +−5 +� +1− +ti +ttotal +��2 +(11) +The results of the ablation study indicate that the combina- +tion of CNN and Transformer in a cross-teaching way shows +superiority over the existing semi-supervised methods. Further- +more, the novel method has the potential to reduce the label +cost by learning from limited data and large-scale unlabeled +data. However, it is observed that achieving SOTA via semi- +supervised approaches remains a significant challenge. +UNet (CNN) +Transformer +Encoder +Transformer +Decoder +Training set +Mini-batch +L +L +L +L +U +U +U +U +L +L: Labeled image. +U: Unlabeled image. +: Stop-gradient. +Swin-UNet (Transformer) +U +L +Label +Figure 20: The model performs the semi-supervised medical image segmenta- +tion task. The regularization scheme between CNN and Transformer is referred +to as Cross Teaching. L denotes the labeled data and U denotes the unlabeled +data. The cross-teaching employs a bidirectional loss function: one path is +from the CNN branch to the Transformer branch, and the other is from the +Transformer to the CNN. A Transformer is applied for complementary training +instead of prediction generation [147]. +Zhou et. +al. +[148] hypothesize that the ability to aggre- +gate contextual information is imperative to improve the per- +formance of medical image analysis. Nonetheless, there is no +ImageNet-scale medical image dataset for pre-training. There- +fore, they investigate a novel self pre-training paradigm based +on Masked Autoencoder (MAE), MAE self pre-training, for +medical image analysis, one of the masked image modeling +(MIM) frameworks [194] [195] [196] [197]. MIM encourages +the framework to restore the masked target by integrating infor- +mation from the context, where the main idea of MIM is mask- +ing and reconstructing: masking a set of image patches before +input into the Transformer and reconstructing these masked +patches at the output. +The pipeline for segmentation with MAE self pre-training +contains two stages. In the first stage (as shown on the left +of Figure 21), ViT is pre-trained with MAE as the encoder. +The input patches are randomly divided into visible ones and +masked ones. The ViT encoder only acts on the visible patches. +Compared to other MIM methods, MAE does not employ mask +tokens in the encoder, which saves time and allows for faster +pre-training. A lightweight Transformer decoder is appended +to reconstruct the full image. The decoder is only an auxiliary +part used for pre-training and will not be applied in downstream +tasks. +In the second stage (as shown on the right of Figure 21), the +pre-trained ViT weights are transferred to initialize the segmen- +tation encoder. Then, the task-specific heads are appended to +perform downstream tasks. The whole segmentation network, +e.g., UNETR, is finetuned to perform the segmentation task. +The experiments, including the MAE pre-training and the +downstream task, are conducted to evaluate the performance of +the proposed method. The results show that MAE can recover +the lost information in the masked input patches. MAE pre- +training enables the model to improve its classification and seg- +mentation performance on medical image analysis tasks, sur- +passing the ImageNet pre-trained model to SOTA. +20 + +Table 3: An overview of the reviewed Transformer-based medical image Segmentation models. +Method +Modality +Organ +Type +Pre-trained Module: Type +Datasets +Metrics +Year +Pure +Swin-Unet +[32] +CT +Multi-organ +2D +ViT: Supervised +1 Synapse [135] +2 ACDC [176] +Dice +2021 +nnFormer +[137] +CT +MRI +Multi-organ +3D +ViT: Supervised +1 Synapse [135] +2 ACDC [176] +Dice +2021 +MISSFormer +[138] +CT +MRI +Multi-organ +2D + +1 Synapse [135] +2 ACDC [176] +Dice +Hausdorff distance +2021 +TransDeepLab +[33] +CT +Dermoscopy +Multi-organ +Skin +2D +ViT: Supervised +1 Synapse [135] +2 ISIC 2017,2018 [166, 177] +3 PH2 [178] +Dice +Hausdorff distance +2022 +Encoder +TransUNet +[24] +CT +MRI +Multi-organ +2D +ViT: Supervised +1 Synapse [135] +2 ACDC [176] +Dice +Hausdorff distance +2021 +TransBTS +[140] +MRI +Brain +3D + +BraTS 19-20 [179, 180, 181] +Dice +Hausdorff distance +2021 +TransFuse +[141] +Colonoscopy +Multi-organ +2D& 3D +ViT: Supervised +1 Synapse [135] +2 ACDC [176] +Dice +Hausdorff distance +2021 +MedT +[142] +Microscopy +Ultrasound +Multi-organ +2D + +1 Brain US (Private) +2 GLAS [182] +3 MoNuSeg [183] +F1 +2021 +UNETR +[143] +CT +MRI +Brain, Spleen +3D + +1 Synapse [135] +2 MSD [184] +Dice +Hausdorff distance +2021 +Swin UNETR +[144] +MRI +Brain +3D +ViT: Supervised +BraTS 21 [185] +Dice +Hausdorff distance +2022 +Skip connection +CoTr +[145] +CT +Multi-organ +3D + +Synapse [135] +Dice +2021 +HiFormer +[37] +MRI +Dermoscopy +Microscopic +Multi-organ +Skin +Cells +2D +ViT: Supervised +CNN: Supervised +1 Synapse [135] +2 ISIC 2017 [166] +3 SegPC 2021 [186, 187, 188] +Dice +Hausdorff distance +2022 +Decoder +SegTran +[139] +Fundus +MRI +X-Colonoscopy +Multi-organ +2D&3D +CNN: Supervised +REFUGE 20 [189] +1 BraTS 19 [179, 180, 181] +2 X-CVC [190] +3 KVASIR [191] +Dice +2021 +Other architectures +T-AutoML +[146] +CT +Liver and lung tumor +3D + +MSD 2019 [192] +Dice +the normalized surface distance (NSD) +2021 +Cross Teaching +[147] +MRI +Multi-organ +2D + +ACDC [176] +Dice +Hausdorff distance +2021 +Self-pretraining with MAE +[148] +CT +MRI +X-ray +Lung +Brain +Multi-organ +3D +ViT: supervised +1 ChestX-ray14 [106] +2 Synapse [192] +3 MSD 2019 [184] +Dice +Hausdorff distance +2022 +4.4. Discussion and Conclusion +This section comprehensively investigates the overview +of around 16 Transformer-based models for medical im- +age segmentation presented in Section 4.1 to Section 4.3. +We provide information on the reviewed segmentation ap- +proaches about the architecture type, modality, organ, input +size, the pre-trained manner, datasets, metrics, and the year +in Table 3. Table 4 also lists the methods along with the +number of parameters, contributions, and highlights. ViT- +based works offer solutions in a broad range of multimodal +tasks of 2D or 3D. Most of the approaches demonstrate +superior results over CNN-based segmentation models on +benchmark medical datasets. +Despite the state-of-the-art performance Transformer-based +networks have achieved, there are some challenges in de- +ploying the Transformer-based models at present. The first +challenge is the high computational burden due to the rela- +tively large number of parameters of the Transformer-based +models [198]. The reason is that the time and space com- +plexity of the attention mechanism is quadratic to the se- +quence length. For example, the CNN-based models such +as U-Net [34] requires 3.7M parameters [142] to reach Dice +Score 74.68 [24]. +However, TransUNet, which achieves +Dice Score 77.48 needs 96.07M [143] parameters. The re- +searchers have to meet the high demand for GPU resources. +Thus, several novel approaches such as Swin Transformer +employed in Swin-Unet [32], volume-based Transformer +utilized in nnFormer [137] and efficient self-attention mod- +ule in MISSFormer [138] are proposed to simplify the com- +putation of the Transformer models. The direction of fa- +cilitating the efficiency of models will play a crucial role +in future research. We also note that most existing meth- +ods require pre-training strategies on the ImageNet dataset +to obtain the pre-trained weights for the following down- +stream tasks. However, the natural image datasets and med- +ical datasets differ dramatically from one another, which +may impact the final performance of extracting the medical +features. Meanwhile, pre-training leads to high computa- +tional costs, which hinders the training of models in prac- +tice. Multiple segmentation networks which can be trained +from scratch on the medical dataset are suggested as the so- +lutions, such as MISSFormer [138]. We expect more ap- +21 + +Table 4: A brief description of the reviewed Transformer-based medical image segmentation models. The unreported number of parameters indicates that the +value was not mentioned in the paper, and the code was unavailable. +Method +# Params +Contributions +Highlights +Pure +Swin-Unet +[32] +- +• Builds a pure Transformer model with symmetric Encoder-Decoder architecture based on Swin- +Transformer block connected via skip connections. +• Proposes patch merging layers and patch expanding layers to perform downsampling and upsampling +without convolution or interpolation operation. +• The results of extensive experiments on multi-organ and multi-modal datasets show the good generalization ability of the model. +• Pre-trained on ImageNet rather than medical image data, which may result in sub-optimal performance. +nnFormer +[137] +158.92M +• Proposes a powerful segmentation model with an interleaved architecture (stem) based on the empirical +combination of self-attention and convolution. +• Proposes a volume-based multi-head self-attention (V-MSA) to reduce computational complexity. +• Volume-based operations help to reduce the computational complexity. +• Pre-trained on ImageNet rather than medical image data, which may result in sub-optimal performance. +MISSFormer +[138] +- +• Proposes the Enhanced Transformer Block based on the Enhanced Mix-FFN and the Efficient Self- +attention module. +• Proposes the Enhanced Transformer Context Bridge built on the Enhanced Transformer Block to model +both the local and global feature representation and fuse multi-scale features. +• The model can be trained from scratch without the pretraining step on ImageNet. +• Less computational burden due to the novel design of the Efficient Self-attention module. +TransDeepLab +[33] +21.14M +• Proposes the encoder-decoder DeepLabv3+ architecture based on Swin-Transformer. +• Proposes the cross-contextual attention to adaptively fuse multi-scale representation. +• The first attempt to combine the Swin-Transformer with DeepLab architecture for medical image segmentation. +• A lightweight model with only 21.14M parameters compared with Swin-Unet[32], the original DeepLab model [193] and TransUNet[24]. +Encoder +TransUNet +[24] +96.07M +• Proposes the first CNN-Transformer hybrid network for medical image segmentation, which establishes +self-attention mechanisms from the perspective of sequence-to-sequence prediction. +• Proposes a cascaded upsampler (CUP) which comprises several upsampling blocks to generate the pre- +diction results. +• TransUNet fully exploits the strong global context encoded from the Transformer and local semantics from the CNN module. +• It presents the generalization ability on multi-modalities. +• The approach allows the segmentation of 2D and 3D medical images. +TransBTS +[140] +Moderate TransBTS: 32.99M +Lightweight TransBTS: 15.14M +• Proposes a novel encoder-decoder framework TransBTS that integrates Transformer with 3D CNN for +MRI Brain Tumor Segmentation. +• The method can model the long-range dependencies not only in spatial but also in the depth dimension for 3D volumetric segmentation. +• TransBTS can be trained on the task-specific dataset without the dependence on pre-trained weights. +• TransBTS is a moderate-size model that outperforms in terms of model complexity with 32.99M parameters and 33G FLOPs. Furthermore, +the vanilla Transformer can be replaced with other variants to reduce the computation complexity. +TransFuse +[141] +TransFuse-S: 26.3M +• Proposes the first parallel-in-branch architecture — TransFuse to capture both low-level global features +and high-level fine-grained semantic details. +• Proposes BiFusion module in order to fuse the feature representation from the Transformer branch with +the CNN branch. +• The architecture does not require very deep nets, which alleviates gradient vanishing and feature diminishing reuse problems. +• It improves performance by reducing parameters and increasing inference speed, allowing deployment on both the cloud and the edge. +• The CNN branch is flexible to use any off-the-shelf CNN network. +4 It can be applied to both 2D and 3D medical image segmentation. +MedT +[142] +1.4M +• Proposes a gated axial-attention model that introduces an additional control mechanism to the self- +attention module. +• Proposes a LoGo (Local-Global) training strategy for boosting segmentation performance by simultane- +ously training a shallow global branch and a deep local branch. +• The proposed method does not require pre-training on large-scale datasets compared to other transform-based models. +• The results of predictions are more precise compared to the full attention model. +UNETR +[143] +92.58M +• Proposes a novel architecture to address the task of 3D volumetric medical image segmentation. +• Proposes a new architecture where the Transformer-based encoder learns long-range dependencies and +the CNN-based decoder utilizes skip connections to merge the outputs of a Transformer at each resolution +level with the upsampling part. +• UNETR shows moderate model complexity while outperforming these Transformer-based and CNN-based models. +• The inference time of UNETR is significantly faster than Transformer-based models. +• They did not use any pre-trained weights for the Transformer backbone. +Swin UNETR +[144] +61.98M +• Proposes a novel segmentation model, Swin UNETR, based on the design of UNETR and Swin Trans- +formers. +• The FLOPs of Swin UNETR significantly grow compared to that of UNETR and TransBTS. +• The Swin Transformer is suitable for the downstream tasks. +Skip connection +CoTr +[145] +46.51M +• Proposes a hybrid framework that bridges a convolutional neural network and a Transformer for accurate +3D medical image segmentation. +• Proposes the deformable Transformer (DeTrans) that employs the multi-scale deformable self-attention +mechanism (MS-DMSA) to model the long-range dependency efficiently. +• The deformable mechanism in CoTr reduces computational and spatial complexities, allowing the network to model high-resolution and +multi-scale feature maps. +HiFormer +[37] +HiFormer-S: 23.25M +HiFormer-B: 25.51M +HiFormer-L: 29.52M +• Proposes a encoder-decoder architecture that bridges a CNN and a Transformer for medical image seg- +mentation. +• Proposes a Double-level Fusion module in the skip connection. +• Fewer parameters and lower computational cost. +Decoder +SegTran +[139] +86.03M +• Proposes a novel Transformer design, Squeeze-and-Expansion Transformer, in which a squeezed attention +block helps regularize the huge attention matrix, and an expansion block learns diversified representations. +• Proposes a learnable sinusoidal positional encoding that imposes a continuity inductive bias for the Trans- +former. +• Compared to U-Net and DeepLabV3+, Segtran has the least performance degradation, showing the best cross-domain generalization when +evaluated on the datasets of drastically different characteristics. +Other architectures +T-AutoML +[146] +16.96M +• Proposes the first automated machine learning algorithm, T-AutoML, which automatically estimates “al- +most” all components of a deep learning solution for lesion segmentation in 3D medical images. +• Proposes a new predictor-based search method in a new search space that searches for the best neural +architecture and the best combination of hyperparameters and data augmentation strategies simultaneously. +• The method is effectively transferable to different datasets. +• The applied AutoML alleviates the need for manual design of network structures. +• The intrinsic limitations of AutoML. +Cross Teaching +[147] +- +• Proposes an efficient regularization scheme for semi-supervised medical image segmentation where the +prediction of a network serves as the pseudo label to supervise the other network. +• The proposed method is the first attempt to apply the Transformer to perform the semi-supervised medical +segmentation utilizing the unlabeled data. +• The training process requires less data cost by semi-supervision. +• The framework contains components with low complexity and simple training strategies compared to other semi-supervised learning methods. +• The proposed semi-supervised segmentation still can not achieve the state-of-the-art (SOTA) compared with the fully-supervised approaches. +Self-pretraining with MAE +[148] +- +• Proposes a self-pre-training paradigm with MAE for medical images where the pre-training process of +the model uses the same data as the target dataset. +• The proposed paradigm demonstrates its effectiveness in limited data scenarios. +proaches to exploring more efficient pre-training strategies +or without pre-training. Furthermore, considering the lim- +ited size of some medical datasets, some approaches propose +semi-supervised technologies or self-pre-training paradigms +to reduce the dataset burden of training or pre-training. Nev- +ertheless, the performance is still not comparable to that of +fully-supervised models. Designing semi-supervised mod- +els with improved accuracy in this direction requires more +attention. +5. Medical Image Reconstruction +3D Medical imaging is a clinical breakthrough and very pop- +ular in medical diagnosis and follow-up after treatment. +In +Computed Tomography (CT), Single Photon Emission Tomog- +raphy (SPECT) and Positron Emission Tomogrpahy (PET), the +imaging process relies on ionizing radiation [214, 215], which +implies a potential risk for the patient [216]. A non-invasive +3D imaging technique is Magnetic Resonance Imaging (MRI), +which does not rely on ionizing radiation. However, image ac- +quisition may take longer and confines the patient in a discom- +forting narrow tube [217]. In order to reconstruct 3D volumet- +ric datasets from the acquired data, Medical image reconstruc- +tion is one of the essential components of 3D medical imaging. +The primary objective of 3D image reconstruction is to gener- +ate high-quality volumetric images for clinical usage at mini- +mal cost and radiation exposure, whilst also addressing poten- +tial artifacts inherent to the physical acquisition process. Image +reconstruction solves an inverse problem that is generally chal- +lenging due to its large-scale and ill-posed nature [218]. +In medical imaging, there are ongoing research efforts to +reduce the acquisition time (i.e. to reduce cost and potential +movement artifacts) as well as radiation dose. However, lower- +ing the radiation dose results in higher noise levels and reduced +contrast, which poses a challenge for 3D image reconstruction. +Vision Transformers (ViTs) have effectively demonstrated +possible solutions to address these challenges. We categorize +the literature in this domain into low dose enhancement, sparse- +view reconstruction, undersampled reconstruction, and super- +resolution reconstruction. This section will overview some of +the SOTA Transformer-based studies that fit into our taxonomy. +Figure 22a and Figure 22b demonstrate our proposed taxonomy +22 + +ViT Encoder +Transformer Decoder +ViT Encoder +. +. +UNETR Decoder +Transfer + +Weights +MAE Self Pre-training +UNETR Segmentation +z3 +z6 +z9 +z12 +[mask token] +[patch feature] +Figure 21: Illustration of MAE self pre-training. First, MAE is pre-trained as +an encoder for ViT. the ViT encoder is fed with a random subset of patches +and the decoder of the Transformer reconstructs the complete image as shown +on the left. Then, the pre-trained ViT weights are transferred to the initialized +segmentation encoder, as shown on the right. Finally, the whole segmenta- +tion network, such as UNETR, is fine-tuned to perform the segmentation task. +[148]. +for this field of study. Figure 22a indicates the diversity of our +taxonomy based on the medical imaging modalities we studied +in this research. Figure 22b endorses the usage of the Trans- +former within the overviewed studies’ pipelines. +5.1. Low Dose Enhancement +Zhang et al. [199] used a very general intuition about image +denoising: the noisy image constructed with high-frequency +and low-frequency counterparts as X = XH + XL in a study, +namely, TransCT. Zhang et al. [199] claim that the noisy im- +age’s low counterpart contains two sub-components of main +image content and weakened image textures, which are entirely +noise-free. They applied a Gaussian filter on the input image +to decompose an image into a high-frequency sub-band and a +low-frequency sub-band. After this, they extracted XLc con- +tent features and XLt latent texture features by applying two +shallow CNNs on the low-frequency counterpart of the input +image. +Simultaneously, they applied a sub-pixel layer on a +high-frequency counterpart to transform it into a low-resolution +image and extracted embedding features (XHf ) by applying a +shallow CNN. Then the resultant latent texture features (XLt) +and corresponding high-frequency representation are fed to the +Transformer for noise removal from a high-frequency repre- +sentation. Ultimately, they reconstruct the high-quality image +piecewise. They showed that the latent texture features are ben- +eficial in screening noise from the high-frequency domain. +Despite the TransCT [199], +Wang et al. +proposed +a convolution-free Token-to-Token vision Transformer-based +Encoder-decoder Dilation network (TED-net) design for CT +image denoising [200]. Their approach is based on a U-Net +encoder-decoder scheme enriched by different modules, i.e., +Basic Vision Transformer, Token-to-Token Dilation (T2TD), +and (Inverse) Cyclic Shift blocks. Consider y ∈ RN×N a clean +natural dose CT image, x ∈ RN×N noisy low dose CT im- +age, and T : RN×N → RN×N is a Transformer-based denois- +ing model. According to the Figure 23 after tokenization of x +and passing through the Vision Transformer block to capture +long-range dependencies and alleviate the absence of local in- +ductive bias in Transformers, they employed Token-to-Token +serialization [219]. Also, they utilized feature re-assembling +with a Cyclic Shift block (CSB) to integrate more information. +Obvious from Figure 23, all of these blocks are replicated in +a symmetric decoder path, but instead of the CSB, the Inverse +Cyclic Shift block (ICSB) is implemented to avoid pixel shifts +in the final denoising results (y = x + T(x)). They reached +SOTA results compared to CNN-based methods and a compet- +itive benchmark with regard to the TransCT [199]. +Luthra et al. [201] proposed a Transformer-based network, +Eformer, to deal with low-dose CT images while concurrently +using the edge enhancement paradigm to deliver more accu- +rate and realistic denoised representations. Their architecture +builds upon the LeWin (Locally-enhanced Window) Trans- +former block [220], which is accompanied by an edge enhance- +ment module. The success of the Swin Transformer [57] in +capturing the long-range dependencies with the window-based +self-attention technique makes it a cornerstone in designing +new Transformer blocks due to its linear computational com- +plexity. LeWin Transformer is one of these blocks that capture +the global contextual information and, due to the presence of a +depth-wise block in its structure, could also capture a local con- +text. Eformer’s first step is through the Sobel edge enhancement +filter. In every encoder-decoder stage, convolutional features +pass through the LeWin Transformer block, and downsampling +and upsampling procedures are done by convolution and de- +convolution layers. +Eformer’s learning paradigm is a resid- +ual learning scheme, meaning it learns the noise representation +rather than a denoised image due to the ease of optimization in +predicting a residual mapping. +Akin to Low Dose CT (LDCT), Low-Dose PET (LDPET) +is preferable to avoid the radiation risk, especially for cancer +patients with a weakened immune system who require multi- +ple PET scans during their treatment at the cost of sacrific- +ing diagnosis accuracy in Standard-Dose PET (SDPET). Luo +et al. +[202] proposed an end-to-end Generative Adversarial +Network (GAN) based method integrated with a Transformer +block, namely 3D Transformer-GAN, to reconstruct SDPET +images from the corresponding LDPET images. To alleviate the +inter-slice discontinuity problem of existing 2D methods, they +designed their network to work with 3D PET data. Analogous +to any GAN network, they used a generator network, encoder- +decoder, with a Transformer placed in the bottleneck of the +generator network to capture contextual information. Due to +the computational overhead of Transformers, they did not build +their proposed method solely on it. Therefore, they were sat- +isfied to place a Transformer counterpart across CNN layers of +the generator to guarantee to extract low-level spatial feature +extraction and global semantic dependencies. They also intro- +duced adversarial loss term to their voxel-wise estimation error +to produce more realistic images. +In contradiction with other works, Zhang et al. [203] pro- +posed leveraging the PET/MRI data simultaneously for denois- +ing low-count PET images, which is a crucial assessment for +cancer treatment. PET scan is an emission Computed Tomog- +raphy (CT) operating by positron annihilation radiation. Due to +23 + + Medical Image + Reconstruction + Computed Tomography (CT) + Low Dose Enhancement + 1. TransCT + 2. TED-net + 3.- Eformer + 4. 3D Transformer- + GAN + 5. STFNet + 6. CTformer + 7. SIST + Sparse-View Reconstruction + 8. DuDoTrans + 9. FIT + 10. CTTR + 11. ARMLUT + Magnetic Resonance Imaging (MRI) + Undersampled Reconstruction + 12. ViT-Rec + 13. T2Net + Super Resolution + Reconstruction + 14. DisCNN-ViT + 15. Cohf-T +(a) Taxonomy structure for medical image reconstruction. Methods in this field are categorized by their functionality in +addressing issues in consensus imaging modalities, not how the Transformer is integrated with the architecture like in +the previous sections. The prefix numbers in the paper’s name in ascending order denote the reference for each study as +follows: 1. [199], 2. [200], 3. [201], 4. [202], 5. [203], 6. [204], 7. [205], 8. [206], 9. [207], 10. [208], 11. [209], 12. +[210], 13. [211], 14. [212], 15. [213]. +Modification +Stage +Presence in +Architecture +Design +Encoder +TransCT +(Zhang et +al., 2021e) +STFNet +(Zhang et +al., 2022) +SIST +(Yang et +al., 2022) +DuDoTrans +(Wang et +al., 2021a) +CTTR +(Shi et +al., 2022a) +Pure +TED-net +(Wang et +al., 2021b) +Eformer +(Luthra et +al., 2021) +CTformer +(Wang et +al., 2022a) +FIT +(Buchholz +and Jug, +2021) +ViT-Rec +(Lin and +Heckel, +2021) +Skip +Connection +Cohf-T +(Fang et +al., 2022) +Other +Architectures +3D T-GAN +(Luo et +al., 2021b) +ARMLUT +(Wu et +al., 2022) +T2Net +(Feng et +al., 2021) +DisCNN- +ViT +(Mahapatra +and Ge, +2021) +(b) We also presented a second taxonomy due to the +presence of the Transformer in the reviewed studies. +Figure 22: An overview of medical image reconstruction taxonomies either as categorizing by the task or the location of using Transformer in an architecture. +Figure 23: An overview of TED-net [200]. Tokenize and DeToken blocks are +invertible operations that apply the process of patch embedding and converting +patches again to image, respectively. TB represents a standard Transformer +block. (I)CSB denotes the (inverse) Cyclic Shift block to modify the feature +map, nonetheless, the reverse operation avoids pixel shifts in the final result. +T2T block represents the Token-to-Token process [219] to improve the spatial +inductive bias of Transformers by merging the neighboring tokens. The Dilated +T2T (T2TD) block is used to refine contextual information further. +the foundation and requirements of PET scans, there is a severe +risk of getting infected with secondary cancer by radiotracers. +So to degrade the side effects of this imaging process, there +are two potential methods: reduction in radiotracer dose and +lessening the patient’s bedtime duration. The aforementioned +approaches, without a doubt, affect the imaging result quality +with decreased contrast to noise ratio and bias in texture. The +traditional low-count PET denoising approaches are based on +Non-Local Means (NLM) [221], Block Matching 3D (BM3D) +[222], and Iterative methods [223], etc., which are firmly in +bond with hyperparameter tuning for new data or result in un- +natural smoothings over denoised images. Zhang et al. [203] +testify that simultaneous PET/MRI could boost one modality in +terms of correct attenuation, motion, and partial volume effects, +and also, due to the high contrast among soft tissues in MRI, the +denoising process of PET images is preferably straightforward. +STFNet [203] is a U-Net based structure with different medica- +tions. They proposed a new Siamese encoder comprising dual +input flow for each modality in the encoding path. To obtain +sufficient features from different modalities, they used the Spa- +tial Adaptive (SA) block, a dual path in each block with the +residual block design, which consists of different consecutive +convolutional blocks and deformable convolution with fusion +modulation. This module aims to learn more contextual fea- +tures from each modality. To leverage global attention, they +used a Transformer to produce a pixel-to-pixel interaction be- +tween the PET and the MRI modality. After this integration, the +fused features are input to the two branches based on residual +convolution blocks for PET denoising. +Wang et al. [204] proposed the enhancement for their previ- +ous work, TED-net [200] convolution-free, solely Transformer- +based network, namely CTformer. From Figure 24, it is ap- +parent that their network is an unsupervised residual learning, +U-Net-like encoder-decoder structure, rather than direct map +learning of LDCT to Natural Dose CT (NDCT). The CTformer +tries to compensate for the Transformers’ deficiency in cap- +turing path inter-boundary information and spatial inductive +bias with token rearrangement, T2T [219]. To do so, analo- +gously like TED-net, they used dilation and cyclic shift blocks +in the Token2Token block to broaden the receptive field to cap- +ture more contextual information and not increase the compu- +tational cost. +Yang et al. [205] were inspired by how sinogram works and +proposed Singoram Inner-Structure Transformer (SIST) (Fig- +ure 25). +This inner structure of the sinogram contains the +unique characteristics of the sinogram domain. To do so, they +mimic the global and local characteristics of sinogram in a +loss function based on sinogram inner-structure, namely Sino- +gram Inner-Structure Loss (SISL). The global inner-structure +loss utilizes conjugate sampling pairs in CT, and local inner- +24 + +64 x 64 +512 × 512 +Tokenize +DeToken +TB +TB +CSB +ICSB +T2TD +T2TD +TB +TB +CSB +ICSB +T2T +T2T +TB512 × 512 +64 × 64 +CTformer +Module A +TB +TB +DeTokenization +TB +TB +Encoder Decoder +TB +Tokenization +29×29 +25×25 +29×29 +25×25 +CTformer +Module D +CTformer +Module C +CTformer +Module B +T2TD +IT2TD +T2TD +IT2TD +512 × 512 +64 × 64 +Figure 24: An overview of CTformer [204]. This structure is analogous to the +TED-net [200] structure, the previous study by the same authors. +Table 5: Comparison result on NIH-AAPM-Mayo [224] dataset in low dose +enhancement task. LDE indicates the Low Dose Enhancement task. +Methods +Task +Dataset +SSIM ↑ +RMSE ↓ +� +Eformer +[201] +LDE +NIH-AAPM-Mayo +[224] +0.9861 +0.0067 +� +TransCT +[199] +LDE +NIH-AAPM-Mayo +[224] +0.923 +22.123 +� +CTformer +[204] +LDE +NIH-AAPM-Mayo +[224] +0.9121 +9.0233 +structure loss considers the second-order sparsity of sinograms. +The amalgamation of these two terms could be beneficial in re- +constructing NDCT images while retaining the noise. Due to +the CT imaging mechanism, each row of the sinogram repre- +sentation denotes the projection at a certain view. Naturally, +this procedure is suitable for leveraging the Transformer block +for modeling the interaction between different projections of di- +verse angles to capture contextual information. Therefore, the +SIST module applies to raw sinogram input and captures struc- +tural information. Afterward, the unified network reconstructs +the high-quality images in a residual policy with the image re- +construction module. +Table 5 represents the benchmark results in the LDCT task +over the NIH-AAPM-Mayo [224] dataset respecting SSIM and +RMSE metrics on overviewed methods in this study. For clar- +ification, TED-net [200] achieved better results than CTformer +[204], but due to two studies originating from the same au- +thors and the resemblance between architectures, we preferred +to mention CTformer to count in the comparison table. This +result endorses the capability of the pure Transformer-based +Eformer [201] method in reconstructing natural dose CT im- +ages. +Head +Tail +Split +𝑆𝑙𝑑 +𝑠𝑙𝑑 +1 +𝑠𝑙𝑑 +𝑃 +… +Multi-Head +Self-Attention +Add & Norm +MLP +Add & Norm +Image +Reconstruction +Module +Total +Loss +Noise +Loss +Image +Loss +Conv1D +Residual +Conv +Conv +Conv +Conv +Conv +Conv +𝑆𝑙𝑑 +𝐼𝑙𝑑 +መ𝑆𝑛𝑜𝑖𝑠𝑒 +መ𝑆 +Minus +Minus +𝐼𝑙𝑑 +መ𝑆 +Image +Loss +Noise +Loss +መ𝐼𝑛𝑜𝑖𝑠𝑒 +መ𝐼 +Sinogram Transformer Module +Inner-Structure Loss +𝑆𝑙𝑑 +Sinogram +Loss +×N +Figure 25: The overall architecture of SIST [205] pipeline. S ld and Ild are +the LDCT sinogram and image, ˆS and ˆI denote the output sinogram and im- +age, ˆS noise and ˆInoise are the sinogram noise and image noise. First, the LDCT +sinogram feed to the Transformer for sinogram domain denoising, then the de- +noised sinogram ˆS input to the image reconstruction module for image domain +denoising. Within the image reconstruction module, the sinogram noise ˆS noise +with the usage of residual CNN block generates image domain ˆInoise. NDCT ˆI, +outputs from applying refinement steps on Ild minus ˆInoise. +5.2. Sparse-View Reconstruction +Due to the customary usage of CT images in medical diag- +nosis, another policy to lessen the side effects of X-ray radia- +tion is acquiring fewer projections, known as sparse-view CT, +which is a very feasible and effective method rather than ma- +nipulating the standard radiation dose [225, 226]. However, +the resultant images from this method suffer from severe arti- +facts, and decreasing the number of projections demands pro- +found techniques to reconstruct high-quality images. Wang et +al. [206] is the first paper that inspected the usage of Trans- +formers in this field which was quite successful, namely DuDo- +Trans. Their intuition was to shed light on the globality nature +of the sinogram sampling process, which the previous CNN ar- +chitectures neglected. DuDoTrans, unlike the conventional iter- +ative methods in this literature, does not provide blocky effects +in reconstructed images. This method simultaneously benefits +from enhanced and raw sinogram streams to restore informa- +tive sinograms via long-range dependency modeling in a super- +vised policy. DuDoTrans from Figure 26 is built on three main +modules, namely Singoram Restoration Transformer (SRT), the +DuDo Consistency layer, and the Residual Image Reconstruc- +tion Module (RIRM). SRT block consists of successive hybrid +Swin Transformer modules and convolutional layers to model +local semantic features and inherent global contextual informa- +tion in the sinogram to produce the enhanced sinogram. +Buchholz et al. [207] presented the Fourier Image Trans- +former (FIT) that operates on the image frequency representa- +tion, especially the Fourier description of the image, which in +their study is known as Fourier Domain Encoding (FDE), that +encodes the entire image at lower resolution. The intuition in +their idea is underlying the CT’s acquisition process physics. +CT utilizes a rotating 1D detector array around the patient body +to calculate the Radon transform [227] of a 2D object, which +leads to a sequence of density measurements at different pro- +jection angles, namely sinogram as a 2D image in which each +column of this representation corresponds to one 1D measure- +25 + +Residual Image Reconstruction Module(RIRM) +Shallow Layer +Deep Feature +Extraction Layers +Recon Layer +DuDo +Consistency Layer +FBP +Norm +SW-MSA +Norm +MLP +Input +Output +Swin-Transformer Module (STM) + Conv +STM +… +STM + Conv + Conv + Conv +… +STM +… +STM + Conv +Sinogram Restoration Transformer (SRT) +Figure 26: DuDoTrans [206] framework for sparse-view CT image reconstruc- +tion. First, the sparse-view sinogram Y maps to a low-quality image �X1 and +other estimation �X2 generated by SRT module’s enhanced sinogram output �Y +followed by DuDo Consistency Layer. Lastly, the predicted estimations are +concatenated and fed to the RIRM module that outputs the CT image of �X in a +supervised manner. +ment. The Filtered Back Projection (FBP) [228, 227] is a re- +construction method to map sinograms to tangible CT images. +FBP is based on the Fourier slice theorem; hence, computing +the 1D Fourier transform of 1D projection and rearranging them +by their projection angle in Fourier space, followed by an in- +verse Fourier transformation, results in a reconstructed 2D CT +image slice. Limiting the number of projections leads to miss- +ing Fourier measurements, which ultimately conduce to recon- +struction artifacts. FIT is the first study that uses a Transformer +to query arbitrary Fourier coefficients and fill the unobserved +Fourier coefficients to conceal or avoid the probable artifacts +in reconstruction within sparse-view CT reconstruction litera- +ture. From Figure 27 this procedure starts with calculating the +FDE of the raw sinogram. To do so, first, the discrete Fourier +transform (DFT) of the sinogram will be calculated. Secondly, +after dropping half of the coefficients on the Fourier rings of +the resultant Fourier representation, it preserves the lower fre- +quency counterparts to recover the lower resolution of the raw +sinogram. Afterward, the complex coefficients convert into 1D +sequences by unrolling the Fourier rings. These complex values +convert to normalized amplitudes and phases. Therefore, each +complex coefficient has its own polar representation, which is +a normalized real-valued matrix with N × 2 entries (N is equal +to half of the DFT coefficients number). A linear layer applies +on this tensor to upsample the feature dimensionality to F +2 . Fi- +nally, a 2D positional encoding concatenates to this tensor and +produces a 2D FDE image with the size of N × F. +Shi et al. +[208] presented a CT reconstruction network +with Transformers (CTTR) for sparse-view CT reconstruction. +In contrast to DuDoTrans [206], CTTR enhances low-quality +reconstructions directly from raw sinograms and focuses on +global features in a simple policy in an end-to-end architecture. +CTTR contains four parts: two CNN-based residual blocks ex- +tracting local features from FBP [228] images reconstruction +and sinograms, an encoder-decoder Transformer for long-range +modeling dependencies, and contextual information between +features, and a CNN block to map features to a high-quality +reconstruction. +Cone-Beam Computed Tomography (CBCT) is a conven- +tional way of dental and maxillofacial imaging; due to its fast +3D imaging qualifications, its popularity has extended to lung +imaging. However, studies approved that its radiation dose is +higher than plain radiographs [230] hence sparse-view CBCT +FC-Loss +MSE-Loss +1D FFT +2D iFFT +Encoder +Decoder +2D FFT +2D FFT +FBP +Figure 27: FIT [207] framework for sparse-view CT reconstruction. FDE rep- +resentation of the sinogram calculates that serves as an input to an encoder +of Transformer design. The decoder predicts the Fourier coefficients from the +encoder’s latent space. The Fourier coefficients of applying the FBP [228] al- +gorithm on sinogram information are fed into a Transformer’s decoder to enrich +the Fourier query representation. A shallow CNN block applies after inverse +FFT to hamper the frequency oscillations. +could be a suitable method to lower radiation dose. +Wu et +al. [209] proposed a novel untrained 3D Transformer-based +architecture, namely ARMLUT, with a multi-level loss func- +tion for CBCT reconstruction. While the Transformer mod- +ule, especially the UNETR [143] in this study, captures long- +range contextual information and enhances the resulting image. +The intuition behind this strategy is Deep Image Prior (DIP) +[231] to succeed in the reconstruction field. From Figure 28a, +ARMLUT is an iterative optimization problem between the Im- +age Reconstructor module and Image Generator module to fit a +CBCT inverse solver without a large number of data or ground +truth images. The multi-level loss function comprises a Mean +Squared Error (MSE) and Perceptual Loss (PL) [232] to recon- +struct smooth and streak artifact-free outputs. The entire frame- +work (Figure 28) has three main counterparts: Image Recon- +structor, Image Generator, and Feature Extractor. Image Re- +constructor uses Feldkamp-Davis-Kress (FDK) algorithm [229] +to produce a low enhanced reconstruction from M-view mea- +surements, and the Image generator module maps the noisy +voxel inputs to reconstruct a regularised image. The Feature +Extractor module applies the VGG-11 pre-trained network on +two representations and produces a PL paradigm. To minimize +the distance between these two reconstructions, ARMLUT uti- +lizes an adaptively re-weight multi-loss technique to stabilize +the convergence of the Transformer in the optimization. +5.3. Undersampled Reconstruction +Magnetic Resonance Imaging (MRI) is a dominant technique +for assistive diagnosis. However, due to the physics behind its +operation, the scanning time can take longer and be very te- +dious, affecting the patient experience and leading to inevitable +artifacts in images [241]. Hence, reducing the number of MRI +measurements can result in faster scan times and artifacts re- +duction due to the patient’s movement at the cost of aliasing +artifacts in the image [217]. +Lin et al. [210] proposed a comprehensive analytical study +to investigate the usage of ViT in a pure (CNN-free modules) +and most straightforward Transformer design. This study is ev- +idence of the prominent effect of ViTs in medical image recon- +26 + +(a) ARMLUT [209] pipeline. The collaboration of three distinct modules—FDK algorithm +[229], prior embedding with Transformer, and VGG-11 network for extracting hierarchical +features—in this pipeline generates the reconstructed CBCT image. Red texts in the figure +denote the variable weights that contribute to the iterative optimization step. +(b) UNETR [143] used as an image generator module in the ARMLUT paradigm. +Figure 28: (a) represents multi-loss untrained network for sparse-view CBCT reconstruction. (b) architecture of UNETR [143], as a Transformer module in a +ARMLUT. +struction. For this work, they adopted the original ViT [22] for +image reconstruction by discarding the classification token and +replacing the classification head with the reconstruction head, +which is comprised of successive Norm and Linear layers to +map the Transformer output to a visual image. They performed +a complete ablation study with different ViT settings, from the +number of stacked Transformers to embedding dimension and +number of heads in Multi-Head Self-Attention (MHSA). Their +results were quite effective and proved that trained ViT on suf- +ficient data from natural images like ImageNet or medical data +could perform better or achieve on-par reconstruction accura- +cies compared to CNN baselines such as U-Net [34]. The pro- +posed design’s distinguished power based on the mean attention +distance metric [242] proves that it effectively mimics the con- +volutional receptive fields and could concurrently capture local +and global dependencies. In addition, they showed that the ViT +benefits from two times faster inference times and fewer mem- +ory requirements compared to the U-Net. +Feng et al. +[211] address the particular issue in this do- +main by designing an end-to-end multi-task learning paradigm +to boost feature learning between two sub-tasks, MRI recon- +struction, and super-resolution, which have a high overlap +with each other named Task Transformer Network (T2Net) is +showed in Figure 29. Their network consists of two branches, +each for a specific task. T2Net utilizes a Transformer between +two branches to share feature representation and transmission. +T2Net applies a convolution layer and EDSR [243] backbone +to extract task-specific features in each task branch. To share +information between two branches and benefit from the inter- +actions of these two task-specific features concerning the na- +ture of the Transformer’s globality, T2Net uses a unique Trans- +former design to learn a generalized representation. Since the +reconstruction branch has more potential in artifact removal ca- +pacity than the super-resolution branch, the task Transformer +module guides the super-resolution branch into high-quality +representation from the reconstruction branch. +The Trans- +former module inherits the query (Q: from super-resolution +branch), key (K: from reconstruction branch), and value (V: +from reconstruction branch) from each scale’s two branches’ +output. It forms three main concepts: relevance embedding, +Transfer attention, and soft attention, which differ from the +original Transformer blocks. Relevance embedding tries to en- +close the correlated features from the reconstruction branch to +the super-resolution branch. Transfer attention aims to transmit +the anatomical and global features between two branches, and +last but not least, soft attention amalgamates features from the +previous two steps. Ultimately, this module lets the whole net- +work transfer and synthesize the representative and anatomical +features to produce a high-quality, artifacts-free representation +from highly undersampled measurements. The experimental +results on two datasets expressed the high potential of this ap- +proach rather than conventional algorithms. +5.4. Super-Resolution Reconstruction +Improving the resolution of images leads to the more detailed +delineation of objects. Increasing the medical image resolu- +tion plays a crucial role in computer-aided diagnosis due to +its rich anatomical and textural representation. Based on the +aforementioned fact and the MRI’s pipeline physics during the +image acquisition process for having high-resolution images, +a patient needs to lie a long time in the MRI tube. Hereupon +lower signal-to-noise ratio and more minor spatial coverage +drawbacks are inevitable [241]. Therefore in this section, we +investigate Transformer-utilized algorithms that try to alleviate +this problem. Of note, due to the analogy between MRI and +super-resolution reconstruction, some studies investigate these +two tasks in conjunction with each other. +Mahapatra et al. [212] proposed the GAN-based model with +structural and textural information preservation done by mul- +tiple loss function terms. +Their pipeline included two pre- +trained modules named feature disentanglement module, a con- +ventional autoencoder, and a Transformer-based feature en- +coder, UNETR [143]. UNETR captures the global and local +context of the original low-resolution image and induces the +27 + +Image +Feature +VGG-11 +Reconstructor +Extractor +FDK +DSConv +X +PL +f1() f2() +f9() f10() +W1 +MSE +W2 +MSE +MSE +. +W10 +MSE +f1(X) f2(X) +fg(X) f10 (X) +Image +Generator +VGG-11 +H +Transformer +DSConv +Ge +b.randomPatch Extraction +8000 +0000 +Patch Flatten +Linear Projection +2×2×2 +3×3×3 +Deconvolution +Convolution +Embedding +1×1×1 +Batch Norm + ReLU +Convolution +1st TB +Layer +Multilayer +Multi-Head +Norm +Perceptron +Attention +3rd TB +6th TB +9th TB +2nd TB +12th TB +Concatenation +Other operations +Transformer Block (TB)Table 6: Medical Image Reconstruction. LDE, SVR, USR, and SRR stand for Low Dose Enhancement, Sparse-View Reconstruction, Undersampled Reconstruc- +tion, and Super Resolution Reconstruction, respectively. † indicates that this network uses a pre-trained perceptual loss (loss network). +Method +Task(s) +Modality +Type +Pre-trained Module: Type +Dataset(s) +Metrics +Year +Pure +TED-net +[200] +LDE +CT +2D + +NIH-AAPM-Mayo Clinical LDCT [224] +SSIM +RMSE +2021 +Eformer +[201] +LDE +CT +2D +† +NIH-AAPM-Mayo Clinical LDCT [224] +PSNR, SSIM +RMSE +2021 +CTformer +[204] +LDE +CT +2D + +NIH-AAPM-Mayo Clinical LDCT [224] +SSIM +RMSE +2022 +FIT +[207] +SVR +CT +2D + +LoDoPaB [233] +PSNR +2021 +ViT-Rec +[210] +USR +MRI +2D +Supervised +fastMRI [234] +SSIM +2021 +Encoder +TransCT +[199] +LDE +CT +2D + +1 NIH-AAPM-Mayo Clinical LDCT [224] +2Private clinical pig head CBCT +RMSE +SSIM +VIF +2021 +STFNet +[203] +LDE +PET +MRI +2D + +Private Dataset +RMSE, PSNR +SSIM, PCC +2022 +SIST +[205] +LDE +CT +2D + +1LDCT Dataset [235] +2Private dataset +PSNR, SSIM +RMSE +2022 +DuDoTrans +[206] +SVR +CT +2D + +NIH-AAPM-Mayo Clinical LDCT [224] +PSNR, SSIM +RMSE +2021 +CTTR +[208] +SVR +CT +2D + +LIDC-IDRI [236] +RMSE, PSNR +SSIM +2022 +Skip Connection +Cohf-T +[213] +SRR +MRI +2D + +1 BraTS2018 [181] +2 IXI [237] +PSNR +SSIM +2022 +Other Architectures +3D T-GAN +[202] +LDE +PET +3D + +Private Dataset +PSNR, SSIM +NMSE +2021 +ARMLUT +[209] +SVR +CT +3D +ViT: Supervised † +1 SPARE Challenge Dataset [238] +2 Walnut dataset [239] +PSNR, SSIM +2022 +T2Net +[211] +USR +SRR +MRI +2D + +1 IXI [237] +2 Private Dataset +PSNR, SSIM +NMSE +2021 +DisCNN-ViT +[212] +SRR +MRI +3D +ViT: Self-Supervised +1fastMRI [234] +2 IXI [237] +PSNR, SSIM +NMSE +2021 +high-resolution image to preserve these contexts too. These +two modules fine-tune on a different medical dataset, and after- +ward, the low-resolution input plus the intermediate generator +produced image feed to these modules. The disentanglement +network contains two autoencoders to learn two counterparts, +latent space, structural and textural space, with fed medical im- +ages. In an end-to-end setting, these two pre-trained assistive +modules help to generate more realistic and structural and tex- +tural preserving high-resolution images by imposing module- +related loss terms such as adversarial loss to constrain for pro- +ducing realistic images and cosine similarity loss for each men- +tioned module. Results on the IXI dataset proved that Mahap- +atra et al.’s [212] network outperformed a couple of the CNN- +based attention mechanism networks and T2Net [211]. +Maintaining structural information during the acquiring +high-resolution images plays a crucial role. Hence, the struc- +ture information is embedded in an image’s high-frequency +counterpart, like in an image’s gradients. In addition, due to +the less time-consuming nature of obtaining MR T1WI (T1 +Weighted Image), it is wise to use it as an inter-modality con- +text prior to producing a high-resolution image. Accordingly, +Fang et al. [213] devised a network to leverage these two con- +cerns in their super-resolution pipeline: Cross-Modality High- +Frequency Transformer (Cohf-T). This network is divided into +two streams, the first stream is applied on low-resolution T2WI, +and the second one manipulates T2WI’s gradient and high- +resolution T1WI. The Cohf-T module interacts between two +streams to embed the prior knowledge in the super-resolution +stream’s features. The Cohf-T module consists of three differ- +ent attention modules: short-distance and long-distance win- +dow attention and inter-modality attention. The first two atten- +tion modules help to model intra-modality dependency. To be +precise, the short-distance window helps recover the local dis- +continuity in boundaries with the help of surrounding structure +28 + +Table 7: A brief description of the reviewed Transformer cooperated in the medical image reconstruction field. +Method +Contributions +Highlights +Low Dose Enhancement +TransCT +[199] +• The proposed prototype was the first successful implementation of a Transformer complement to CNN in +the Low Dose CT reconstruction domain by exploring its revenue within high-frequency and low-frequency +counterparts. +• The Transformer effectively could learn the embedded texture representation from the noisy counterpart +• This paradigm is not convolution-free and uses Transformer as a complement +TED-net +[200] +• Convolution-free U-Net based Transformer model +•Introduced Dialted Token-to-Token-based token serialization for an improved receptive field in Transform- +ers +• Using Cyclic Shift block for feature refinement in tokenization +Eformer +[201] +• Incorporate the learnable Sobel filters into the network for preserving edge reconstruction and improve +the overall performance of the network +• Conduct diverse experiments to validate that the residual learning paradigm is much more effective than +other learning techniques such as deterministic learning approaches +• Successfully imposed the Sobel-Feldman generated low-level edge features with intermediate network +layers for better performance +• To guarantee the convergence of the network, they used the Multi-scale Perceptual (MSP) loss alongside +Mean Squared error (MSE) to hinder the generation of disfavored artifacts +3D T-GAN +[202] +• It is a 3D-based method rather than conventional 2D methods +• First LDPET enhancement study that leveraged from Transformer to model long-range contextual infor- +mation +• To produce more reliable images with generator they used an adversarial loss to make the data distribution +same as real data +STFNet +[203] +• Proposed a dual input U-Net-based denoising structure for low-count PET images with excessive MRI +modality contribution +• Used a Transformer block as a hybrid add-on for feature fusion to make a pixel-to-pixel translation of +PET and MRI modalities +• In comparison with the U-Net and residual U-Net structures due to the different training strategy which is +roughly named Siamese structure has a low computational burden and simplified network +• This network successfully handled the disparity and nonuniformity of shape and modality of PET and +MRI +• The visual results of denoised images testify that the proposed method could recover the detail of texture +more clearly than other networks +CTformer +[204] +• Convolution-free, computational efficient design +• Introduce a new inference mechanism to address the boundary artifacts +• Proposed interpretability method to follow each path resultant attention map through the model to under- +stand how the model denoising +• Alleviate receptive filed deficiency with the token rearrangement +SIST +[205] +• Proposed inner-structure loss to mimic the physics of the functionality of sinogram processing by CT +devices to restrain the noise +• Extracting the long-range dependencies between distinct sinogram angles of views via Transformer +• Utilizing the image reconstruction module to alleviate the artifacts that could happen in sinogram domain +denoising by transferring the sinogram noise into the image domain +• Image domain loss back-propagates into the sinogram domain for complementary optimization +Sparse-View Reconstruction +DuDoTrans +[206] +• To cope with the global nature of the sinogram sampling process introduced, the SRT module, a hybrid +Transformer-CNN to capture long-range dependencies +• Utilizing a dual domain model to simultaneously enrich raw sinograms and reconstruct ct images with +both enhanced and raw sinograms +• To compensate for the drift error between raw and enhanced sinogram representation employs DuDo +Consistency Layer +• Utilizing a residual learning paradigm for image-domain reconstruction +• Fewer parameters in comparison with other structures, e.g., DuDoNet [240] with better performance +FIT +[207] +• Introduced the Fourier Domain Encoding to encode the image to a lower resolution representation for +feeding to the encoder-decoder Transformer for reconstructing the sparse-view CT measurements +• Introduced the Fourier coefficient loss as a multiplicative combination of amplitude loss and phase loss in +the complex domain +CTTR +[208] +• Introduced the encoder-decoder Transformer pipeline to utilize dual-domain information, raw sinogram, +and primary reconstruction of CT via FBP [228] algorithm for sparse-view CT measurement reconstruction +• In contrast to DuDoTrans [206], CTTR directly utilizes raw sinograms to enhance reconstruction perfor- +mance +ARMLUT +[209] +• Proposed a paradigm for CT image reconstruction in a non-trainable manner +• Extending the most DIP research on 2D to 3D medical imaging scenario +• Optimising the large-scale 3D Transformer with only one reference data in an unsupervised manner +• Stabilising the iterative optimization of multi-loss untrained Transformer via re-weighting technique +Undersampled Reconstruction +ViT-Rec +[210] +• This study investigated the advantage of pure Transformer framework, ViT, in fastMRI reconstruction +problem in comparison with the baseline U-Net +• ViT benefits from less inference time and memory consumption compared to the U-Net +• Utilizing pre-training weights, e.g., ImageNet extensively improves the performance of ViT in the low- +data regime for fastMRI reconstruction, a widespread concept in the medical domain +• ViTs that accompany pre-training weights demonstrate more robust performance toward anatomy shifts +T•Net +[211] +• Introduce the first Transformer utilized multi-task learning network in the literature +• Designed the task Transformer for maintaining and feature transforming between branches in the network +• Outperformed the sequentially designed networks for simultaneous MRI reconstruction and super- +resolution with T•Net +• Used the same backbone for feature extraction in branches, however, the purpose of the branches diverse +Super Resolution Reconstruction +DisCNN-ViT +[212] +• Using a Transformer-based network to capture global contextual cues and amalgamate them with CNN’s +local information results in the superior quality of high-resolution images in super-resolution literature +• Creating realistic images is just not a burden on an adversarial loss function, in addition, multiple loss +functions incorporate extra constraints that preserve anatomical and textural information in the begotten +image +•Multi prerequisite steps are required to train the experiments; however, the super-resolution step is a +straightforward end-to-end network +• Need fine-tuning steps for two disentanglement and UNETR networks +• The computational burden of UNETR is high and could use new efficient Transformer designed networks +Cohf-T +[213] +• Leverage the high-resolution T1WI due to its rich structural information for super-resolving T2-weighted +MR images +• Introduced the high-frequency structure prior and intra-modality and inter-modality attention paradigms +within the Cohf-T framework +• Assess prior knowledge into super-resolution paradigm successfully +• Competitive number of FLOPS in reaching SOTA PSNR results in comparison with other attention-based +networks +• End-to-end pipeline for training the network +information, and the long-distance window can capture the tex- +tural and structural patterns for enhanced results. Due to the +discrepancy in intensity levels between T1WI and T2WI, it is +vital to make an alignment between these two domains, and +Fang et al. [213] presented a Feature Alignment (FA) module +to reduce the cross-modality representation gap. They com- +pared their results with T2Net [211] and MTrans [245], which +outperformed both approaches by ∼ 1% in terms of PSNR. +5.5. Discussion and Conclusion +In this section, we outline different Transformer-based ap- +proaches for medical image reconstruction and present +a detailed taxonomy of reconstruction approaches. +We +overviewed 15 studies that profit from the Transformer de- +sign to compensate for the deficiency of CNN’s limited re- +ceptive field. We investigate each study in depth and repre- +sent Table 6 for detailed information about the dataset, uti- +lized metrics, modality, and objective tasks. In Table 7, we +provide the main contribution of each study and the promi- +nent highlight of each method. +Most of the studies in this domain use the original Trans- +former as a plug-and-play module in their design and only +a limited number of studies utilize hierarchical and effi- +cient Transformers. However, the criteria for using multi- +scale and hierarchical architectures are generally important +for dense prediction tasks, e.g. image reconstruction, and +should be considered further. Also, another direction to fol- +low for future research could be to investigate the influence +of using pre-training weights on Transformers due to the +need for a large amount of data for better convergence re- +sults in Transformers, which contradicts the nature of the +medical domain, due to the scarceness of annotated medical +data. +In addition, we noticed that most of the studies focus on +MRI and CT image reconstruction tasks. So there is a need +29 + +Conv +Conv +Upsample +long skip connection +long skip connection +Element-wise sum +super-resolution branch +reconstruction branch +ˆxLR +x′LR +x′ +HRB +SR1 +HRB +SR2 +HRB +SRN +HRB +Rec1 +HRB +Rec2 +HRB +RecN +Htt +1 +Htt +2 +Htt +N +F 0 +SR +F 0 +Rec +F 1 +SR +F 1 +Rec +F 2 +SR +F 2 +Rec +F N +SR +F N +Rec +F 1 +T T +F 2 +T T +F N +T T +(a) An overview of T2Net [244], a multi-task learning framework that consists of a super- +resolution branch and reconstruction branch. The reconstruction branch embraces the +stronger capability of artifact removal therefore, the task Transformer module is fed with +the reconstruction branch. +Transfer Attention +Relevance +Embeding +Concanate +Conv +Soft Attention +Element-wise +multiplication +Conv +Q +K +V +F i +SR +F i +Rec +F i +Rec ↑↓ +T +S +C +Z +FT T +(b) inner design of proposed Htt task Transformer module. Analogous to Figure 2, the +design of T2 module follows the naive design with some modifications, and in contrast to +seminal design, all Q, K, and V entities do not originate from the same representation—Q +comes from the super-resolution branch and the rest from the reconstruction branch. +Figure 29: An overview of T2Net [244]. (a) Multi-Task T2Net pipeline and (b) +Task Transformer Module—T2 Module +Conv +5xRRDB +Conv +5xRRDB +Conv +5xRRDB +MLP +MLP +MLP +Conv +Cross-modality High-frequency Transformer (Cohf-T) +FA +Cohf-T +5xRRDB +𝐈𝑖𝑛 +Conv +𝐑𝑐 +𝐅0 +𝐄1 +𝐅1 +𝐅𝑠0 +𝐅𝑐0 +𝐅𝑠1 +Cohf-T +5xRRDB +𝐄2 +𝐅2 +𝐅3 +𝐏1 +𝐄4 +𝐅4 +Output Gate +𝐏4 +𝐏2 +ത𝐅𝑠1 +𝐑𝑠 +Conv +𝐅𝑐0 +Cohf-T +5xRRDB +𝐄3 +𝐏3 +𝐅𝑐0 +𝐓1 +Input Gate +addition +concatenation +broadcast element-wise +product +Sigmoid function +𝐈𝑜𝑢𝑡 +𝐑𝑜𝑢𝑡 +Short-distance +Window Attention +Long-distance +Window Attention +Inter-modality +Attention +Figure 30: The pipeline of Cohf-T [213] consists of three main branches with +the corresponding input modalities as follows: Iin, Rs, and Rc denote the +low-resolution T2WI, the gradient of low-resolution T2WI and high-resolution +T1WI, respectively. A fully-convolutional branch for density-domain super- +resolution, a Transformer-based branch for restoring high-frequency signals in +the gradient domain, and a guidance branch for extracting priors from the T1 +modality. Conv, RRDB and MLP represent a 3 × 3 convolution operation and +residual-in-residual dense block and multi-layer perceptron, respectively. +for evaluating the applicability of these methods on other +modalities, too. +6. Medical Image Synthesis +In this section, we will overview several instances of Trans- +formers in the medical image synthesis task. The scarcity of +medical data and the high cost of acquisition processes make +this task very valuable in the medical field. Some studies aim +to synthesize missing slices from MRI and CT sequences. In +addition, some methods target capturing the structural infor- +mation in diverse modalities, e.g., CT to MRI image-to-image +translation and vice versa. Figure 31 shows our taxonomy for +the image-synthesized methods. +6.1. Intra-Modality +The main objective of the intra-modality methods is to syn- +thesize high-quality images using low-quality samples from + Medical Image Synthesis + Intra-Modality + 1. PTNet + 2. ResViT + 3. MMT + Inter-Modality + 2. ResViT + 4. CyTran + 5. VTGAN +Figure 31: An overview of ViTs in medical image synthesis. Methods are +categorized by target and source modality. The prefix numbers in the paper’s +name in ascending order denote the reference for each study as follows: 1. +[246], 2. [247], 3. [248], 4. [249], 5. [250]. +the same modality. In this respect, several Transformer-based +approaches are presented to formulate the synthesis task as +a sequence-to-sequence matching problem to generate fine- +grained features. In this section, we will briefly present some +recent samples [246]. +Brain development monitoring is a de facto standard in pre- +dicting later risks; hence it is critical to screen brain biomark- +ers via available imaging tools from early life stages. Due to +this concern and the nature of MRI and subjective infants’ rest- +lessness, it is not relatively straightforward to take all the MR +modalities during the MRI acquisition. Zhang et al. [246] pro- +posed a Pyramid Transformer Net (PTNet) as a tool to re- +construct realistic T1WI images from T2WI. This pipeline is an +end-to-end Transformer-based U-Net-like and multi-resolution +structure network utilizing an efficient Transformer, Performer +[251], in its encoder (PE) and decoder (PD). Analogously to +original U-Net [34], they used skip connection paths for pre- +serving fine-grained features and accurate localization features +for reconstruction. Moreover, the paradigm’s two-level pyra- +midal design helps the network capture local and global infor- +mation in a multi-resolution fashion. They achieved the SOTA +results on the dHCP [252] dataset compared with the flagship +GAN-based image generation method pix2pix (HD) [253, 254]. +Dalmaz et al. [247] introduced a conditional generative ad- +versarial network based on the cooperation of Transformers +and CNN operators, namely ResViT. This paradigm addresses +the issue of needing to rebuild separate synthesis models for +varying source-target modality settings and represents a unified +framework as a single model for elevating its practicality. The +ResViT (Figure 32) pervasively refers to the generator of its +pipeline, whereby it leverages a hybrid pipeline of residual con- +volutional operations and Transformer blocks that enable ef- +fective aggregation of local and long-range dependencies. The +discriminator is based on a conditional PatchGAN framework +[253]. Utilizing standalone Transformer architectures (e.g., PT- +Net [246]) in pixel-to-pixel tasks is quite challenging due to +the quadratic complexity, which limits its usage to fixed-size +patches that hamper its effectiveness. From Figure 32, it is +30 + +Figure 32: The ResViT [247] framework for multi-modal medical image syn- +thesis. The bottleneck of this encode-decoder comprises successively residual +Transformers and residual convolutions layers for synergistically capturing the +fine-grained global and local context. +evident that residual Transformer blocks stacked successively, +known as aggregated residual Transformer (ART) blocks, in the +bottleneck of the encoder-decoder design of the generator to ex- +tract the hidden contextual information of input features. The +primary motivation of ART blocks is to learn an integrated rep- +resentation that combines contextual, local, and hybrid local- +contextual features underhood from the input flow. Channel +Compression (CC) module recalibrates the concatenated fea- +tures from the previous ART block and Transformer module to +select the most discriminant representations. Due to the cas- +cade of Transformers in design, to decrease the model com- +plexity and computational burden, ResViT utilizes weight shar- +ing strategy among projection tensors for Query, Key, value, +and attention heads besides weight matrices for multi-layer per- +ceptron operation. +The superiority of this method has been +proved over several MRI datasets in multi-contrast MRI synthe- +sis and MRI to CT experiments with high PSNR and SSIM met- +rics over the conventional SOTA methods, e.g., pGAN [255], +SAGAN [256], pix2pix [253] and PTNet [246]. +Likewise, Liu et al. [248] addressed the issue of missing +contrasts in MRI imaging and proposed a multi-contrast multi- +scale Transformer (MMT) framework to handle the unavail- +ability of this information by synthesizing the existing con- +trasts as a means to substitute the missing data. To achieve +efficient contrast synthetization, the task is considered as a seq- +to-seq problem, in which the model learns to generate missing +contrasts by leveraging the existing contrast in the following +manner: A Swin multi-contrast Transformer encoder is imple- +mented that creates hierarchical representation from the input +MRI image. Then, a Swin Transformer-based architecture de- +codes the provided representation at multiple scales to perform +medical image synthesis. Both the encoder and decoder are +composed of two sequential swin blocks that capture contrast +dependencies effectively. Conducted experiments on the IXI +[237] and BraTS [185] datasets demonstrated MMT’s advan- +tage compared to previous methods. +6.2. Inter-Modality +Unlike the intra-modality strategies, the inter-modality meth- +ods are designed to learn the mapping function between two +different modalities. This approach allows the network to con- +vert the samples from the base modality into a new modality +and leverage the generated samples in the training process for +the sake of performance gain. In this section, we will elaborate +on two Transformer-based [249, 250] strategies. +Several medical conditions may prevent patients from re- +ceiving intravenous contrast agents while getting CT screen- +ing. However, the contrast agent is crucial in assisting medi- +cal professionals in identifying some specific lesions. There- +fore, CyTran [249] is proposed as an unsupervised generative +adversarial convolutional Transformer for translating between +contrast and non-contrast CT scans and image alignment of +contrast-enhanced CT scans to non-enhanced. Its unsupervised +part is also derived from its cyclic loss. CyTran is composed +of three main modules: I) A downsample CNN-based mod- +ule designed for handling high-resolution images, II) A con- +volutional Transformer module tailored for incorporating both +local and global features, and III) An upsampling module de- +veloped to revert the transformation of the downsampling block +through transpose-convolution. Additionally, the authors intro- +duce a new dataset, Coltea-Lung-CT-100W, comprised of 100 +3D anonymized triphasic lung CT scans of female patients. +Q +K +V + Multi-head attention +Norm layer +Pointwise convolution +Convolutional projection +Convolutional transformer block +Upsampling block +Downsampling block +T +Z* +Figure 33: An overview illustration of CyTran [249]. An input image is fed +through a downsampling block to extract its features and make it compatible +with high-resolution images. The output then passes through a convolutional +Transformer block to enrich features by capturing local and global informa- +tion. In the final step, enriched features are upsampled to the image size using +transpose-convolution. +Furthermore, Kamran et al. [250] trained a ViT-based gener- +ative adversarial network (VTGAN) in a semi-supervised fash- +ion on the Fundus Fluorescein Angiography (FFA) dataset pro- +vided by [258] via incorporating multiple modules, including +residual blocks as generators for coarse and fine image gener- +ation, and two ViT architectures consisting of identical trans- +former encoder blocks for concurrent retinal abnormality clas- +sification and FA image synthesis. +6.3. Discussion and Conclusion +This section covers the adoption of ViT architectures in +medical image synthesis applications. We explored the pro- +31 + +Input Patch Embeddings +Outru +Layer Norm +ReLU +Downsampler +Multi-Head + Self-Attention +Patch Flattening +ReLU +Trainable Linear Projection +Residua +BatchNorm +CNN +3x3 conv +Positional Encoding +Layer Norm +ReLU +BatchNorm +Transformer Encoder +3x3 conv +Multi-Layer +123456789101112 +Perceptron +Patch Deflattening +3x3 conv +Upsampler +channe +1x1 conv +compressior +3x3 conv +Concatenation +Output Patch EmbeddingsTable 8: An overview of the reviewed Transformer-based medical image synthesizing approaches. +Method +Concept(s) +Modality +Type +Pre-trained Module: Type +Dataset(s) +Metrics +Year +Pure +PTNet +[246] +Intra-Modality +MRI +2D + +dHCP dataset [252] +SSIM +PSNR +2021 +MMT +[248] +Intra-Modality +Inter-Modality +MRI +2D + +1 IXI [237] +2 BraTS [185] +SSIM +PSNR +LPIPS +2022 +Bottleneck +ResViT +[247] +Intra-Modality +Inter-Modality +CT +MRI +2D +ViT: Supervised +1 IXI [237] +2 BraTS [179, 180, 181] +3 Multi-modal pelvic MRI-CT [257] +PSNR +SSIM +2021 +CyTran +[249] +Inter-Modality +CT +2D +3D + +Coltea-Lung-CT-100W [249] +MAE +RMSE +SSIM +2022 +Decoder +VTGAN +[250] +Inter-Modality +Angiography +2D + +Fundus Fluorescein Angiography [258] +Fr´echet inception distance +Kernel Inception distance +2021 +Table 9: A brief description of the reviewed Transformer-based medical image synthesizing models. +Method +Contributions +Highlights +Pure +PTNet +[246] +• Introduced the pure Transformer-based network with linear computational complexity for image- +synthesizing context +• Practical inference time around 30 image/s +MMT +[248] +• Proposed a pure Transformer-based architecture that incorporates Swin Transformer blocks to perform +missing data imputation by leveraging the existing MRI contrasts. +• Conducted experiments on the IXI and BraTS datasets to perform qualitative and quantitative analysis +and confirm their model’s efficiency. +• Since the attention mechanism can be utilized to pinpoint influential features in the model’s reasoning +and decision-making, the attention scores of the Transformer decoder in MMT make it interpretable by +capturing information in different contrasts that play an important role in generating the output sequence. +• The framework can be applied to a variety of medical analysis tasks, including image segmentation and +cross-modality synthesis. +Bottleneck +ResViT +[247] +• First conditional adversarial model for medical image-to-image translation with hybrid CNN-Transformer +generator +• Introduced a new module, ART block, for simultaneously capturing localization and contextual informa- +tion +• Utilized weight sharing strategy among Transformers to hinder the computational overhead and lessen +the model complexity +• An end-to-end design for the synthesized model that generalizes through multiple settings of source-target +modalities, e.g., one-to-one and many-to-one tasks +CyTran +[249] +• Proposing a generative adversarial convolutional Transformer for two tasks of image translation and image +registration. +• Introducing a new dataset, named Coltea-Lung-CT-100W, comprised of 100 3D anonymized triphasic +lung CT scans of female patients. +• The presented method can handle high-resolution images due to its hybrid structure +• Utilized style transfer techniques to improve alignment between contrast and non-contrast CT scans +Decoder +VTGAN +[250] +• Proposed a synthesis model for the task of fundus-to-angiogram that incorporates ViT architecture in the +decoder section of the system to concurrently classify retinal abnormalities and synthesize FA images. +• Prepared experimental data based on quantitative and qualitative metrics regarding the model’s general- +ization ability under the influence of spatial and radial transformations. +• Has the potential to be adopted as a tool for tracking disease progression. +• The system is designed to operate on non-invasive and low-cost fundus data to generate FA images. +posed methods based on two synthesis approaches: (1) inter- +modality, in which the target modality is synthesized in a +way that it encapsulates crucial diagnostic features from +different source images; and (2) intra-modality, with the +objective of yielding target images with better quality by +integrating information from lower resolution source im- +ages. To demonstrate their effectiveness, these approaches +usually rely on SSIM, PSNR, and LPIPS as the evaluation +metrics, since they are designed to measure the similarity +between images. We also reviewed a ViT-based synthesis +model [250] that operates in a decoder fashion for the task +of fundus-to-angiogram translation with different evaluation +measurements, including Fr´echet Inception Distance (FID) +and Kernel Inception Distance (KID). We have additionally +provided the architectural type, modality, input size, training +setting, datasets, metrics, and year for every medical regis- +tration technique analyzed in Table 8. Furthermore, Table 9 +lists the contributions and highlights of the proposed works. +In particular, with the scarcity of works with ViT implemen- +tations and the recent advancement in the medical synthesis +field with Transformer models, we believe that these sys- +tems require more research effort to be put into them. For +example, Transformers have much room for improvement +to generate more realism and high-quality synthesized med- +ical images. One way to achieve this is by incorporating +more detailed anatomy and physiology features using more +efficient and effective attention mechanisms. Additionally, +while much of the current research in this area has focused +on 2D medical images and CT and MRI modalities, there is +potential to apply these techniques to other types of medical +images, including 3D and microscopy images. +7. Medical Image Detection +Object detection remains one of the challenging problems in +computer vision, especially detection in the medical image do- +main has its own challenges. Current state-of-the-art architec- +tures which work on 2D natural images use Vision Transform- +ers. The Vision Transformers used in the detection task can be +classified into two Transformer backbones and detection Trans- +formers. In addition, the Transformer module can be used in a +hybrid manner. Detection Transformers generally represent an +32 + + Medical Image + Detection + Backbone + 1. TR-Net + 2. RDFNet + 3. + CellCentroidFormer + Neck + 4. COTR + 5. CT-CAD + 6. Spine-Transformer + Head + 7. Focused Decoder +Figure 34: An overview of Transformers in medical image detection. Methods are classified into the backbone, neck, and head according to the positions of the +Transformers in their architecture. The prefix numbers in the paper’s name in ascending order denote the reference for each study as follows: 1. [259], 2. [260], +3. [261], 4. [145], 5. [262], 6. [263], 7. [264]. +end-to-end detection pipeline with an encoder-decoder struc- +ture, while the Transformer backbone solely utilizes the Trans- +former encoder for feature refinement. In order to increase de- +tection performance, object detectors combine variants of vi- +sion Transformers with classical convolutional neural networks +(CNNs). +Quite recently, Carion et al. introduced the concept of DETR +[163], which forms a foundation for Detection Transformers. +DETR uses a ResNet backbone to create a lower-resolution rep- +resentation of the input images. Even though this approach +achieves very good 2D detection results, comparable to the +R-CNN backbone, high computational complexity is a down- +side of this method. +The deformable DETR [23] approach +has improved DETR’s detection performance overcoming the +problem of high computational complexity. Many recent ap- +proaches have tried to improve DETR’s detection concept over +time. Efficient DETR [265] eliminated DETR’s requirement for +iterative refinement. Conditional DETR [266] introduced the +concept of a conditional cross-attention module. DN-DETR +[267] introduced a denoising strategy, and DINO [268] im- +proved many aspects, such as denoising training, etc. Recently, +some studies performed experiments on 2D medical data such +as [269], [270] etc. However, only very few attempts tried to +adapt it to 3D. Spine Transformer was proposed by Tao et al. +[263] for sphere-based vertebrae detection. Another approach +in 3D detection was proposed by Ma et al. [259], which in- +troduced a novel Transformer that combines convolutional lay- +ers and Transformer encoders for automatically detecting coro- +nary artery stenosis in Coronary CT angiography (CCTA). An +approach to better extract complex tooth decay features was +proposed by [260]. For end-to-end polyp detection, Shen et +al. [271] proposed an approach which was based on the DETR +model. Kong et al. have proposed the approach CT-CAD [262], +context-aware Transformers for end-to-end chest abnormality +detection. The pros and cons of different approaches are sum- +marized in Table 11. Table 10, indicates other details such as +modalities, organs, datasets, metrics, etc. Some of the afore- +mentioned detection papers in the medical image domain are +summarized in this section. +7.1. Backbone +This section explains Transformer networks using only the +Transformer encoder layers for object detection. The work pro- +posed by Ma et al. [259] uses a Transformer network (TR-Net) +for identifying stenosis. A leading threat to the lives of cardio- +vascular disease patients globally is Coronary Artery Disease +(CAD). Hence, the automatic detection of CAD is quite signif- +icant and is considered a challenging task in clinical medicine. +The complexity of coronary artery plaques, which results in +CAD, makes the detection of coronary artery stenosis in Coro- +nary CT angiography (CCTA) challenging. +Figure 35: Proposed architecture of TR-Net model [259]. +The architecture introduces a Transformer and combines the +feature extraction capability of convolutional layers and Trans- +former encoders. TR-Net can easily analyze the semantic in- +formation of the sequences and can generate the relationship +between image information in each position of a multiplayer +reformatted (MPR) image. This model can effectively detect +stenosis based on both local and global features. The CNN eas- +ily extracts the local semantic information from images, and +the Transformer captures global semantic details more easily. +A 3D-CNN is employed to capture the local semantic features +33 + +Input +3D-CNN +TransformerStructure +Output +non-significant +stenosis +Transformer +CNN +Encoder +Transformer +CNN +Encoder +significant +stenosis +CNN +Flattening +CCTA +max +Transformer +Encoder +Centerline of +CoronaryArtery +Transformer +CNN +Encoder +XT +MPRImage +featuremaps +orderembeddingfrom each position in an MPR image. After this step, the Trans- +former encoders are mainly used to analyze feature sequences. +The main advantage here is that this helps in mining the depen- +dency of local stenosis on each position of the coronary artery. +The architecture of the TR-Net is given in Figure 35. One part +of the figure indicates the 3D-CNN. This module extracts the +local features. The other part indicates the Transformer encoder +structure. This module associates the local feature maps of each +position. This module also helps in analyzing the dependency +between different positions, which in turn is helpful for classi- +fying the significant stenosis at each position. The CNN part +mainly has two main advantages: it prevents the overfitting of +semantic information and improves the model’s efficiency. The +input to the network architecture is the coronary artery MPR +image. +The 3D-CNN module has four sequentially connected sub- +structures, which consist of a convolutional kernel of size +3 × 3 × 3, a non-linear ReLU layer and a 2 × 2 × 2 max-pooling +layer. The number of filters is 16 in the first part, and in sub- +sequent parts, the number of filters is double the number in the +previous part. Since Transformers have 1D vector sequences +as input, the feature maps are flattened. The Transformer in +the proposed architecture consists of 12 Transformer encoders. +Each Transformer encoder mainly consists of two sub-blocks - +multi-head self-attention (MSA) and the feed-forward network +(FFN), which are connected sequentially. Layer normal (LN) +and residual connections are employed before and after two +sub-blocks. In order to ensure the consistency of the encoders, +the size of the input is made the same as the size of the output. +The output of the previous encoder is given as input to the next +encoder. In the final layer, the embeddings are fed into softmax +classifiers to detect significant stenosis. +RDFNet approach proposed by Jiang et al. [260] basically +incorporates the Transformer mechanism in order to better ex- +tract the complex tooth decay features. The incorporation of the +Transformer has improved the detection accuracy. The main +three modules of the network are the backbone, neck, and pre- +diction modules. The backbone module is mainly used to ex- +tract the features from caries images. In the backbone module, +the focus operation is a slicing operation that could easily re- +place the convolution operation and reduce the loss of feature +information. The C3Modified layer is a convolution module ac- +tivated by the FReLU function, which extracts complex visual- +spatial information of the caries images. SPP [272] module +has a spatial pyramid structure that could expand the percep- +tual field, which intern fuses the local and global features and +enhance the feature maps. After the SPP structure, RDFNet +appends an improved Transformer-encoder module to improve +the feature extraction capability. The main functionality of the +neck module is to mainly fuse the feature maps of different sizes +and extract high-level semantic structures. This module mainly +uses the structure of the feature pyramid network (FPN) pro- +posed in [273], and path aggregation network (PAN) proposed +in [274]. The FPN approach is employed in a top-down fash- +ion, and PAN is performed in a bottom-up fashion to generate +the feature pyramids. In order to prevent information loss, fea- +ture fusion is performed using both bottom-up and top-down +approaches. An improved C3Modified convolutional module +is adopted into the neck module to better extract the seman- +tic features of caries images. The high-level features generated +by the neck module are used by the prediction module, which +in turn is used to classify and regress the location and class of +the objects. To overcome the problems of the single-stage de- +tection method, which has quite a low detection accuracy, it +mainly has three detection heads for detecting large, medium, +and small objects. As Transformers have proved to have strong +feature extraction capability, in order to extract complex fea- +tures, they utilized the Transformer model. To better extract the +features, three Transformer encoders were stacked together. To +simplify the model, the authors removed the original normaliza- +tion layer from the Transformer encoder. In order to extract the +deep features, the feature map was fed into this structure. For +each head, the attention values were calculated independently +and later concatenated. +Wagner et al. [261] proposed a novel hybrid cell detection +approach (CellCentroidFormer) in microscopic images that +combines the advantages of vision Transformers (ViTs) and +convolutional neural networks (CNNs). A CNN model pre- +trained on the ImageNet dataset is mainly used for extracting +the features and reducing the amount of training data, and the +concept of transfer learning is also proposed. Authors show +that the combined use of convolutional and Transformer layers +is advantageous as the convolutional layers can focus on the +local information (cell centroids), and the Transformer layers +can focus on the global information ( overall shapes of a cell). +The proposed centroid-based approach represents the cells as +ellipses and is trainable in an end-to-end fashion. Four differ- +ent 2D microscopic datasets were used for experimental eval- +uations, and the results outperformed the fully convolutional +architecture-based methods. Figure 36 shows the architecture. +The encoder is then folded into a 3D tensor, which is afterward +Figure 36: Proposed architecture of CellCentroidFormer model [261]. +concatenated with the input tensor. The MobileViT block is a +lightweight alternative to the actual encoder-decoder approach +using a Transformer [21]. Due to the multi-head self-attention +layers, the MobileViT block causes a much higher computa- +tional complexity than convolutional layers. To not increase the +computational complexity excessively, the MobileViT blocks +are combined in the neck part of the proposed model. Layer +34 + + 384 x 384 x 1 +384 x 384 x 3 +384 x 384 x 2 +Neck +Backbone +Heads +MobileViT +Bilinear +Layer +2D +Block +Upsampling +Normalization +Convolutionnormalization is added for regularization and to allow higher +learning rates. The backbone module of our proposed model +is the EfficientNetV2S [275] CNN model. This block mainly +consists of six high-level blocks, out of which five blocks are +used to extract image features. To use the advantage of trans- +fer learning, the backbone module is initialized with weights +learned from training on ImageNet. This, in turn, reduces the +amount of required training data. The EfficientNetV2S [275] +CNN models are generally optimized for a fixed input size. +Therefore the input images need to be resized to this input +size. The cells are represented mainly by the centroid, width, +and height parameters. Mainly, two fully convolutional heads +are used to predict these cell parameters in the paper. These +heads contain 2D convolution, batch normalization, and bilin- +ear upsampling layers. More MobileViT blocks are not used as +it will increase the computational complexity. Later convolu- +tional layers have a bigger receptive field which helps in cap- +turing the global information [276] effectively. The first convo- +lutional head predicts a heatmap for detecting the cell centroids, +and the second head is used for predicting the cell dimensions. +The output dimensions of this model are 384 × 384. The au- +thors use one decoder of the Dual U-Net to predict the centroid +heatmap, and the second branch predicts the dimensions of the +detected cells. The shapes of the cells are focused on by the +Transformer layers in the network. +7.2. Head +Detection Transformers based on Transformer encoder- +decoder architecture require a large amount of training data to +deliver the highest performance. However, this is not feasible in +the medical domain, where access to labeled data is limited. To +address this problem, for the detection of 3D anatomical struc- +tures from the human body, Wittmann et al. [264] proposed a +detection Transformer network with a focused decoder. This +network considers the relative position of the anatomical struc- +tures and thus requires less training data. The focused decoder +uses an anatomical region atlas to deploy query anchors to fo- +cus on the relevant anatomical structures. The proposed net- +work omits the Transformer encoder network and consists of +only Transformer decoder blocks. The authors show that in +3D datasets, avoiding the encoder can reduce the complexity of +modeling relations with a self-attention module. +The model architecture contains a backbone network for fea- +ture extraction, a focus decoder network for providing well- +defined detection results, a classification network to predict the +classes, and a bounding box regression network to output the +best possible bounding box. The feature extraction backbone +network is a feature pyramid network (FPN) inspired by the +RetinaNet [277]. Features from the second layer (P2) are flat- +tened before being given as input to the focus decoder. A spe- +cific anatomical region atlas [278] containing regions of interest +(RoI) is determined for each dataset. Then to each RoI, uni- +formly spaced query anchors are placed, and a dedicated object +query is assigned to each. Such an object query will restrict the +focus decoder network to predict solely within their respective +RoI. +The focused decoder network contains a self-attention mod- +ule, a focused cross-attention module, and a feedforward net- +work (FFN). The self-attention module encodes strong posi- +tional inter-dependencies among object queries. The focused +cross-attention module matches the input sequence to object +queries to regulate the individual feature map for prediction via +attention. The FFN network then enables richer feature repre- +sentation. Also, residual skip connections and normalizations +are used to increase gradient flow. The classification network +consists of a single fully-connected layer, and the bounding box +regression network consists of three layers. The bounding box +predictions are combined with query anchors to get the bound- +ing box together with class-specific confidence scores. The net- +work is trained to predict 27 candidates predictions per class. +Dynamic labeling with the help of generalized intersection over +union (GIoU) is created during training to get 27 predictions. +During inference, the prediction with the highest confidence +score indicates the best candidate. The model is trained end- +to-end with the above GIoU loss, binary cross-entropy loss for +the classification network, and L1 loss for the bounding box +predictions. +7.3. Neck +Detection methods using region-based approaches need to +generate anchor boxes to encode their prior knowledge and use +a non-maximum suppression to filter the resulting bounding +boxes after prediction. These pre-and post-processing steps re- +markably reduce the detection performance. To bypass these +surrogate tasks, Carion et al. [163] proposed Detection Trans- +former (DETR), which views the object detection task as a +direct set prediction problem using an encoder-decoder archi- +tecture using Transformers. The self-attention mechanism of +the Transformers, which explicitly models all pairwise inter- +actions between elements in a sequence, helps to predict the +set of detections with absolute prediction boxes directly from +the image rather than using an anchor. For the end-to-end de- +tection of polyp lesions, Shen et al. [271] proposed a convo- +lution in Transformer (COTR) network based on the DETR +model. COTR consists of 4 main layers: 1) a CNN backbone +network used for extracting features, 2) Transformer encoder +layers embedded with convolutional layers used for feature en- +coding and reconstruction, 3) Transformer decoder layers used +for object querying, and 4) a feed-forward network used for +detecting prediction. Embedding convolutional layers into the +Transformer encoder layer leads to convergence acceleration +compared to the slow convergence of the DETR model. The +CNN backbone uses a model pre-trained with ResNet18 [63] +architecture for feature extraction. This layer converts input +medical images to a high-level feature map. The authors then +use a 1 × 1 convolution to reduce the channel dimensions. In +the Transformer encoder layers, they used six convolution-in- +Transformer encoders to collapse this spatial structure into a +sequence. Then they use a convolution layer to reconstruct the +sequential layer back to the spatial one. In the encoder layer, +each Transformer has a standard architecture with a multi-head +self-attention module and a feed-forward network. To the in- +put of each attention layer, a positional embedding [21] is also +introduced. In the Transformer decoder layers, they used six +decoders which follow the standard architecture of the Trans- +former except that it also decodes object queries in parallel. +35 + +Each object query will correspond to a particular object in the +image. The decoders take these object queries with position +embeddings as well as output embeddings from the encoder +network and convert them into decoded embeddings. +Then +they used a feed-forward network with two fully connected lay- +ers for converting these decoded embeddings into object pre- +dictions. The first fully connected layer is a box regression +layer to predict object location, and the second one is a box- +classification layer to predict object scores. Therefore, the ob- +ject queries are independently decoded into box coordinates +and classes by the feed-forward network, which results in fi- +nal predictions, including object and no object (background) +predictions. This model transforms the object detection prob- +lem into a direct set prediction problem by training end-to-end +by calculating bipartite matching loss (Hungarian algorithm) +between predictions and ground truth for each query. If the +number of queries exceeds the number of objects in the image, +the remaining boxes are annotated as no object class. Thus, +the model is trained to predict output for each query as an ob- +ject or no object detection. For the class prediction, they used +negative log-likelihood loss, and for the bounding box local- +ization, they used an L1 loss with generalized intersection over +union (GIOU) [279] loss. The experiments demonstrated that +the proposed model achieved comparable performance against +state-of-the-art methods. +Many deep learning detection methods lack using context- +relevant information for improved accuracy, and they also gen- +erally suffer from slower convergence issues and high compu- +tational costs. The proposed CT-CAD [262], context-aware +Transformers for end-to-end chest abnormality detection, ad- +dress these problems. The model consists of two main modules: +1) a context-aware feature extractor module for enhancing the +features, and 2) a deformable Transformer detector module for +detection prediction and to accelerate the convergence speed. +The context-aware feature extractor network uses a ResNet50 +backbone, dilated context encoding (DCE) blocks, and posi- +tional encoding structure. The deformable Transformer detec- +tor contains a Transformer encoder-decoder architecture and +a feed-forward network. The proposed design of the context- +aware feature extractor is inspired by the feature fusion scheme +from DetectoRS [280] which is based on the Feature Pyramid +Networks (FPN) [281]. The feature fusion scheme iteratively +enhances the features of the FPN to powerful feature represen- +tations. Likewise, the DCE blocks enhance the features ex- +tracted from the ResNet50 backbone by expanding the recep- +tive fields to fuse multiscale context information using dilated +convolution filters of different sizes. This powerful feature map +benefits in detecting objects across various scales. Inspired by +YOLOF [282] the DCE block uses dilated convolution and skip +connections to achieve a larger receptive field and acquire more +local context information. Finally, all the features from differ- +ent DCE blocks computed at different scales are summed up to +get the feature map for the output. +The proposed design of the deformable Transformer detector +contains single-scale and multi-head attention properties. The +deformable attention block attends to a small set of key sam- +pling points, thus allowing the Transformer to focus on the fea- +ture space and accelerate the convergence. The authors used +six encoder and decoder layers with positional encoding to ob- +tain the decoder outputs. The outputs from the decoder are the +number of abnormalities detected and the dimension of the de- +coder layers. Finally, a feed-forward network is used to out- +put the category classification and location regression results. +The model is trained end-to-end with a combination of bound- +ing box loss and classification (cross-entropy) loss. The authors +adopted GIoU [283] to balance the loss between large and small +object bounding boxes. +The attention module in the detection Transformers com- +putes similarity scores between elements of each input data +to identify complex dependencies within these data. Calcu- +lating similarities of all possible positional pairs in the in- +put data scales quadratically with the number of positions and +thus becomes computationally very expensive. For this reason, +the Transformer-based object detection model from 3D images +has never been applied. +Tao et al. +[263] proposed a novel +Transformer-based 3D object detection model as a one-to-one +set prediction problem for the automatic detection of verte- +brae in arbitrary Field-Of-View (FOV) scans, called the Spine- +Transformers. Here the authors used a one-to-one set-based +global loss that compels a unique prediction for preserving the +sequential order of different levels of vertebrae and eliminated +bipartite matching between ground truth and prediction. The +main modules of the Spine-Transformer are (1) a backbone +network to extract features, (2) a light-weighted Transformer +encoder-decoder network using positional embeddings and a +skip connection, and (3) two feed-forward networks for detec- +tion prediction. The authors used a ResNet50 [63] architec- +ture without the last SoftMax layer as the backbone network to +extract high-level features. These features are passed through +a 1 × 1 × 1 convolutional layer to reduce the channel dimen- +sions and then flattened to get a feature sequence to feed as the +input for the Transformer network. The light-weighted Trans- +former encoder-decoder network contains only a two-layer en- +coder and two-layer decoder to balance between feature res- +olution and memory constraint. In both the encoder and de- +coder layers of the network, learnable positional embeddings +are used. The authors found that using a skip connection across +the Transformer encoder-decoder network will help in the prop- +agation of context and gradient information during training and +thus improves performance. The two feed-forward networks +are then used to predict the existence of the objects and regress +their coordinates. The authors also proposed a sphere-based +bounding box detector to replace the rectangular-based bound- +ing box to introduce rotational invariance called InSphere de- +tector. The Spine-Transformer is trained end-to-end with fixed- +size patch images to predict all the vertebrae objects in parallel +by forcing one-to-one matching. Binary cross-entropy loss is +used as classification loss, and to enforce the order of the pre- +dicted vertebrae objects, an edge loss is introduced, which is +an L1 distance loss introduced between the centers of the top +and bottom neighborhood vertebrae objects. For better local- +ization accuracy of the bounding sphere detection, the authors +used generalized inception-over-union (GIoU) [279] loss. The +results of this model showed superior results to all the state- +36 + +Table 10: An overview of the reviewed Transformer-based medical image detection approaches. +Method +Modality +Organ +Type +Pre-trained Module: Type +Datasets +Metrics +Year +Backbone +TR-Net [259] +MPR +Heart (Coronary Artery) +3D +Supervised +Private dataset +Accuracy, +Sensitivity, +Specificity, +PPV, NPV, +F1-score +2021 +RDFNet [260] +Dental Caries +Teeth +2D +Supervised +Private dataset +Precision, +Recall, +mAP@0:5 +2021 +CellCentroidFormer [261] +Microscopy +Cells +2D +Supervised +1 Fluo-N2DL-HeLa (HeLa) [284] , +2 Fluo-N2DH-SIM+ (SIM+) [284], +3 Fluo-N2DH-GOWT1 (GOWT1) [284] , +4 PhC-C2DH-U373 (U373) [284]. +Mean-IoU, +SSIM +2022 +Head +Focused decoder [264] +CT +Multi-organ +3D +Semi-Supervised +1 VISCERAL anatomy benchmark [285], +2 AMOS22 challenge [286]. +mAP, +AP50, +AP75 +2022 +Neck +COTR [163] +Colonoscopy +Colon +2D +Supervised +1 CVC-ClinicDB [287], +2 ETIS-LARIB [288], +3 CVC-ColonDB [289]. +Precision, +Sensitivity, +F1-score +2021 +CT-CAD [262] +X-ray +Chest +2D +Supervised +1 Vinbig Chest X-Ray dataset [290] +2 ChestXDet-10 dataset [291] +AP50 +2021 +Spine-Transformer [263] +CT +Vertebra +3D +Supervised +1 VerSe 2019 [292], +2 MICCAI-CSI 2014 [293], +3 Private dataset +Id-Rate, +L-Error +2021 +of-the-art methods. The authors also claim that by using a 3D +CNN-based landmark regression [299], the localization accu- +racy can be further improved. +7.4. Discussion and Conclusion +In this chapter, several well-known Transformer architec- +tures are analyzed to address the automatic detection chal- +lenge. Based on the Transformer model contribution to the +network structure, we grouped the set of literature work into +the backbone, neck, or head strategies and for each cate- +gory, we provided sample works. In this respect, the core +idea behind each network design along with the pros and +cons of the strategies are highlighted in the summary tables. +Vision Transformers have been shown to make more accu- +rate diagnoses compared to traditional methods of analyzing +medical images. These deep learning models can be trained +on large datasets, such as ImageNet, and fine-tuned on medi- +cal image datasets to improve their performance in detecting +abnormalities in X-rays, CT scans, and MRIs. By incorpo- +rating information from multiple modalities, Transformers +can further enhance their ability to identify and detect rare +or subtle abnormalities in medical images. Many medical +images are often taken over time, and incorporating tempo- +ral information into the model can improve its performance. +For example, the model can be designed to take into account +the temporal evolution of diseases or conditions. Overall, +Transformers have demonstrated their capabilities to signifi- +cantly improve the accuracy and efficiency of medical image +analysis, leading to advances in healthcare. +8. Medical Image Registration +Medical image registration is the task of transforming a set +of two or more images of an organ or a biological process taken +with different poses, time stamps, or modalities (e.g., CT and +MRI) into a geometrically aligned and spatially corresponding +image that can be utilized for medical analysis. The transfor- +mation can be discovered by solving an optimization problem +that maximizes the similarity between the images to be regis- +tered [300]. A pair-wise registration of two MRI brain scans is +shown in Figure 37 for illustration. +Moving Image (M) +Fixed Image (F) +Spatial Transformation +Registered Image +Figure 37: An example of pair-wise medical image registration. The goal of +image registration is to geometrically align the moving image with the target or +fixed image by performing the spatial transformation. +Despite remarkable advancements in the quality of medical +imaging techniques that aid professionals in better visualiza- +tion and analysis of image data, a prominent challenge prevails +37 + +Table 11: A brief description of the reviewed Transformer-based medical image detection models. The unreported number of parameters indicates that the value +was not mentioned in the paper. +Method +# Params +Contributions +Highlights +Backbone +TR-Net +[259] +- +• This work is the first attempt to detect coronary artery stenosis more accu- +rately by employing Transformers. +• To detect significant stenosis, local and global features are effectively inte- +grated into this approach, which has resulted in more accurate results. +• While detecting significant stenosis, the TR-Net architecture is capa- +ble of combining the information of local areas near stenoses and the +global information of coronary artery branches. +• Compared to state-of-the-art methods, the TR-Net model has better +results on multiple indicators. +• The shallow CNN layer prevents the overfitting of semantic informa- +tion and improves the overall efficiency. +• The gain in performance comes with a trade-off in the number of pa- +rameters, which affects the computational complexity. +RDFNet +[260] +- +• An image dataset of caries is created, which is annotated by professional +dentists. +• For better extraction of the complex features of dental caries, the Transformer +mechanism is incorporated. +• In order to increase the inference speed significantly, the FReLU activation +function is adopted. +• Compared with existing approaches, the accuracy and speed of caries +detection are better. +• Method is applicable to portable devices. +• The method does not work really well when the illumination of the +oral image is insufficient. +• Even though detection accuracy and speed are improved compared to +the original approach, the detection speed is not the fastest. +CellCentroidFormer +[261] +11.5M +• A novel deep learning approach that combines the self-attention of Trans- +formers and the convolution operation of convolutional neural networks is pro- +posed. +• A centroid-based cell detection method, denoting the cells as ellipses is pro- +posed. +• Pseudocoloring in combination with pre-trained backbones shows im- +proved cell detection performance. +• The model outperforms other state-of-the-art fully convolutional one- +stage detectors on four microscopy datasets, despite having a lower +number of parameters. +• Larger output strides worsen the performance. +Head +Focused Decoder +[264] +VISCERAL Dataset - 41.8M +AMOS22 Dataset - 42.6M +• First detection Transformer model for 3D anatomical structure detection. +• Introduced a focused decoder to focus the predictions on RoI. +• Better results compared to existing detection models using a Trans- +former network like DETR [163] and deformable DETR [23]. +• Comparable results to the RetinaNet [277]. +• Varying anatomical fields of view (FoVs) can affect the robustness of +the model. +Neck +COTR +[145] +- +• Proposed a convolution layer embedded into the Transformer encoder for +better feature reconstruction and faster convergence compared to DETR. +• COTR has comparable results with state-of-the-art methods like Mask +R-CNN [294] and MDeNet-plus [295]. +• This approach produces low confidence for a particular type of lesion. +CT-CAD +[262] +- +• Proposed a context-aware feature extractor, which enhances the receptive +fields to encode multi-scale context-relevant information. +• Proposed a deformable Transformer detector that attends to a small set of key +sampling locations and then the Transformers can focus to feature subspace and +accelerate the convergence speed. +• CT-CAD outperforms the existing methods in Cascade R-CNN [296], +YoLo [297], and DETR [23]. +• CT-CAD is capable to detect hard cases, such as nodules that are ig- +nored by Faster R-CNN. +• Compared to the ChestXDet-10 dataset, this model has a lower perfor- +mance on the Vinbig Chest X-Ray dataset which has higher categories +of abnormalities with more complex patterns. +Spine-Transformers +[263] +- +• Proposed a 3D object detection model based on the Transformer’s architec- +ture. +• Proposed a one-to-one set global loss that enforces unique prediction and +preserves the sequential order of vertebrae. +• Proposed a Sphere-based bounding box to enforce rotational invariance. +• Obtained better results for all datasets compared to state-of-the-art +methods. +• The model has a higher Id-Rate on both the datasets, but a higher +L-Error compared to the benchmark by [298]. +in developing a system capable of effective integration of visual +data that captures useful information from original images with +high precision. Most registration procedures take into account +the whole image as input by utilizing global information for +spatial transformation, which leads to inefficient and slow inte- +gration of data. Furthermore, the collection process of medical +images for training is slow and toilsome, performance degrades +due to the presence of outliers, and local maxima entail neg- +ative effects on performance during optimization [301, 302]. +The emergence of deep learning methods alleviated these prob- +lems by automatic extraction of features utilizing convolutional +neural networks (CNN), optimizing a global function, and im- +proving registration accuracy. For instance, Balakrishnan et al. +[303] utilized a CNN to achieve unsupervised deformable reg- +istration by treating it as a parametric function to be optimized +during training. Furthermore, Chen et al. [304] presented an +unsupervised CNN-based registration algorithm to produce an- +thropomorphic phantoms. However, there are still limitations +in capturing long-range spatial correspondence in CNN-based +frameworks [305, 42]. +Fueled by the strong ability of Transformers to model long- +range dependencies and detect global information [306, 27, +307], they have gained the attention of researchers in the med- +ical image registration domain in recent years. In this section, +we review Transformer-based methods in medical image reg- +istration that ameliorate the aforementioned shortcomings of +previous systems by utilizing the self-attention mechanism. We +have organized the relevant approaches based on their type of +registration: +(a) Deformable registration, which employs an optimization +algorithm to tune the transformation model, is a way that +maximizes the similarity measure function for the images +of interest [308]; +(b) Rigid registration, which achieves correspondence by +maintaining the relative distance between each pair of +points between the patient’s anatomy images [308]. +38 + + Medical Image + Registration + Rigid + 1. SVoRT + Deformable + 2. ViT-V-Net + 3. TransMorph + 4. DTN + 5. XMorpher + Affine + 6. C2FViT +Figure 38: Taxonomy of Transformer-based image registration based on their +transformation type. We use the prefix numbers in the figure in ascending order +and reference the corresponding paper as follows: 1. [307], 2. [309], 3. [310], +4. [311], 5. [312], 6. [313]. +(c) Affine registration, which contains the same operations as +rigid registration plus non-isometric scaling. +8.1. Deformable Registration +Most existing Transformer-based algorithms focus on de- +formable transformation to perform medical image registra- +tion. +Vit-V-Net [309] is the earliest work that incorporates +Transformers to perform medical image registration in a self- +supervised fashion. It is inspired by the integration of vision +Transformer-based segmentation methods with convolutional +neural networks to enhance the localization information recov- +ered from the images. Unlike previous research that employed +2D images for spatial correspondence, Vit-V-net stepped to- +wards utilizing ViT [22] as the first study for volumetric med- +ical image registration (i.e., 3D image registration). As illus- +trated in Figure 39, the images are first encoded into high-level +feature representations by implementing multiple convolution +blocks; then, these features get split into P patches in the ViT +block. Next, the patches are mapped to a D-dimensional em- +bedding space to provide patch embeddings, which are then +integrated with learnable positional encodings to retain posi- +tional information. +Next, these patches are passed into the +encoder block of the Transformer, followed by multiple skip +connections to retain localization information, and then de- +coded employing a V-Net style decoder [314]. Finally, a spa- +tial Transformer [315] warps the moving image by utilizing +the final output of the network. TransMorph [310] extended +ViT-V-Net and proposed a hybrid Transformer ConvNet frame- +work that utilizes the Swin Transformer [57] as the encoder +and a ConvNet as the decoder to provide a dense displacement +field. Like ViT-V-Net, it employed long skip connections to +retain the flow of localization information that may enhance +registration accuracy. The output of the network, which is a +nonlinear warping function, gets applied to the moving image +with the deformation field utilizing the spatial transformation +function proposed in [315]. An affine transformation Trans- +former network is incorporated to align the moving image with +the fixed image before feeding it to the deformable registra- +tion network. This work also proposed two variants of Trans- +Morph: diffeomorphic TransMorph (TransMorph-diff) to fa- +cilitate topology-preserving deformations and Bayesian Trans- +Morph (TransMorph-Bayes) to promote a well-calibrated reg- +istration uncertainty estimate. +Figure 39: Overview of ViT-V-Net. Multiple convolution blocks encode images +into high-level features, which the Vit block splits into patches. These patches +are then mapped to D-dimensional patch embeddings that get integrated with +learnable positional encodings to retain positional information. Next, these +patches are passed into the Transformer encoder block, followed by multiple +skip connections to retain localization information, and decoded using a V-Net +style decoder. Using the network’s final output, a spatial Transformer warps the +moving image. Figure taken from [309]. +Likewise, Zhang et al. [311] introduced the dual Transformer +network (DTN) framework to perform diffeomorphic registra- +tion. It is composed of a CNN-based 3D U-Net encoder [299] +for the embedding of separate and concatenated volumetric im- +ages and a dual Transformer to capture the cross-volume depen- +dencies. One of the Transformers is responsible for modeling +the inter- and intra-image dependencies, and the other one han- +dles the modeling of the global dependencies by employing the +self-attention mechanism. The concatenation of the generated +features from these Transformers results in enhanced feature +embeddings, which are utilized by the CNN-based decoder to +provide a diffeomorphic deformation field. The evaluation of +the framework was conducted on the brain MRI scans of the +OASIS dataset [316], which substantiates their improvements +in diffeomorphic registration compared to the existing deep- +learning-based approaches. +Furthermore, XMorpher [312] put emphasis on the signifi- +cance of backbone architectures in feature extraction and match +of pair-wise images, and proposed a novel full Transformer net- +work as the backbone, which consists of two parallel U-Net +structures [34] as the sub-networks with their convolutions re- +placed by the introduced Cross Attention Transformer for fea- +ture extraction of moving and fixed images, and cross-attention- +based fusion modules that utilize these features for generating +the feature representation of moving-fixed correspondence and +fine-grained multi-level semantic information that contributes +to a fine registration. +8.2. Affine Registration +To perform affine medical image registration with Trans- +formers, Mok et al. +[313] proposed C2FViT, a coarse-to- +fine vision Transformer that performs affine registration, a ge- +ometric transformation that preserves points, straight lines, and +planes while registering 3D medical images. Former studies +have relied on CNN-based affine registration that focuses on lo- +cal misalignment or global orientation [322, 323], which limits +the modeling of long-range dependencies and hinders high gen- +eralizability. C2FVit, as the first work that takes into account +the non-local dependencies between medical images, leverages +vision Transformers instead of CNNs for 3D registration. As +39 + +(T'MH'2) +(16, H, W,L) +1/2 +(16, H,W,L) +(3, H, W,L) +(ze) +1/4 +ze) + +Transformer +(N, D) +Moving Image +8t) +Image +Transformer +Conv-up Block +Layer +Conv. Block +Conv. Block +%) +3D Conv. Layer +3D Conv. Layer +Target Image +Conv. Block +3D Conv. Layer +Instance Norm. +Concatenation +3D Conv. Layer +Instance Norm. +3D Conv. + +Instance Norm. +3D Conv. + +Leaky ReLU +Leaky ReLU +Leaky ReLU +Leaky ReLU +Leaky ReLU +Instance Norm +Reshape +Leaky ReLU +Upsample +Vision +Conv-up +Max-pool +Deformed I +Conv. +Spatial +3D +Conv Block +Conv-up Block +Loss (L)Table 12: An overview of the reviewed Transformer-based medical image registration approaches. +Method +Modality +Organ +Type +Datasets +Metrics +Year +Deformable +ViT-V-Net +[309] +MRI +Brain +3D +Private Dataset +Dice +2021 +TransMorph +[310] +MRI +CT +Brain +Chest-Abdomen-Pelvis region +3D +1 IXI [237] +2 T1-weighted brain MRI scans from Johns Hopkins University +3 Chest-Abdomen-Pelvis CT [317] +Dice +% of |JΦ| ≤ 0 +SSIM +2021 +DTN +[311] +MRI +Brain +3D +OASIS [316] +Dice +|JΦ| ≤ 0 +2021 +XMorpher +[312] +CT +MRI +Heart +3D +1 MM-WHS 2017 [318] +2 ASOCA [319] +Dice +% of |JΦ| ≤ 0 +2022 +Affine +C2FViT +[313] +MRI +Brain +3D +1 OASIS [316] +2 LPBA [320] +Dice +Hausdorff distance +2022 +Rigid +SVoRT +[307] +MRI +Brain +3D +FeTA [321] +PSNR +SSIM +2022 +Table 13: A brief description of the reviewed Transformer-based medical image registration techniques. +Method +Contributions +Highlights +Deformable +ViT-V-Net +[309] +• Contributed to the medical image registration domain as the first work to exploit ViTs to develop a volu- +metric (3D) registration. +• Integrated Transformers with CNNs to build a hybrid architecture for self-supervised brain MRI registra- +tion +• Employed a hybrid architecture to incorporate long-range and local information in the registration process. +• Attempted to preserve the localization data with the help of long skip connections between the encoder and decoder stages. +TransMorph +[310] +• Proposed a Transformer-based unsupervised registration approach for affine and deformable objective. +• Conducted experiments on two brain MRI datasets and in a phantom-to-CT registration task to demon- +strate their superior performance compared to traditional approaches. +• They additionally proposed two distinguishable versions of their model: a diffeomorphic variant to facilitate the topology-preserving defor- +mations and a Bayesian variant to promote a well-calibrated registration uncertainty estimate. +• Studied the effect of receptive fields by comparing TransMorph with CNNs, and addressed that while the receptive field of ConvNets only +increases with the layer depth, their presented model takes into account the whole image at each due to the self-attention mechanism. +DTN +[311] +• Proposed a dual Transformer architecture to capture semantic correspondence of anatomical structures. +• The suggested DTN demonstrated remarkable results in diffeomorphic registration and atlas-based seg- +mentation of multi-class anatomical structures. +• The Dual Transformer is capable of reducing the negative Jacobian determinant while preserving the atlas-based registration quality. +• The qualitative and quantitative analysis of their method on the OASIS dataset indicates that diffeomorphic registration fields are effective. +XMorpher +[312] +• Devised a deformable registration system consisting of dual parallel feature extraction networks which +facilitate the association of representative features between moving and fixed images. +• Proposed cross-attention Transformer that establishes spatial correspondences through computation of +bilateral information in the attention mechanism. +• Promotes visual superiority by presenting fine-grained visual results in terms of boundary smoothness, adjacent regions’ resolution quality, +and deformation grid polishness. +• Demonstrated the model’s great diagnostic potential by conducting experiments with different training regimes. +Affine +C2FViT +[313] +Presented a method in order to learn the global affine registration by taking advantage of the strong long- +range dependency recognition and locality of the hybrid Transformer and the multi-resolution strategy. +The suggested training framework can be extended to a number of parametric-based registration approaches by removing or scaling the geo- +metrical transformation matrices. +Rigid +SVoRT +[307] +• Devised an approach for the task of Volumetric reconstruction of fetal brains based on Transformer archi- +tectures. +• Employed a Transformer network trained on artificially sampled 2D MR slices that estimates the under- +lying 3D volume from the input slices to more accurately predict transformation. +• Experimental procedures on the FeTA dataset [321] represented the model’s ability in high-quality volumetric reconstruction. +• The volumetric reconstruction associated with the transformations of the proposed method displays higher visual quality. +Figure 40: The model has L stages with convolutional patch embedding lay- +ers and N Transformer encoder blocks to learn the optimal affine registration +matrix. In each stage, fixed and moving images are downsampled and con- +catenated, then passed to the convolutional patch embedding layer to produce +image patch embeddings. The Transformer then produces the input feature em- +bedding from the embeddings [313]. +depicted in Figure 40, the model is split into L stages, each con- +taining a convolutional patch embedding layer and Ni Trans- +former encoder blocks (i indicates the stage number), intending +to learn the optimal affine registration matrix. In each stage, the +fixed and moving images are downsampled and concatenated +with each other, then the new representation gets passed to the +convolutional patch embedding layer to produce image patch +embeddings. Next, the Transformer receives the embeddings +and produces the feature embedding of the input. Conducted +experiments on OASIS [316] and LPBA [320] demonstrated +their superior performance compared to existing CNN-based +affine registration techniques in terms of registration accuracy, +robustness, and generalisation ability. +8.3. Rigid Registration +SVoRT [307] addressed the necessity of slice-to-volume reg- +istration before volumetric reconstruction for the task of volu- +metric fetal brains reconstruction, and employed a Transformer +network trained on artificially sampled 2D MR slices that learns +to predict slice transformation based on the information gained +from other slices. The model also estimates the underlying 3D +volume from the input slices to promote higher accuracy in +transformation prediction. The superiority of their proposed +method in terms of registration accuracy and reconstruction +based on the evaluation on synthetic data and their experiments +on real-world MRI scans demonstrated the ability of the model +in high-quality volumetric reconstruction. +8.4. Discussion and Conclusion +According to the research discussed in this section, vision +Transformers are prominent tools in image registration tasks +40 + + Patch Embedding + Patch Embedding +Convolutional + Convolutional + Encoder +Transformer +Transfor +Encoder +rmer +MLP +MLP +MLP +Head +Head +Headdue to their training capability on large-scale data, which +is made feasible by parallel computing and self-attention +mechanisms. Leveraging Transformers to encourage bet- +ter global dependency identification improves registration in +terms of dice scores and Jacobian matrix determinant com- +pared to CNNs. +To mitigate the burden of quadratic complexity when pro- +cessing images at high-resolution and modelling local rela- +tionships, reviewed studies usually employ CNNs to provide +feature maps or dense displacement fields [309, 310, 311]. +C2FViT [313] disregarded convolutional networks and im- +plemented convolutional patch embeddings to promote lo- +cality. +However, in deformably registering medical con- +tent, XMorpher recently demonstrated the power of cross- +attention in better capturing spatial relevancy without a CNN +implementation [312], and SVoRT purely utilized Trans- +formers to perform rigid registration [307]. +The notable experimental attempts on brain MRI scan data, +such as OASIS [316] and FeTA [321], show the importance +of accurate automatic registration for neuroimaging data. +One particular work [312] proposed to evaluate their regis- +tration on images of cardiac region datasets including MM- +WHO-2017 [318] and ASOCA [319]. To further clarify the +modality type used in the aforementioned proposed meth- +ods, all works conducted their evaluations on 3D or volu- +metric imaging modalities. +Based on the brief review of Transformer-based medical im- +age registration research, we believe that other regions of in- +terest (ROI) such as neurons, retina, and neck area are worth +exploring to facilitate diagnostic operations in different do- +mains with more precise registration models. +We have also specified the architectural type, modality, or- +gan, data size, training paradigm, datasets, metrics, and +year for each medical registration technique reviewed in Ta- +ble 12. Furthermore, Table 13 provides a list of the contri- +butions and highlights of the proposed works. +9. Medical Report Generation +Medical report generation focuses on producing comprehen- +sive captions and descriptions pivoting on medical images for +diagnostic purposes. Designing automatic methods capable of +performing this task can alleviate tedious and time-consuming +work in producing medical reports and promote medical au- +tomation [324]. Recently, advancements in deep learning have +brought the attention of researchers to employing an intelligent +system capable of understanding the visual content of an im- +age and describing its comprehension in natural language for- +mat [325]. Research efforts in improving this area can be em- +ployed in medical imaging by implementing systems capable +of providing descriptions and captions (i.e., generating medical +reports) concerning medical images. These captioning systems +usually utilize encoder-decoder models that encode medical im- +ages and decode their understandings to provide diagnostic in- +formation in a natural language format. +Despite the success of deep learning, limitations including +reliability on an immense amount of data, unbalanced data in +radiology datasets (e.g., IU X-ray chest X-Ray [326]), and the +black box nature of DL models entail challenges in medical +report generation [327]. The success of Transformer models +in many vision-and-language tasks has drawn the attention of +researchers in the medical report generation domain to the em- +ployment of this architecture. In this section, we discuss ap- +proaches that utilize Transformers to promote effective capture +of long-range context dependencies and better report genera- +tion. As illustrated in Figure 41, the following is our taxonomy +of these systems according to the mechanism by which they +produce accurate and reliable clinical reports: +(a) Reinforcement Learning-based. +The ultimate goal of a +medical report generation system is to provide clinically +accurate and reliable reports. In reinforcement learning, the +MRG system is considered an agent with the objective of +maximizing clinical accuracy based on the feedback given +by the reward signal, which is directly calculated by the +evaluation metric score (e.g., CIDEr [328]). +(b) Graph-based. Radiology reports are typically composed of +a long finding section with multiple sentences that make +report generation a challenging task. Therefore, the inclu- +sion of prior information is beneficial for facilitating the +generation of long narratives from visual data. +Knowl- +edge graphs, which are powerful models that can capture +domain-specific information in a structured manner, can be +used to exploit prior information for medical report gener- +ation [329, 330, 331]. +(c) Memory-based. Memory is a resource through which im- +portant information is recorded. +In designing a proper +MRG system, it is crucial to store vital and diagnostic in- +formation that can benefit the generation process by incor- +porating prior knowledge and experience. Hence, config- +uring a memory mechanism with Transformers as a report +generation framework facilitates longer and more coherent +text generation by sharing information gained through the +process [332, 333]. +(d) Other Systems. +Systems that introduce different ideas +from previous categories to improve clinical accuracy, such +as curriculum learning, contrastive learning, and alternate +learning, belong to this group. +9.1. Reinforcement Learning-based Systems +The first work to implement a Transformer architecture for +medical report generation is RTMIC [334]. It used the rein- +forcement learning strategy in training to mitigate the problem +of exposure bias prevailing in Seq2Seq models [358]. In their +approach, the original images are fed into a DenseNet [335] as +the region detector to extract bottom-up visual features. These +features are then passed into a visual encoder to generate visual +representations from the detected regions, which the caption- +ing detector then utilizes to generate captions for the specified +regions. The proposed method was experimented on the IU +X-Ray dataset [326] and achieved state-of-the-art results. In- +tegration of RL and Transformers was also applied in surgical +41 + +Table 14: An overview of the reviewed Transformer-based Medical Report Generation approaches. +Method +Modality +Organ +Type +Visual Backbone +Datasets +Metrics +Year +Reinforcement Learning +RTMIC +[334] +X-ray +Lung +2D +DenseNet-121 [335] +IU Chest X-ray [326] +BLEU [336] +CIDEr [328] +2019 +SIG +[337] +Ultrasound +Colonoscopy +Multi-organ +3D +ResNet-101 [63] +DAISI [338] +BLUE [336] +Meteor [339] +CIDEr [328] +ROUGE [340] +SPICE [341] +2021 +Graph +KERP +[330] +X-ray +Lung +2D +DenseNet-121 [335] +1 IU Chest X-ray [326] +2 CX-CHR (private dataset) +BLEU [336] +CIDEr [328] +ROUGE [340] +2019 +PPKED +[342] +X-ray +Lung +2D +ResNet-152 [63] +1 IU Chest X-ray [326] +2 MIMIC-CXR [343] +BLUE [336] +Meteor [339] +CIDEr [328] +ROUGE [340] +2021 +Memory +MDT +[332] +X-ray +Lung +2D +ResNet-121 [63] +1 IU Chest X-ray [326] +2 MIMIC-CXR [343] +BLEU [336] +Meteor [339] +ROUGE [340] +2020 +AlignTransformer +[344] +X-ray +Lung +2D +ResNet-50 [63] +1 IU Chest X-ray [326] +2 MIMIC-CXR [343] +BLEU [336] +Meteor [339] +ROUGE [340] +2021 +M2 TR. progressive +[345] +X-ray +Lung +2D +DenseNet-121 [335] +1 IU Chest X-ray [326] +2 MIMIC-CXR [343] +BLEU [336] +Meteor [339] +ROUGE [340] +2021 +MDT-WCL +[346] +X-ray +Lung +2D +ResNet [63] +1 MIMIC-ABN [347] +2 MIMIC-CXR [343] +BLUE [336] +Meteor [339] +ROUGE [340] +2021 +CMN +[333] +X-ray +Lung +2D +ResNet-101 [63] +1 IU Chest X-ray [326] +2 MIMIC-CXR [343] +BLEU [336] +Meteor [339] +ROUGE [340] +2022 +Other +CRG +[348] +X-ray +Lung +2D +DenseNet-121 [335] +MIMIC-CXR [343] +BLUE [336] +Meteor [339] +CIDEr [328] +ROUGE [340] +2020 +Medical-VLBERT +[349] +CT +X-ray +Lung +2D +DenseNet-121 [335] +1 Chinese Covid-19 CT [350] +2 CX-CHR (private dataset) +BLUE [336] +CIDEr [328] +ROUGE [340] +2021 +CDGPT2 +[351] +X-ray +Lung +2D +DenseNet-121 [335] +IU chest X-ray [326] +BLUE [336] +Meteor [339] +CIDEr [328] +ROUGE [340] +2021 +CGI +[352] +X-ray +Lung +2D +DenseNet-121 [335] +1 MIMIC-CXR [343] +2 IU chest X-ray [326] +BLUE [336] +Meteor [339] +ROUGE [340] +2021 +CGRG +[353] +X-ray +Lung +2D +ResNet-101 [63] +1 IU Chest X-ray [326] +2 COV-CTR [354] +BLUE [336] +Meteor [339] +ROUGE [340] +2021 +instruction generation since the joint understanding of surgical +activity along with modeling relations linking visual and tex- +tual data is a challenging task. Zhang et al. [337] employed a +Transformer-backboned encoder-decoder architecture and ap- +plied the self-critical reinforcement learning [359] approach to +optimize the CIDEr score [328] as the reward. Their approach +surpasses existing models in performance on the DAISI dataset +[338] with caption evaluation metrics applied to the model. +42 + +Table 15: A brief summary of the reviewed Transformer-based medical report generation methods. +Method +Contributions +Highlights +Reinforcement Learning +RTMIC +[334] +• Presented a novel Hierarchical Reinforced Transformer for producing comprehensible, informative med- +ical reports by training through reinforcement learning-based training. +• The initial attempt at incorporating Transformers to develop a medical report generation system. +• Utilized reinforcement learning to ameliorate the exposure bias problem. +• Enhanced clinical report coherence by employing Transformers to capture long-range dependencies. +• The selected metric (CIDEr) as a reward signal is not designed for the medical domain [355]. +SIG +[337] +• Generated surgical instructions from multiple clinical domains by utilizing a Transformer-based Encoder- +decoder architecture. +• The proposed method is able to produce multimodal dependencies, form pixel-wise patterns, and develop textual associations for masked +self-attention decoder. +• Utilizing self-critical reinforcement learning to perform optimization increased the performance of surgical instruction generation. +• The selected metric (CIDEr) as a reward signal is not designed for the medical domain [355]. +Graph +KERP +[330] +• Developed an MRG system using a hybrid retrieval-generation technique that unifies standard retrieval- +based and recent visual text generation methods. +• Introduced Graph Transformer (GTR) as the first research to employ an attention mechanism to convert +different data types formulated as a graph. +• Aligning the generated reports with abnormality attention maps by providing location reference facilitates medical diagnosis. +• Since KERP is designed based on abnormality detection, it may disregard other valuable information [352]. +PPKED +[342] +• Proposed a three-module system that mimics the working habits of radiologists by extracting abnormal +regions, encoding prior information, and distilling the useful knowledge to generate accurate reports. +• Provides abnormal descriptions and locations to facilitate medical diagnosis. +• Capable of extracting relevant information from the explored posterior and prior multi-domain knowledge. +• Some mistakes, such as duplicate reports and inaccurate descriptions, are present in the generated reports. [356] +Memory +MDT +[332] +• Introduced Memory-Driven Transformer for radiology report generation +• Developed a relational memory to retain essential knowledge gathered through the previous generations. +• To facilitate medical diagnosis, visual-textual attention mappings were incorporated to capture correspondence with essential medical terms. +• Dataset imbalance with dominating normal findings hinders the model’s generalizability. +AlignTransformer +[344] +• Introduced an MRG framework that mitigates the problem of data bias by hierarchically aligning visual +abnormality regions and illness tags in an iterative fashion. +• Conducted experiments on MIMIC-CXR and IU-Xray datasets and demonstrated the capability of the model in ameliorating the data bias +problem +M2 TR. progressive +[345] +• Developed a progressive text generation model for medical report generation by incorporating high-level +concepts into the process of generation. +• The division of report generation into two steps enhanced the performance in terms of language generation and clinical efficacy metrics. +• The progressive generation process increases the false positive rate by including abnormality mentions in negation mode. [355]. +MDT-WCL +[346] +• Introduced the contrastive learning technique into chest X-ray report generation by proposing a weakly +supervised approach that contrasts report samples against each other to better identify abnormal findings. +• Optimization with contrastive loss facilitates generalizability in comparison to constrastive retrieval-based methods. +CMN +[333] +• Cross-modal memory networks were introduced to improve report generation based on encoder-decoder +architectures by incorporating a shared memory to capture multi-modal alignment. +• Capable of properly aligning data from radiological images and texts to aid in the preparation of more precise reports in terms of clinical +accuracy. +Other +CRG +[348] +• Formulated the problem in two steps: (1) a report generation phase incorporating a standard language +generation objective to train a Transformer model, and (2) a sampling phase that includes sampling a report +from the model and extracting clinical observations from it. +• Transformers’ ability to provide more coherent and fluent reports was demonstrated. +• Due to the biased nature of the dataset caused by dominant normal findings, the algorithm tends to generate reports that lack essential +descriptions of abnormal sections [357]. +Medical-VLBERT +[349] +• Proposed a framework as the first work that generates medical reports for the COVID-19 CT scans +• Devised an alternate learning strategy to minimize the inconsistencies between the visual and textual data. +• Alleviated the shortage of COVID-19 data by employing the transfer learning strategy. +• Capable of effective terminology prediction +• Overreliance on predetermined terminologies undermines robustness and generalizability. +CDGPT2 +[351] +• Presented a conditioning mechanism that to improve radiology report generation in terms of word-overlap +metrics and time complexity. +• Utilized a pre-trained GPT2 conditioned on visual and weighted semantic features to promote faster +training, eliminate vocabulary selection, and handle punctuation. +• The first study to employ semantic similarity metrics to quantitatively analyze medical report generation +results. +• Conditioning mechanism tackled punctuations, vocabulary collection, and reduced training duration. +• The architecture does not require modification to be trained on distinct data sets. +• Incorporating semantic similarity in addition to word overlap metrics improved medical report evaluation. +4 The model’s generalization ability and robustness against over-fitting are both hindered when the size of the dataset is small. +CGI +[352] +• Provides cohesive and precise X-ray reports in a fully differentiable manner by dividing the report gener- +ation system into a classifier, generator, and interpreter. +• Their conducted experiments revealed that incorporating additional scans besides clinical history can be +beneficial in providing higher-quality X-ray reports. +• Flexibility in processing additional input data, such as clinical documents and extra scans, which also contributes to performance improve- +ment. +• The model doesn’t provide vital information, such as illness orientation and time-series correlations, which facilitates more reliable reports. +CGRG +[353] +• Presented a Transformer-based method that estimates report uncertainty to develop a more reliable MRG +system and facilitate diagnostic decision-making. +• Introduced the Sentence Matched Adjusted Semantic Similarity (SMAS) to capture vital and relevant +features in radiology report generation more effectively. +• Assessing visual and textual uncertainties leads to more reliable reports in medical diagnosis. +• Measuring uncertainties can properly provide correlated confidence between various reports, which is beneficial to aiding radiologists in +clinical report generation [357]. +This work’s key difference from others is that their model is +proposed to generate instructions instead of descriptions. +9.2. Graph-based Systems +In graph-based medical report generation, Li et al. [330] pro- +posed KERP, a Graph Transformer implementation to generate +robust graph structures from visual features that are extracted +by a DenseNet [335] backbone. This approach is composed +of three modules: Encode, Retrieve and Paraphrase. First, it +constructs an abnormality graph by converting the visual fea- +tures extracted from the medical images via an encoder mod- +ule. Next, a sequence of templates is retrieved considering the +detected abnormalities by utilizing a retrieve module. Subse- +quently, the terms of the produced templates are paraphrased +into a report by employing the paraphrase module. The KERP’s +workflow is illustrated in Figure 42. +Additionally, Liu et al. [342] addressed the visual and tex- +tual data biases and their consequences in generating radiology +reports and proposed the PPKED framework to alleviate these +challenges. Their work introduced three modules to perform re- +port generation: (1) Prior Knowledge Explorer (PrKE), which +obtains relevant prior information for the input images; (2) Pos- +terior Knowledge Explorer (PoKE), which extracts the poste- +rior information, including the abnormal regions of the medi- +cal image; and (3) Multi-domain Knowledge Distiller (MKD), +which distills the obtained information from the previous mod- +ules to perform the final report generation. PPKED then formu- +lated the problem by employing the presented modules in the +following manner: PoKE first extracts the image features cor- +responding to the relevant disease topics by taking the visual +features extracted by ResNet-152 [63] from the input image +and abnormal topic word embeddings as the input. Next, the +PrKE module filters the prior knowledge from the introduced +prior working experience (a BERT encoder) and prior medical +knowledge component that is relevant to the abnormal regions +of the input image by utilizing the output of the PoKE module. +Next, the MKD module generates the final medical report by +using this obtained information, which is implemented based +on the decoder part of the Transformers equipped with Adap- +tive Distilling Attention. +43 + + Medical Report + Generation + RL-based + 1. RTMIC + 2. SIG + Memory-based + 3. MDT + 4. AlignTransformer + 5. M2 TR Progressive + 6. MDT-WCL + 7. CMN + Graph-based + 8. KERP + 9. PPKED + Other Systems + 10. CRG + 11. Medical-VLBERT + 12. CDGPT2 + 13. CGI + 14. CCRG +Figure 41: Taxonomy of Transformer-based medical report generation approaches based on the mechanism by which they generate clinical reports. We reference +the papers in ascending order corresponding to their prefix number: 1. [334], 2. [337], 3. [332], 4. [344], 5. [345], 6. [346]. 7. [333], 8. [330], 9. [342], 10. +[348], 11. [349], 12. [351], 13. [352], 14. [353]. +Pleural +effusion +Consolidation +Encode +GTRi2g +Retrieve +GTRg2s +Paraphrase +GTRgs2s +GTRg2g +Abnormality graph +Disease graph +Visual feature +CNN +Templates +Report +Degenerative changes in the spine. +No pleural effusion. +There is hyperexpansion of the lungs +suggesting underlying emphysema. +No focal airspace consolidation. +Heart size is normal. +Emphysema +Degenerative +disease +Degenerative change +of spine (0.66) +Focal airspace +consolidation +(0.01) +Hyperexpansion of +lungs (0.78) +Enlarged +heart size +(0.04) +Tortuous +aorta (0.12) +Low lung +volumes (0.00) +0.03 +0.19 +0.12 +0.00 +Figure 42: Using an encoder module, KERP creates an abnormality graph from +the extracted visual features. Then, a retrieval module retrieves a sequence +of templates based on detected abnormalities. Next, the paraphrase module +paraphrases the templates’ terms into a report [330]. +9.3. Memory-based Systems +Concerning the development of systems that rely on a mem- +ory mechanism to generate medical reports, Chen et al. [332] +presented a Memory-Driven Transformer (MDT), a model suit- +able for the generation of long informative reports and one of +the first works on the MIMIC-CXR dataset [343]. MDT em- +ploys a relational memory to exploit characteristics prevailing +in reports of similar images, and then the memory is incorpo- +rated into the decoder section of the Transformer by implement- +ing a memory-driven conditional layer normalization (MCLN). +Likewise, Nooralahzadeh et al. [345] introduced M2 TR. +progressive, a report generation approach that utilizes curricu- +lum learning, which is a strategy of training machine learning +models by starting with easy samples and gradually increasing +the samples’ difficulty [360]. Instead of directly generating full +reports from medical images, their work formulates the prob- +lem into two steps: first, the Meshed-Memory Transformer (M2 +TR.) [361], as a powerful image captioning model, receives the +visual features extracted by a DenseNet [335] backbone and +generates high-level global context. Second, BART [362], as +a Transformer-based architecture, encodes these contexts with +a bidirectional encoder and decodes its output using a left-to- +right decoder into coherent reports. The overview of the pro- +cess is depicted in Figure 43. +Additionally, You et al. [344] proposed AlignTransformer, +a framework composed of two modules: Align Hierarchical At- +tention (AHA) and Multi-Grained Transformer (MGT). In their +approach, first visual features and disease tags are extracted +Figure 43: Workflow of the M2 Tr. Progressive framework. The task is ac- +complished in two stages: First, the Meshed-Memory Transformer (M2 TR.) +receives visual features extracted by a DenseNet [335] backbone and generates +high-level global context. Second, the BART [362] architecture encodes con- +texts with a bidirectional encoder and decodes them with a left-to-right decoder +to produce coherent reports [345]. +from the medical image by an image encoder, then they get +aligned hierarchically to obtain multi-grained disease-grounded +visual features in the AHA module. The obtained grounded fea- +tures are capable of tackling the data bias problem by promot- +ing a better representation of abnormal sections. Next, these +grounded visual features are exploited by an adaptive exploit- +ing attention (AEA) [361] mechanism in the MGT module for +the generation of the medical reports. They also justified their +model’s efficiency through the manual evaluation of clinical ra- +diologists. +In MDT-WCL [346], the problem is approached with a +weakly supervised contrastive loss, which lends more weight +to the reports that are semantically close to the target reports, +and a memory-driven Transformer is adopted as the backbone +model to store key information in its memory module. To aid +the contrastive learning during training, after clustering the re- +ports into groups with the K-Means algorithm, each report is +assigned a label corresponding to its cluster, and the semanti- +cally closed ones are considered to be in the same cluster. +Although previous approaches have achieved promising re- +sults, they lack the ability to generate mappings between im- +ages and texts to align visual-textual information and assist +medical diagnosis. In order to facilitate visual-textual align- +ment, the Cross-modal Memory Network (CMN) [333] ex- +tended encoder-decoder methods by utilizing a shared memory +for better alignment of information between images and texts. +44 + +High-level Context +Report +Visual +Compared to prior examination, +Backbone +PosITIVE improved bilateral airspace opaci. +there is a significant improvement +POSITIVE minimal streaky opaci. +in aeration bilaterally,with improved +UNCERTAIN airspace disease. +bilateral airspace opacities. +Currently, +UNCERTAIN atelecta. +there are only minimal streaky opacities +Visual +NEGATIVE large focal pneumothorax effusions +in the bilateral midlung,which may +Language +identified consolidate. +Language +represent mild residual +airspace disease, +Model +NEGATIVE consolidations identified pneumothorax. +Model +atelectasis, or underlying changes +NEGATIVE consolidations definite pleural +of chronic lung disease.No large focal +identified effusion. +consolidations,pneumothorax,or definite +NEGATIVE +contour. stable mediastinal silhouette +pleural effusions identified. +The mediastinal +Visual +silhouette is stable and within normal +Backbone +limits for size and contour.No acute osseous +abnormality is identified.It uses a pre-trained ResNet [63] as the visual extractor to out- +put visual features, then passes them to the cross-modal mem- +ory network that utilizes a matrix to store information where +each row represents the embedding of information linking im- +ages and texts. To access the stored information aligning the +modalities, memory querying and responding are implemented +in a multi-threaded manner. +9.4. Other Systems +Other MRG systems focus on solving the problem with dif- +ferent ideas. Lovelace et al. [348] proposed a generation frame- +work composed of two stages. In the first stage, a Transformer +model is adopted to map the input image features extracted by +a DenseNet-121 [335] to contextual annotations and learn re- +port generation. In the second stage, a procedure is introduced +to differentiably sample a clinical report from the Transformer +decoder and obtain observational clinical information from the +sample. This differentiability is further employed to fine-tune +the model for improving clinical coherence by applying their +differentiable CheXpert to the sampled reports. Fueled by re- +cent progress in explainable artificial intelligence and the in- +troduction of algorithms that attempt to provide interpretable +prediction in DL-based systems, Likewise, in CDGPT2 [351], +the medical image is passed into a Chexnet [363] to provide +localizations of 14 types of diseases from the images as vi- +sual features. To implement better semantic features, the model +was fine-tuned as a multi-label classification problem to extract +manual tags from the IU-Xray dataset [326] by replacing the +final layer of the model with a layer containing 105 neurons to +produce 105 tags. The vector representation of the tags is then +fed into a pre-trained distilGPT2 [364] as the decoder to gener- +ate medical reports. Moreover, Wang et al. [353] presented a +confidence-guided report generation (CGRG) approach to sup- +port reliability in report generation by quantifying visual and +textual uncertainties. It’s comprised of an auto-encoder that +reconstructs images, a Transformer encoder that encodes the +input visual feature extracted by ResNet-101 [63], and a Trans- +former decoder for report generation. Visual uncertainty is ob- +tained by the AutoEncoder, which acts as a guide for the visual +feature extractor, and textual uncertainty is quantified based on +the introduced Sentence Matched Adjusted Semantic Similar- +ity (SMAS) which captures the similarity between the gener- +ated reports. These uncertainties are further utilized to aid the +model optimization process. +The recent outbreak of COVID-19, one of the deadliest pan- +demics, has influenced the research community to alleviate the +tedious and time-consuming work of producing medical re- +ports. VL-BERT, [365] as an extension of BERT, [36] can be +employed as an intelligent medical report generation system to +expedite the diagnosis process. Medical-VLBERT [349] in- +troduced VL-BERT to the medical report generation domain. +It defines the problem as a two-step procedure: First, it uti- +lizes two distinct VL-BERTs as terminology encoders to pro- +duce terminology-related features (textual and visual), and then +these features are fed into a shared language decoder to produce +medical textbooks and reports. The proposed method takes into +account predefined terminology word embeddings that repre- +sent medical domain knowledge. These embeddings are paired +distinctly with two other embeddings as an input to the en- +coders: textbook embeddings, which are generated by employ- +ing a lookup table, and spatial feature embeddings (termed ”vi- +sual context”) that are extracted from medical images by im- +plementing DenseNet-121 [335]. The encoders then integrate +this pairwise information separately to produce textual and vi- +sual terminological features. Subsequently, a shared language +decoder is trained by utilizing an alternate approach to properly +exchange the knowledge captured by the encoders. +Furthermore, in the work of Nguyen et al. [352], a clas- +sification, generation, and interpretation framework (CGI) is +proposed to address clinical accuracy. Each term of the frame- +work’s name represents a different module to perform the task. +The classification module learns how to discover diseases and +generate their embeddings, which consist of an image and +text encoder to extract the global visual features from medi- +cal images and obtain text-summarized embeddings from clini- +cal documents. The generation module is a Transformer model +that takes the disease embeddings as input and generates med- +ical reports from them. The interpretation module then takes +these reports for evaluation and fine-tuning. +9.5. Discussion and Conclusion +This section offers a systematic review of the Trans- +former architectures configured for medical report gener- +ation. Compared to previous sections that reviewed ViT- +based frameworks to tackle different medical tasks and prob- +lems, this section focuses mostly on using standard Trans- +formers as the core of a medical report generation sys- +tem. A common theme prevailing in these systems is to +solve the problem with an encoder-decoder architecture sup- +ported by a CNN-based visual backbone. As mentioned in +previous sections, the self-attention mechanism undermines +the representation of low-level details. On the other hand, +since medical reports consist of long and multiple sentences, +Transformers are of great significance to model long-term +dependencies, which assists clinically accurate report gen- +eration [366, 352]. +To exploit the power of both CNNs +and Transformers simultaneously, state-of-the-art MRG sys- +tems usually embed CNNs along with Transformers in their +frameworks [334, 351, 353]. We have provided information +in Table 14 on the reviewed report generation methods con- +cerning their architectural type, modality, organ, pre-trained +strategy, datasets, metrics, and year. +Table 15 contains +summarized information about the methodologies, includ- +ing their contributions and highlights. In addition, it should +be noted that several survey publications have been pub- +lished in this field of medicine [327, 355, 367], and the most +recent one provided a technical overview of Transformer- +based clinical report generation [28]. +We approach our +review differently by distinguishing the proposed methods +based on the mechanism they used to support the prevailing +concerns such as long and coherent text generation, reliabil- +ity, and visual-textual biases. +The ultimate goal of these frameworks is to increase clin- +ical accuracy to expedite the diagnosis process and reduce +45 + +the workloads in radiology professions [348, 352]. Numer- +ous works have attempted to facilitate diagnostic decision- +making by aligning correlated sections of medical image +and textual report that provide valuable information for de- +tecting abnormalities [344, 333]. Also, multiple studies em- +phasized the importance of universal knowledge, and de- +signed a system to incorporate prior information for detect- +ing disease [330, 332]. Some research effort was also put +into better representation learning by contrasting normal and +abnormal samples against each other in representation space +by utilizing a contrastive loss as the objective [346]. One re- +cent work was inspired by curriculum learning to imitate the +order of the human learning process [345]. +Overall, we believe that MRG systems need more research +and progression to be robustly incorporated in a practical +setting. +10. Open Challenges and Future Perspectives +So far, we discussed the application of Transformers (espe- +cially vision Transformers) and reviewed state-of-the-art mod- +els in medical image analysis. +Even though their effective- +ness is exemplified in previous sections by delicately present- +ing their ideas and analyzing the significant aspects that were +addressed in their proposed methods, there is still room for +improvement in many areas to devise a more practical and +medically accurate system by leveraging Transformers. Conse- +quently, we discuss the challenges and future directions hoping +to help researchers gain insight into the limitations and develop +more convenient automatic medical systems based on Trans- +formers. +10.1. Explainability +Fueled by recent progress in XAI (explainable artificial intel- +ligence) and the introduction of algorithms that attempt to pro- +vide interpretable prediction in DL-based systems, researchers +are putting effort into incorporating XAI methods into con- +structing Transformer-based models to promote a more reliable +and understandable system in different areas, including medi- +cal analysis [368, 369]. Existing approaches usually highlight +important regions of the medical image that contribute to the +model prediction by employing attention maps [370, 40]. Fur- +thermore, Vision Transformers (ViTs) have the ability to pro- +vide attention maps that indicate the relevant correlations be- +tween the regions of the input and the prediction. However, +the challenge of numerical instabilities in using propagation- +based XAI methods such as LRP [371] and the vagueness of +the attention maps, which leads to inaccurate token associations +[75, 372], makes interpretable ViTs an open research opportu- +nity in computer vision, especially in medical image analysis. +We believe that including interpretable vision Transformers, +such as ViT-NeT [372], in various medical applications can pro- +mote user-friendly predictions and facilitate decision-making in +the diagnosis of medical conditions, and is a promising direc- +tion in medical research problems. +10.2. Richer Feature Representation +An effective and suitable representation space is substantially +influential in building medical analysis systems. Transformers +have demonstrated their efficiency in obtaining global informa- +tion and capturing long-term dependencies in many areas, such +as Natural Language Processing (NLP), Computer Vision, and +Speech Recognition [306], and CNNs have proven to be effec- +tive in extracting local context from visual data [373]. However, +this locality usually enables these networks to capture rich local +texture representation [374, 375] and lacks model global depen- +dency. As a result, many approaches stack Transformers along +with CNNs to leverage both local and global information si- +multaneously in clinical applications (e.g., medical report gen- +eration) [344, 348, 50]. Recent studies stated that the single- +scale representation of ViTs hinders improvement in dense pre- +diction tasks, so a multi-scaled feature representation is imple- +mented which achieves better performance in computer vision +tasks, including image classification, object detection, and im- +age segmentation [376, 377]. Generalizing this idea to medical +applications of ViTs to facilitate devising a clinically suitable +system can be considered as future work. +10.3. Video-based analysis +There has been an increasing interest in the vision commu- +nity in extending ViT architectures to video recognition tasks. +Recently, a handful of papers have integrated standard Trans- +formers with their models in AI-assisted dynamic clinical tasks +[378, 379, 380, 381]. However, the scarcity of the proposed ap- +proaches puts video-based medical analysis in an infancy stage +and open for future investigations. Another potential research +direction is to explore the power of video vision Transformer +variants, such as Video Swin Transformer [382], in clinical +video understanding and to facilitate automatic robotic surgery. +10.4. High Computational Complexity +The robustness of Transformer models in layouts that im- +plement large numbers of parameters is one of their strengths. +While this is a beneficial trait that makes it possible to train +models of enormous scale, it leads to the requirement of large +resources for training and inferencing [27]. Particularly dis- +advantageous to medical image analysis is that expanding the +use of ViTs for pretraining in new tasks and datasets comes +with substantial expenses and burdens. Additionally, gathering +medical samples can be difficult and the dataset scale is often +limited. For instance, according to empirical studies in [22], +pretraining a ViT-L/16 model on the large-scale dataset of Ima- +geNet takes approximately 30 days employing a standard cloud +TPUv3 with 8 cores. As a result, a notable number of papers +utilized the pre-trained weights of ViT models to exploit the +transfer learning strategy to alleviate training load [44, 24, 43], +but in some cases, such as dealing with volumetric medical im- +ages, where transfer learning doesn’t demonstrate any improve- +ments [143, 309], the pretraining process is necessary to cap- +ture domain-specific features for generalization and better per- +formance. Ultimately, designing effective Transformer systems +with fewer parameters while maintaining optimality in terms of +clinical accuracy and robustness is a preferable research direc- +tion. +46 + +10.5. Transformer-based Registration +As reviewed in Section 8, the idea of employing Transform- +ers to support efficient medical image registration has become +popular in recent years. The ability of the self-attention mecha- +nism assists the learning of long-term visual correlations since +their unlimited receptive field promotes a more accurate under- +standing of the spatial relationship between moving and fixed +images [309, 310]. However, registration systems composed of +Transformer architectures are still in their infancy and require +more research effort to be put into them. +10.6. Data-Driven Predictions +With supervised learning as a popular fashion in building in- +telligent systems, the model learns features based on the pro- +vided annotations that are suitable to accomplish a specific +task, which hinders generalizability. +In other words, super- +vised learning modifies the bias-variance trade-off in favor of +the strong inductive biases that lead to making assumptions as a +means to aid the model in learning a particular task quicker and +with higher sample efficiency. However, these hard assump- +tions sacrifice adaptability to other settings and unseen datasets, +and the model learns to accomplish its task without having an +innate understanding of the data. To tackle this issue, unsu- +pervised regimes enable the algorithms to act as general de- +scriptors and capture features that will assist them in perform- +ing efficiently in a wide range of tasks. Similarly, in medical +image analysis, adopting Transformer networks with unsuper- +vised learning algorithms promotes robustness and generaliz- +ability to other datasets and tasks. +10.7. Medical Software Ecosystems +A future direction for advancing in the automatic medical +analysis is to provide an open-source environment that contains +libraries suitable for solving multiple medical tasks and chal- +lenges with Transformer architectures. Developers can further +contribute to the ecosystem by updating and adding additional +tasks, bringing novelty, and proposing ideas to enhance perfor- +mance and accuracy [132]. Companies and organizations can +support the system by preparing the necessary computational +resources and hardware requirements. Sample of software pro- +totypes in this direction are nnU-Net [383], Ivadomed [384], +and preliminary works such as [133], which provides an end- +to-end pipeline for implementing deep models on medical data. +11. Discussion and Conclusion +In this paper, we presented a comprehensive encyclopedic +review of the applications of Transformers in medical imag- +ing. First, we provided preliminary information regarding the +Transformer structures and the idea behind the self-attention +mechanism in the introduction and background sections. Start- +ing from Section 3, we reviewed the literature on Transformer +architecture in diverse medical imaging tasks, namely, classifi- +cation, segmentation, detection, reconstruction, synthesis, reg- +istration, and clinical report generation. For each application, +we provided a taxonomy and high-level abstraction of the core +techniques employed in these models along with the SOTA ap- +proaches. We also provided comparison tables to highlight the +pros and cons, network parameters, type of imaging modality +they are considering, organ, and the metrics they are using. Fi- +nally, we outlined possible avenues for future research direc- +tions. +Acknowledgments This work was funded by the German Re- +search Foundation (Deutsche Forschungsgemeinschaft, DFG) +under project number 191948804. +We thank Johannes +Stegmaier for his contribution to the proofreading of this docu- +ment. +References +[1] J. Arevalo, F. A. Gonz´alez, R. Ramos-Poll´an, J. L. +Oliveira, M. A. G. Lopez, Representation learning for +mammography mass lesion classification with convolu- +tional neural networks, Computer methods and programs +in biomedicine 127 (2016) 248–257. +[2] E. Nasr-Esfahani, S. Samavi, N. Karimi, S. M. R. +Soroushmehr, M. H. Jafari, K. Ward, K. Najarian, +Melanoma detection by analysis of clinical images us- +ing convolutional neural network, in: 2016 38th An- +nual International Conference of the IEEE Engineering +in Medicine and Biology Society (EMBC), IEEE, 2016, +pp. 1373–1376. +[3] M. Asadi-Aghbolaghi, R. Azad, M. Fathy, S. 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Cohen-Adad, Ivadomed: A med- +ical imaging deep learning toolbox, arXiv preprint +arXiv:2010.09984 (2020). +64 + diff --git a/0tE1T4oBgHgl3EQf4wXL/content/tmp_files/load_file.txt b/0tE1T4oBgHgl3EQf4wXL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4d89c72f026b267015724119dea629259066547d --- /dev/null +++ b/0tE1T4oBgHgl3EQf4wXL/content/tmp_files/load_file.txt @@ -0,0 +1,6362 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf,len=6361 +page_content='Advances in Medical Image Analysis with Vision Transformers: A Comprehensive Review Reza Azad1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Amirhossein Kazerouni2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Moein Heidari2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Ehsan Khodapanah Aghdam3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Amirali Molaei4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Yiwei Jia1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Abin Jose1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Rijo Roy1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Dorit Merhof†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='6 1Institute of Imaging and Computer Vision,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' RWTH Aachen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Aachen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Germany 2School of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Iran University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Tehran,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Iran 3Department of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Shahid Beheshti University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Tehran,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Iran 4School of Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Iran University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Tehran,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Iran 5Institute of Image Analysis and Computer Vision,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Faculty of Informatics and Data Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' University of Regensburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Regensburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Germany 6Fraunhofer Institute for Digital Medicine MEVIS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Bremen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Germany Abstract The remarkable performance of the Transformer architecture in natural language processing has recently also triggered broad interest in Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Among other merits, Transformers are witnessed as capable of learning long-range dependen- cies and spatial correlations, which is a clear advantage over convolutional neural networks (CNNs), which have been the de facto standard in Computer Vision problems so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Thus, Transformers have become an integral part of modern medical image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In this review, we provide an encyclopedic review of the applications of Transformers in medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Specifi- cally, we present a systematic and thorough review of relevant recent Transformer literature for different medical image analysis tasks, including classification, segmentation, detection, registration, synthesis, and clinical report generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For each of these applications, we investigate the novelty, strengths and weaknesses of the different proposed strategies and develop taxonomies highlighting key properties and contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Further, if applicable, we outline current benchmarks on different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Finally, we summarize key challenges and discuss different future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, we have provided cited papers with their corresponding implementations in https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='com/mindflow-institue/Awesome-Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Keywords: Transformers, Medical Image Analysis, Vision Transformers, Deep Neural Networks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Introduction Convolutional neural networks (CNNs) have been an inte- gral part of research in the field of medical image analysis for many years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' By virtue of convolutional filters whose primary function is to learn and extract necessary features from medi- cal images, a wealth of research has been dedicated to CNNs ranging from tumor detection and classification [1], detection of skin lesions [2, 3, 4] to segmentation of intervertebral discs [5, 6], brain tumor segmentation [7, 8], to name only a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CNNs have also contributed significantly to the analysis of dif- ferent imaging modalities in clinical medicine, including X-ray radiography, computed tomography (CT), magnetic resonance imaging (MRI), ultrasound (US), and digital pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' De- spite their outstanding performance, CNNs suffer from concep- tual limitations and are innately unable to model explicit long- distance dependencies due to the limited receptive field of con- volution kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Moreover, the convolutional operator suffers from the fact that at inference time, it applies fixed weights re- gardless of any changes to the visual input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To mitigate the aforementioned problems, there have been great research ef- forts to integrate attention mechanisms, which can be regarded as a dynamic weight adjustment process based on input fea- tures to the seminal CNN-based structures to improve the non- local modeling capability [9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To this end, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [12] designed a non-local flexible building block, which can be plugged into multiple intermediate convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SENet [13] suggested a channel attention squeeze-and-excitation (SE) block, which collects global information in order to recalibrate each channel accordingly, in order to create a more robust rep- resentation [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Inspired by this line of research, there has been an overwhelming influx of models with attention variants proposed in the medical imaging field [15, 16, 17, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Al- though these attention mechanisms allow the modeling of full image contextual information, as the computational complex- ity of these approaches typically grows quadratically with re- spect to spatial size, they imply an intensive computational bur- den, thus making them inefficient in the case of medical im- ages that are dense in pixel resolution [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Moreover, despite the fact that the combination of the attention mechanism with the convolutional operation leads to systematic performance gains, these models inevitably suffer from constraints in learn- ing long-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transformers [21] have demon- strated exemplary performance on a broad range of natural lan- guage processing (NLP) tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', machine translation, text †Corresponding author: Dorit Merhof, Tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' : +49 (941) 943-68509, E-mail: dorit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='merhof@ur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' January 10, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='03505v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='CV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='9 Jan 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Transformers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Classification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Pure Transformers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Hybrid Models ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Segmentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Pure Transformers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Hybrid Models ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Other Architectures ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Reconstruction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Low Dose Enhancement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Sparse-View Reconstruction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Undersampled Reconstruction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Super Resolution Reconstruction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Report Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Reinforcement Learning-based Systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Graph-based Systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Memory-based Systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Other Systems ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Registration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Deformable Registration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Affine Registration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Rigid Registration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Detection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Backbone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Neck ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Head ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Synthesizing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Intra-Modality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Inter-Modality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Figure 1: Overview of the applications covered in this review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' classification, and question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Inspired by the emi- nent success of Transformer architectures in the field of NLP, they have become a widely applied technique in modern Com- puter Vision (CV) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Since the establishment of Vision- Transformers (ViTs) [22], Transformers proved to be valid al- ternatives to CNNs in diverse tasks ranging from image recog- nition [22], object detection [23], image segmentation [24] to video understanding [25] and image super-resolution [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As a central piece of the Transformer, the self-attention mecha- nism comes with the ability to model relationships between el- ements of a sequence, thereby learning long-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Moreover, Transformers allow for large-scale pre-training for specific downstream tasks and applications and are capable of dealing with variable-length inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The immense interest in Transformers has also spurred research into medical imaging applications (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Being dominant in reputable top- tier medical imaging conferences and journals, it is extremely challenging for researchers and practitioners to keep up with the rate of innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The rapid adoption of Transformers in the medical imaging field necessitates a comprehensive summary and outlook, which is the main scope of this review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Specif- ically, this review provides a holistic overview of the Trans- former models developed for medical imaging and image anal- ysis applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We provide a taxonomy of the network de- sign, highlight the major strengths and deficiencies of the exist- ing approaches and introduce the current benchmarks in each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We inspect several key technologies that arise from the various medical imaging applications, including medical im- age segmentation, medical image registration, medical image reconstruction, and medical image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' So far, review papers related to Transformers do not concentrate on applica- tions of Transformers in the medical imaging and image analy- sis domain [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The few literature reviews that do focus on the medical domain [28, 29], despite being very comprehensive, do not necessarily discuss the drawbacks and merits of each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In our work, we explicitly cover this aspect and also provide a taxonomy that comprises the imaging modality, organ of interest, and type of training procedure each paper has se- lected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' More specifically, in Section 3 (Medical Image Classi- fication), we comprehensively elaborate on the most promising networks along with their key ideas, limitations, the number of parameters, and the specific classification task they are address- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In Section 4 (Medical Image Segmentation), we analyze network architectures in terms of their design choice and pro- pose a detailed taxonomy to categorize each network to provide insight for the reader to understand the current limitations and progress in segmentation networks based on the Transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In Section 5 (Medical Image Reconstruction), we take a different perspective to categorize networks based on their network structure and the imaging modality they are built upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We categorize the synthesis methods in Section 6 based on their objective (intra-modality or inter-modality) and then provide detailed information regarding the network archi- tecture, parameters, motivations, and highlights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the sections related to detection (Section 7), registration (Section 8), and re- port generation (Section 9) we briefly summarize the state-of- the-art (SOTA) networks and provide detailed information re- garding the network architectures, advantages, and drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Moreover, due to the swift development of the field, we believe that the community requires a more recent overview of the lit- erature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We hope this work will point out new research options and provide a guideline for researchers and initiate further inter- est in the vision community to leverage the potential of Trans- former models in the medical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Our major contributions are as follows: We systematically and comprehensively review the ap- plications of Transformers in the medical imaging do- main and provide a comparison and analysis of SOTA ap- proaches for each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Specifically, more than 200 papers 2 Linear Projection of Flattened Patches Patch + Position Embedding Extra learnable [class] embedding Embedded Patches LN Multi-Head Self-Attention LN MLP Class Benign Malignant Transformer Encoder MLP Head Figure 2: Architecture of the Vision Transformer as proposed in [22] and the detailed structure of the Vision Transformer encoder block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the Vision Transformer, sequential image patches are used as the input and processed using a Transformer encoder to produce the final classification output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' are covered in a hierarchical and structured manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Our work provides a taxonomized (Figure 1), in-depth analysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' task-specific research progress and limita- tions), as well as a discussion of various aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Finally, We discuss challenges and open issues and also identify new trends, raise open questions and identify fu- ture directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Paper Organizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The remaining sections of the paper are organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In Section 2, we provide an overview of the key components of the well-established Transformer ar- chitecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Moreover, this section clarifies the categorization of neural network variants in terms of the position where the Transformer is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Section 3 to Section 9 comprehen- sively review the applications of Transformers in diverse med- ical imaging tasks as depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For each task, we propose a taxonomy to characterize technical innovations and major use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Section 10 presents open challenges and future perspectives of the field as a whole, while finally, Section 11 concludes this work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Background In this section, we first provide an overview of the over- all architecture of the Transformer module and the key ideas behind its feasible design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then, we outline a general taxon- omy of Transformer-based models, characterized by their core techniques of using Transformers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', whether they are purely Transformer-based, or whether the Transformer module is ei- ther used in the encoder, decoder, bottleneck, or skip connec- tion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transformers The original Transformer [21] was first applied to the task for machine translation as a new attention-driven building block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The vanilla Transformer consists of an encoder and a decoder, each of which is a stack of L tandem of consecutive identi- cal blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Transformer module is convolutional-free and solely based on the self-attention mechanism or attention mech- anism in short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Specifically, these attention blocks are neu- ral network layers that relate different positions of a single se- quence to compute the sequence’s representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Since the es- tablishment of Transformer models, they have attained remark- able performance in diverse natural language processing tasks [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Inspired by this, Dosovitskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' proposed the Vision Transformer (ViT) [22] model as illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' When trained on large datasets, for instance, JFT-300M, ViT outper- forms the then state-of-the-art, namely ResNet-based models like BiT [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In their approach, an image is turned into fixed- sized patches before being flattened into vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These vectors are then passed through a trainable linear projection layer that maps them into N vectors with the dimensionality of D × N is the number of patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The outputs of this stage are referred to as patch embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To preserve the positional information present within each patch, they add positional embeddings to the patch embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition to this, a trainable class em- bedding is also appended to the patch embeddings before going through the Transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Transformer encoder is comprised of multiple Transformer encoder blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' There are one multi-head self-attention (MSA) block and an MLP block in each Transformer encoder block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The activations are first normalized using LayerNorm (LN) before going into these blocks in the Transformer encoder block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, there 3 are skip connections before the LN that add a copy of these ac- tivations to the corresponding MSA or MLP block outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the end, there is an MLP block used as a classification head that maps the output to class predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The self-attention mech- anism is a key defining characteristic of Transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Hence, we start by introducing the core principle of the atten- tion mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Self-Attention In a self-attention layer (Figure 3a), the input vector is firstly transformed into three separate vectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', the query vector q, the key vector k, and the value vector v with a fixed dimen- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These vectors are then packed together into three different weight matrices, namely WQ, WK, and WV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A common form of Q, K, and V can be formulated as Equation (1) for an input X K = WKX, Q = WQX, V = WVX, (1) where WK, WQ, and WV refers to the learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The scaled dot-product attention mechanism is then formulated as Attention(Q, K, V) = Softmax �QKT √dk � V, (2) where √dk is a scaling factor, and a softmax operation is ap- plied to the generated attention weights to translate them into a normalized distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Multi-Head Self-Attention The multi-head self-attention (MHSA) mechanism (Fig- ure 3b) has been proposed [21] to model the complex relation- ships of token entities from different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Specifically, the MHSA block helps the model to jointly attend to information from multiple representation sub-spaces, as the modeling ca- pability of the single-head attention block is quite coarse The process of MHSA can be formulated as MultiHead(Q, K, V) = [Concat (head1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' , headh)]WO, (3) where headi = Attention � QWQ i , KWK i , VWV i � , and WO indi- cates a linear mapping function to combine multi-head repre- sentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Note that h is a hyper-parameter set to h = 8 in the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transformer modes While the Transformer was originally introduced with an encoder-decoder pipeline, many modern architectures gener- ally exploit the Transformer architecture in different fashions, which generally depend on the target application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The usage of Transformers in vision tasks can broadly be classified into pure and hybrid designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Pure Transformers Due to the deficiency of CNN-based architectures in learning global and long-range semantic information interactions, which stems from the locality of convolution operation, a cohort study has investigated the purely Transformer-based models without any convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These models usually consist of encoder, MatMul Scale SoftMax MatMul (a) Self-Attention Linear Linear Scaled Dot-Product Attention Linear Concat Linear (b) Multi-Head Self-Attention Figure 3: (a) The process of self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (b) Multi-head attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The MSA consists of multiple SA blocks (heads) concatenated together channel-wise as proposed in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' bottleneck, decoder, and skip connections directly built upon the ViT or its variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In this criteria, there are usually multiple multi-head self-attention modules in both encoding and decod- ing sections that allow the decoder to utilize information from the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Examples of such methods are the Swin-Unet [32] and the TransDeepLab [33] networks which, as their name sug- gests, try to model the seminal U-Net [34], and DeepLab [35] architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transformer: Hybrid The hybrid Transformer models usually modify the base CNN structure by replacing the encoder or decoder modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Encoder: Encoder-only models such as the seminal BERT [36] are designed to make a single prediction per input or a sin- gle prediction for an entire input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the computer vision era, these models are applicable for classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Moreover, as utilizing a pure Transformer can result in lim- ited localization capacity stemming from inadequate low-level features, many cohort studies try to combine CNN and Trans- former in the encoding section [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Such a design can enhance finer details by recovering localized spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Decoder: Transformers can also be used in a decoding fash- ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Such a causal model is typically used for generation tasks such as language modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Besides that, the modification can apply to the skip connections of the decoder module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Skip con- nection is a widely-used technique to improve the performance and the convergence of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It can also serve as a modulating mechanism between the encoder and the de- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To effectively provide low-level spatial information for the decoding path, the idea of exploiting Transformers in de- signing skip connections has emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This notable idea can lead to finer feature fusion and recalibration while guaranteeing the aggregation scheme of using both high-level and low-level features [24, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 4 Medical Image Classification Pure Hybrid Original ViT Structure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransMed Other ViTs 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' LAT 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' DT-MIL Original ViT Structure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Covid-Transformer 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ViT-BUS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ViT-vs-CNN 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' FESTA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' XViTCOS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MIL-ViT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' COVID-VIT Other ViTs 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' POC-Former 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' RadioTransformer 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' COVID-VOLO 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 3DMET 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Femur-ViT 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Hybrid-Covid-ViT 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' HATNet 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransMIL 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' GTP Figure 4: Taxonomy of ViT-based approaches in medical image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Methods are categorized based on their proposed architecture into pure and hybrid methods, in which they adopt the vanilla ViT or present a new type of vision Transformer in medical image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Notably, we utilize the prefix numbers in the paper’s name in ascending order and denote the reference for each study as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [38], 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [39], 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [40], 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [41], 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [42], 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [43], 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [44], 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [45], 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [46], 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [47], 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [48], 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [49], 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [50], 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [51], 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [52], 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [53], 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [54], 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [55], 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Medical Image Classification Image classification is still one of the challenging problems in computer vision, which aids in segregating extensive quan- tities of data into meaningful categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Vision Transformers (ViT) have recently demonstrated outstanding results in vari- ous image classification tasks and offer significant advantages over conventional CNNs [57, 58, 59, 60, 61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These advan- tages include long-range relationships, adaptive modeling, and attention maps that yield intuition on what the model deems more important inside an image [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Due to these alluring ad- vantages, there is rising interest in building Transformer-based models for medical image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Therefore, highly pre- cise classification is becoming increasingly vital for facilitating clinical care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In this section, we exhaustively examine ViTs in medical im- age classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As illustrated in Figure 4, we have broadly classified these methods based on the role ViT plays in their architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These categories include pure Transformers and Hybrid Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Generally, a vision Transformer-based clas- sification architecture consists of three modules: (1) a back- bone for capturing input features, (2) an encoder for model- ing the information, and (3) a classification head for generat- ing output based on the specified task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Therefore, the Trans- former can be adopted in each module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, some works, including Lesion Aware Transformer (LAT) [52] and De- formable Transformer for Multi-Instance Learning (DT-MIL) [53], take a different approach and utilize encoder-decoder structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' LAT proposes a unified encoder-decoder system for Diabetic Retinopathy (DR) grading, and DT-MIL introduces a Transformer-based encoder-decoder architecture for classi- fying histopathological images, where the deformable Trans- former was embraced for the encoder part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the following, we will go into great depth on both hybrid and pure models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Pure Transformers Since the emergence of Transformers, there has been a grow- ing debate regarding whether it is time to entirely switch from CNNs to Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Matsoukas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [42] conduct a se- ries of experiments to answer this critical question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They take ResNet50 [63] and the DeiT-S [64] models to represent CNN and ViT models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They train each of these two models in 3 different fashions: a) randomly-initialized weights, b) pre-trained on ImageNet (transfer learning), and c) pre- training on the target dataset in a self-supervised scheme using DINO [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Their findings show that when utilizing random initialization, ViTs are inferior to CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the case of transfer learning, the results are similar for both models, with ViT being superior for two out of three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Additionally, ViT per- forms better when self-supervision on the target data is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They conclude that Vision Transformers, indeed, are suitable replacements for CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transformers have had a profound effect on medical devel- opment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Researchers have thoroughly investigated adopting the ViT in medical image classification tasks since its introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, the limited number of medical images has hindered Transformers from replicating their success in medical image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ViT-BUS [43] studies the use of ViTs in medi- cal ultrasound (US) image classification for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They propose to transfer pre-trained ViT models based on the breast US dataset to compensate for the data-hunger of ViTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Eval- uated results on B [66], BUSI [67], and B+BUSI datasets in- dicate the predominance of attention-based ViT models over CNNs on US datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Likewise, COVID-Transformer [44] utilizes ViT-L/16 to detect COVID from Non-COVID based on CXR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Due to the limitation of sufficient data, they intro- duce a balanced dataset containing 30K chest X-ray images for multi-class classification and 20K images for binary classifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The published dataset is created by merging datasets [68], [69], and [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They fine-tune the model on the dataset with 5 a custom MLP block on top of ViT to classify chest x-ray (CXR) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Moreover, COVID-Transformer exploits the GradCAM Map [71] to visualize affected lung areas that are significant for disease prediction and progression to display the model interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Similarly, Mondal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [40] present xViTCOS for detecting COVID-19 in CTs and CXRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' xViT- COS employs a model that has been pre-trained on ImageNet- 21k [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Nevertheless, the training data capacity might over- shadow the generalization performance of the pre-trained ViT to transfer the knowledge from the learned domain to the tar- get domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' By training the model on the COVIDx-CT-2A dataset [73], a moderately-sized dataset, xViTCOS overcomes this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, due to the shortage of the insufficient amount of CXR images, the pre-trained ViT model is fine- tuned using the CheXpert dataset [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, xViTCOS leverages the Gradient Attention Rollout algorithm [75] to vi- sually demonstrate the model’s prediction on the input image for clinically interpretable and explainable visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In experiments using COVID CT-2A and their custom-collected Chest X-ray dataset, xViTCOS significantly outperforms con- ventional COVID-19 detection approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MIL-VT [39] sim- ilarly suggests pre-training the Transformer on a fundus image large dataset beforehand, initialized by the pre-trained weight of ImageNet, then fine-tuning it on the downstream retinal dis- ease classification task in order to encourage the model to learn global information and achieve generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Unlike previ- ous approaches, they apply some modifications to the vanilla ViT structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the classic ViT, embedded features are ne- glected for classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' instead, only the class token, which retains the summarization of embedded features’ information, is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [39] propose a novel multiple-instance learn- ing (MIL)-head module to exploit those embedded features to complement the class token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This head comprises three sub- modules that attach to the ViT in a plug-and-play manner: 1) the MIL embedding submodule that maps the feature embed- dings to a low-dimensional embedding vector, 2) the attention aggregation submodule that outputs a spatial weight matrix for the low-dimensional patch embeddings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' this weight matrix is then applied to the low-dimensional embeddings to ascertain each instance’s importance, 3) the MIL classifier submodule that determines the probability of each class through aggre- gated features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the downstream task, both MLP and MIL heads use the weighted cross-entropy loss function for train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The outputs of both heads are then weight-averaged for the inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Results indicate the effectiveness of the proposed training strategy and the MIL-head module by dra- matically boosting the performance over APTOS2019 [76] and RFMiD2020 [77] datasets when compared to CNN-based base- lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In contrast to the previous 2D-based methods that employ transfer learning, COVID-ViT [38] proposes training ViT to classify COVID and non-COVID cases using 3D CT lung im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Given that a COVID volume may contain non-COVID 2D slices, COVID-ViT applies a slice voting mechanism after the ViT classification result in which the subject is categorized as having COVID if more than a certain percentage of slices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', 25%) are predicted to be COVID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The findings reported for the MIA-COVID19 competition [78] confirm that ViT outperforms CNN-based approaches such as DenseNet [79] in identifying COVID from CT images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Besides the remarkable accuracy of Transformers compared to CNNs, one of their major drawbacks is their high computa- tional cost, thereby making them less effective for real-world applications, such as detecting COVID-19 in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In light of the prevalence of COVID-19, the rapid diagnosis will be ben- eficial for starting the proper course of medical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CXR and lung CT scans are the most common imaging techniques employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, CT imaging is a time-consuming process, and using CXR images is unreliable in identifying COVID-19 in the early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, vision Transformers are compu- tationally expensive to deploy on mobile devices for real-time COVID-19 classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Therefore, Perera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [45] present a lightweight Point-of-Care Transformer (POCFormer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The compactness of POCFormer allows for real-time diagnosis of COVID-19 utilizing commercially accessible POC ultrasound devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' POCFormer reduces the complexity of the vanilla ViT self-attention mechanism from quadratic to linear using Lin- former [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The results display the superiority of POCFormer in the real-time detection of COVID-19 over the CNN-based SOTAs on the POCUS dataset [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, despite the great potential shown by ViTs in Im- ageNet classification, their performance is still lower than the latest SOTA CNNs without additional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These Transform- ers mainly focus on a coarse level by adopting a self-attention mechanism to establish global dependency between input to- kens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, relying only on a coarse level restricts the Transformer’s ability to achieve higher performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Thus, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [47] leverage a pre-trained version of VOLO for an X-ray COVID-19 classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' VOLO [82] first encodes fine-level information into the token representations through proposed outlook attention, alleviating the limitations of Transformers that require a large amount of data for training, and second ag- gregates the global features via self-attention at the coarse level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Through the outlook attention mechanism, VOLO dynamically combines fine-level features by treating each spatial coordinate (i, j) as the center of a K × K local window and calculating its similarity with all its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The findings indicate that fine- tuning VOLO on Dataset-1 [83] leads to 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='67% top1 accuracy on Dataset-1 test cases and 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='98% top1 accuracy on unseen Dataset-2 [84], which demonstrates the generality of the ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, accessible labeled images have considerably influenced research on the use of Transformers to diagnose COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Considering the shortage of labeled data, data shar- ing between hospitals is needed so as to create a viable central- ized dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, such collaboration is challenging due to privacy concerns and patient permission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Motivated by Fed- erated Learning (FL) and Split Learning (SL), Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [41] present a Federated Split Task-Agnostic (FESTA) framework that uses ViT for multi-task learning of classification, detection, and segmentation of COVID-19 CXR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' FESTA benefits from the decomposable modular design of ViT to train heads and tails via clients and share the server-side Transformer body across clients to aggregate extracted features and process each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The embedded features from the body Transformer are 6 then passed to their task-specific tail on the client side to pro- duce the final prediction (Figure 5(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 5(b) illustrates the single-task learning scheme and (c) the multi-task learning scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In multi-task learning, heads, tails, and a task-agnostic Transformer body are first jointly trained for 6000 rounds (see Figure 5(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then, heads and tails are fine-tuned according to the desired specific task while freezing the weights of the Trans- former body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' FESTA merits from 220000 decentralized CXR images and attains competitive results compared to the data- centralized training approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The experimental results also demonstrate the stable generalization performance of FESTA, where multi-task learning enhances the performance of the in- dividual tasks through their mutual effect during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 5: Overview of the FESTA framework [41], which utilizes ViT for multi-task learning of COVID-19 CXR classification, detection, and segmenta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (a) FESTA leverages ViT’s decomposable modular design to train heads (H) and tails (T ) via clients while sharing the server-side Transformer body (B) between clients to integrate retrieved features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Final predictions are then derived by feeding embedded features to their task-specific tails on the client side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (b) illustrates the single-task learning scheme, and (c) two steps multi-task learning scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The former is trained for 12000 rounds, while the latter un- dergoes two training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' First, the whole parts train in 6000 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then by freezing the weights of the Transformer body, the heads and tails are fine-tuned for 6000 steps based on the desired specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Most attention-based networks utilized for detection and classification rely on the neural network to learn the neces- sary regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Bhattacharya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [46] in Radio- Transformer argue that in certain applications, utilizing ex- perts’ opinions can prove beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Specifically, they apply this notion to leverage radiologists’ gaze patterns while di- agnosing different diseases on medical images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' then, using a teacher-student architecture, they teach a model to pay attention to regions of an image that a specialist is most likely to examine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The teacher and the student networks consist of two main com- ponents: global and focal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The global component learns coarse representation while the focal module works on low-level fea- tures, and both these segments are comprised of Transformer blocks with shifting windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, the global and focal components are interconnected using two-way lateral connec- tions to form the global-focal module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' this is to address the inherent attention gap between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The teacher network is first directly pre-trained on human visual attention maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then, the entire model is trained for different downstream tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', object detection and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, the authors propose a self-supervised Visual Attention Loss (VAL) that in- corporates both GIoU and MSE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The student network is trained to predict probability values for different classes and at- tention regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These attention regions are then compared to those obtained from the teacher model, and the weights are op- timized using VAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Visual Attention Loss TEACHER STUDENT Global-Focal Global-Focal Human Visual Attention Training Input Image Visual Attention Predicted Attention TEACHER Global-Focal HVAT Disease Classification Eye gaze points Figure 6: Overview of RadioTransformer [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Human Visual Attention Train- ing (HVAT) block first uses radiologists’ visual observations of chest radio- graphs to train a global-focal teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The pre-trained teacher network is then utilized to distill the teacher’s knowledge to a global-focal student net- work through visual attention loss, enabling the student to learn visual infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Following the teacher-student strategy and incorporating radiologist visual examinations leads to an improvement in the classification of disease on chest radiographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Hybrid Models In spite of the vision Transformers’ ability to model global contextual representations, the self-attention mechanism un- dermines the representation of low-level details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CNN- Transformer hybrid approaches have been proposed to ame- liorate the problem above by encoding both global and local features using the locality of CNNs and the long-range depen- dency of Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransMed [48] proposes a hybrid CNN-Transformer net- work that leverages the locality of CNNs and the long-range dependency character of Transformers for parotid gland tumor and knee injury classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Multimodal medical images pri- marily have long-range interdependencies, and improving per- formance requires an effective fusion strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransMed pro- poses a novel image fusion strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Firstly, three neighboring 2D slices of a multimodal image are overlaid to create three- channel images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then, each image is partitioned into K × K patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This fusion approach allows the following network to learn mutual information from images of different modali- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Patch tokens are fed into a CNN network to capture their low-level features and generate patch embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The classic ViT is then used to determine the relationship between patch se- quences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransMed’s final results verify the effectiveness of hy- brid models in classifying multimodal medical images by out- performing all its counterparts by a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransMed-S enhances average accuracy on the PGT dataset by about 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1% over its nearest counterpart, BoTNet [86], while requiring fewer parameters and FLOP count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Comparably, Tanzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [50] de- velop a new CAD system (Femur-ViT) based on Vision Trans- 7 (a) (b) Single-task learning scheme (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' classification) Classification Single body Client 1 Hcls Training for 12,000 rounds cls 6 6 Client 1 : Client 6 Task-agnostic Hcls Transformer cls .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Body Client 6 B Segmentation Hcls 6 Client 7 Task-agnostic seg Transformer (c) Multi-task learning scheme Body Client 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Hseg Step 2: Training for 6,oo0 rounds (body fixed) Step 1: Training for 6,oo0 rounds (body learnable) B 6 6 6 Client 1 Client 1 Detection cls cls Task-agnostic Task-agnostic Client 9 Tap Transformer Transformer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} 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' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='w2b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Word-to-bag attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Figure 7: The overall architecture of [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' HATNet hierarchically divides an input image into n × m words, which are then fed into the CNN encoder to provide word-level representations for each bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then by performing a bottom-up decoding strategy and applying a linear classifier, breast biopsy classification results are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Notably, bag-to-image attention has the same procedure as word-to-bag attention, shown in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' formers for diagnosing femoral fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' First, YOLOv3 [87] is utilized to detect and crop the left and right femur regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Afterward, a CNN (InceptionV3 [88]) and a hierarchical CNN (different InceptionV3 networks in cascade) [89] are applied to the dataset, and the results serve as baselines for the classifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then, they use a modified ViT to classify seven differ- ent fracture types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Finally, a clustering approach is proposed as an evaluation technique for the ViT encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This study highlights the power of using ViT models for medical image classification and the ability of the proposed CAD system to significantly increase clinicians’ diagnostic accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 3DMeT [49] proposes applying a 3D medical image Transformer for assessing knee cartilage defects in three grades: grade 0 (no de- fect), grade 1 (mild defect), and grade 2 (severe defect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Primar- ily, using medical 3D volumes as an input to the Transformer is computationally expensive, thereby making it impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 3DMeT resolves the high computational cost problem by re- placing conventional linear embedding with 3D convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The weights of convolutional layers are adopted using the teacher-student training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 3DMeT takes an exponen- tial moving average from the first one/few-layer(s) of the CNN teacher’s weights and uses it as convolutional layers’ weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This method enables Transformers to be compatible with small medical datasets and to benefit from CNNs’ spatial inductive biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Lastly, the Transformer and CNN teacher’s outputs are combined in order to derive the classification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Operating Transformers over Whole Slide Images (WSIs) is computationally challenging since WSI is a gigapixel im- age that retains the original structure of the tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MIL and CNN backbones have demonstrated practical tools for acting on WSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MIL is a weakly supervised learning approach that enables deep learning methods to train high-resolution images like WSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Since annotating such images at the pixel level is impractical, MIL proposes to divide an input WSI into a bag of instances and assign a single label to the bag of each im- age based on pathology diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The bag has a positive label if it contains at least one positive instance, and it is consid- ered negative if all the instances in the bag are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then CNN backbones are employed to down-sample and extract the features of each instance and allow Transformers to operate according to the generated feature maps and currently avail- able hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Therefore, DT-MIL [53] proposes to compress WSIs into compact feature images by embedding each patch of the original WSI into a super-pixel at its corresponding posi- tion using EfficientNet-B0 [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The resulting thumbnail image feed into a 1 × 1 Conv for feature reduction, followed by a de- formable Transformer encoder that aggregates instance repre- sentations globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A similar approach is adopted by Holistic ATtention Network (HATNet) [55], where they first divide an input image into n non-overlapping bags, each broken down into m non-overlapping words (or patches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' n × m words are fed into the CNN encoder to obtain word-level representations for each bag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' HATNet aims to develop a computer-aided di- agnosis system to help pathologists in reducing breast cancer detection errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' According to the World Health Organization (WHO), breast cancer is the most frequent non-skin cancer in women, accounting for one out of every four new female can- cers annually [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As illustrated in Figure 7, HATNet follows a bottom-up decoding strategy such that it first performs multi- head attention to words in a word-to-word attention block, then considers the relationship between words and bags in word-to- bag attention, followed by bag-to-bag attention to attain inter- bag representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The acquired bag features are then ag- gregated in bag-to-image attention to build image-level repre- sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A linear classifier is ultimately applied to achieve the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, unlike most MIL methods that take all the instances in each bag independent and identically distributed [92, 93, 94], TransMIL [54] suggests that it is es- sential to consider the correlation between different instances and explore both morphological and spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Two Transformer layers address the morphological information, and a conditional position encoding layer named Pyramid Position Encoding Generator (PPEG) addresses the spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed PPEG module has two merits: 1) It handles posi- tional encoding of sequences with a variant number of instances by using group convolution over the 2D reshaped patch tokens, 8 and 2) It enriches the features of tokens by capturing more con- text information through convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In contrast to conven- tional iid-based MIL methods requiring many epochs to con- verge, TransMIL converges two to three times faster by using morphological and spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransMIL also outper- forms all the latest MIL methods [95, 96, 97, 98, 92] in terms of accuracy and AUC by a significant margin in binary and multi- ple classification tasks and exhibits the superiority of taking the correlation between different instances into account and con- sidering both morphological and spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Previous methods mainly rely on weakly supervised learn- ing or dividing WSIs into image patches and using supervised learning to assess the overall disease grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Nevertheless, these approaches overlook WSI contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Thus, Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [56] propose a Graph-based Vision Transformer (GTP) framework for predicting disease grade using both morphologi- cal and spatial information at the WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The graph term allows for the representation of the entire WSI, and the Transformer term allows for computationally efficient WSI-level analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The input WSI is first divided into patches, and those that con- tain more than 50% of the background are eliminated and not considered for further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Selected patches are fed forward through a contrastive learning-based patch embedding module for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A graph is then built via a graph construction module utilizing patch embeddings as nodes of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the graph Transformer section, a graph convolution layer followed by a mincut pooling layer [99] is applied first to learn and enrich node embeddings and then lessen the number of Transformer input tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Since the graph adjacency matrix contains spatial information of nodes, by adding an adjacency matrix to node features, GTP obviates the need for adding ex- tra learnable positional embeddings to nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The final Trans- former layer predicts the WSI-level class label for three lung tu- mor classes: Normal, LUAD, and LSCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' GTP also introduces a graph-based class activation mapping (GraphCAM) technique that highlights the class-specific regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' GraphCAM exploits attention maps from multi-head self-attention (MHSA) blocks in the Transformer layer and maps them to the graph space to create a heatmap for the predicted class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The experiments show that GTP performs as a superior interpretable and effi- cient framework for classifying WSI images while considering morphological and spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Diabetic Retinopathy (DR) is an eye disorder that can cause impaired vision and sightlessness by damaging blood vessels in the retina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Most deep-learning approaches view lesion dis- covery and DR grading as independent tasks that may produce suboptimal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In contrast to conventional methods, LAT [52] proposes a unified encoder-decoder structure that com- prises a pixel relation-based encoder to capture the image con- text information and a lesion filter-based decoder to discover le- sion locations, which the whole network jointly optimized and complemented during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The encoder is particularly in charge of modeling the pixel correlations, and the Transformer- based decoder part is formulated as a weakly supervised lo- calization problem to detect lesion regions and categories with only DR severity level labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, LAT proposes two novel mechanisms to improve the effectiveness of lesion-aware filters: 1) Lesion region importance mechanism, g(·|Φ), to de- termine the contribution of each lesion-aware feature, and 2) Lesion region diversity mechanism to diversify and compact lesion-aware features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The former is a linear layer followed by a sigmoid activation function that generates importance weights for lesion-aware features, and the latter adopts a triplet loss [100] to encourage lesion filters to find diverse lesion regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the DR grading branch, LAT presents a DR grading classi- fication module that calculates a global consistency loss based on the lesion-aware features, indicated as h(·|σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Eventually, the final DR grading prediction is achieved by calculating the cross-entropy loss between the predicted labels obtained from the fusion of g(·|Φ) and h(·|σ) and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The total loss is the aggregation of cross-entropy loss, global consistency loss, and triplet loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Visual results of LAT regarding the lesion discovery are depicted in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Ground-truth LAT CAM Figure 8: LAT [52] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CAM [101] visual comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The ground truth con- sists of microaneurysms, hemorrhages, soft exudates, and hard exudates, which are colored as green, yellow, green, and blue dots, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Discussion and Conclusion Section 3 thoroughly outlines 19 distinctive Transformer- based models in medical image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We have cat- egorized the introduced models based on their architectures into hybrid and pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These approaches differ according to whether they adhere to the original structure of the vanilla ViT or provide a new variant of the vision Transformer that can be applied to medical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, we have presented details on the studied classification methods re- garding their architecture type, modality, organ, pre-trained strategy, datasets, metrics, and the year of publication in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Additional descriptions of the methods, including their model size, contributions, and highlights, are described in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As is evident in the storyline of this section, we have dis- cussed methods in each paragraph regarding the underlying problems in medical image classification and introduced so- lutions and how they address such issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, the need for more research on these problems is crucial to making such approaches widely applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 9 Table 1: An overview of the reviewed Transformer-based medical image classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Method Modality Organ Type Pre-trained Module: Type Datasets Metrics Year Pure ViT-vs-CNN [42] Dermoscopy Mammograms Multi-organ 2D ViT: Self-supervised & Supervised 1APTOS-2019 [76] 2ISIC-2019 [102] 3CBIS-DDSM [103] Kappa Recall ROC-AUC 2021 ViT-BUS [43] Ultrasound Breast 2D ViT: Supervised 1B [66] 2BUSI [67] ACC AUC 2021 POCFormer [45] Ultrasound Chest 2D \x17 POCUS [81] Recall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' F1 SP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ACC 2021 MIL-VT [39] Fundus Eye 2D ViT: Supervised 1Private Dataset 2APTOS-2019 [76] 3RFMiD-2020 [77] Recall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' F1 ACC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' AUC Precision 2021 COVID-VIT [38] CT Chest 3D \x17 MIA-COV19 [78] ACC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' F1 2021 xViTCOS [40] X-ray CT Chest 2D ViT: Supervised 1COVID CT-2A [73] 2CheXpert [74] Recall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' F1 Precision SP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' NPV 2021 FESTA [41] X-ray Chest 2D ViT: Supervised 1Four Private Datasets 2CheXpert [74],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 3BIMCV [104] 4Brixia [105],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 5NIH [106] 6SIIM-ACR [107],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 7RSNA [108] Recall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' F1 SP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' AUC 2021 COVID-Transformer [44] X-ray Chest 2D ViT: Supervised 1[68],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 2[70] 3[69] Recall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' F1 ACC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' AUC Precision 2021 COVID-VOLO [47] X-ray Chest 2D ViT: Supervised 1[83] 2[84] ACC 2021 RadioTransformer [46] X-ray Chest 2D ViT: Supervised 1RSNA [109],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 2Cell Pneumonia [110] 3COVID-19 Radiography [83,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 111] 4NIH [106],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 5VinBigData [112] 6SIIM-FISABIO-RSNA [113] 7RSNA-MIDRC [114,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 115] 8TCIA-SBU COVID-19 [116,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 117] Recall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' F1 ACC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' AUC Precision 2022 Hybrid TransMIL [54] Microscopy Multi-organ 2D CNN: Supervised 1Camelyon16 [118] 2TCGA-NSCLC [119,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 120] 3TCGA-RCC[121] ACC AUC 2021 LAT [52] Fundus Eye 2D CNN: Supervised 1Messidor-1[122] 2Messidor-2[123] 3EyePACKS[124] AUC & Kappa 2021 TransMed [48] MRI Ear Knee 3D ViT: Supervised CNN: Supervised 1PGT [48] 2MRNET [125] Precision ACC 2021 3DMeT [49] MRI Knee 3D CNN: Supervised Private dataset ACC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Recall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' F1 2021 Hybrid-COVID-ViT [51] X-ray Chest 2D CNN: Supervised CheXpert [74] AUC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ACC SP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SE 2021 Femur-ViT [50] X-ray Femur 2D ViT: Supervised CNN: Unsupervised Private dataset Recall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' F1 Precision,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ACC 2022 DT-MIL [53] Microscopy Lung Breast 2D CNN: Supervised 1CPTAC-LUAD [116] 2BREAST-LNM [53] Recall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' F1 AUC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' precision 2021 GTP [56] Microscopy Lung 2D CNN: Self-supervised 1NLST [126],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 2CPTAC [127] 3TCGA [128] Precision,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Recall SP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ACC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' AUC 2022 HATNet [55] Microscopy Breast 2D CNN: Supervised Breast Biopsy WSI Dataset [129] ACC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ROC-AUC F1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SE 2022 Data availability in the medical domain is one of the most challenging aspects of developing Transformer-based mod- els since Transformer models are known for being data- hungry to generalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Reasons for data scarcity in the med- ical field can be referred to as privacy concerns of patients, the time-consuming and costly process of annotation, and the need for expert staff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To this end, the use of genera- tive models [130, 131, 132] and their integration with Trans- former models can become prominent since they are capable of creating synthetic data that is comparable to genuine data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, another way to attack this problem is by utiliz- ing federated learning, such as [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Nevertheless, there is still room for improvement when it comes to privacy con- cerns since, in federated learning, communication between the client and server is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Despite their SOTA performance, Transformer-based net- 10 Table 2: A brief description of the reviewed Transformer-based medical image classification models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The unreported number of parameters indicates that the value was not mentioned in the paper, and the code was unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Method # Params Contributions Highlights Pure ViT-vs-CNN [42] 22M They investigate three different weight initialization approaches on three medical datasets: (1) random initialization, (2) transfer learning using supervised ImageNet pre-trained weights, and (3) self-supervised pretraining on the target dataset using DINO [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Their final verdict is that ViTs can replace CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Utilize three different training schemes on three different datasets to conclude whether ViT can replace CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They repeat their processes five times to be certain of the outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Comparing only two models, DeiT-S and Resnet-50, on only three datasets cannot generalize the conclusion of the superiority of each of the Transformer and CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ViT-BUS [43] ViT-Ti/16: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='53M ViT-S/32: 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='87M ViT-B/32: 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='44M Proposes the use of ViT for the classification of breast ultrasound images for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transferring pre-trained ViT models based on small ultrasound datasets yields much higher accuracy than CNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' POCFormer [45] Binary CLS: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='8M Multiclass CLS: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='9M Proposes a lightweight Transformer architecture that uses lung ultrasound images for real-time detection of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' POCFormer can perform in real-time and be deployed to portable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' POCFormer can be used for rapid mass testing of COVID-19 due to its compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MIL-VT [39] 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='12M Proposes to first pre-train the Vision Transformer on a fundus image large dataset and then fine-tune it on the downstream task of the retinal disease classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Introduces the MIL-VT framework with a novel Multiple Instance Learning(MIL)-head to effectively utilize embedding features to improve the ViT performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The MIL head can significantly enhance the performance by easily attaching to the Transformer in a plug-and-play manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MIL-VT efficiently exploits the embedding features overlooked in the final prediction of the ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' COVID-VIT [38] 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='81M Offers utilizing ViT to classify COVID and non-COVID patients using 3D CT lung images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' COVID-ViT performs better in classifying COVID from Non-COVID compared to DenseNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The reported result is not enough to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They only compare ViT with DenseNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' xViTCOS [40] 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='99 Proposes and explores using ViT for detecting COVID-19 from CXR and CT images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 2 xViTCOS makes use of the Gradient Attention Rollout algorithm [75] for visualization and clinical interpretability of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Uses a heatmap plot to demonstrate the model’s explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' FESTA [41] Body: 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='37M (CLS) Head: 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='31M, Tail: 2k (SEG) Head: 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='04M, Tail: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='39M (DET) Head: 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='09M, Tail: 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='77M Proposes a Federated Split Task-Agnostic (FESTA) framework that leverages ViT to merit from federated learning and split learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They use multi-task learning to classify, detect, and segment COVID-19 CXR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed FESTA Transformer improved individual task performance when combined with multi-task learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Experimental results demonstrate stable generalization and SOTA performance of FESTA in the external test dataset even under non- independent and non-identically distributed (non-IID) settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' FESTA eliminates the need for data exchange between health centers while maintaining data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Using FESTA in the industry may not be safe because it may encounter external attacks on the server that may lose the parameters of the entire network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, using this method may jeopardize patient information through privacy attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Authors did not evaluate the robustness of their approach to difficulties through communication, stragglers, and fault tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' COVID-Transformer [44] ViT-L/16: 307M COVID-Transformer investigates using ViT for detecting COVID-19 from CXR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' COVID-Transformer introduces a new balanced chest X-ray dataset containing 30K images for multi- class classification and 20K for binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Uses a heatmap plot to demonstrate the model’s explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' COVID-VOLO [47] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3M Proposes fine-tuning the pre-trained VOLO [82] model for COVID-19 diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Using VOLO enables capturing both fine-level and coarse-level features resulting in higher performance in COVID-19 binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' RadioTransformer [46] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='93M Presents a novel global-focal RadioTransformer architecture, including Transformer blocks with shifting windows, to improve diagnosis accuracy by leveraging the knowledge of experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Introduces an innovative technique for training student networks by utilizing visual attention regions generated by teacher networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Outperform counterpart backbones on multiple datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Model’s explainability Hybrid TransMIL [54] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='67M Presents a Transformer-based Multiple Instance Learning (MIL) approach that uses both morphological and spatial information for weakly supervised WSI classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes to consider the correlation between different instances of WSI instead of assuming them inde- pendently and identically distributed Proposes a CNN-based PPEG module for conditional position encoding, which is adaptive to the number of tokens in the corresponding sequence Converges two to three times faster than SOTA MIL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed method can be employed for unbalanced/balanced and binary/multiple classification with great visualization and interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransMIL is adaptive for positional encoding as token numbers in the sequences changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It needs further improvement to handle higher magnification than ×20 of WSIs - Higher magnification means longer sequences, which in turn require more memory and computational costs to process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' LAT [52] Proposes a unified Transformer-based encoder-decoder structure capable of DR grading and lesion detec- tion simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes a Transformer-based decoder to formulate lesion discovery as a weakly supervised lesion local- ization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes lesion region importance mechanism to determine the importance of lesion-aware features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes lesion region diversity mechanism to diversify and compact lesion-aware features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Unlike most approaches that confront lesion discovery and diabetic retinopathy grading tasks independently, which may generate suboptimal results, the proposed encoder-decoder structure is jointly optimized for both tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Despite existing methods that only perform well for discovering explicit lesion regions, LAT can also detect less dense lesion areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed LAT is capable of identifying Grades 0 and 1, which are hard to distinguish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransMed [48] TransMed-T: 17M TransMed-S: 43M TransMed-B: 110M TransMed-L: 145M Proposes a hybrid CNN-Transformer network for multimodal medical image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes a novel image fusion strategy for 3D MRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransMed achieves much higher accuracy in classifying parotid tumors and knee injuries than CNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Requiring fewer computational resources compared to SOTA CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 3DMeT [49] Replaces conventional linear embedding with 3D convolution layers to reduce the computational cost of using 3D volumes as the Transformer’s inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Obtains weights for 3D convolution layers by using a teacher-student training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed method makes the Transformers capable of using 3D medical images as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 3DMeT Uses significantly fewer computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Adopting CNN as a teacher assists in inheriting CNN’s spatial inductive biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Hybrid-COVID-ViT [51] Proposes a Vision Transformer that embeds features for high-level COVID-19 diagnosis classification using a backbone trained to spot low-level abnormalities in CXR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Different from SOTA models, the proposed model does not use the ImageNet pre-trained weights while archiving significantly better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They examine the interpretability of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Femur-ViT [50] Investigates using ViT for classifying femur fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' proposes using unsupervised learning to evaluate the ViT results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Achieves SOTA results compared to the CNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' GTP [56] Proposes a graph-based vision Transformer (GTP) framework for predicting disease grade using both morphological and spatial information at the WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes a graph-based class activation mapping (GraphCAM) method that captures regional and contex- tual information and highlights the class-specific regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They use a self-supervised contrastive learning approach to extract more robust and richer patch features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They exploit a mincut pooling layer [99] before the vision Transformer layer to lessen the number of Transformer input tokens and reduce the model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In contrast to SOTA approaches, the proposed GTP can operate on the entire WSI by taking advantage of graph representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, GTP can efficiently classify disease gade by leveraging a vision Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed GTP is interpretable so that it can identify salient WSI areas associated with the Transformer output class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' GTP obviates the need for adding extra learnable positional embeddings to nodes by using the graph adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It enables diminishing the complexity of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed GTP takes both morphological and spatial information into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' DT-MIL [53] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='88M Presents a novel embedded-space MIL approach incorporated with an encoder-decoder Transformer for histopathological image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Encoding is done with a deformable Transformer, and decoding with a classic ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' An efficient method to render a huge WSI is proposed, which encodes the WSI into a position-encoded feature image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed method selects the most discriminative instances simultaneously by utilizing associated attention weights and calibrating instance features using the deformable self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed method efficiently embeds instances’ position relationships and context information into bag embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' An extensive analysis of four different bag-embedding modules is presented on two datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' HATNet [55] (w/ MobileNetv2): 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='59M (w/ ESPNetv2): 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='58M (w/ MNASNet): 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='47M Presents a novel end-to-end hybrid method for classifying histopathological images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' HATNet surpasses the bag-of-words models by following a bottom-up strategy and taking into account inter-word, word-to-bag, inter-bag, and bag-to-image representations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (word → bag → image) works still face challenges in deploying their models in the real world due to computational limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As shown in Table 2, most approaches have a high number of parame- ters which provokes a serious problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Different novel ap- proaches have been introduced to reduce the quadratic com- plexity of self-attention, which can be leveraged in the med- ical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, though ViTs have shown im- pressive capabilities in ImageNet classification, their per- formance is still lower than the latest SOTA CNNs with- out additional data [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Hence, existing methods mostly follow pre-training strategies on the ImageNet dataset to build the pre-trained weights for the subsequent downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, despite the enhancement, the domain of natural images is significantly different from medical data, thereby may restrict the performance of further improve- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Therefore, we believe efficient Transformers will con- siderably influence the future research of Transformer-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Medical Image Segmentation Medical segmentation is a significant sub-field of image seg- mentation in digital image processing [133].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It aims to ex- tract features from a set of regions partitioned from the en- tire image and segment the key organs simultaneously, which can assist physicians in making an accurate diagnosis in prac- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' X-ray, positron emission tomography (PET), computed 11 tomography (CT), magnetic resonance imaging (MRI), and ul- trasound are common imaging modalities used to collect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The CNN-based U-Net [34, 133] has been the main choice in this field due to its effective performance and high accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Nevertheless, it cannot extract long-range dependencies in high-dimensional and high-resolution medical images [134].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Therefore, the flexible combination of the U-Net structure with Transformers become a prevalent solution to the segmentation problem at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Take the multi-organ segmentation task as an example: some networks can achieve state-of-the-art multi- organ segmentation performance on the Synapse dataset (as shown in Figure 9) for abdominal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 9: Transformer-based models can perform image segmentation on med- ical image datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure a and c illustrate two 2D slices of raw images with the labels from Synapse dataset [135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure b and d show the 3D visualiza- tion of the labeled organs from different angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These images were generated with MITK Workbench [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In this section, we present the application of ViTs in seg- mentation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' First, we divide the approaches into two cate- gories: pure Transformers and hybrid Transformers, where the pure Transformer denotes the use of the multiple multi-head self-attention modules in both the encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Hy- brid architecture-based approaches fuse the ViTs with convolu- tion modules as the encoder, bottleneck, decoder, or skip con- nection part to leverage information about the global context and local details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, we review some methods with other architectures that propose several novel manners for self- supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 10 demonstrates the different di- rections of the methods employing Transformers in the U-Net architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Pure Transformers In this section, we review several networks referred to as pure Transformers, which employ Transformer blocks in both the encoding and the decoding paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Despite the great suc- cess of CNN-based approaches in medical segmentation tasks, these models still have limitations in learning long-range se- mantic information of medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The authors proposed Swin-Unet, a symmetric Encoder-Decoder architecture moti- vated by the hierarchical Swin Transformer [57], to improve segmentation accuracy and robust generalization capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In contrast to the closest approaches [142, 141, 140, 149] using integrations of CNN with Transformer, Swin-Unet explores the possibility of pure Transformer applied to medical image seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As shown in Figure 11, Swin-Unet consists of encoder, bot- tleneck, decoder, and skip connections utilizing the Swin Trans- former block with shifted windows as the basic unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For the encoder, the sequence of embeddings transformed from im- age patches is fed into multiple Swin Transformer blocks and patch merging layers, with Swin Transformer blocks perform- ing feature learning, and patch merging layers downsampling the feature resolution and unifying the feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The designed bottleneck comprises two consecutive Swin Trans- former blocks to learn the hierarchical representation from the encoder with feature resolution and dimension unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Swin Transformer blocks and patch-expanding layers con- struct the symmetric Transformer-based decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In contrast to the patch merging layers in the encoder, each patch expanding layer is responsible for upsampling the feature maps into double resolutions and halving the corresponding feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The final reshaped feature maps pass through a linear projec- tion to produce the pixel-wise segmentation outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Inspired by the U-Net, the framework also employs skip connections to combine multi-scale features with the upsampled features at various resolution levels to reduce the loss of fine-grained con- textual information caused by down-sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In contrast to the CNN-based methods showing over- segmentation issues, the proposed U-shape pure Transformer presents better segmentation performance resulting from learn- ing both local and long-range dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Compared to the previous methods [150, 24], the HD evaluation metric of Swin- Unet shows an improvement in accuracy for better edge predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The experiments on the Synapse multi-organ CT dataset and ACDC dataset from MRI scanners also demonstrate the ro- bustness and generalization ability of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Compared to Swin-Unet and DS-TransUNet, nnFormer [137] proposed by Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' preserves the superior perfor- mance of convolution layers for local detail extraction and em- ploys a hierarchical structure to model multi-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It utilizes the volume-based multi-head self-attention (V-MSA) and the shifted version (SV-MSA) in the Transformer blocks instead of processing 2D slices of the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The overall ar- chitecture of nnFormer is composed of an encoder and a de- coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Each stage in the encoder and decoder consists of a Transformer block applying V-MSA and SV-MSA and a suc- cessive upsampling or downsampling block built upon convo- lution layers, which is referred to as the interleaved architec- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' V-MSA conducts self-attention within 3D local volumes instead of 2D local windows to reduce the computational com- plexity by approximately 98% and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='5% on the Synapse and ACDC datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' nnFormer is first pre-trained on the ImageNet dataset and uti- lizes symmetrical initialization to reuse the pre-trained weights of the encoder in the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The results of experiments that compare nnFormer with prior Transformer-based [24, 32] and CNN-based arts [151] illustrate nnFormer makes significant progress on the segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 12 a b c d Medical Image Segmentation Pure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Swin-Unet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' nnFormer 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MISSFormer 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransDeepLab Hybrid Decoder 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SegTran Encoder 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Trans-UNet 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransBTS 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransFuse 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MedT 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' UNETR 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Swin UNETR Skip Connection 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CoTr 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' HiFormer Other Architectures 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' T-AutoML 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Cross Teaching 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Self-pretraining with MAE Figure 10: An overview of ViTs in medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Methods are classified into the pure Transformer, hybrid Transformer, and other architectures according to the positions of the Transformers in the entire architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The prefix numbers of the methods denote 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [32], 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [137], 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [138], 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [33], 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [139], 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [24], 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [140], 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [141], 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [142], 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [143], 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [144], 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [145], 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [37], 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [146], 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [147], 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [148].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 11: The architecture of the Swin-Unet [32] which follows the U-Shape structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It contains the encoder, the bottleneck and the decoder part which are built based on the Swin Transformer block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The encoder and the decoder are connected with skip connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Although the recent Transformer-based methods improve the problem that CNN methods cannot capture long-range depen- dencies, they show the limitation of the capability of modeling local details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Some methods directly embedded the convolu- tion layers between fully-connected layers in the feed-forward network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Such structure supplements the low-level information but limits the discrimination of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' propose MISSFormer [138], a hierarchical encoder-decoder network, which employs the Transformer block named Enhanced Trans- former Block and equips the Enhanced Transformer Context Bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Enhanced Transformer Block utilizes a novel efficient self-attention module that illustrates the effectiveness of spa- tial reduction for better usage of the high-resolution map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The original multi-head self-attention can be formulated as follows: Attention(Q, K, V) = S oftmax( QKT √dhead )V, (4) where Q, K, and V refer to query, key and value respectively and have the same shape of N × C, dhead denotes the number of heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The computational complexity is O(N2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In efficient self-attention, the K and V are reshaped by a spatial reduction ratio R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Take K for example: new K = Reshape(N R ,C · R)W(C · R,C) (5) K is first resized from N × C to N R × (C · R) and then pro- jected linearly to restore the channel depth from C · R to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The computational cost reduces to O( N2 R ) accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Further- more, the structure of the Enhanced Mix Feed-forward network (Mix-FFN) extended from [152] introduces recursive skip con- nections to make the model more expressive and consistent with each recursive step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The U-shaped architecture of the MISSFormer contains the encoder and decoder built on the Enhanced Transformer blocks connected with an enhanced Transformer context bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Multi-scale features produced from the encoder are flattened and concatenated together and passed through the Enhanced Transformer Context Bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The pipeline of the Enhanced Transformer Context Bridge is based on the Enhanced Trans- former Block to fuse the hierarchical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The output of the bridge is split and recovered to each original spatial dimension to pass through the corresponding stage in the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The results of experiments show a robust capacity of the method to capture more discriminative details in medical image segmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It is worth mentioning that MISSFormer trained from scratch even outperforms state-of-the-art methods pre-trained on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The results in Figure 12 show that the performance of MISSFormer for prediction and segmentation of edges in pure 13 W H × 48 Patch Partition Linear Projection X W x H xClass 4 4 Linear Embedding Patch Expanding W ×H ×C(4x) Skip Connection Swin Transformer 1/4 Swin Transformer W H x 4 4 Block x2 Block x2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 4 Patch Merging Patch Expanding Skip Connection 1/8 W H Swin Transformer Swin Transformer W H ×2C ×2C 8 8 Block x2 Block x2 8 8 Patch Merging Patch Expanding Skip Connection 1/16 W H Swin Transformer Swin Transformer W H ×4C ×4C 16＇16 Block x2 Block x2 16＇16 4 Patch Merging Patch Expanding Encoder Decoder W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='.H Swin Transformer Swin Transformer ×8C 32＇～32 Block x1 Block x1 BottleneckTransformer network structures is more accurate compared to TransUNet and Swin-Unet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Comparing MISSFormer and MISSFormer-S (MISSFormer without bridge), MISSFormer has fewer segmentation errors because the bridge is effective for integrating multi-scale information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 12: A visual comparison with the state-of-the-art approaches on Synapse dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Above the red line shows the successful cases of segmentation, and below the red line are the failed cases with relatively large errors [138] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Inspired by the notable DeepLabv3 [153] which utilizes the Atrous Spatial Pyramid Pooling (ASPP) to learn multi-scale feature representations and depth-wise separable convolution to reduce the computational burden, the authors propose Trans- DeepLab [33] to combine the DeepLab network with the Swin- Transformer blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Applying the Swin-Transformer module with windows of multiple sizes enables the fusion of multi- scale information with a lightweight model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransDeepLab is a pure Transformer-based DeepLabv3+ ar- chitecture, as shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The model builds a hi- erarchical architecture based on the Swin-Transformer mod- ules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransDeepLab first employs N stacked Swin-Transformer blocks to model the input embedded images into a deep-level feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 2D medical images are first to split into non- overlapping patches of dimension C and size 4 × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The ensu- ing Swin-Transformer block learns local semantic information and global contextual dependencies of the sequence of patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then, the authors introduce windows with different sizes to pro- cess the output of the Swin-Transformer and fuse the resulting multi-scale feature layers, which are then passed through Cross Contextual attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This design, referred to as Spatial Pyra- mid Pooling (SSPP) block, replaces the original Atrous Spa- tial Pyramid Pooling (ASPP) module exploited in DeepLabV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A cross-contextual attention mechanism is utilized to explore the multi-scale representation after fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This attention mod- ule applies channel attention and spatial attention to the out- put from windows of each size (from each layer of the spatial pyramid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Finally, in the decoder part, the low-level features from the encoder are concatenated with the multi-scale features extracted by Cross Contextual Attention after bilinear upsam- pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The last two Swin-Transformer blocks and the patch ex- panding module generate the final prediction masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Stacked N blocks … … Cross Contextual Attention Multi-scale Representation Encoder Low-level Features Concat Upsample Decoder Idea: Pure transformer model to model DeepLab3 model with additional attention mechanism 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Using swim transformer strategy to reduce time complexity 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transformer structure to better model long-range contextual dependency 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Novel structure using transformer Swin Transformer Block × 𝟐 Patch Merging Patch Partition Linear Embedding 2 × 2 Window 7 × 7 Window 𝐼𝑚𝑎𝑔𝑒 𝑃𝑜𝑜𝑙 Swin Transformer Block × 𝟐 Patch Expanding … … Figure 13: The overview architecture of TransDeepLab, which comprises encoder and decoder built on Swin-Transformer blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It is the pure Transformer-based extension of DeepLabv3++ [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Hybrid Models Hybrid Transformers concatenate Transformer blocks with convolution layers to extract local details and long-range de- pendencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We further classify this category into Transformer: Encoder, Transformer: Decoder and Transformer: skip con- nection according to the position of the combined module in the U-Net architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transformer: Encoder Starting with TransUNet [24], multiple methods in the med- ical image segmentation field adopt the self-attention mecha- nism in the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transformers have developed as an alternative architecture for modeling global context that exclusively relies on attention mechanisms instead of convolution operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, its in- ner global self-attention mechanism induces missing low-level details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Direct upsampling cannot retrieve the local informa- tion, which results in inaccurate segmentation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The au- thors propose the TransUNet architecture, a hybrid approach that integrates CNN-Transformer hybrid as the encoder and cascaded upsampler as the decoder, combining the advantages of Transformer and U-Net to boost the segmentation perfor- mance by recovering localized spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The framework of the TransUNet is illustrated in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed encoder initially employs CNN as a feature ex- tractor to build a feature map for the Transformer input layer, rather than the Transformer directly projecting the raw tok- enized picture patches to latent embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In this way, the intermediate CNN feature maps of different resolutions can be saved and utilized in the following process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For the decoder, the Cascaded Upsampler (CUP) is proposed to replace naive bilinear upsampling, applying several upsam- pling blocks to decode the hidden feature and output the final segmentation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Finally, the hybrid encoder and the CUP constitute the overall architecture with skip connections to fa- cilitate feature aggregation at different resolution levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This strategy can compensate for the loss of local fine-grained de- tails caused by the Transformer encoder and merge the encoded 14 (b) MISSFormer (a) GroundTruth (c) MISSFormer_S (d) Swin-Unet (e) TransUnetglobal information with the local information contained in in- termediate CNN feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The experiments show that TransUNet significantly outper- forms the model consisting of pure Transformer encoder and naive upsampling, as well as the ViT-hybrid model without skip connections [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Comparisons with prior work [34, 150] also demonstrate the superiority of TransUNet over competing CNN-based approaches in terms of both qualitative visualiza- tion and the quantitative evaluation criteria (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='average DSC and HD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransUNet integrates the benefits of both high-level global contextual information and low-level details as an alter- native approach for medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' reshape 1/4 1/8 1/2 Conv3x3, ReLU Upsample Segmentation head (n_patch, D) (D, H/16, W/16) (512, H/16, W/16) (256, H/8, W/8) (128, H/4, W/4) (64, H/2, W/2) (16, H, W) Transformer Layer … (n = 12) Hidden Feature Linear Projection CNN Hidden Feature Downsample Feature Concatenation Transformer Layer Embedded Sequence 𝒙𝒑𝟏, 𝒙𝒑𝟐, … , 𝒙𝒑𝑵 Layer Norm MSA Layer Norm MLP + + 𝒛𝟏 (a) (b) Figure 14: The overview architecture of the TransUNet [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Transformer layers are employed in the encoder part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The schematic of the Transformer is shown on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [140] propose the encoder-decoder architecture, TransBTS, which leverages Transformer on learning global contextual information and merits the 3D CNN for modeling local details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In contrast to the concurrent Transformer-based model [24], which analyzes 3D medical volumetric data in a slice-by-slice manner, TransBTS also explores the local fea- tures along the depth dimension by processing all the image slices at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The network encoder initially employs a 3D CNN to capture volumetric spatial features, simultaneously downsampling the input 3D images, yielding compact volumetric feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Each feature map is projected into a token and fed into the Transformer encoder to investigate the global relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The full-resolution segmentation maps are generated by the 3D CNN decoder after the progressive upsampling while us- ing the feature embedding from the Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For the en- coder part, TransBTS first utilizes the 3 × 3 × 3 convolution blocks with downsampling to process the 3D input medical im- age data, which boosts the effective embedding of rich local 3D features a cross spatial and depth dimensions into the low- resolution feature representation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They apply a linear projec- tion to the feature representation F to obtain the sequence f, which is then integrated with position embeddings, as the input for the Transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Transformer encoder con- sists of multiple Transformer layers, each of which comprises a Multi-Head Attention(MHA) block and a Feed-Forward Net- work(FFN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The output sequence of the Transformer encoder passes through the feature mapping module to be reshaped to a 4D feature map Z of the same dimension as F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The approach employs cascaded upsampling and convolution blocks to pro- gressively restore the segmentation predictions at the original resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, skip connections combine the fine- grained details of local information with the decoder modules, resulting in more accurate segmentation masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The authors conduct comparisons between the proposed TransBTS and the closest method TransUNet [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransUNet essentially processes 3D medical images slice by slice, while TransBTS is a 3D model that explores the continuous interac- tion through the depth dimension by processing a 3D medical image in a single pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In contrast to TransUNet, which adopts pre-trained ViT models on other large-scale datasets, TransBTS is trained on the dataset for the specified task without relying on pre-trained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The framework is evaluated on the Brain Tumor Segmenta- tion (BraTS) 2019 challenge and 2020 challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Compared to the 3D U-Net baseline, TransBTS achieves a significant en- hancement in segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The prediction results indicate the improved accuracy and the superiority of modeling long-range dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Previous approaches [141] primarily focus on replacing convolution operation with Transformer layers or consecu- tively stacking the two together to address the inherent lack of pure Transformer-based models to learn local information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In this study, the authors propose a new strategy-TransFuse which consists of the CNN-based encoder branch and the Transformer-based branch in parallel fused with the proposed BiFusion module, thus further exploring the benefits of CNNs and Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The construction of the Transformer branch is designed in the typical encoder-decoder manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The input images are first split into non-overlapped patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The linear embedding layer then projects the flattened patches into the raw embedding sequence which is added to the learnable position embedding of the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The obtained embeddings are fed into the Transformer encoder, which comprises L layers of MSA and MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The output of the last layer of the Trans- former encoder is passed through layer normalization to obtain the encoded sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The decoder part utilizes the same progressive upsampling (PUP) approach as SETR [154].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The encoded sequence is first reshaped back to a sequence of 2D feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then they employ two stacked upsampling-convolution layers to restore the feature scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The feature maps with different spatial res- olutions generated by each upsampling-convolution layer are retained for the subsequent fusion operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For the CNN branch, the approach discards the last layer of the traditional CNNs architecture and combines the information extracted from the CNNs with the global contextual features obtained from the Transformer branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A shallower model is yielded as a result of this design, avoiding the requirement for extremely deep models that exhaust resources to get long-range depen- dencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For instance, there are five blocks in a typical ResNet- based network where only the outputs of the 4th, 3rd, and 2nd layers are saved for the following fusion with the feature maps from the Transformer branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The BiFusion module is proposed to fuse the features ex- tracted from the two branches mentioned above to predict the segmentation results of medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The global features 15 from the Transformer branch are boosted by the channel atten- tion proposed in SE-Block [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Meanwhile, the feature maps from the CNN branch are filtered by the spatial attention which is adopted in CBAM [155] block to suppress the irrelevant and noisy part and highlight local interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then the Hadamard product is applied to the features from the two branches to learn the interaction between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They concatenate the interaction feature bi with attended features ˆti and ˆgi and feed the results through a residual block to produce the feature f i, which suc- cessfully models both the global and local features at the orig- inal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Finally, the segmentation prediction is gener- ated by integrating the f i from different BiFusion modules via the attention-gated (AG) [156] skip connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They evaluate the performance of three variants of Trans- Fuse on four segmentation tasks with different imaging modal- ities and target sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransFuse-S is constructed with ResNet-34 (R34) and 8-layer DeiT-Small (DeiT-S) [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Be- sides, TransFuse-L is composed of Res2Net-50 and 10-layer DeiT-Base (DeiT-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransFuse-L* is implemented based on ResNetV2-50 and ViT-B [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For polyp segmenta- tion, Transfuse-S/L outperforms significantly the CNN base- line models with fewer parameters and faster running time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransFuse-L* also achieves the best performance among the previous SOTA Transformer-based methods with a faster speed for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It runs at 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3 FPS and about 12% faster than TransUNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The experiments for other segmentation tasks also show the superiority of the segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Despite the powerful results of applying Transformers to seg- mentation tasks [157, 154], the dilemma is that properly train- ing existing Transformer-based models requires large-scale datasets, whereas the number of images and labels available for medical image segmentation is relatively limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To over- come the difficulty, MedT [142] proposes a gated position- sensitive axial attention mechanism where the introduced gates are learnable parameters to enable the model to be applied to a dataset of arbitrary size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, they suggested a Local- Global(LoGo) training strategy to improve the segmentation performance by operating on both the original image and the local patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The main architecture of MedT, as shown in Figure 15 (a), is composed of 2 branches: a shallow global branch that works on the original resolution of the entire image, and a deep local branch that acts on the image patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Two encoder blocks and two decoder blocks comprise the global branch, which is suf- ficient to model long-range dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the local branch, the original image is partitioned into 16 patches and each patch is feed-forwarded through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The output feature maps are re-sampled based on their locations to obtain the out- put feature maps of the branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then the results generated from both branches are added and fed into a 1 × 1 convolution layer to produce the output segmentation mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The LoGo training strategy enables the global branch to concentrate on high-level information and allows the local branch to learn the finer inter- actions between pixels within the patch, resulting in improved segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 15 (b) and (c) illustrates the gated axial Transformer layer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' which is used as the main building block in MedT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Global Branch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Local Branch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='WV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='WK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='WQ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='rQ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='yY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Gates ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Positional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Embeddings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Weights ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Multiplication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Addition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Gated Axial Attention Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Resample ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Patches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Patches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Figure 15: Overview of the MedT [142] architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The network uses the LoGo strategy for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The upper global branch utilizes the first fewer blocks of the Transformer layers to encode the long-range dependency of the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the local branch, the images are converted into small patches and then fed into the network to model the local details within each patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The output of the local branch is re-sampled relying on the location information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Finally, a 1 × 1 convolution layer fuses the output feature maps from the two branches are to generate the final segmentation mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' the feed-forward structure in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They introduced four learnable gates GV1,GV2,GQ,GK ∈ R that control the amount of informa- tion the positional embeddings supply to key, query, and value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Based on whether a relative positional encoding is learned ac- curately or not, the gate parameters will be assigned weights either converging to 1 or some lower value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The gated mecha- nism can control the impact of relative positional encodings on the encoding of non-local context and allows the model to work well on any dataset regardless of size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Unlike the fully-attended baseline [157], MedT trained on even smaller datasets outperforms the convolutional baseline and other Transformer-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, improve- ments in medical segmentation are also observed since the pro- posed method takes into account pixel-level dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In contrast to multiple proposed methods [154, 24, 142, 141] that investigate the task of 2D medical image segmentation, UNETR [143] proposes a novel Transformer-based architec- ture for 3D segmentation which employs the Transformer as the encoder to learn global contextual information from the volumetric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, unlike the previous frameworks proposed for 3D medical image segmentation [145, 140], the encoded feature from the Transformer of this proposed model is directly connected to a CNN-based decoder via skip connec- tions at different resolution levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The U-shaped UNETR com- prises a stack of Transformers as the encoder and a decoder coupling with it by skip connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They begin by generating the 1D sequence of patches by splitting the 3D input volume in a non-overlapping manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The flattened input patches are then passed through a linear projection layer to yield K dimen- sional patch embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They attach a 1D learnable positional embedding to each patch embedding taking into account the spatial information of the extracted patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' After the embed- ding layer, the global multi-scale representation is captured us- ing Transformer blocks composed of multi-head self-attention 16 modules and multilayer perceptron layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They resize and project the sequence representation extracted from the Trans- former at different resolutions for use in the decoder in order to retrieve spatial information of the low-level details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the expanding pattern of the framework, the proposed CNN-based decoder combines the output feature of different resolutions from the Transformer with upsampled feature maps to properly predict the voxel-wise segmentation mask at the original input resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The paper claims UNETR achieves new state-of-the-art per- formance on all organs compared against CNN [158, 35, 159, 160] and competing for Transformer-based [145, 24, 154] base- lines on BTCV dataset, with significant improvement in perfor- mance on small organs in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, it outperforms the closest methodologies on brain tumor and spleen segmen- tation tasks in MSD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' UNETR shows the superiority of learning both global dependencies and fine-grained local rela- tionships in medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 16 presents qualitative segmentation comparisons for brain tumor segmentation on the MSD dataset between UNETR [143], TransBTS [140], CoTr [145] and U-Net [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It can be seen that the details of the brain tumor are captured well by UNETR [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Ground Truth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='86 UNet TransBTS CoTr UNETR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='83 Figure 16: Comparison of visualization of brain tumor segmentation on the MSD dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The whole tumor (WT) includes a combination of red, blue, and green regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The union of red and blue regions demonstrates the tumor core (TC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The green regions indicate the enhanced tumor core (ET) [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As opposed to other methods which attempted to utilize the Transformer module as an additional block beside the CNN- based components in the architectures, UNETR [143] lever- ages the Transformer as the encoder instead of the CNN-based encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Swin Transformer [57] is a hierarchical visual Transformer featuring an efficient shift-window partitioning scheme for computing self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Inspired by these two ap- proaches, a novel model termed Swin Unet Transformer (Swin UNETR) [144] is proposed for brain tumor segmentation in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed framework applies a U-shape architecture with the Swin Transformers as the encoder and a CNN-based mod- ule as the decoder connected to the encoder via skip connec- tions at different resolution levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The model initially converts 3D MRI images with four channels to non-overlapping patches and creates windows of a specific size with a patch partition layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Swin UNETR encoder is composed of 4 stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Each stage comprises 2 Transformer blocks and a patch merging layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the Transformer blocks, the self-attention is computed with a shifted windowing mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Swin UNETR employs the windows of size M × M × M to partition the 3D token with resolution of H′ × W′ × D′ into regions of ⌈ H′ M × W′ M × D′ M ⌉ at layer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The partitioned window regions are then shifted by (⌊ M 2 ⌋, ⌊ M 2 ⌋, ⌊ M 2 ⌋) voxels at the following l + 1 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The patch merging layer after the Transformer components reduces the resolution of feature maps by a factor of two and concatenates them to form a feature embedding with the doubled dimension- ality of the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For the decoder of the architecture, the output feature repre- sentations of the bottleneck are reshaped and passed through the residual block containing two convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The subse- quent deconvolutional layer increases the resolution of feature maps by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The outputs are then concatenated with the outputs of the previous stage and fed into another residual block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' After the resolutions of the feature maps are restored to the original H′ ×W′ × D′, a head is utilized to generate the final segmentation predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The authors conduct the experiments to compare Swin UN- ETR against the previous methodologies SegResNet [161], nn- UNet [151]and TransBTS [140] in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The results demonstrate that the proposed model has prominence as one of the top-ranking approaches in the BraTS 2021 challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It is due to the better capability of learning multi-scale con- textual information and modeling long-range dependencies by Swin Transformers in comparison to regular Transformers with a fixed resolution of windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transformer: Decoder Another direction is to modify the decoder of the U-shape structure to aggregate the Transformer-CNN-based modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the Segtran framework [139], Squeeze-and-Expansion Transformer is proposed to ”squeeze” the attention matrix and aggregate multiple sets of contextualized features from the out- put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A novel Learnable Sinusoidal Position Encoding is also employed to impose the continuity inductive bias for images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Segtran consists of five components: a CNN backbone to extract image features, 2) input/output feature pyramids to do upsampling, 3) the Learnable Sinusoidal Positional Encod- ing, 4) Squeeze-and-Expansion Transformer layers to contex- tualize features, and 5) a segmentation head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The pretrained CNN backbone is first utilized to learn feature maps from the input medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Since the input features to Transformers are of a low spatial resolution, the authors increase their spa- tial resolutions with an input Feature Pyramid Network (FPN) [162] to upsample the feature maps by bilinear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then the proposed Learnable Sinusoidal Positional Encoding is added to the visual features to inject spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In contrast to the previous two mainstream PE schemes [163, 22], the new positional embedding vector, a combination of sine and cosine functions of linear transformations of (x, y), brings in the continuity bias with adaptability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The equation of the encoding strategy varies gradually with pixel coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Thus, close 17 units receive similar positional encodings, increasing the atten- tion weights between them towards higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The encod- ing vectors generated from the addition of positional encodings and visual features are then fed into the Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The novel Transformer architecture combines Squeezed At- tention Block (SAB) [164] with an Expanded Attention Block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Here this method employs the Induced Set Attention Block (ISAB) proposed by [164] as a squeezed attention block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Squeezed Attention Block computes attention between the in- put and inducing points and compresses the attention matrices to lower rank matrices, reducing noises and overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Expanded Attention Block (EAB), a mixture-of-experts model, outputs Nm sets of complete contextualized features from Nm modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Each mode is an individual single-head Transformer and shares the same feature space with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' That is as opposed to multi-head attention in which each head outputs an exclusive feature subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' All features are then aggregated into one set using dynamic mode attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The dynamic mode at- tention can be obtained by doing a linear transformation of each mode feature and taking softmax over all the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Compared with representative existing methods in the exper- iments, Segtran consistently achieved the highest segmentation accuracy and exhibited good cross-domain generalization capa- bilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' skip connection conv + max pool 2x2 up-conv 2x2 Downsampling path Upsampling path Figure 17: Segtran network extracts image features using a CNN backbone and combines the features with the position encoding of pixels flattened into a series of local feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Multiple squeezed and extended transform layers are stacked to process the local feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Finally, an output FPN after the Transformer upsamples the features to generate the final prediction [139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transformer: Skip Connection In this section, Transformer blocks are incorporated into the skip connections to facilitate the transmission of detailed infor- mation from the encoder to the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Although Transformer-based methods overcome the limi- tation of capturing long-range dependency, they present ex- treme computational and spatial complexity in analyzing high- resolution volumetric image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Some studies [163, 24] em- ploy hybrid structures, fusing CNN with Transformer in an at- tempt to reduce the training requirement on huge datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The recent approach, TransUNet [24], shows good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, it is difficult to optimize the model due to the inner self-attention mechanism of the vanilla Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' First, it takes a long time to train the attention, which is caused by ini- tially distributing attention uniformly to each pixel within the salient regions [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Second, a vanilla Transformer struggles to handle multi-scale and high-resolution feature maps due to its high computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Motivated by this, [145] proposes a novel encoder-decoder framework, CoTr, which bridges CNN and Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The architecture exploits CNN to learn feature representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' An efficient deformable self-attention mechanism in the Trans- former is designed to model the global context from the ex- tracted feature maps, which reduces the computational com- plexity and enables the model to process high-resolution fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' And the final segmentation results are generated by the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As shown in Figure 18, the DeTrans-encoder consists of an input-to-sequence layer and multiple DeTrans Layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The input-to-sequence layer first flattens the feature maps at differ- ent resolutions extracted from CNN-encoder into 1D sequences {fl}L l=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then the corresponding 3D positional encoding se- quence pl is supplemented with each of the flattened sequences fl to complement the spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The combined se- quence is fed as the input into the DeTrans Layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Each of the DeTrans Layers is a composition of an MS-DMSA and a Feed-Forward Network (FFN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In contrast to the self-attention mechanism which casts attention to all the possible locations, the proposed MS-DMSA layer only attends to a small set of key sampling locations around a reference location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As a re- sult, it can achieve faster convergence and lower computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The skip connection is utilized after each DeTrans Layer for preserving the low-level details of local information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The output of the DeTrans-encoder is successively upsampled by the pure CNN encoder to restore the original resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Be- sides, they apply skip connections and a deep supervision strat- egy to add fine-grained details and auxiliary losses to the pre- diction outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The results of experiments indicate CoTr with the hybrid ar- chitecture has superiority of performance over the models with pure CNN encoder or pure Transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It also outper- forms other hybrid methods like TransUNet [24] in processing multi-scale 3D medical images with reduced parameters and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' HiFormer [37] is proposed to aggregate a fusion module in the skip connections to learn richer representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Fig- ure 19 demonstrates the end-to-end network structure of the strategy that incorporates the global dependencies learned with the Swin Transformer and the detailed local features extracted by the CNN modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The encoder is composed of two hierar- chical CNN, Swin Transformer modules and the novel Double- Level Fusion module (DLF module).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' First, medical images are fed into a CNN module to obtain a local fine-grained se- mantic representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' After the CNN layer catches the shal- low feature layers, HiFormer introduces the Swin Transformer modules to complement the global feature information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Swin Transformer module employs windows of different sizes to learn the dependencies between multiple scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To reuse 18 DeTrans Layer DeTrans Layer .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' F F F R R R DeTrans Layer Positional encoding Upsampling Flatten Reshape CNN-encoder DeTrans-encoder Decoder MS-DMSA Feed Forward Reference point Layer Norm Layer Norm Deformable Transformer Layer Figure 18: Overview of the CoTr [145] architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It is composed of a CNN- encoder, a DeTrans-encoder and a decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The CNN-encoder models the lo- cal information of the input images and provides the outputs at each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The outputs of different resolutions are flattened, fused and passed through the Deformable Transformer Layers along with positional encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The decoder reshapes the processed sequences from the DeTrans-encoder and produces the final predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' the shallow and deep multi-scale feature information in the en- coder, HiFormer designs a novel skip connection module, the DLF module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The deep-level semantic and shallow-level lo- calization information are fed into the DLF module and fused by the cross-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Finally, both generated fea- ture maps are passed into the decoder to produce the final segmentation prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The experiments conducted on the Synapse dataset [135], SegPC [165], and ISIC 2017 dataset [166] demonstrate the superiority of the learning abil- ity of HiFormer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Moreover, the lightweight model with fewer parameters also exceeds CNN-based methods and previous Transformer-based approaches with lower computational com- plexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 𝐶𝑜𝑛𝑣 1 × 1 Patch merging Patch merging DLF Module Conv Block Segmentation Head Transformer Encoder × 𝑺 GAP Cross Attention Transformer Encoder × 𝑳 GAP 𝐶𝑜𝑛𝑣 1 × 1 𝐶𝑜𝑛𝑣 1 × 1 𝐇/𝟒 × 𝐖/𝟒, 𝐃 𝐇/𝟖 × 𝐖/𝟖, 𝟐𝐃 𝐇/𝟏𝟔 × 𝐖/𝟏𝟔, 𝟒𝐃 𝐇/𝟒 × 𝐖/𝟒, 𝐃′ 𝐇/𝟖 × 𝐖/𝟖, 𝟐𝐃′ 𝐇/𝟏𝟔 × 𝐖/𝟏𝟔, 𝟒𝑫′ 𝑯 × 𝑾 × 3 ConvUp Ps Pl × 6 × 2 × 2 ConvUp Figure 19: HiFormer comprises the CNN-Transformer encoder, the CNN-based decoder and the Double-Level Fusion Module (DLF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='The feature layers of the shallowest level pl and of the deepest level ps are fed into the DLF module for the fusion of hierarchical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Blue blocks and orange blocks refer to Swin Transformer and CNN modules, respectively [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Other Architectures Most ViT-based models rely on pre-training of large natu- ral image datasets to obtain pre-weights and then solve down- stream tasks by transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Several works explore train- ing in a self-supervised or semi-supervised manner to effi- ciently utilize medical image datasets of limited size or datasets without manual labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, some approaches apply Transformers to seek the design of architectures that implement medical image segmentation, instead of using the Transformers to act directly on the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Unlike the previously proposed methods that employ the Transformers to act directly on the medical image for feature extraction, this method [146] adopts the AutoML for automat- ically designing the network architecture without much human heuristics or assumptions, where the Transformer is applied to encode the embedding vector regarding the architecture config- urations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The approach reduces the workload of algorithm de- sign by automatically estimating ”almost” all the components of the framework instead of manually designing for the network and training strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' That improves the model performance of segmentation simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed Transformer-based T-AutoML inspired by SpineNet [167] leverages neural architecture search (NAS) with a larger search space to optimize the selection of the network connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This framework can connect the feature maps at different spatial levels of the network with another one arbitrar- ily, compared with the previous methods that only search for the encoder-decoder U-shape networks [168, 169, 170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The can- didates of different blocks in the network consist of 3D residual blocks, 3D bottleneck blocks, and 3D axial-attention blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The residual blocks and bottleneck blocks are effective in alle- viating the vanishing gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The axial-attention blocks are applied to model the long-range dependency in the 2D medi- cal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Another upsampling layer (linear interpolation) is utilized at the end of the architecture to produce the results of feature maps at the original volume size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To search for the optimal architecture and training configu- ration, the authors first encode the necessary components in the search space to form a one-dimensional vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The search space contains candidates of different configurations with re- gard to data augmentation, learning rates, learning rate sched- ulers, loss function, the optimizer, the number and spatial reso- lution of blocks, and block types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' After the obtainment of the encoding vector v, the proposed new predictor predicts the binary relation of validation accuracy values between vi and v j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The predictor employs the Trans- former encoder to encode the vector v of varying lengths into feature maps of a fixed resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then the feature maps are passed through the Multiple FC layers to generate the binary relation predictions denoted as GTvi,vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Since the predictor is designed for ranking the vectors with respect to the accuracy values and estimating the relations, the actual values of the pre- dicted accuracy are not necessary to be calculated for each vec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Thus, the new predictor requires less overall training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The experiments indicate that the proposed method can achieve the state of the art(SOTA) in lesion segmentation tasks and shows superiority in transferring to different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Despite the promising results achieved by the CNNs and Transformer-based methods with large-scale images, these ap- proaches require expert labeling at the pixel/voxel level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Expen- sive time costs and manual annotation limit the size of the med- ical image dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Due to this dilemma, the proposed semi- supervised segmentation [147] provides a low-cost and practi- cal scheme, called Cross Teaching between CNN and Trans- former, to train effective models using a little amount of cor- 19 rectly labeled data and a large amount of unlabeled or coarsely labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Inspired by the existing works [171, 172, 173] for semi- supervised learning which introduce perturbation at different levels and encourage prediction to be consistent during the training stage, the designed cross teaching introduces the per- turbation in both learning paradigm-level and output-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As shown in Figure 20, each image within the training set con- taining labeled and unlabeled images is fed into two different learning paradigms: a CNN and a Transformer respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For the unlabeled dataset with raw images, the cross teaching scheme allows the cross supervision between a CNN ( f c φ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=')) and a Transformer(f t φ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' )), which aims at integrating the properties of the Transformer modeling the long-range dependency and CNN t0 learn local information in the output level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The unlabeled data initially passes through a CNN and a Transformer respectively to generate predictions pc i and pt i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' pc i = f c φ(xi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' pt i = f t φ(xi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (6) Then the pseudo labels plc i and plt i are produced in this manner: plc i = argmax(pt i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' plt i = argmax(pc i );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (7) The pseudo label plc i used for the CNN training is generated by the Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Similarly, the CNN model provides pseudo labels for Transformer training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The cross-teaching loss for the unlabeled data is defined as follows: Lctl = LDice(pc i , plc i ) ������������������������ supervision for CNNs + LDice(pt i, plt i) ���������������������� supervision for Transformers (8) which is a bidirectional loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' One direction of the data stream is from the CNN to the Transformer, and the other di- rection is from the Transformer to the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For the labeled data, the CNN and Transformer are supervised by the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The commonly-used supervised loss functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' the cross-entropy loss and Dice loss, are employed to update model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Lsup = Lce(pi, yi) + LDice(pi, yi) (9) where pi , yi represent the prediction and the label of image xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' And the overall objective combining the cross-teaching branch and supervised branch is defined as: Ltotal = Lsup + λLctl (10) where λ is a weight factor, which is defined by time- dependent Gaussian warming up function [174, 175]: λ(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1 · e � −5 � 1− ti ttotal ��2 (11) The results of the ablation study indicate that the combina- tion of CNN and Transformer in a cross-teaching way shows superiority over the existing semi-supervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Further- more, the novel method has the potential to reduce the label cost by learning from limited data and large-scale unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, it is observed that achieving SOTA via semi- supervised approaches remains a significant challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' UNet (CNN) Transformer Encoder Transformer Decoder Training set Mini-batch L L L L U U U U L L: Labeled image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' U: Unlabeled image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' : Stop-gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Swin-UNet (Transformer) U L Label Figure 20: The model performs the semi-supervised medical image segmenta- tion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The regularization scheme between CNN and Transformer is referred to as Cross Teaching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' L denotes the labeled data and U denotes the unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The cross-teaching employs a bidirectional loss function: one path is from the CNN branch to the Transformer branch, and the other is from the Transformer to the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A Transformer is applied for complementary training instead of prediction generation [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Zhou et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [148] hypothesize that the ability to aggre- gate contextual information is imperative to improve the per- formance of medical image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Nonetheless, there is no ImageNet-scale medical image dataset for pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' There- fore, they investigate a novel self pre-training paradigm based on Masked Autoencoder (MAE), MAE self pre-training, for medical image analysis, one of the masked image modeling (MIM) frameworks [194] [195] [196] [197].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MIM encourages the framework to restore the masked target by integrating infor- mation from the context, where the main idea of MIM is mask- ing and reconstructing: masking a set of image patches before input into the Transformer and reconstructing these masked patches at the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The pipeline for segmentation with MAE self pre-training contains two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the first stage (as shown on the left of Figure 21), ViT is pre-trained with MAE as the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The input patches are randomly divided into visible ones and masked ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The ViT encoder only acts on the visible patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Compared to other MIM methods, MAE does not employ mask tokens in the encoder, which saves time and allows for faster pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A lightweight Transformer decoder is appended to reconstruct the full image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The decoder is only an auxiliary part used for pre-training and will not be applied in downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the second stage (as shown on the right of Figure 21), the pre-trained ViT weights are transferred to initialize the segmen- tation encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then, the task-specific heads are appended to perform downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The whole segmentation network, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', UNETR, is finetuned to perform the segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The experiments, including the MAE pre-training and the downstream task, are conducted to evaluate the performance of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The results show that MAE can recover the lost information in the masked input patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MAE pre- training enables the model to improve its classification and seg- mentation performance on medical image analysis tasks, sur- passing the ImageNet pre-trained model to SOTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 20 Table 3: An overview of the reviewed Transformer-based medical image Segmentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Modality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Organ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Pre-trained Module: Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Datasets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Metrics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Pure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Swin-Unet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[32] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='CT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Multi-organ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2D ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='CT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='MRI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Multi-organ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ViT: Supervised ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1 Synapse [135] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2 ACDC [176] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Dice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='MISSFormer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[138] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='CT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='MRI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Multi-organ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='\x17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1 Synapse [135] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2 ACDC [176] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Dice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Hausdorff distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='TransDeepLab ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[33] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='CT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Dermoscopy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Multi-organ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Skin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ViT: Supervised ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1 Synapse [135] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2 ISIC 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2018 [166,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 177] 3 PH2 [178] Dice Hausdorff distance 2022 Encoder TransUNet [24] CT MRI Multi-organ 2D ViT: Supervised 1 Synapse [135] 2 ACDC [176] Dice Hausdorff distance 2021 TransBTS [140] MRI Brain 3D \x17 BraTS 19-20 [179,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 181] Dice Hausdorff distance 2021 TransFuse [141] Colonoscopy Multi-organ 2D& 3D ViT: Supervised 1 Synapse [135] 2 ACDC [176] Dice Hausdorff distance 2021 MedT [142] Microscopy Ultrasound Multi-organ 2D \x17 1 Brain US (Private) 2 GLAS [182] 3 MoNuSeg [183] F1 2021 UNETR [143] CT MRI Brain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Spleen 3D \x17 1 Synapse [135] 2 MSD [184] Dice Hausdorff distance 2021 Swin UNETR [144] MRI Brain 3D ViT: Supervised BraTS 21 [185] Dice Hausdorff distance 2022 Skip connection CoTr [145] CT Multi-organ 3D \x17 Synapse [135] Dice 2021 HiFormer [37] MRI Dermoscopy Microscopic Multi-organ Skin Cells 2D ViT: Supervised CNN: Supervised 1 Synapse [135] 2 ISIC 2017 [166] 3 SegPC 2021 [186,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 187,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 188] Dice Hausdorff distance 2022 Decoder SegTran [139] Fundus MRI X-Colonoscopy Multi-organ 2D&3D CNN: Supervised REFUGE 20 [189] 1 BraTS 19 [179,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 181] 2 X-CVC [190] 3 KVASIR [191] Dice 2021 Other architectures T-AutoML [146] CT Liver and lung tumor 3D \x17 MSD 2019 [192] Dice the normalized surface distance (NSD) 2021 Cross Teaching [147] MRI Multi-organ 2D \x17 ACDC [176] Dice Hausdorff distance 2021 Self-pretraining with MAE [148] CT MRI X-ray Lung Brain Multi-organ 3D ViT: supervised 1 ChestX-ray14 [106] 2 Synapse [192] 3 MSD 2019 [184] Dice Hausdorff distance 2022 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Discussion and Conclusion This section comprehensively investigates the overview of around 16 Transformer-based models for medical im- age segmentation presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1 to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We provide information on the reviewed segmentation ap- proaches about the architecture type, modality, organ, input size, the pre-trained manner, datasets, metrics, and the year in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Table 4 also lists the methods along with the number of parameters, contributions, and highlights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ViT- based works offer solutions in a broad range of multimodal tasks of 2D or 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Most of the approaches demonstrate superior results over CNN-based segmentation models on benchmark medical datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Despite the state-of-the-art performance Transformer-based networks have achieved, there are some challenges in de- ploying the Transformer-based models at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The first challenge is the high computational burden due to the rela- tively large number of parameters of the Transformer-based models [198].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The reason is that the time and space com- plexity of the attention mechanism is quadratic to the se- quence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For example, the CNN-based models such as U-Net [34] requires 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='7M parameters [142] to reach Dice Score 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='68 [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, TransUNet, which achieves Dice Score 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='48 needs 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='07M [143] parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The re- searchers have to meet the high demand for GPU resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Thus, several novel approaches such as Swin Transformer employed in Swin-Unet [32], volume-based Transformer utilized in nnFormer [137] and efficient self-attention mod- ule in MISSFormer [138] are proposed to simplify the com- putation of the Transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The direction of fa- cilitating the efficiency of models will play a crucial role in future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We also note that most existing meth- ods require pre-training strategies on the ImageNet dataset to obtain the pre-trained weights for the following down- stream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, the natural image datasets and med- ical datasets differ dramatically from one another, which may impact the final performance of extracting the medical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Meanwhile, pre-training leads to high computa- tional costs, which hinders the training of models in prac- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Multiple segmentation networks which can be trained from scratch on the medical dataset are suggested as the so- lutions, such as MISSFormer [138].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We expect more ap- 21 Table 4: A brief description of the reviewed Transformer-based medical image segmentation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The unreported number of parameters indicates that the value was not mentioned in the paper, and the code was unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Method # Params Contributions Highlights Pure Swin-Unet [32] Builds a pure Transformer model with symmetric Encoder-Decoder architecture based on Swin- Transformer block connected via skip connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes patch merging layers and patch expanding layers to perform downsampling and upsampling without convolution or interpolation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The results of extensive experiments on multi-organ and multi-modal datasets show the good generalization ability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Pre-trained on ImageNet rather than medical image data, which may result in sub-optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' nnFormer [137] 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='92M Proposes a powerful segmentation model with an interleaved architecture (stem) based on the empirical combination of self-attention and convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes a volume-based multi-head self-attention (V-MSA) to reduce computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Volume-based operations help to reduce the computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Pre-trained on ImageNet rather than medical image data, which may result in sub-optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MISSFormer [138] Proposes the Enhanced Transformer Block based on the Enhanced Mix-FFN and the Efficient Self- attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes the Enhanced Transformer Context Bridge built on the Enhanced Transformer Block to model both the local and global feature representation and fuse multi-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The model can be trained from scratch without the pretraining step on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Less computational burden due to the novel design of the Efficient Self-attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransDeepLab [33] 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='14M Proposes the encoder-decoder DeepLabv3+ architecture based on Swin-Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes the cross-contextual attention to adaptively fuse multi-scale representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The first attempt to combine the Swin-Transformer with DeepLab architecture for medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A lightweight model with only 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='14M parameters compared with Swin-Unet[32], the original DeepLab model [193] and TransUNet[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Encoder TransUNet [24] 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='07M Proposes the first CNN-Transformer hybrid network for medical image segmentation, which establishes self-attention mechanisms from the perspective of sequence-to-sequence prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes a cascaded upsampler (CUP) which comprises several upsampling blocks to generate the pre- diction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransUNet fully exploits the strong global context encoded from the Transformer and local semantics from the CNN module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It presents the generalization ability on multi-modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The approach allows the segmentation of 2D and 3D medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransBTS [140] Moderate TransBTS: 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='99M Lightweight TransBTS: 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='14M Proposes a novel encoder-decoder framework TransBTS that integrates Transformer with 3D CNN for MRI Brain Tumor Segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The method can model the long-range dependencies not only in spatial but also in the depth dimension for 3D volumetric segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransBTS can be trained on the task-specific dataset without the dependence on pre-trained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransBTS is a moderate-size model that outperforms in terms of model complexity with 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='99M parameters and 33G FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, the vanilla Transformer can be replaced with other variants to reduce the computation complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransFuse [141] TransFuse-S: 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3M Proposes the first parallel-in-branch architecture — TransFuse to capture both low-level global features and high-level fine-grained semantic details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes BiFusion module in order to fuse the feature representation from the Transformer branch with the CNN branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The architecture does not require very deep nets, which alleviates gradient vanishing and feature diminishing reuse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It improves performance by reducing parameters and increasing inference speed, allowing deployment on both the cloud and the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The CNN branch is flexible to use any off-the-shelf CNN network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 4 It can be applied to both 2D and 3D medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MedT [142] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='4M Proposes a gated axial-attention model that introduces an additional control mechanism to the self- attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes a LoGo (Local-Global) training strategy for boosting segmentation performance by simultane- ously training a shallow global branch and a deep local branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed method does not require pre-training on large-scale datasets compared to other transform-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The results of predictions are more precise compared to the full attention model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' UNETR [143] 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='58M Proposes a novel architecture to address the task of 3D volumetric medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes a new architecture where the Transformer-based encoder learns long-range dependencies and the CNN-based decoder utilizes skip connections to merge the outputs of a Transformer at each resolution level with the upsampling part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' UNETR shows moderate model complexity while outperforming these Transformer-based and CNN-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The inference time of UNETR is significantly faster than Transformer-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They did not use any pre-trained weights for the Transformer backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Swin UNETR [144] 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='98M Proposes a novel segmentation model, Swin UNETR, based on the design of UNETR and Swin Trans- formers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The FLOPs of Swin UNETR significantly grow compared to that of UNETR and TransBTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Swin Transformer is suitable for the downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Skip connection CoTr [145] 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='51M Proposes a hybrid framework that bridges a convolutional neural network and a Transformer for accurate 3D medical image segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes the deformable Transformer (DeTrans) that employs the multi-scale deformable self-attention mechanism (MS-DMSA) to model the long-range dependency efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The deformable mechanism in CoTr reduces computational and spatial complexities, allowing the network to model high-resolution and multi-scale feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' HiFormer [37] HiFormer-S: 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='25M HiFormer-B: 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='51M HiFormer-L: 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='52M Proposes a encoder-decoder architecture that bridges a CNN and a Transformer for medical image seg- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes a Double-level Fusion module in the skip connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Fewer parameters and lower computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Decoder SegTran [139] 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='03M Proposes a novel Transformer design, Squeeze-and-Expansion Transformer, in which a squeezed attention block helps regularize the huge attention matrix, and an expansion block learns diversified representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes a learnable sinusoidal positional encoding that imposes a continuity inductive bias for the Trans- former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Compared to U-Net and DeepLabV3+, Segtran has the least performance degradation, showing the best cross-domain generalization when evaluated on the datasets of drastically different characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Other architectures T-AutoML [146] 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='96M Proposes the first automated machine learning algorithm, T-AutoML, which automatically estimates “al- most” all components of a deep learning solution for lesion segmentation in 3D medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposes a new predictor-based search method in a new search space that searches for the best neural architecture and the best combination of hyperparameters and data augmentation strategies simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The method is effectively transferable to different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The applied AutoML alleviates the need for manual design of network structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The intrinsic limitations of AutoML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Cross Teaching [147] Proposes an efficient regularization scheme for semi-supervised medical image segmentation where the prediction of a network serves as the pseudo label to supervise the other network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed method is the first attempt to apply the Transformer to perform the semi-supervised medical segmentation utilizing the unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The training process requires less data cost by semi-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The framework contains components with low complexity and simple training strategies compared to other semi-supervised learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed semi-supervised segmentation still can not achieve the state-of-the-art (SOTA) compared with the fully-supervised approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Self-pretraining with MAE [148] Proposes a self-pre-training paradigm with MAE for medical images where the pre-training process of the model uses the same data as the target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed paradigm demonstrates its effectiveness in limited data scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' proaches to exploring more efficient pre-training strategies or without pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, considering the lim- ited size of some medical datasets, some approaches propose semi-supervised technologies or self-pre-training paradigms to reduce the dataset burden of training or pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Nev- ertheless, the performance is still not comparable to that of fully-supervised models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Designing semi-supervised mod- els with improved accuracy in this direction requires more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Medical Image Reconstruction 3D Medical imaging is a clinical breakthrough and very pop- ular in medical diagnosis and follow-up after treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In Computed Tomography (CT), Single Photon Emission Tomog- raphy (SPECT) and Positron Emission Tomogrpahy (PET), the imaging process relies on ionizing radiation [214, 215], which implies a potential risk for the patient [216].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A non-invasive 3D imaging technique is Magnetic Resonance Imaging (MRI), which does not rely on ionizing radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, image ac- quisition may take longer and confines the patient in a discom- forting narrow tube [217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In order to reconstruct 3D volumet- ric datasets from the acquired data, Medical image reconstruc- tion is one of the essential components of 3D medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The primary objective of 3D image reconstruction is to gener- ate high-quality volumetric images for clinical usage at mini- mal cost and radiation exposure, whilst also addressing poten- tial artifacts inherent to the physical acquisition process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Image reconstruction solves an inverse problem that is generally chal- lenging due to its large-scale and ill-posed nature [218].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In medical imaging, there are ongoing research efforts to reduce the acquisition time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' to reduce cost and potential movement artifacts) as well as radiation dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, lower- ing the radiation dose results in higher noise levels and reduced contrast, which poses a challenge for 3D image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Vision Transformers (ViTs) have effectively demonstrated possible solutions to address these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We categorize the literature in this domain into low dose enhancement, sparse- view reconstruction, undersampled reconstruction, and super- resolution reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This section will overview some of the SOTA Transformer-based studies that fit into our taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 22a and Figure 22b demonstrate our proposed taxonomy 22 ViT Encoder Transformer Decoder ViT Encoder .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' UNETR Decoder Transfer Weights MAE Self Pre-training UNETR Segmentation z3 z6 z9 z12 [mask token] [patch feature] Figure 21: Illustration of MAE self pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' First, MAE is pre-trained as an encoder for ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' the ViT encoder is fed with a random subset of patches and the decoder of the Transformer reconstructs the complete image as shown on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then, the pre-trained ViT weights are transferred to the initialized segmentation encoder, as shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Finally, the whole segmenta- tion network, such as UNETR, is fine-tuned to perform the segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [148].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' for this field of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 22a indicates the diversity of our taxonomy based on the medical imaging modalities we studied in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 22b endorses the usage of the Trans- former within the overviewed studies’ pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Low Dose Enhancement Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [199] used a very general intuition about image denoising: the noisy image constructed with high-frequency and low-frequency counterparts as X = XH + XL in a study, namely, TransCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [199] claim that the noisy im- age’s low counterpart contains two sub-components of main image content and weakened image textures, which are entirely noise-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They applied a Gaussian filter on the input image to decompose an image into a high-frequency sub-band and a low-frequency sub-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' After this, they extracted XLc con- tent features and XLt latent texture features by applying two shallow CNNs on the low-frequency counterpart of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Simultaneously, they applied a sub-pixel layer on a high-frequency counterpart to transform it into a low-resolution image and extracted embedding features (XHf ) by applying a shallow CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then the resultant latent texture features (XLt) and corresponding high-frequency representation are fed to the Transformer for noise removal from a high-frequency repre- sentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Ultimately, they reconstruct the high-quality image piecewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They showed that the latent texture features are ben- eficial in screening noise from the high-frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Despite the TransCT [199], Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' proposed a convolution-free Token-to-Token vision Transformer-based Encoder-decoder Dilation network (TED-net) design for CT image denoising [200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Their approach is based on a U-Net encoder-decoder scheme enriched by different modules, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', Basic Vision Transformer, Token-to-Token Dilation (T2TD), and (Inverse) Cyclic Shift blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Consider y ∈ RN×N a clean natural dose CT image, x ∈ RN×N noisy low dose CT im- age, and T : RN×N → RN×N is a Transformer-based denois- ing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' According to the Figure 23 after tokenization of x and passing through the Vision Transformer block to capture long-range dependencies and alleviate the absence of local in- ductive bias in Transformers, they employed Token-to-Token serialization [219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Also, they utilized feature re-assembling with a Cyclic Shift block (CSB) to integrate more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Obvious from Figure 23, all of these blocks are replicated in a symmetric decoder path, but instead of the CSB, the Inverse Cyclic Shift block (ICSB) is implemented to avoid pixel shifts in the final denoising results (y = x + T(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They reached SOTA results compared to CNN-based methods and a compet- itive benchmark with regard to the TransCT [199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Luthra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [201] proposed a Transformer-based network, Eformer, to deal with low-dose CT images while concurrently using the edge enhancement paradigm to deliver more accu- rate and realistic denoised representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Their architecture builds upon the LeWin (Locally-enhanced Window) Trans- former block [220], which is accompanied by an edge enhance- ment module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The success of the Swin Transformer [57] in capturing the long-range dependencies with the window-based self-attention technique makes it a cornerstone in designing new Transformer blocks due to its linear computational com- plexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' LeWin Transformer is one of these blocks that capture the global contextual information and, due to the presence of a depth-wise block in its structure, could also capture a local con- text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Eformer’s first step is through the Sobel edge enhancement filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In every encoder-decoder stage, convolutional features pass through the LeWin Transformer block, and downsampling and upsampling procedures are done by convolution and de- convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Eformer’s learning paradigm is a resid- ual learning scheme, meaning it learns the noise representation rather than a denoised image due to the ease of optimization in predicting a residual mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Akin to Low Dose CT (LDCT), Low-Dose PET (LDPET) is preferable to avoid the radiation risk, especially for cancer patients with a weakened immune system who require multi- ple PET scans during their treatment at the cost of sacrific- ing diagnosis accuracy in Standard-Dose PET (SDPET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [202] proposed an end-to-end Generative Adversarial Network (GAN) based method integrated with a Transformer block, namely 3D Transformer-GAN, to reconstruct SDPET images from the corresponding LDPET images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To alleviate the inter-slice discontinuity problem of existing 2D methods, they designed their network to work with 3D PET data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Analogous to any GAN network, they used a generator network, encoder- decoder, with a Transformer placed in the bottleneck of the generator network to capture contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Due to the computational overhead of Transformers, they did not build their proposed method solely on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Therefore, they were sat- isfied to place a Transformer counterpart across CNN layers of the generator to guarantee to extract low-level spatial feature extraction and global semantic dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They also intro- duced adversarial loss term to their voxel-wise estimation error to produce more realistic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In contradiction with other works, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [203] pro- posed leveraging the PET/MRI data simultaneously for denois- ing low-count PET images, which is a crucial assessment for cancer treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' PET scan is an emission Computed Tomog- raphy (CT) operating by positron annihilation radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Due to 23 Medical Image Reconstruction Computed Tomography (CT) Low Dose Enhancement 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransCT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TED-net 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='- Eformer 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 3D Transformer- GAN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' STFNet 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CTformer 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SIST Sparse-View Reconstruction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' DuDoTrans 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' FIT 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CTTR 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ARMLUT Magnetic Resonance Imaging (MRI) Undersampled Reconstruction 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ViT-Rec 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' T2Net Super Resolution Reconstruction 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' DisCNN-ViT 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Cohf-T (a) Taxonomy structure for medical image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Methods in this field are categorized by their functionality in addressing issues in consensus imaging modalities, not how the Transformer is integrated with the architecture like in the previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The prefix numbers in the paper’s name in ascending order denote the reference for each study as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [199], 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [200], 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [201], 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [202], 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [203], 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [204], 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [205], 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [206], 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [207], 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [208], 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [209], 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [210], 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [211], 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [212], 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [213].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Modification Stage Presence in Architecture Design Encoder TransCT (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', 2021e) STFNet (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', 2022) SIST (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', 2022) DuDoTrans (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', 2021a) CTTR (Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', 2022a) Pure TED-net (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', 2021b) Eformer (Luthra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', 2021) CTformer (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', 2022a) FIT (Buchholz and Jug, 2021) ViT-Rec (Lin and Heckel, 2021) Skip Connection Cohf-T (Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', 2022) Other Architectures 3D T-GAN (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', 2021b) ARMLUT (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', 2022) T2Net (Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', 2021) DisCNN- ViT (Mahapatra and Ge, 2021) (b) We also presented a second taxonomy due to the presence of the Transformer in the reviewed studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 22: An overview of medical image reconstruction taxonomies either as categorizing by the task or the location of using Transformer in an architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 23: An overview of TED-net [200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Tokenize and DeToken blocks are invertible operations that apply the process of patch embedding and converting patches again to image, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TB represents a standard Transformer block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (I)CSB denotes the (inverse) Cyclic Shift block to modify the feature map, nonetheless, the reverse operation avoids pixel shifts in the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' T2T block represents the Token-to-Token process [219] to improve the spatial inductive bias of Transformers by merging the neighboring tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Dilated T2T (T2TD) block is used to refine contextual information further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' the foundation and requirements of PET scans, there is a severe risk of getting infected with secondary cancer by radiotracers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' So to degrade the side effects of this imaging process, there are two potential methods: reduction in radiotracer dose and lessening the patient’s bedtime duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The aforementioned approaches, without a doubt, affect the imaging result quality with decreased contrast to noise ratio and bias in texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The traditional low-count PET denoising approaches are based on Non-Local Means (NLM) [221], Block Matching 3D (BM3D) [222], and Iterative methods [223], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', which are firmly in bond with hyperparameter tuning for new data or result in un- natural smoothings over denoised images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [203] testify that simultaneous PET/MRI could boost one modality in terms of correct attenuation, motion, and partial volume effects, and also, due to the high contrast among soft tissues in MRI, the denoising process of PET images is preferably straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' STFNet [203] is a U-Net based structure with different medica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They proposed a new Siamese encoder comprising dual input flow for each modality in the encoding path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To obtain sufficient features from different modalities, they used the Spa- tial Adaptive (SA) block, a dual path in each block with the residual block design, which consists of different consecutive convolutional blocks and deformable convolution with fusion modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This module aims to learn more contextual fea- tures from each modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To leverage global attention, they used a Transformer to produce a pixel-to-pixel interaction be- tween the PET and the MRI modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' After this integration, the fused features are input to the two branches based on residual convolution blocks for PET denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [204] proposed the enhancement for their previ- ous work, TED-net [200] convolution-free, solely Transformer- based network, namely CTformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' From Figure 24, it is ap- parent that their network is an unsupervised residual learning, U-Net-like encoder-decoder structure, rather than direct map learning of LDCT to Natural Dose CT (NDCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The CTformer tries to compensate for the Transformers’ deficiency in cap- turing path inter-boundary information and spatial inductive bias with token rearrangement, T2T [219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To do so, analo- gously like TED-net, they used dilation and cyclic shift blocks in the Token2Token block to broaden the receptive field to cap- ture more contextual information and not increase the compu- tational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [205] were inspired by how sinogram works and proposed Singoram Inner-Structure Transformer (SIST) (Fig- ure 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This inner structure of the sinogram contains the unique characteristics of the sinogram domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To do so, they mimic the global and local characteristics of sinogram in a loss function based on sinogram inner-structure, namely Sino- gram Inner-Structure Loss (SISL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The global inner-structure loss utilizes conjugate sampling pairs in CT, and local inner- 24 64 x 64 512 × 512 Tokenize DeToken TB TB CSB ICSB T2TD T2TD TB TB CSB ICSB T2T T2T TB512 × 512 64 × 64 CTformer Module A TB TB DeTokenization TB TB Encoder Decoder TB Tokenization 29×29 25×25 29×29 25×25 CTformer Module D CTformer Module C CTformer Module B T2TD IT2TD T2TD IT2TD 512 × 512 64 × 64 Figure 24: An overview of CTformer [204].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This structure is analogous to the TED-net [200] structure, the previous study by the same authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Table 5: Comparison result on NIH-AAPM-Mayo [224] dataset in low dose enhancement task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' LDE indicates the Low Dose Enhancement task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Methods Task Dataset SSIM ↑ RMSE ↓ � Eformer [201] LDE NIH-AAPM-Mayo [224] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='9861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='0067 � TransCT [199] LDE NIH-AAPM-Mayo [224] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='923 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='123 � CTformer [204] LDE NIH-AAPM-Mayo [224] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='9121 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='0233 structure loss considers the second-order sparsity of sinograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The amalgamation of these two terms could be beneficial in re- constructing NDCT images while retaining the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Due to the CT imaging mechanism, each row of the sinogram repre- sentation denotes the projection at a certain view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Naturally, this procedure is suitable for leveraging the Transformer block for modeling the interaction between different projections of di- verse angles to capture contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Therefore, the SIST module applies to raw sinogram input and captures struc- tural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Afterward, the unified network reconstructs the high-quality images in a residual policy with the image re- construction module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Table 5 represents the benchmark results in the LDCT task over the NIH-AAPM-Mayo [224] dataset respecting SSIM and RMSE metrics on overviewed methods in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For clar- ification, TED-net [200] achieved better results than CTformer [204], but due to two studies originating from the same au- thors and the resemblance between architectures, we preferred to mention CTformer to count in the comparison table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This result endorses the capability of the pure Transformer-based Eformer [201] method in reconstructing natural dose CT im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Head Tail Split 𝑆𝑙𝑑 𝑠𝑙𝑑 1 𝑠𝑙𝑑 𝑃 … Multi-Head Self-Attention Add & Norm MLP Add & Norm Image Reconstruction Module Total Loss Noise Loss Image Loss Conv1D Residual Conv Conv Conv Conv Conv Conv 𝑆𝑙𝑑 𝐼𝑙𝑑 መ𝑆𝑛𝑜𝑖𝑠𝑒 መ𝑆 Minus Minus 𝐼𝑙𝑑 መ𝑆 Image Loss Noise Loss መ𝐼𝑛𝑜𝑖𝑠𝑒 መ𝐼 Sinogram Transformer Module Inner-Structure Loss 𝑆𝑙𝑑 Sinogram Loss ×N Figure 25: The overall architecture of SIST [205] pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' S ld and Ild are the LDCT sinogram and image, ˆS and ˆI denote the output sinogram and im- age, ˆS noise and ˆInoise are the sinogram noise and image noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' First, the LDCT sinogram feed to the Transformer for sinogram domain denoising, then the de- noised sinogram ˆS input to the image reconstruction module for image domain denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Within the image reconstruction module, the sinogram noise ˆS noise with the usage of residual CNN block generates image domain ˆInoise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' NDCT ˆI, outputs from applying refinement steps on Ild minus ˆInoise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Sparse-View Reconstruction Due to the customary usage of CT images in medical diag- nosis, another policy to lessen the side effects of X-ray radia- tion is acquiring fewer projections, known as sparse-view CT, which is a very feasible and effective method rather than ma- nipulating the standard radiation dose [225, 226].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, the resultant images from this method suffer from severe arti- facts, and decreasing the number of projections demands pro- found techniques to reconstruct high-quality images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [206] is the first paper that inspected the usage of Trans- formers in this field which was quite successful, namely DuDo- Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Their intuition was to shed light on the globality nature of the sinogram sampling process, which the previous CNN ar- chitectures neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' DuDoTrans, unlike the conventional iter- ative methods in this literature, does not provide blocky effects in reconstructed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This method simultaneously benefits from enhanced and raw sinogram streams to restore informa- tive sinograms via long-range dependency modeling in a super- vised policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' DuDoTrans from Figure 26 is built on three main modules, namely Singoram Restoration Transformer (SRT), the DuDo Consistency layer, and the Residual Image Reconstruc- tion Module (RIRM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SRT block consists of successive hybrid Swin Transformer modules and convolutional layers to model local semantic features and inherent global contextual informa- tion in the sinogram to produce the enhanced sinogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Buchholz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [207] presented the Fourier Image Trans- former (FIT) that operates on the image frequency representa- tion, especially the Fourier description of the image, which in their study is known as Fourier Domain Encoding (FDE), that encodes the entire image at lower resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The intuition in their idea is underlying the CT’s acquisition process physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CT utilizes a rotating 1D detector array around the patient body to calculate the Radon transform [227] of a 2D object,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' which leads to a sequence of density measurements at different pro- jection angles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' namely sinogram as a 2D image in which each column of this representation corresponds to one 1D measure- 25 Residual Image Reconstruction Module(RIRM) Shallow Layer Deep Feature Extraction Layers Recon Layer DuDo Consistency Layer FBP Norm SW-MSA Norm MLP Input Output Swin-Transformer Module (STM) Conv STM … STM Conv Conv Conv … STM … STM Conv Sinogram Restoration Transformer (SRT) Figure 26: DuDoTrans [206] framework for sparse-view CT image reconstruc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' First, the sparse-view sinogram Y maps to a low-quality image �X1 and other estimation �X2 generated by SRT module’s enhanced sinogram output �Y followed by DuDo Consistency Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Lastly, the predicted estimations are concatenated and fed to the RIRM module that outputs the CT image of �X in a supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Filtered Back Projection (FBP) [228, 227] is a re- construction method to map sinograms to tangible CT images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' FBP is based on the Fourier slice theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' hence, computing the 1D Fourier transform of 1D projection and rearranging them by their projection angle in Fourier space, followed by an in- verse Fourier transformation, results in a reconstructed 2D CT image slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Limiting the number of projections leads to miss- ing Fourier measurements, which ultimately conduce to recon- struction artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' FIT is the first study that uses a Transformer to query arbitrary Fourier coefficients and fill the unobserved Fourier coefficients to conceal or avoid the probable artifacts in reconstruction within sparse-view CT reconstruction litera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' From Figure 27 this procedure starts with calculating the FDE of the raw sinogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To do so, first, the discrete Fourier transform (DFT) of the sinogram will be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Secondly, after dropping half of the coefficients on the Fourier rings of the resultant Fourier representation, it preserves the lower fre- quency counterparts to recover the lower resolution of the raw sinogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Afterward, the complex coefficients convert into 1D sequences by unrolling the Fourier rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These complex values convert to normalized amplitudes and phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Therefore, each complex coefficient has its own polar representation, which is a normalized real-valued matrix with N × 2 entries (N is equal to half of the DFT coefficients number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A linear layer applies on this tensor to upsample the feature dimensionality to F 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Fi- nally, a 2D positional encoding concatenates to this tensor and produces a 2D FDE image with the size of N × F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [208] presented a CT reconstruction network with Transformers (CTTR) for sparse-view CT reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In contrast to DuDoTrans [206], CTTR enhances low-quality reconstructions directly from raw sinograms and focuses on global features in a simple policy in an end-to-end architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CTTR contains four parts: two CNN-based residual blocks ex- tracting local features from FBP [228] images reconstruction and sinograms, an encoder-decoder Transformer for long-range modeling dependencies, and contextual information between features, and a CNN block to map features to a high-quality reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Cone-Beam Computed Tomography (CBCT) is a conven- tional way of dental and maxillofacial imaging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' due to its fast 3D imaging qualifications, its popularity has extended to lung imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, studies approved that its radiation dose is higher than plain radiographs [230] hence sparse-view CBCT FC-Loss MSE-Loss 1D FFT 2D iFFT Encoder Decoder 2D FFT 2D FFT FBP Figure 27: FIT [207] framework for sparse-view CT reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' FDE rep- resentation of the sinogram calculates that serves as an input to an encoder of Transformer design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The decoder predicts the Fourier coefficients from the encoder’s latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Fourier coefficients of applying the FBP [228] al- gorithm on sinogram information are fed into a Transformer’s decoder to enrich the Fourier query representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A shallow CNN block applies after inverse FFT to hamper the frequency oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' could be a suitable method to lower radiation dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [209] proposed a novel untrained 3D Transformer-based architecture, namely ARMLUT, with a multi-level loss func- tion for CBCT reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' While the Transformer mod- ule, especially the UNETR [143] in this study, captures long- range contextual information and enhances the resulting image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The intuition behind this strategy is Deep Image Prior (DIP) [231] to succeed in the reconstruction field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' From Figure 28a, ARMLUT is an iterative optimization problem between the Im- age Reconstructor module and Image Generator module to fit a CBCT inverse solver without a large number of data or ground truth images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The multi-level loss function comprises a Mean Squared Error (MSE) and Perceptual Loss (PL) [232] to recon- struct smooth and streak artifact-free outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The entire frame- work (Figure 28) has three main counterparts: Image Recon- structor, Image Generator, and Feature Extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Image Re- constructor uses Feldkamp-Davis-Kress (FDK) algorithm [229] to produce a low enhanced reconstruction from M-view mea- surements, and the Image generator module maps the noisy voxel inputs to reconstruct a regularised image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Feature Extractor module applies the VGG-11 pre-trained network on two representations and produces a PL paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To minimize the distance between these two reconstructions, ARMLUT uti- lizes an adaptively re-weight multi-loss technique to stabilize the convergence of the Transformer in the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Undersampled Reconstruction Magnetic Resonance Imaging (MRI) is a dominant technique for assistive diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, due to the physics behind its operation, the scanning time can take longer and be very te- dious, affecting the patient experience and leading to inevitable artifacts in images [241].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Hence, reducing the number of MRI measurements can result in faster scan times and artifacts re- duction due to the patient’s movement at the cost of aliasing artifacts in the image [217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [210] proposed a comprehensive analytical study to investigate the usage of ViT in a pure (CNN-free modules) and most straightforward Transformer design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This study is ev- idence of the prominent effect of ViTs in medical image recon- 26 (a) ARMLUT [209] pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The collaboration of three distinct modules—FDK algorithm [229], prior embedding with Transformer, and VGG-11 network for extracting hierarchical features—in this pipeline generates the reconstructed CBCT image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Red texts in the figure denote the variable weights that contribute to the iterative optimization step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (b) UNETR [143] used as an image generator module in the ARMLUT paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 28: (a) represents multi-loss untrained network for sparse-view CBCT reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (b) architecture of UNETR [143], as a Transformer module in a ARMLUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For this work, they adopted the original ViT [22] for image reconstruction by discarding the classification token and replacing the classification head with the reconstruction head, which is comprised of successive Norm and Linear layers to map the Transformer output to a visual image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They performed a complete ablation study with different ViT settings, from the number of stacked Transformers to embedding dimension and number of heads in Multi-Head Self-Attention (MHSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Their results were quite effective and proved that trained ViT on suf- ficient data from natural images like ImageNet or medical data could perform better or achieve on-par reconstruction accura- cies compared to CNN baselines such as U-Net [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The pro- posed design’s distinguished power based on the mean attention distance metric [242] proves that it effectively mimics the con- volutional receptive fields and could concurrently capture local and global dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, they showed that the ViT benefits from two times faster inference times and fewer mem- ory requirements compared to the U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [211] address the particular issue in this do- main by designing an end-to-end multi-task learning paradigm to boost feature learning between two sub-tasks, MRI recon- struction, and super-resolution, which have a high overlap with each other named Task Transformer Network (T2Net) is showed in Figure 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Their network consists of two branches, each for a specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' T2Net utilizes a Transformer between two branches to share feature representation and transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' T2Net applies a convolution layer and EDSR [243] backbone to extract task-specific features in each task branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To share information between two branches and benefit from the inter- actions of these two task-specific features concerning the na- ture of the Transformer’s globality, T2Net uses a unique Trans- former design to learn a generalized representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Since the reconstruction branch has more potential in artifact removal ca- pacity than the super-resolution branch, the task Transformer module guides the super-resolution branch into high-quality representation from the reconstruction branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Trans- former module inherits the query (Q: from super-resolution branch), key (K: from reconstruction branch), and value (V: from reconstruction branch) from each scale’s two branches’ output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It forms three main concepts: relevance embedding, Transfer attention, and soft attention, which differ from the original Transformer blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Relevance embedding tries to en- close the correlated features from the reconstruction branch to the super-resolution branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transfer attention aims to transmit the anatomical and global features between two branches, and last but not least, soft attention amalgamates features from the previous two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Ultimately, this module lets the whole net- work transfer and synthesize the representative and anatomical features to produce a high-quality, artifacts-free representation from highly undersampled measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The experimental results on two datasets expressed the high potential of this ap- proach rather than conventional algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Super-Resolution Reconstruction Improving the resolution of images leads to the more detailed delineation of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Increasing the medical image resolu- tion plays a crucial role in computer-aided diagnosis due to its rich anatomical and textural representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Based on the aforementioned fact and the MRI’s pipeline physics during the image acquisition process for having high-resolution images, a patient needs to lie a long time in the MRI tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Hereupon lower signal-to-noise ratio and more minor spatial coverage drawbacks are inevitable [241].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Therefore in this section, we investigate Transformer-utilized algorithms that try to alleviate this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Of note, due to the analogy between MRI and super-resolution reconstruction, some studies investigate these two tasks in conjunction with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [212] proposed the GAN-based model with structural and textural information preservation done by mul- tiple loss function terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Their pipeline included two pre- trained modules named feature disentanglement module, a con- ventional autoencoder, and a Transformer-based feature en- coder, UNETR [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' UNETR captures the global and local context of the original low-resolution image and induces the 27 Image Feature VGG-11 Reconstructor Extractor FDK DSConv X PL f1() f2() f9() f10() W1 MSE W2 MSE MSE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' W10 MSE f1(X) f2(X) fg(X) f10 (X) Image Generator VGG-11 H Transformer DSConv Ge b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='randomPatch Extraction 8000 0000 Patch Flatten Linear Projection 2×2×2 3×3×3 Deconvolution Convolution Embedding 1×1×1 Batch Norm + ReLU Convolution 1st TB Layer Multilayer Multi-Head Norm Perceptron Attention 3rd TB 6th TB 9th TB 2nd TB 12th TB Concatenation Other operations Transformer Block (TB)Table 6: Medical Image Reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' LDE, SVR, USR, and SRR stand for Low Dose Enhancement, Sparse-View Reconstruction, Undersampled Reconstruc- tion, and Super Resolution Reconstruction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' † indicates that this network uses a pre-trained perceptual loss (loss network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Method Task(s) Modality Type Pre-trained Module: Type Dataset(s) Metrics Year Pure TED-net [200] LDE CT 2D \x17 NIH-AAPM-Mayo Clinical LDCT [224] SSIM RMSE 2021 Eformer [201] LDE CT 2D \x17† NIH-AAPM-Mayo Clinical LDCT [224] PSNR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SSIM RMSE 2021 CTformer [204] LDE CT 2D \x17 NIH-AAPM-Mayo Clinical LDCT [224] SSIM RMSE 2022 FIT [207] SVR CT 2D \x17 LoDoPaB [233] PSNR 2021 ViT-Rec [210] USR MRI 2D Supervised fastMRI [234] SSIM 2021 Encoder TransCT [199] LDE CT 2D \x17 1 NIH-AAPM-Mayo Clinical LDCT [224] 2Private clinical pig head CBCT RMSE SSIM VIF 2021 STFNet [203] LDE PET MRI 2D \x17 Private Dataset RMSE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' PSNR SSIM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' PCC 2022 SIST [205] LDE CT 2D \x17 1LDCT Dataset [235] 2Private dataset PSNR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SSIM RMSE 2022 DuDoTrans [206] SVR CT 2D \x17 NIH-AAPM-Mayo Clinical LDCT [224] PSNR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SSIM RMSE 2021 CTTR [208] SVR CT 2D \x17 LIDC-IDRI [236] RMSE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' PSNR SSIM 2022 Skip Connection Cohf-T [213] SRR MRI 2D \x17 1 BraTS2018 [181] 2 IXI [237] PSNR SSIM 2022 Other Architectures 3D T-GAN [202] LDE PET 3D \x17 Private Dataset PSNR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SSIM NMSE 2021 ARMLUT [209] SVR CT 3D ViT: Supervised † 1 SPARE Challenge Dataset [238] 2 Walnut dataset [239] PSNR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SSIM 2022 T2Net [211] USR SRR MRI 2D \x17 1 IXI [237] 2 Private Dataset PSNR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SSIM NMSE 2021 DisCNN-ViT [212] SRR MRI 3D ViT: Self-Supervised 1fastMRI [234] 2 IXI [237] PSNR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SSIM NMSE 2021 high-resolution image to preserve these contexts too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These two modules fine-tune on a different medical dataset, and after- ward, the low-resolution input plus the intermediate generator produced image feed to these modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The disentanglement network contains two autoencoders to learn two counterparts, latent space, structural and textural space, with fed medical im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In an end-to-end setting, these two pre-trained assistive modules help to generate more realistic and structural and tex- tural preserving high-resolution images by imposing module- related loss terms such as adversarial loss to constrain for pro- ducing realistic images and cosine similarity loss for each men- tioned module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Results on the IXI dataset proved that Mahap- atra et al.’s [212] network outperformed a couple of the CNN- based attention mechanism networks and T2Net [211].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Maintaining structural information during the acquiring high-resolution images plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Hence, the struc- ture information is embedded in an image’s high-frequency counterpart, like in an image’s gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, due to the less time-consuming nature of obtaining MR T1WI (T1 Weighted Image), it is wise to use it as an inter-modality con- text prior to producing a high-resolution image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Accordingly, Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [213] devised a network to leverage these two con- cerns in their super-resolution pipeline: Cross-Modality High- Frequency Transformer (Cohf-T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This network is divided into two streams, the first stream is applied on low-resolution T2WI, and the second one manipulates T2WI’s gradient and high- resolution T1WI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Cohf-T module interacts between two streams to embed the prior knowledge in the super-resolution stream’s features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Cohf-T module consists of three differ- ent attention modules: short-distance and long-distance win- dow attention and inter-modality attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The first two atten- tion modules help to model intra-modality dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To be precise, the short-distance window helps recover the local dis- continuity in boundaries with the help of surrounding structure 28 Table 7: A brief description of the reviewed Transformer cooperated in the medical image reconstruction field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Method Contributions Highlights Low Dose Enhancement TransCT [199] The proposed prototype was the first successful implementation of a Transformer complement to CNN in the Low Dose CT reconstruction domain by exploring its revenue within high-frequency and low-frequency counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='The Transformer effectively could learn the embedded texture representation from the noisy counterpart ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='This paradigm is not convolution-free and uses Transformer as a complement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='TED-net ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[200] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Convolution-free U-Net based Transformer model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Introduced Dialted Token-to-Token-based token serialization for an improved receptive field in Transform- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Using Cyclic Shift block for feature refinement in tokenization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Eformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[201] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Incorporate the learnable Sobel filters into the network for preserving edge reconstruction and improve ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='the overall performance of the network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Conduct diverse experiments to validate that the residual learning paradigm is much more effective than ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='other learning techniques such as deterministic learning approaches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Successfully imposed the Sobel-Feldman generated low-level edge features with intermediate network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='layers for better performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='To guarantee the convergence of the network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' they used the Multi-scale Perceptual (MSP) loss alongside ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Mean Squared error (MSE) to hinder the generation of disfavored artifacts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3D T-GAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[202] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='It is a 3D-based method rather than conventional 2D methods ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='First LDPET enhancement study that leveraged from Transformer to model long-range contextual infor- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='mation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='To produce more reliable images with generator they used an adversarial loss to make the data distribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='same as real data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='STFNet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[203] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Proposed a dual input U-Net-based denoising structure for low-count PET images with excessive MRI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='modality contribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Used a Transformer block as a hybrid add-on for feature fusion to make a pixel-to-pixel translation of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='PET and MRI modalities ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='In comparison with the U-Net and residual U-Net structures due to the different training strategy which is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='roughly named Siamese structure has a low computational burden and simplified network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='This network successfully handled the disparity and nonuniformity of shape and modality of PET and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='MRI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='The visual results of denoised images testify that the proposed method could recover the detail of texture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='more clearly than other networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='CTformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[204] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Convolution-free,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' computational efficient design ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Introduce a new inference mechanism to address the boundary artifacts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Proposed interpretability method to follow each path resultant attention map through the model to under- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='stand how the model denoising ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Alleviate receptive filed deficiency with the token rearrangement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='SIST ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[205] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Proposed inner-structure loss to mimic the physics of the functionality of sinogram processing by CT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='devices to restrain the noise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Extracting the long-range dependencies between distinct sinogram angles of views via Transformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Utilizing the image reconstruction module to alleviate the artifacts that could happen in sinogram domain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='denoising by transferring the sinogram noise into the image domain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Image domain loss back-propagates into the sinogram domain for complementary optimization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Sparse-View Reconstruction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='DuDoTrans ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[206] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='To cope with the global nature of the sinogram sampling process introduced,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' the SRT module,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' a hybrid Transformer-CNN to capture long-range dependencies Utilizing a dual domain model to simultaneously enrich raw sinograms and reconstruct ct images with both enhanced and raw sinograms To compensate for the drift error between raw and enhanced sinogram representation employs DuDo Consistency Layer Utilizing a residual learning paradigm for image-domain reconstruction Fewer parameters in comparison with other structures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' DuDoNet [240] with better performance FIT [207] Introduced the Fourier Domain Encoding to encode the image to a lower resolution representation for feeding to the encoder-decoder Transformer for reconstructing the sparse-view CT measurements Introduced the Fourier coefficient loss as a multiplicative combination of amplitude loss and phase loss in the complex domain CTTR [208] Introduced the encoder-decoder Transformer pipeline to utilize dual-domain information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' raw sinogram,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' and primary reconstruction of CT via FBP [228] algorithm for sparse-view CT measurement reconstruction In contrast to DuDoTrans [206],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CTTR directly utilizes raw sinograms to enhance reconstruction perfor- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='mance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ARMLUT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[209] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Proposed a paradigm for CT image reconstruction in a non-trainable manner ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Extending the most DIP research on 2D to 3D medical imaging scenario ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Optimising the large-scale 3D Transformer with only one reference data in an unsupervised manner ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Stabilising the iterative optimization of multi-loss untrained Transformer via re-weighting technique ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Undersampled Reconstruction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ViT-Rec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[210] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='This study investigated the advantage of pure Transformer framework,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ViT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' in fastMRI reconstruction problem in comparison with the baseline U-Net ViT benefits from less inference time and memory consumption compared to the U-Net Utilizing pre-training weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ImageNet extensively improves the performance of ViT in the low- data regime for fastMRI reconstruction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' a widespread concept in the medical domain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ViTs that accompany pre-training weights demonstrate more robust performance toward anatomy shifts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='T•Net ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[211] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Introduce the first Transformer utilized multi-task learning network in the literature ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Designed the task Transformer for maintaining and feature transforming between branches in the network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Outperformed the sequentially designed networks for simultaneous MRI reconstruction and super- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='resolution with T•Net ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Used the same backbone for feature extraction in branches,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' however,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' the purpose of the branches diverse Super Resolution Reconstruction DisCNN-ViT [212] Using a Transformer-based network to capture global contextual cues and amalgamate them with CNN’s local information results in the superior quality of high-resolution images in super-resolution literature Creating realistic images is just not a burden on an adversarial loss function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' in addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' multiple loss functions incorporate extra constraints that preserve anatomical and textural information in the begotten image Multi prerequisite steps are required to train the experiments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' however,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' the super-resolution step is a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='straightforward end-to-end network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Need fine-tuning steps for two disentanglement and UNETR networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='The computational burden of UNETR is high and could use new efficient Transformer designed networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Cohf-T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[213] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Leverage the high-resolution T1WI due to its rich structural information for super-resolving T2-weighted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='MR images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Introduced the high-frequency structure prior and intra-modality and inter-modality attention paradigms ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='within the Cohf-T framework ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Assess prior knowledge into super-resolution paradigm successfully ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Competitive number of FLOPS in reaching SOTA PSNR results in comparison with other attention-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='networks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='End-to-end pipeline for training the network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' and the long-distance window can capture the tex- tural and structural patterns for enhanced results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Due to the discrepancy in intensity levels between T1WI and T2WI, it is vital to make an alignment between these two domains, and Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [213] presented a Feature Alignment (FA) module to reduce the cross-modality representation gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They com- pared their results with T2Net [211] and MTrans [245], which outperformed both approaches by ∼ 1% in terms of PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Discussion and Conclusion In this section, we outline different Transformer-based ap- proaches for medical image reconstruction and present a detailed taxonomy of reconstruction approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We overviewed 15 studies that profit from the Transformer de- sign to compensate for the deficiency of CNN’s limited re- ceptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We investigate each study in depth and repre- sent Table 6 for detailed information about the dataset, uti- lized metrics, modality, and objective tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In Table 7, we provide the main contribution of each study and the promi- nent highlight of each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Most of the studies in this domain use the original Trans- former as a plug-and-play module in their design and only a limited number of studies utilize hierarchical and effi- cient Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, the criteria for using multi- scale and hierarchical architectures are generally important for dense prediction tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' image reconstruction, and should be considered further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Also, another direction to fol- low for future research could be to investigate the influence of using pre-training weights on Transformers due to the need for a large amount of data for better convergence re- sults in Transformers, which contradicts the nature of the medical domain, due to the scarceness of annotated medical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, we noticed that most of the studies focus on MRI and CT image reconstruction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' So there is a need 29 Conv Conv Upsample long skip connection long skip connection Element-wise sum super-resolution branch reconstruction branch ˆxLR x′LR x′ HRB SR1 HRB SR2 HRB SRN HRB Rec1 HRB Rec2 HRB RecN Htt 1 Htt 2 Htt N F 0 SR F 0 Rec F 1 SR F 1 Rec F 2 SR F 2 Rec F N SR F N Rec F 1 T T F 2 T T F N T T (a) An overview of T2Net [244],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' a multi-task learning framework that consists of a super- resolution branch and reconstruction branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The reconstruction branch embraces the stronger capability of artifact removal therefore, the task Transformer module is fed with the reconstruction branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transfer Attention Relevance Embeding Concanate Conv Soft Attention Element-wise multiplication Conv Q K V F i SR F i Rec F i Rec ↑↓ T S C Z FT T (b) inner design of proposed Htt task Transformer module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Analogous to Figure 2, the design of T2 module follows the naive design with some modifications, and in contrast to seminal design, all Q, K, and V entities do not originate from the same representation—Q comes from the super-resolution branch and the rest from the reconstruction branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 29: An overview of T2Net [244].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (a) Multi-Task T2Net pipeline and (b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Task Transformer Module—T2 Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='5xRRDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='5xRRDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='5xRRDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='MLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='MLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='MLP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Cross-modality High-frequency Transformer (Cohf-T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='FA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Cohf-T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='5xRRDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐈𝑖𝑛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐑𝑐 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐅0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐄1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐅1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐅𝑠0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐅𝑐0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐅𝑠1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Cohf-T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='5xRRDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐄2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐅2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐅3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐏1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐄4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐅4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Output Gate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐏4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐏2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ത𝐅𝑠1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐑𝑠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐅𝑐0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Cohf-T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='5xRRDB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐄3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐏3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐅𝑐0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐓1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Input Gate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='addition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='concatenation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='broadcast element-wise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='product ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Sigmoid function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐈𝑜𝑢𝑡 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='𝐑𝑜𝑢𝑡 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Short-distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Window Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Long-distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Window Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Inter-modality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Figure 30: The pipeline of Cohf-T [213] consists of three main branches with ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='the corresponding input modalities as follows: Iin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Rs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' and Rc denote the low-resolution T2WI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' the gradient of low-resolution T2WI and high-resolution T1WI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A fully-convolutional branch for density-domain super- resolution, a Transformer-based branch for restoring high-frequency signals in the gradient domain, and a guidance branch for extracting priors from the T1 modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Conv, RRDB and MLP represent a 3 × 3 convolution operation and residual-in-residual dense block and multi-layer perceptron, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' for evaluating the applicability of these methods on other modalities, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Medical Image Synthesis In this section, we will overview several instances of Trans- formers in the medical image synthesis task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The scarcity of medical data and the high cost of acquisition processes make this task very valuable in the medical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Some studies aim to synthesize missing slices from MRI and CT sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, some methods target capturing the structural infor- mation in diverse modalities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', CT to MRI image-to-image translation and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 31 shows our taxonomy for the image-synthesized methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Intra-Modality The main objective of the intra-modality methods is to syn- thesize high-quality images using low-quality samples from Medical Image Synthesis Intra-Modality 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' PTNet 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ResViT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MMT Inter-Modality 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ResViT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CyTran 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' VTGAN Figure 31: An overview of ViTs in medical image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Methods are categorized by target and source modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The prefix numbers in the paper’s name in ascending order denote the reference for each study as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [246], 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [247], 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [248], 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [249], 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [250].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' the same modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In this respect, several Transformer-based approaches are presented to formulate the synthesis task as a sequence-to-sequence matching problem to generate fine- grained features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In this section, we will briefly present some recent samples [246].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Brain development monitoring is a de facto standard in pre- dicting later risks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' hence it is critical to screen brain biomark- ers via available imaging tools from early life stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Due to this concern and the nature of MRI and subjective infants’ rest- lessness, it is not relatively straightforward to take all the MR modalities during the MRI acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [246] pro- posed a Pyramid Transformer Net (PTNet) as a tool to re- construct realistic T1WI images from T2WI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This pipeline is an end-to-end Transformer-based U-Net-like and multi-resolution structure network utilizing an efficient Transformer, Performer [251], in its encoder (PE) and decoder (PD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Analogously to original U-Net [34], they used skip connection paths for pre- serving fine-grained features and accurate localization features for reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Moreover, the paradigm’s two-level pyra- midal design helps the network capture local and global infor- mation in a multi-resolution fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They achieved the SOTA results on the dHCP [252] dataset compared with the flagship GAN-based image generation method pix2pix (HD) [253, 254].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Dalmaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [247] introduced a conditional generative ad- versarial network based on the cooperation of Transformers and CNN operators, namely ResViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This paradigm addresses the issue of needing to rebuild separate synthesis models for varying source-target modality settings and represents a unified framework as a single model for elevating its practicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The ResViT (Figure 32) pervasively refers to the generator of its pipeline, whereby it leverages a hybrid pipeline of residual con- volutional operations and Transformer blocks that enable ef- fective aggregation of local and long-range dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The discriminator is based on a conditional PatchGAN framework [253].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Utilizing standalone Transformer architectures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', PT- Net [246]) in pixel-to-pixel tasks is quite challenging due to the quadratic complexity, which limits its usage to fixed-size patches that hamper its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' From Figure 32, it is 30 Figure 32: The ResViT [247] framework for multi-modal medical image syn- thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The bottleneck of this encode-decoder comprises successively residual Transformers and residual convolutions layers for synergistically capturing the fine-grained global and local context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' evident that residual Transformer blocks stacked successively, known as aggregated residual Transformer (ART) blocks, in the bottleneck of the encoder-decoder design of the generator to ex- tract the hidden contextual information of input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The primary motivation of ART blocks is to learn an integrated rep- resentation that combines contextual, local, and hybrid local- contextual features underhood from the input flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Channel Compression (CC) module recalibrates the concatenated fea- tures from the previous ART block and Transformer module to select the most discriminant representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Due to the cas- cade of Transformers in design, to decrease the model com- plexity and computational burden, ResViT utilizes weight shar- ing strategy among projection tensors for Query, Key, value, and attention heads besides weight matrices for multi-layer per- ceptron operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The superiority of this method has been proved over several MRI datasets in multi-contrast MRI synthe- sis and MRI to CT experiments with high PSNR and SSIM met- rics over the conventional SOTA methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', pGAN [255], SAGAN [256], pix2pix [253] and PTNet [246].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Likewise, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [248] addressed the issue of missing contrasts in MRI imaging and proposed a multi-contrast multi- scale Transformer (MMT) framework to handle the unavail- ability of this information by synthesizing the existing con- trasts as a means to substitute the missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To achieve efficient contrast synthetization, the task is considered as a seq- to-seq problem, in which the model learns to generate missing contrasts by leveraging the existing contrast in the following manner: A Swin multi-contrast Transformer encoder is imple- mented that creates hierarchical representation from the input MRI image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then, a Swin Transformer-based architecture de- codes the provided representation at multiple scales to perform medical image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Both the encoder and decoder are composed of two sequential swin blocks that capture contrast dependencies effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Conducted experiments on the IXI [237] and BraTS [185] datasets demonstrated MMT’s advan- tage compared to previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Inter-Modality Unlike the intra-modality strategies, the inter-modality meth- ods are designed to learn the mapping function between two different modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This approach allows the network to con- vert the samples from the base modality into a new modality and leverage the generated samples in the training process for the sake of performance gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In this section, we will elaborate on two Transformer-based [249, 250] strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Several medical conditions may prevent patients from re- ceiving intravenous contrast agents while getting CT screen- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, the contrast agent is crucial in assisting medi- cal professionals in identifying some specific lesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' There- fore, CyTran [249] is proposed as an unsupervised generative adversarial convolutional Transformer for translating between contrast and non-contrast CT scans and image alignment of contrast-enhanced CT scans to non-enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Its unsupervised part is also derived from its cyclic loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CyTran is composed of three main modules: I) A downsample CNN-based mod- ule designed for handling high-resolution images, II) A con- volutional Transformer module tailored for incorporating both local and global features, and III) An upsampling module de- veloped to revert the transformation of the downsampling block through transpose-convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Additionally, the authors intro- duce a new dataset, Coltea-Lung-CT-100W, comprised of 100 3D anonymized triphasic lung CT scans of female patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Q K V Multi-head attention Norm layer Pointwise convolution Convolutional projection Convolutional transformer block Upsampling block Downsampling block T Z* Figure 33: An overview illustration of CyTran [249].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' An input image is fed through a downsampling block to extract its features and make it compatible with high-resolution images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The output then passes through a convolutional Transformer block to enrich features by capturing local and global informa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the final step, enriched features are upsampled to the image size using transpose-convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, Kamran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [250] trained a ViT-based gener- ative adversarial network (VTGAN) in a semi-supervised fash- ion on the Fundus Fluorescein Angiography (FFA) dataset pro- vided by [258] via incorporating multiple modules, including residual blocks as generators for coarse and fine image gener- ation, and two ViT architectures consisting of identical trans- former encoder blocks for concurrent retinal abnormality clas- sification and FA image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Discussion and Conclusion This section covers the adoption of ViT architectures in medical image synthesis applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We explored the pro- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='31 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Input Patch Embeddings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Outru ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Layer Norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Downsampler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Multi-Head ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Self-Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Patch Flattening ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Trainable Linear Projection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Residua ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='CNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3x3 conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Positional Encoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Layer Norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ReLU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='BatchNorm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Transformer Encoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3x3 conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Multi-Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='123456789101112 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Perceptron ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Patch Deflattening ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3x3 conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Upsampler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='channe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1x1 conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='compressior ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3x3 conv ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Concatenation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Output Patch EmbeddingsTable 8: An overview of the reviewed Transformer-based medical image synthesizing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Method Concept(s) Modality Type Pre-trained Module: Type Dataset(s) Metrics Year Pure PTNet [246] Intra-Modality MRI 2D \x17 dHCP dataset [252] SSIM PSNR 2021 MMT [248] Intra-Modality Inter-Modality MRI 2D \x17 1 IXI [237] 2 BraTS [185] SSIM PSNR LPIPS 2022 Bottleneck ResViT [247] Intra-Modality Inter-Modality CT MRI 2D ViT: Supervised 1 IXI [237] 2 BraTS [179,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 181] 3 Multi-modal pelvic MRI-CT [257] PSNR SSIM 2021 CyTran [249] Inter-Modality CT 2D 3D \x17 Coltea-Lung-CT-100W [249] MAE RMSE SSIM 2022 Decoder VTGAN [250] Inter-Modality Angiography 2D \x17 Fundus Fluorescein Angiography [258] Fr´echet inception distance Kernel Inception distance 2021 Table 9: A brief description of the reviewed Transformer-based medical image synthesizing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Method Contributions Highlights Pure PTNet [246] Introduced the pure Transformer-based network with linear computational complexity for image- synthesizing context Practical inference time around 30 image/s MMT [248] Proposed a pure Transformer-based architecture that incorporates Swin Transformer blocks to perform missing data imputation by leveraging the existing MRI contrasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Conducted experiments on the IXI and BraTS datasets to perform qualitative and quantitative analysis and confirm their model’s efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Since the attention mechanism can be utilized to pinpoint influential features in the model’s reasoning and decision-making, the attention scores of the Transformer decoder in MMT make it interpretable by capturing information in different contrasts that play an important role in generating the output sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The framework can be applied to a variety of medical analysis tasks, including image segmentation and cross-modality synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Bottleneck ResViT [247] First conditional adversarial model for medical image-to-image translation with hybrid CNN-Transformer generator Introduced a new module, ART block, for simultaneously capturing localization and contextual informa- tion Utilized weight sharing strategy among Transformers to hinder the computational overhead and lessen the model complexity An end-to-end design for the synthesized model that generalizes through multiple settings of source-target modalities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', one-to-one and many-to-one tasks CyTran [249] Proposing a generative adversarial convolutional Transformer for two tasks of image translation and image registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Introducing a new dataset, named Coltea-Lung-CT-100W, comprised of 100 3D anonymized triphasic lung CT scans of female patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The presented method can handle high-resolution images due to its hybrid structure Utilized style transfer techniques to improve alignment between contrast and non-contrast CT scans Decoder VTGAN [250] Proposed a synthesis model for the task of fundus-to-angiogram that incorporates ViT architecture in the decoder section of the system to concurrently classify retinal abnormalities and synthesize FA images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Prepared experimental data based on quantitative and qualitative metrics regarding the model’s general- ization ability under the influence of spatial and radial transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Has the potential to be adopted as a tool for tracking disease progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The system is designed to operate on non-invasive and low-cost fundus data to generate FA images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' posed methods based on two synthesis approaches: (1) inter- modality, in which the target modality is synthesized in a way that it encapsulates crucial diagnostic features from different source images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' and (2) intra-modality, with the objective of yielding target images with better quality by integrating information from lower resolution source im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To demonstrate their effectiveness, these approaches usually rely on SSIM, PSNR, and LPIPS as the evaluation metrics, since they are designed to measure the similarity between images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We also reviewed a ViT-based synthesis model [250] that operates in a decoder fashion for the task of fundus-to-angiogram translation with different evaluation measurements, including Fr´echet Inception Distance (FID) and Kernel Inception Distance (KID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We have additionally provided the architectural type, modality, input size, training setting, datasets, metrics, and year for every medical regis- tration technique analyzed in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, Table 9 lists the contributions and highlights of the proposed works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In particular, with the scarcity of works with ViT implemen- tations and the recent advancement in the medical synthesis field with Transformer models, we believe that these sys- tems require more research effort to be put into them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For example, Transformers have much room for improvement to generate more realism and high-quality synthesized med- ical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' One way to achieve this is by incorporating more detailed anatomy and physiology features using more efficient and effective attention mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Additionally, while much of the current research in this area has focused on 2D medical images and CT and MRI modalities, there is potential to apply these techniques to other types of medical images, including 3D and microscopy images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Medical Image Detection Object detection remains one of the challenging problems in computer vision, especially detection in the medical image do- main has its own challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Current state-of-the-art architec- tures which work on 2D natural images use Vision Transform- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Vision Transformers used in the detection task can be classified into two Transformer backbones and detection Trans- formers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, the Transformer module can be used in a hybrid manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Detection Transformers generally represent an 32 Medical Image Detection Backbone 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TR-Net 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' RDFNet 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CellCentroidFormer Neck 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' COTR 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CT-CAD 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Spine-Transformer Head 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Focused Decoder Figure 34: An overview of Transformers in medical image detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Methods are classified into the backbone, neck, and head according to the positions of the Transformers in their architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The prefix numbers in the paper’s name in ascending order denote the reference for each study as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [259], 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [260], 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [261], 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [145], 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [262], 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [263], 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [264].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' end-to-end detection pipeline with an encoder-decoder struc- ture, while the Transformer backbone solely utilizes the Trans- former encoder for feature refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In order to increase de- tection performance, object detectors combine variants of vi- sion Transformers with classical convolutional neural networks (CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Quite recently, Carion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' introduced the concept of DETR [163], which forms a foundation for Detection Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' DETR uses a ResNet backbone to create a lower-resolution rep- resentation of the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Even though this approach achieves very good 2D detection results, comparable to the R-CNN backbone, high computational complexity is a down- side of this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The deformable DETR [23] approach has improved DETR’s detection performance overcoming the problem of high computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Many recent ap- proaches have tried to improve DETR’s detection concept over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Efficient DETR [265] eliminated DETR’s requirement for iterative refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Conditional DETR [266] introduced the concept of a conditional cross-attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' DN-DETR [267] introduced a denoising strategy, and DINO [268] im- proved many aspects, such as denoising training, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Recently, some studies performed experiments on 2D medical data such as [269], [270] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, only very few attempts tried to adapt it to 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Spine Transformer was proposed by Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [263] for sphere-based vertebrae detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Another approach in 3D detection was proposed by Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [259], which in- troduced a novel Transformer that combines convolutional lay- ers and Transformer encoders for automatically detecting coro- nary artery stenosis in Coronary CT angiography (CCTA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' An approach to better extract complex tooth decay features was proposed by [260].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For end-to-end polyp detection, Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [271] proposed an approach which was based on the DETR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' have proposed the approach CT-CAD [262], context-aware Transformers for end-to-end chest abnormality detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The pros and cons of different approaches are sum- marized in Table 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Table 10, indicates other details such as modalities, organs, datasets, metrics, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Some of the afore- mentioned detection papers in the medical image domain are summarized in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Backbone This section explains Transformer networks using only the Transformer encoder layers for object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The work pro- posed by Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [259] uses a Transformer network (TR-Net) for identifying stenosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A leading threat to the lives of cardio- vascular disease patients globally is Coronary Artery Disease (CAD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Hence, the automatic detection of CAD is quite signif- icant and is considered a challenging task in clinical medicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The complexity of coronary artery plaques, which results in CAD, makes the detection of coronary artery stenosis in Coro- nary CT angiography (CCTA) challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 35: Proposed architecture of TR-Net model [259].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The architecture introduces a Transformer and combines the feature extraction capability of convolutional layers and Trans- former encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TR-Net can easily analyze the semantic in- formation of the sequences and can generate the relationship between image information in each position of a multiplayer reformatted (MPR) image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This model can effectively detect stenosis based on both local and global features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The CNN eas- ily extracts the local semantic information from images, and the Transformer captures global semantic details more easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A 3D-CNN is employed to capture the local semantic features 33 Input 3D-CNN TransformerStructure Output non-significant stenosis Transformer CNN Encoder Transformer CNN Encoder significant stenosis CNN Flattening CCTA max Transformer Encoder Centerline of CoronaryArtery Transformer CNN Encoder XT MPRImage featuremaps orderembeddingfrom each position in an MPR image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' After this step, the Trans- former encoders are mainly used to analyze feature sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The main advantage here is that this helps in mining the depen- dency of local stenosis on each position of the coronary artery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The architecture of the TR-Net is given in Figure 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' One part of the figure indicates the 3D-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This module extracts the local features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The other part indicates the Transformer encoder structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This module associates the local feature maps of each position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This module also helps in analyzing the dependency between different positions, which in turn is helpful for classi- fying the significant stenosis at each position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The CNN part mainly has two main advantages: it prevents the overfitting of semantic information and improves the model’s efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The input to the network architecture is the coronary artery MPR image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The 3D-CNN module has four sequentially connected sub- structures, which consist of a convolutional kernel of size 3 × 3 × 3, a non-linear ReLU layer and a 2 × 2 × 2 max-pooling layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The number of filters is 16 in the first part, and in sub- sequent parts, the number of filters is double the number in the previous part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Since Transformers have 1D vector sequences as input, the feature maps are flattened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Transformer in the proposed architecture consists of 12 Transformer encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Each Transformer encoder mainly consists of two sub-blocks - multi-head self-attention (MSA) and the feed-forward network (FFN), which are connected sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Layer normal (LN) and residual connections are employed before and after two sub-blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In order to ensure the consistency of the encoders, the size of the input is made the same as the size of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The output of the previous encoder is given as input to the next encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the final layer, the embeddings are fed into softmax classifiers to detect significant stenosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' RDFNet approach proposed by Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [260] basically incorporates the Transformer mechanism in order to better ex- tract the complex tooth decay features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The incorporation of the Transformer has improved the detection accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The main three modules of the network are the backbone, neck, and pre- diction modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The backbone module is mainly used to ex- tract the features from caries images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the backbone module, the focus operation is a slicing operation that could easily re- place the convolution operation and reduce the loss of feature information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The C3Modified layer is a convolution module ac- tivated by the FReLU function, which extracts complex visual- spatial information of the caries images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SPP [272] module has a spatial pyramid structure that could expand the percep- tual field, which intern fuses the local and global features and enhance the feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' After the SPP structure, RDFNet appends an improved Transformer-encoder module to improve the feature extraction capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The main functionality of the neck module is to mainly fuse the feature maps of different sizes and extract high-level semantic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This module mainly uses the structure of the feature pyramid network (FPN) pro- posed in [273], and path aggregation network (PAN) proposed in [274].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The FPN approach is employed in a top-down fash- ion, and PAN is performed in a bottom-up fashion to generate the feature pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In order to prevent information loss, fea- ture fusion is performed using both bottom-up and top-down approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' An improved C3Modified convolutional module is adopted into the neck module to better extract the seman- tic features of caries images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The high-level features generated by the neck module are used by the prediction module, which in turn is used to classify and regress the location and class of the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To overcome the problems of the single-stage de- tection method, which has quite a low detection accuracy, it mainly has three detection heads for detecting large, medium, and small objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As Transformers have proved to have strong feature extraction capability, in order to extract complex fea- tures, they utilized the Transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To better extract the features, three Transformer encoders were stacked together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To simplify the model, the authors removed the original normaliza- tion layer from the Transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In order to extract the deep features, the feature map was fed into this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For each head, the attention values were calculated independently and later concatenated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Wagner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [261] proposed a novel hybrid cell detection approach (CellCentroidFormer) in microscopic images that combines the advantages of vision Transformers (ViTs) and convolutional neural networks (CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A CNN model pre- trained on the ImageNet dataset is mainly used for extracting the features and reducing the amount of training data, and the concept of transfer learning is also proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Authors show that the combined use of convolutional and Transformer layers is advantageous as the convolutional layers can focus on the local information (cell centroids), and the Transformer layers can focus on the global information ( overall shapes of a cell).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed centroid-based approach represents the cells as ellipses and is trainable in an end-to-end fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Four differ- ent 2D microscopic datasets were used for experimental eval- uations, and the results outperformed the fully convolutional architecture-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 36 shows the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The encoder is then folded into a 3D tensor, which is afterward Figure 36: Proposed architecture of CellCentroidFormer model [261].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' concatenated with the input tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The MobileViT block is a lightweight alternative to the actual encoder-decoder approach using a Transformer [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Due to the multi-head self-attention layers, the MobileViT block causes a much higher computa- tional complexity than convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To not increase the computational complexity excessively, the MobileViT blocks are combined in the neck part of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Layer 34 384 x 384 x 1 384 x 384 x 3 384 x 384 x 2 Neck Backbone Heads MobileViT Bilinear Layer 2D Block Upsampling Normalization Convolutionnormalization is added for regularization and to allow higher learning rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The backbone module of our proposed model is the EfficientNetV2S [275] CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This block mainly consists of six high-level blocks, out of which five blocks are used to extract image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To use the advantage of trans- fer learning, the backbone module is initialized with weights learned from training on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This, in turn, reduces the amount of required training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The EfficientNetV2S [275] CNN models are generally optimized for a fixed input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Therefore the input images need to be resized to this input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The cells are represented mainly by the centroid, width, and height parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Mainly, two fully convolutional heads are used to predict these cell parameters in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These heads contain 2D convolution, batch normalization, and bilin- ear upsampling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' More MobileViT blocks are not used as it will increase the computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Later convolu- tional layers have a bigger receptive field which helps in cap- turing the global information [276] effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The first convo- lutional head predicts a heatmap for detecting the cell centroids, and the second head is used for predicting the cell dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The output dimensions of this model are 384 × 384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The au- thors use one decoder of the Dual U-Net to predict the centroid heatmap, and the second branch predicts the dimensions of the detected cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The shapes of the cells are focused on by the Transformer layers in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Head Detection Transformers based on Transformer encoder- decoder architecture require a large amount of training data to deliver the highest performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, this is not feasible in the medical domain, where access to labeled data is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To address this problem, for the detection of 3D anatomical struc- tures from the human body, Wittmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [264] proposed a detection Transformer network with a focused decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This network considers the relative position of the anatomical struc- tures and thus requires less training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The focused decoder uses an anatomical region atlas to deploy query anchors to fo- cus on the relevant anatomical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed net- work omits the Transformer encoder network and consists of only Transformer decoder blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The authors show that in 3D datasets, avoiding the encoder can reduce the complexity of modeling relations with a self-attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The model architecture contains a backbone network for fea- ture extraction, a focus decoder network for providing well- defined detection results, a classification network to predict the classes, and a bounding box regression network to output the best possible bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The feature extraction backbone network is a feature pyramid network (FPN) inspired by the RetinaNet [277].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Features from the second layer (P2) are flat- tened before being given as input to the focus decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A spe- cific anatomical region atlas [278] containing regions of interest (RoI) is determined for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then to each RoI, uni- formly spaced query anchors are placed, and a dedicated object query is assigned to each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Such an object query will restrict the focus decoder network to predict solely within their respective RoI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The focused decoder network contains a self-attention mod- ule, a focused cross-attention module, and a feedforward net- work (FFN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The self-attention module encodes strong posi- tional inter-dependencies among object queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The focused cross-attention module matches the input sequence to object queries to regulate the individual feature map for prediction via attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The FFN network then enables richer feature repre- sentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Also, residual skip connections and normalizations are used to increase gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The classification network consists of a single fully-connected layer, and the bounding box regression network consists of three layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The bounding box predictions are combined with query anchors to get the bound- ing box together with class-specific confidence scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The net- work is trained to predict 27 candidates predictions per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Dynamic labeling with the help of generalized intersection over union (GIoU) is created during training to get 27 predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' During inference, the prediction with the highest confidence score indicates the best candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The model is trained end- to-end with the above GIoU loss, binary cross-entropy loss for the classification network, and L1 loss for the bounding box predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Neck Detection methods using region-based approaches need to generate anchor boxes to encode their prior knowledge and use a non-maximum suppression to filter the resulting bounding boxes after prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These pre-and post-processing steps re- markably reduce the detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To bypass these surrogate tasks, Carion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [163] proposed Detection Trans- former (DETR), which views the object detection task as a direct set prediction problem using an encoder-decoder archi- tecture using Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The self-attention mechanism of the Transformers, which explicitly models all pairwise inter- actions between elements in a sequence, helps to predict the set of detections with absolute prediction boxes directly from the image rather than using an anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For the end-to-end de- tection of polyp lesions, Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [271] proposed a convo- lution in Transformer (COTR) network based on the DETR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' COTR consists of 4 main layers: 1) a CNN backbone network used for extracting features, 2) Transformer encoder layers embedded with convolutional layers used for feature en- coding and reconstruction, 3) Transformer decoder layers used for object querying, and 4) a feed-forward network used for detecting prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Embedding convolutional layers into the Transformer encoder layer leads to convergence acceleration compared to the slow convergence of the DETR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The CNN backbone uses a model pre-trained with ResNet18 [63] architecture for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This layer converts input medical images to a high-level feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The authors then use a 1 × 1 convolution to reduce the channel dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the Transformer encoder layers, they used six convolution-in- Transformer encoders to collapse this spatial structure into a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then they use a convolution layer to reconstruct the sequential layer back to the spatial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the encoder layer, each Transformer has a standard architecture with a multi-head self-attention module and a feed-forward network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To the in- put of each attention layer, a positional embedding [21] is also introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the Transformer decoder layers, they used six decoders which follow the standard architecture of the Trans- former except that it also decodes object queries in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 35 Each object query will correspond to a particular object in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The decoders take these object queries with position embeddings as well as output embeddings from the encoder network and convert them into decoded embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then they used a feed-forward network with two fully connected lay- ers for converting these decoded embeddings into object pre- dictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The first fully connected layer is a box regression layer to predict object location, and the second one is a box- classification layer to predict object scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Therefore, the ob- ject queries are independently decoded into box coordinates and classes by the feed-forward network, which results in fi- nal predictions, including object and no object (background) predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This model transforms the object detection prob- lem into a direct set prediction problem by training end-to-end by calculating bipartite matching loss (Hungarian algorithm) between predictions and ground truth for each query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' If the number of queries exceeds the number of objects in the image, the remaining boxes are annotated as no object class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Thus, the model is trained to predict output for each query as an ob- ject or no object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For the class prediction, they used negative log-likelihood loss, and for the bounding box local- ization, they used an L1 loss with generalized intersection over union (GIOU) [279] loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The experiments demonstrated that the proposed model achieved comparable performance against state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Many deep learning detection methods lack using context- relevant information for improved accuracy, and they also gen- erally suffer from slower convergence issues and high compu- tational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed CT-CAD [262], context-aware Transformers for end-to-end chest abnormality detection, ad- dress these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The model consists of two main modules: 1) a context-aware feature extractor module for enhancing the features, and 2) a deformable Transformer detector module for detection prediction and to accelerate the convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The context-aware feature extractor network uses a ResNet50 backbone, dilated context encoding (DCE) blocks, and posi- tional encoding structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The deformable Transformer detec- tor contains a Transformer encoder-decoder architecture and a feed-forward network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed design of the context- aware feature extractor is inspired by the feature fusion scheme from DetectoRS [280] which is based on the Feature Pyramid Networks (FPN) [281].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The feature fusion scheme iteratively enhances the features of the FPN to powerful feature represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Likewise, the DCE blocks enhance the features ex- tracted from the ResNet50 backbone by expanding the recep- tive fields to fuse multiscale context information using dilated convolution filters of different sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This powerful feature map benefits in detecting objects across various scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Inspired by YOLOF [282] the DCE block uses dilated convolution and skip connections to achieve a larger receptive field and acquire more local context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Finally, all the features from differ- ent DCE blocks computed at different scales are summed up to get the feature map for the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed design of the deformable Transformer detector contains single-scale and multi-head attention properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The deformable attention block attends to a small set of key sam- pling points, thus allowing the Transformer to focus on the fea- ture space and accelerate the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The authors used six encoder and decoder layers with positional encoding to ob- tain the decoder outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The outputs from the decoder are the number of abnormalities detected and the dimension of the de- coder layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Finally, a feed-forward network is used to out- put the category classification and location regression results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The model is trained end-to-end with a combination of bound- ing box loss and classification (cross-entropy) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The authors adopted GIoU [283] to balance the loss between large and small object bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The attention module in the detection Transformers com- putes similarity scores between elements of each input data to identify complex dependencies within these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Calcu- lating similarities of all possible positional pairs in the in- put data scales quadratically with the number of positions and thus becomes computationally very expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For this reason, the Transformer-based object detection model from 3D images has never been applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [263] proposed a novel Transformer-based 3D object detection model as a one-to-one set prediction problem for the automatic detection of verte- brae in arbitrary Field-Of-View (FOV) scans, called the Spine- Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Here the authors used a one-to-one set-based global loss that compels a unique prediction for preserving the sequential order of different levels of vertebrae and eliminated bipartite matching between ground truth and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The main modules of the Spine-Transformer are (1) a backbone network to extract features, (2) a light-weighted Transformer encoder-decoder network using positional embeddings and a skip connection, and (3) two feed-forward networks for detec- tion prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The authors used a ResNet50 [63] architec- ture without the last SoftMax layer as the backbone network to extract high-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These features are passed through a 1 × 1 × 1 convolutional layer to reduce the channel dimen- sions and then flattened to get a feature sequence to feed as the input for the Transformer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The light-weighted Trans- former encoder-decoder network contains only a two-layer en- coder and two-layer decoder to balance between feature res- olution and memory constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In both the encoder and de- coder layers of the network, learnable positional embeddings are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The authors found that using a skip connection across the Transformer encoder-decoder network will help in the prop- agation of context and gradient information during training and thus improves performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The two feed-forward networks are then used to predict the existence of the objects and regress their coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The authors also proposed a sphere-based bounding box detector to replace the rectangular-based bound- ing box to introduce rotational invariance called InSphere de- tector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Spine-Transformer is trained end-to-end with fixed- size patch images to predict all the vertebrae objects in parallel by forcing one-to-one matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Binary cross-entropy loss is used as classification loss, and to enforce the order of the pre- dicted vertebrae objects, an edge loss is introduced, which is an L1 distance loss introduced between the centers of the top and bottom neighborhood vertebrae objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For better local- ization accuracy of the bounding sphere detection, the authors used generalized inception-over-union (GIoU) [279] loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The results of this model showed superior results to all the state- 36 Table 10: An overview of the reviewed Transformer-based medical image detection approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Method Modality Organ Type Pre-trained Module: Type Datasets Metrics Year Backbone TR-Net [259] MPR Heart (Coronary Artery) 3D Supervised Private dataset Accuracy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Sensitivity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Specificity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' PPV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' NPV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' F1-score 2021 RDFNet [260] Dental Caries Teeth 2D Supervised Private dataset Precision,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Recall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' mAP@0:5 2021 CellCentroidFormer [261] Microscopy Cells 2D Supervised 1 Fluo-N2DL-HeLa (HeLa) [284] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 2 Fluo-N2DH-SIM+ (SIM+) [284],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 3 Fluo-N2DH-GOWT1 (GOWT1) [284] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 4 PhC-C2DH-U373 (U373) [284].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Mean-IoU, SSIM 2022 Head Focused decoder [264] CT Multi-organ 3D Semi-Supervised 1 VISCERAL anatomy benchmark [285], 2 AMOS22 challenge [286].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' mAP, AP50, AP75 2022 Neck COTR [163] Colonoscopy Colon 2D Supervised 1 CVC-ClinicDB [287], 2 ETIS-LARIB [288], 3 CVC-ColonDB [289].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Precision, Sensitivity, F1-score 2021 CT-CAD [262] X-ray Chest 2D Supervised 1 Vinbig Chest X-Ray dataset [290] 2 ChestXDet-10 dataset [291] AP50 2021 Spine-Transformer [263] CT Vertebra 3D Supervised 1 VerSe 2019 [292], 2 MICCAI-CSI 2014 [293], 3 Private dataset Id-Rate, L-Error 2021 of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The authors also claim that by using a 3D CNN-based landmark regression [299], the localization accu- racy can be further improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Discussion and Conclusion In this chapter, several well-known Transformer architec- tures are analyzed to address the automatic detection chal- lenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Based on the Transformer model contribution to the network structure, we grouped the set of literature work into the backbone, neck, or head strategies and for each cate- gory, we provided sample works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In this respect, the core idea behind each network design along with the pros and cons of the strategies are highlighted in the summary tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Vision Transformers have been shown to make more accu- rate diagnoses compared to traditional methods of analyzing medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These deep learning models can be trained on large datasets, such as ImageNet, and fine-tuned on medi- cal image datasets to improve their performance in detecting abnormalities in X-rays, CT scans, and MRIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' By incorpo- rating information from multiple modalities, Transformers can further enhance their ability to identify and detect rare or subtle abnormalities in medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Many medical images are often taken over time, and incorporating tempo- ral information into the model can improve its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For example, the model can be designed to take into account the temporal evolution of diseases or conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Overall, Transformers have demonstrated their capabilities to signifi- cantly improve the accuracy and efficiency of medical image analysis, leading to advances in healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Medical Image Registration Medical image registration is the task of transforming a set of two or more images of an organ or a biological process taken with different poses, time stamps, or modalities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', CT and MRI) into a geometrically aligned and spatially corresponding image that can be utilized for medical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The transfor- mation can be discovered by solving an optimization problem that maximizes the similarity between the images to be regis- tered [300].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A pair-wise registration of two MRI brain scans is shown in Figure 37 for illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Moving Image (M) Fixed Image (F) Spatial Transformation Registered Image Figure 37: An example of pair-wise medical image registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The goal of image registration is to geometrically align the moving image with the target or fixed image by performing the spatial transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Despite remarkable advancements in the quality of medical imaging techniques that aid professionals in better visualiza- tion and analysis of image data, a prominent challenge prevails 37 Table 11: A brief description of the reviewed Transformer-based medical image detection models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The unreported number of parameters indicates that the value was not mentioned in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Method # Params Contributions Highlights Backbone TR-Net [259] This work is the first attempt to detect coronary artery stenosis more accu- rately by employing Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To detect significant stenosis, local and global features are effectively inte- grated into this approach, which has resulted in more accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' While detecting significant stenosis, the TR-Net architecture is capa- ble of combining the information of local areas near stenoses and the global information of coronary artery branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Compared to state-of-the-art methods, the TR-Net model has better results on multiple indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The shallow CNN layer prevents the overfitting of semantic informa- tion and improves the overall efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The gain in performance comes with a trade-off in the number of pa- rameters, which affects the computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' RDFNet [260] An image dataset of caries is created, which is annotated by professional dentists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For better extraction of the complex features of dental caries, the Transformer mechanism is incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In order to increase the inference speed significantly, the FReLU activation function is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Compared with existing approaches, the accuracy and speed of caries detection are better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Method is applicable to portable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The method does not work really well when the illumination of the oral image is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Even though detection accuracy and speed are improved compared to the original approach, the detection speed is not the fastest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CellCentroidFormer [261] 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='5M A novel deep learning approach that combines the self-attention of Trans- formers and the convolution operation of convolutional neural networks is pro- posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A centroid-based cell detection method, denoting the cells as ellipses is pro- posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Pseudocoloring in combination with pre-trained backbones shows im- proved cell detection performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The model outperforms other state-of-the-art fully convolutional one- stage detectors on four microscopy datasets, despite having a lower number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Larger output strides worsen the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Head Focused Decoder [264] VISCERAL Dataset - 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='8M AMOS22 Dataset - 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='6M First detection Transformer model for 3D anatomical structure detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Introduced a focused decoder to focus the predictions on RoI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Better results compared to existing detection models using a Trans- former network like DETR [163] and deformable DETR [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Comparable results to the RetinaNet [277].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Varying anatomical fields of view (FoVs) can affect the robustness of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Neck COTR [145] Proposed a convolution layer embedded into the Transformer encoder for better feature reconstruction and faster convergence compared to DETR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' COTR has comparable results with state-of-the-art methods like Mask R-CNN [294] and MDeNet-plus [295].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This approach produces low confidence for a particular type of lesion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CT-CAD [262] Proposed a context-aware feature extractor, which enhances the receptive fields to encode multi-scale context-relevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposed a deformable Transformer detector that attends to a small set of key sampling locations and then the Transformers can focus to feature subspace and accelerate the convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CT-CAD outperforms the existing methods in Cascade R-CNN [296], YoLo [297], and DETR [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CT-CAD is capable to detect hard cases, such as nodules that are ig- nored by Faster R-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Compared to the ChestXDet-10 dataset, this model has a lower perfor- mance on the Vinbig Chest X-Ray dataset which has higher categories of abnormalities with more complex patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Spine-Transformers [263] Proposed a 3D object detection model based on the Transformer’s architec- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposed a one-to-one set global loss that enforces unique prediction and preserves the sequential order of vertebrae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposed a Sphere-based bounding box to enforce rotational invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Obtained better results for all datasets compared to state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The model has a higher Id-Rate on both the datasets, but a higher L-Error compared to the benchmark by [298].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' in developing a system capable of effective integration of visual data that captures useful information from original images with high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Most registration procedures take into account the whole image as input by utilizing global information for spatial transformation, which leads to inefficient and slow inte- gration of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, the collection process of medical images for training is slow and toilsome, performance degrades due to the presence of outliers, and local maxima entail neg- ative effects on performance during optimization [301, 302].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The emergence of deep learning methods alleviated these prob- lems by automatic extraction of features utilizing convolutional neural networks (CNN), optimizing a global function, and im- proving registration accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For instance, Balakrishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [303] utilized a CNN to achieve unsupervised deformable reg- istration by treating it as a parametric function to be optimized during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [304] presented an unsupervised CNN-based registration algorithm to produce an- thropomorphic phantoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, there are still limitations in capturing long-range spatial correspondence in CNN-based frameworks [305, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Fueled by the strong ability of Transformers to model long- range dependencies and detect global information [306, 27, 307], they have gained the attention of researchers in the med- ical image registration domain in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In this section, we review Transformer-based methods in medical image reg- istration that ameliorate the aforementioned shortcomings of previous systems by utilizing the self-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We have organized the relevant approaches based on their type of registration: (a) Deformable registration, which employs an optimization algorithm to tune the transformation model, is a way that maximizes the similarity measure function for the images of interest [308];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (b) Rigid registration, which achieves correspondence by maintaining the relative distance between each pair of points between the patient’s anatomy images [308].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 38 Medical Image Registration Rigid 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SVoRT Deformable 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ViT-V-Net 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransMorph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' DTN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' XMorpher Affine 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' C2FViT Figure 38: Taxonomy of Transformer-based image registration based on their transformation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We use the prefix numbers in the figure in ascending order and reference the corresponding paper as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [307], 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [309], 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [310], 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [311], 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [312], 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [313].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (c) Affine registration, which contains the same operations as rigid registration plus non-isometric scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Deformable Registration Most existing Transformer-based algorithms focus on de- formable transformation to perform medical image registra- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Vit-V-Net [309] is the earliest work that incorporates Transformers to perform medical image registration in a self- supervised fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It is inspired by the integration of vision Transformer-based segmentation methods with convolutional neural networks to enhance the localization information recov- ered from the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Unlike previous research that employed 2D images for spatial correspondence, Vit-V-net stepped to- wards utilizing ViT [22] as the first study for volumetric med- ical image registration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', 3D image registration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As illus- trated in Figure 39, the images are first encoded into high-level feature representations by implementing multiple convolution blocks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' then, these features get split into P patches in the ViT block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Next, the patches are mapped to a D-dimensional em- bedding space to provide patch embeddings, which are then integrated with learnable positional encodings to retain posi- tional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Next, these patches are passed into the encoder block of the Transformer, followed by multiple skip connections to retain localization information, and then de- coded employing a V-Net style decoder [314].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Finally, a spa- tial Transformer [315] warps the moving image by utilizing the final output of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransMorph [310] extended ViT-V-Net and proposed a hybrid Transformer ConvNet frame- work that utilizes the Swin Transformer [57] as the encoder and a ConvNet as the decoder to provide a dense displacement field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Like ViT-V-Net, it employed long skip connections to retain the flow of localization information that may enhance registration accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The output of the network, which is a nonlinear warping function, gets applied to the moving image with the deformation field utilizing the spatial transformation function proposed in [315].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' An affine transformation Trans- former network is incorporated to align the moving image with the fixed image before feeding it to the deformable registra- tion network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This work also proposed two variants of Trans- Morph: diffeomorphic TransMorph (TransMorph-diff) to fa- cilitate topology-preserving deformations and Bayesian Trans- Morph (TransMorph-Bayes) to promote a well-calibrated reg- istration uncertainty estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 39: Overview of ViT-V-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Multiple convolution blocks encode images into high-level features, which the Vit block splits into patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These patches are then mapped to D-dimensional patch embeddings that get integrated with learnable positional encodings to retain positional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Next, these patches are passed into the Transformer encoder block, followed by multiple skip connections to retain localization information, and decoded using a V-Net style decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Using the network’s final output, a spatial Transformer warps the moving image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure taken from [309].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Likewise, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [311] introduced the dual Transformer network (DTN) framework to perform diffeomorphic registra- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It is composed of a CNN-based 3D U-Net encoder [299] for the embedding of separate and concatenated volumetric im- ages and a dual Transformer to capture the cross-volume depen- dencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' One of the Transformers is responsible for modeling the inter- and intra-image dependencies, and the other one han- dles the modeling of the global dependencies by employing the self-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The concatenation of the generated features from these Transformers results in enhanced feature embeddings, which are utilized by the CNN-based decoder to provide a diffeomorphic deformation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The evaluation of the framework was conducted on the brain MRI scans of the OASIS dataset [316], which substantiates their improvements in diffeomorphic registration compared to the existing deep- learning-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' XMorpher [312] put emphasis on the signifi- cance of backbone architectures in feature extraction and match of pair-wise images,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' and proposed a novel full Transformer net- work as the backbone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' which consists of two parallel U-Net structures [34] as the sub-networks with their convolutions re- placed by the introduced Cross Attention Transformer for fea- ture extraction of moving and fixed images,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' and cross-attention- based fusion modules that utilize these features for generating the feature representation of moving-fixed correspondence and fine-grained multi-level semantic information that contributes to a fine registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Affine Registration To perform affine medical image registration with Trans- formers, Mok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [313] proposed C2FViT, a coarse-to- fine vision Transformer that performs affine registration, a ge- ometric transformation that preserves points, straight lines, and planes while registering 3D medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Former studies have relied on CNN-based affine registration that focuses on lo- cal misalignment or global orientation [322, 323], which limits the modeling of long-range dependencies and hinders high gen- eralizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' C2FVit, as the first work that takes into account the non-local dependencies between medical images, leverages vision Transformers instead of CNNs for 3D registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=" As 39 (T'MH'2) (16, H, W,L) 1/2 (16, H,W,L) (3, H, W,L) (ze) 1/4 ze) + Transformer (N, D) Moving Image 8t) Image Transformer Conv-up Block Layer Conv." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Block Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Block %) 3D Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Layer 3D Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Layer Target Image Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Block 3D Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Layer Instance Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Concatenation 3D Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Layer Instance Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 3D Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' + Instance Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 3D Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' + Leaky ReLU Leaky ReLU Leaky ReLU Leaky ReLU Leaky ReLU Instance Norm Reshape Leaky ReLU Upsample Vision Conv-up Max-pool Deformed I Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Spatial 3D Conv Block Conv-up Block Loss (L)Table 12: An overview of the reviewed Transformer-based medical image registration approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Modality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Organ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Datasets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Metrics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Deformable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ViT-V-Net ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[309] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='MRI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Brain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Private Dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Dice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='TransMorph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[310] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='MRI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='CT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Brain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Chest-Abdomen-Pelvis region ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1 IXI [237] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2 T1-weighted brain MRI scans from Johns Hopkins University ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3 Chest-Abdomen-Pelvis CT [317] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Dice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='% of |JΦ| ≤ 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='SSIM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='DTN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[311] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='MRI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Brain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='OASIS [316] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Dice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='|JΦ| ≤ 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='XMorpher ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[312] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='CT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='MRI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Heart ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1 MM-WHS 2017 [318] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2 ASOCA [319] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Dice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='% of |JΦ| ≤ 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2022 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Affine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='C2FViT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[313] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='MRI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Brain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1 OASIS [316] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2 LPBA [320] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Dice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Hausdorff distance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2022 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Rigid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='SVoRT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[307] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='MRI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Brain ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='FeTA [321] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='PSNR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='SSIM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2022 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Table 13: A brief description of the reviewed Transformer-based medical image registration techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Method Contributions Highlights Deformable ViT-V-Net [309] Contributed to the medical image registration domain as the first work to exploit ViTs to develop a volu- metric (3D) registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Integrated Transformers with CNNs to build a hybrid architecture for self-supervised brain MRI registra- tion Employed a hybrid architecture to incorporate long-range and local information in the registration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Attempted to preserve the localization data with the help of long skip connections between the encoder and decoder stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' TransMorph [310] Proposed a Transformer-based unsupervised registration approach for affine and deformable objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Conducted experiments on two brain MRI datasets and in a phantom-to-CT registration task to demon- strate their superior performance compared to traditional approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They additionally proposed two distinguishable versions of their model: a diffeomorphic variant to facilitate the topology-preserving defor- mations and a Bayesian variant to promote a well-calibrated registration uncertainty estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Studied the effect of receptive fields by comparing TransMorph with CNNs, and addressed that while the receptive field of ConvNets only increases with the layer depth, their presented model takes into account the whole image at each due to the self-attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' DTN [311] Proposed a dual Transformer architecture to capture semantic correspondence of anatomical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The suggested DTN demonstrated remarkable results in diffeomorphic registration and atlas-based seg- mentation of multi-class anatomical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Dual Transformer is capable of reducing the negative Jacobian determinant while preserving the atlas-based registration quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The qualitative and quantitative analysis of their method on the OASIS dataset indicates that diffeomorphic registration fields are effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' XMorpher [312] Devised a deformable registration system consisting of dual parallel feature extraction networks which facilitate the association of representative features between moving and fixed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Proposed cross-attention Transformer that establishes spatial correspondences through computation of bilateral information in the attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Promotes visual superiority by presenting fine-grained visual results in terms of boundary smoothness, adjacent regions’ resolution quality, and deformation grid polishness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Demonstrated the model’s great diagnostic potential by conducting experiments with different training regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Affine C2FViT [313] Presented a method in order to learn the global affine registration by taking advantage of the strong long- range dependency recognition and locality of the hybrid Transformer and the multi-resolution strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The suggested training framework can be extended to a number of parametric-based registration approaches by removing or scaling the geo- metrical transformation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Rigid SVoRT [307] Devised an approach for the task of Volumetric reconstruction of fetal brains based on Transformer archi- tectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Employed a Transformer network trained on artificially sampled 2D MR slices that estimates the under- lying 3D volume from the input slices to more accurately predict transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Experimental procedures on the FeTA dataset [321] represented the model’s ability in high-quality volumetric reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The volumetric reconstruction associated with the transformations of the proposed method displays higher visual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Figure 40: The model has L stages with convolutional patch embedding lay- ers and N Transformer encoder blocks to learn the optimal affine registration matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In each stage, fixed and moving images are downsampled and con- catenated, then passed to the convolutional patch embedding layer to produce image patch embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The Transformer then produces the input feature em- bedding from the embeddings [313].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' depicted in Figure 40, the model is split into L stages, each con- taining a convolutional patch embedding layer and Ni Trans- former encoder blocks (i indicates the stage number), intending to learn the optimal affine registration matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In each stage, the fixed and moving images are downsampled and concatenated with each other, then the new representation gets passed to the convolutional patch embedding layer to produce image patch embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Next, the Transformer receives the embeddings and produces the feature embedding of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Conducted experiments on OASIS [316] and LPBA [320] demonstrated their superior performance compared to existing CNN-based affine registration techniques in terms of registration accuracy, robustness, and generalisation ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Rigid Registration SVoRT [307] addressed the necessity of slice-to-volume reg- istration before volumetric reconstruction for the task of volu- metric fetal brains reconstruction, and employed a Transformer network trained on artificially sampled 2D MR slices that learns to predict slice transformation based on the information gained from other slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The model also estimates the underlying 3D volume from the input slices to promote higher accuracy in transformation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The superiority of their proposed method in terms of registration accuracy and reconstruction based on the evaluation on synthetic data and their experiments on real-world MRI scans demonstrated the ability of the model in high-quality volumetric reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Discussion and Conclusion According to the research discussed in this section, vision Transformers are prominent tools in image registration tasks 40 Patch Embedding Patch Embedding Convolutional Convolutional Encoder Transformer Transfor Encoder rmer MLP MLP MLP Head Head Headdue to their training capability on large-scale data, which is made feasible by parallel computing and self-attention mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Leveraging Transformers to encourage bet- ter global dependency identification improves registration in terms of dice scores and Jacobian matrix determinant com- pared to CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To mitigate the burden of quadratic complexity when pro- cessing images at high-resolution and modelling local rela- tionships, reviewed studies usually employ CNNs to provide feature maps or dense displacement fields [309, 310, 311].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' C2FViT [313] disregarded convolutional networks and im- plemented convolutional patch embeddings to promote lo- cality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, in deformably registering medical con- tent, XMorpher recently demonstrated the power of cross- attention in better capturing spatial relevancy without a CNN implementation [312], and SVoRT purely utilized Trans- formers to perform rigid registration [307].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The notable experimental attempts on brain MRI scan data, such as OASIS [316] and FeTA [321], show the importance of accurate automatic registration for neuroimaging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' One particular work [312] proposed to evaluate their regis- tration on images of cardiac region datasets including MM- WHO-2017 [318] and ASOCA [319].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To further clarify the modality type used in the aforementioned proposed meth- ods, all works conducted their evaluations on 3D or volu- metric imaging modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Based on the brief review of Transformer-based medical im- age registration research, we believe that other regions of in- terest (ROI) such as neurons, retina, and neck area are worth exploring to facilitate diagnostic operations in different do- mains with more precise registration models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We have also specified the architectural type, modality, or- gan, data size, training paradigm, datasets, metrics, and year for each medical registration technique reviewed in Ta- ble 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, Table 13 provides a list of the contri- butions and highlights of the proposed works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Medical Report Generation Medical report generation focuses on producing comprehen- sive captions and descriptions pivoting on medical images for diagnostic purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Designing automatic methods capable of performing this task can alleviate tedious and time-consuming work in producing medical reports and promote medical au- tomation [324].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Recently, advancements in deep learning have brought the attention of researchers to employing an intelligent system capable of understanding the visual content of an im- age and describing its comprehension in natural language for- mat [325].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Research efforts in improving this area can be em- ployed in medical imaging by implementing systems capable of providing descriptions and captions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', generating medical reports) concerning medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These captioning systems usually utilize encoder-decoder models that encode medical im- ages and decode their understandings to provide diagnostic in- formation in a natural language format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Despite the success of deep learning, limitations including reliability on an immense amount of data, unbalanced data in radiology datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', IU X-ray chest X-Ray [326]), and the black box nature of DL models entail challenges in medical report generation [327].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The success of Transformer models in many vision-and-language tasks has drawn the attention of researchers in the medical report generation domain to the em- ployment of this architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In this section, we discuss ap- proaches that utilize Transformers to promote effective capture of long-range context dependencies and better report genera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As illustrated in Figure 41, the following is our taxonomy of these systems according to the mechanism by which they produce accurate and reliable clinical reports: (a) Reinforcement Learning-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The ultimate goal of a medical report generation system is to provide clinically accurate and reliable reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In reinforcement learning, the MRG system is considered an agent with the objective of maximizing clinical accuracy based on the feedback given by the reward signal, which is directly calculated by the evaluation metric score (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', CIDEr [328]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (b) Graph-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Radiology reports are typically composed of a long finding section with multiple sentences that make report generation a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Therefore, the inclu- sion of prior information is beneficial for facilitating the generation of long narratives from visual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Knowl- edge graphs, which are powerful models that can capture domain-specific information in a structured manner, can be used to exploit prior information for medical report gener- ation [329, 330, 331].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (c) Memory-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Memory is a resource through which im- portant information is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In designing a proper MRG system, it is crucial to store vital and diagnostic in- formation that can benefit the generation process by incor- porating prior knowledge and experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Hence, config- uring a memory mechanism with Transformers as a report generation framework facilitates longer and more coherent text generation by sharing information gained through the process [332, 333].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (d) Other Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Systems that introduce different ideas from previous categories to improve clinical accuracy, such as curriculum learning, contrastive learning, and alternate learning, belong to this group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Reinforcement Learning-based Systems The first work to implement a Transformer architecture for medical report generation is RTMIC [334].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It used the rein- forcement learning strategy in training to mitigate the problem of exposure bias prevailing in Seq2Seq models [358].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In their approach, the original images are fed into a DenseNet [335] as the region detector to extract bottom-up visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These features are then passed into a visual encoder to generate visual representations from the detected regions, which the caption- ing detector then utilizes to generate captions for the specified regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed method was experimented on the IU X-Ray dataset [326] and achieved state-of-the-art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In- tegration of RL and Transformers was also applied in surgical 41 Table 14: An overview of the reviewed Transformer-based Medical Report Generation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Modality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Organ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Visual Backbone ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Datasets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Metrics ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Reinforcement Learning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='RTMIC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[334] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='X-ray ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Lung ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='DenseNet-121 [335] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='IU Chest X-ray [326] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='BLEU [336] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='CIDEr [328] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2019 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='SIG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[337] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Ultrasound ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='X-ray ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Lung ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='DenseNet-121 [335] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1 Chinese Covid-19 CT [350] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2 CX-CHR (private dataset) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='BLUE [336] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='CIDEr [328] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ROUGE [340] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='CDGPT2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[351] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='X-ray ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Lung ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='DenseNet-121 [335] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='IU chest X-ray [326] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='BLUE [336] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Meteor [339] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='CIDEr [328] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ROUGE [340] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='CGI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[352] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='X-ray ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Lung ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='DenseNet-121 [335] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1 MIMIC-CXR [343] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2 IU chest X-ray [326] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='BLUE [336] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Meteor [339] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ROUGE [340] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='CGRG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='[353] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='X-ray ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Lung ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ResNet-101 [63] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1 IU Chest X-ray [326] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2 COV-CTR [354] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='BLUE [336] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='Meteor [339] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='ROUGE [340] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2021 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='instruction generation since the joint understanding of surgical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='activity along with modeling relations linking visual and tex- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='tual data is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [337] employed a Transformer-backboned encoder-decoder architecture and ap- plied the self-critical reinforcement learning [359] approach to optimize the CIDEr score [328] as the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Their approach surpasses existing models in performance on the DAISI dataset [338] with caption evaluation metrics applied to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 42 Table 15: A brief summary of the reviewed Transformer-based medical report generation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Method Contributions Highlights Reinforcement Learning RTMIC [334] Presented a novel Hierarchical Reinforced Transformer for producing comprehensible, informative med- ical reports by training through reinforcement learning-based training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The initial attempt at incorporating Transformers to develop a medical report generation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Utilized reinforcement learning to ameliorate the exposure bias problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Enhanced clinical report coherence by employing Transformers to capture long-range dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The selected metric (CIDEr) as a reward signal is not designed for the medical domain [355].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SIG [337] Generated surgical instructions from multiple clinical domains by utilizing a Transformer-based Encoder- decoder architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed method is able to produce multimodal dependencies, form pixel-wise patterns, and develop textual associations for masked self-attention decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Utilizing self-critical reinforcement learning to perform optimization increased the performance of surgical instruction generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The selected metric (CIDEr) as a reward signal is not designed for the medical domain [355].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Graph KERP [330] Developed an MRG system using a hybrid retrieval-generation technique that unifies standard retrieval- based and recent visual text generation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Introduced Graph Transformer (GTR) as the first research to employ an attention mechanism to convert different data types formulated as a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Aligning the generated reports with abnormality attention maps by providing location reference facilitates medical diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Since KERP is designed based on abnormality detection, it may disregard other valuable information [352].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' PPKED [342] Proposed a three-module system that mimics the working habits of radiologists by extracting abnormal regions, encoding prior information, and distilling the useful knowledge to generate accurate reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Provides abnormal descriptions and locations to facilitate medical diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Capable of extracting relevant information from the explored posterior and prior multi-domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Some mistakes, such as duplicate reports and inaccurate descriptions, are present in the generated reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [356] Memory MDT [332] Introduced Memory-Driven Transformer for radiology report generation Developed a relational memory to retain essential knowledge gathered through the previous generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To facilitate medical diagnosis, visual-textual attention mappings were incorporated to capture correspondence with essential medical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Dataset imbalance with dominating normal findings hinders the model’s generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' AlignTransformer [344] Introduced an MRG framework that mitigates the problem of data bias by hierarchically aligning visual abnormality regions and illness tags in an iterative fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Conducted experiments on MIMIC-CXR and IU-Xray datasets and demonstrated the capability of the model in ameliorating the data bias problem M2 TR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' progressive [345] Developed a progressive text generation model for medical report generation by incorporating high-level concepts into the process of generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The division of report generation into two steps enhanced the performance in terms of language generation and clinical efficacy metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The progressive generation process increases the false positive rate by including abnormality mentions in negation mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [355].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MDT-WCL [346] Introduced the contrastive learning technique into chest X-ray report generation by proposing a weakly supervised approach that contrasts report samples against each other to better identify abnormal findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Optimization with contrastive loss facilitates generalizability in comparison to constrastive retrieval-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CMN [333] Cross-modal memory networks were introduced to improve report generation based on encoder-decoder architectures by incorporating a shared memory to capture multi-modal alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Capable of properly aligning data from radiological images and texts to aid in the preparation of more precise reports in terms of clinical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Other CRG [348] Formulated the problem in two steps: (1) a report generation phase incorporating a standard language generation objective to train a Transformer model, and (2) a sampling phase that includes sampling a report from the model and extracting clinical observations from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transformers’ ability to provide more coherent and fluent reports was demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Due to the biased nature of the dataset caused by dominant normal findings, the algorithm tends to generate reports that lack essential descriptions of abnormal sections [357].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Medical-VLBERT [349] Proposed a framework as the first work that generates medical reports for the COVID-19 CT scans Devised an alternate learning strategy to minimize the inconsistencies between the visual and textual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Alleviated the shortage of COVID-19 data by employing the transfer learning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Capable of effective terminology prediction Overreliance on predetermined terminologies undermines robustness and generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CDGPT2 [351] Presented a conditioning mechanism that to improve radiology report generation in terms of word-overlap metrics and time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Utilized a pre-trained GPT2 conditioned on visual and weighted semantic features to promote faster training, eliminate vocabulary selection, and handle punctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The first study to employ semantic similarity metrics to quantitatively analyze medical report generation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Conditioning mechanism tackled punctuations, vocabulary collection, and reduced training duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The architecture does not require modification to be trained on distinct data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Incorporating semantic similarity in addition to word overlap metrics improved medical report evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 4 The model’s generalization ability and robustness against over-fitting are both hindered when the size of the dataset is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CGI [352] Provides cohesive and precise X-ray reports in a fully differentiable manner by dividing the report gener- ation system into a classifier, generator, and interpreter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Their conducted experiments revealed that incorporating additional scans besides clinical history can be beneficial in providing higher-quality X-ray reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Flexibility in processing additional input data, such as clinical documents and extra scans, which also contributes to performance improve- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The model doesn’t provide vital information, such as illness orientation and time-series correlations, which facilitates more reliable reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CGRG [353] Presented a Transformer-based method that estimates report uncertainty to develop a more reliable MRG system and facilitate diagnostic decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Introduced the Sentence Matched Adjusted Semantic Similarity (SMAS) to capture vital and relevant features in radiology report generation more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Assessing visual and textual uncertainties leads to more reliable reports in medical diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Measuring uncertainties can properly provide correlated confidence between various reports, which is beneficial to aiding radiologists in clinical report generation [357].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This work’s key difference from others is that their model is proposed to generate instructions instead of descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Graph-based Systems In graph-based medical report generation, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [330] pro- posed KERP, a Graph Transformer implementation to generate robust graph structures from visual features that are extracted by a DenseNet [335] backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This approach is composed of three modules: Encode, Retrieve and Paraphrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' First, it constructs an abnormality graph by converting the visual fea- tures extracted from the medical images via an encoder mod- ule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Next, a sequence of templates is retrieved considering the detected abnormalities by utilizing a retrieve module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Subse- quently, the terms of the produced templates are paraphrased into a report by employing the paraphrase module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The KERP’s workflow is illustrated in Figure 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Additionally, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [342] addressed the visual and tex- tual data biases and their consequences in generating radiology reports and proposed the PPKED framework to alleviate these challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Their work introduced three modules to perform re- port generation: (1) Prior Knowledge Explorer (PrKE), which obtains relevant prior information for the input images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' (2) Pos- terior Knowledge Explorer (PoKE), which extracts the poste- rior information, including the abnormal regions of the medi- cal image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' and (3) Multi-domain Knowledge Distiller (MKD), which distills the obtained information from the previous mod- ules to perform the final report generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' PPKED then formu- lated the problem by employing the presented modules in the following manner: PoKE first extracts the image features cor- responding to the relevant disease topics by taking the visual features extracted by ResNet-152 [63] from the input image and abnormal topic word embeddings as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Next, the PrKE module filters the prior knowledge from the introduced prior working experience (a BERT encoder) and prior medical knowledge component that is relevant to the abnormal regions of the input image by utilizing the output of the PoKE module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Next, the MKD module generates the final medical report by using this obtained information, which is implemented based on the decoder part of the Transformers equipped with Adap- tive Distilling Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 43 Medical Report Generation RL-based 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' RTMIC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' SIG Memory-based 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MDT 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' AlignTransformer 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' M2 TR Progressive 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MDT-WCL 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CMN Graph-based 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' KERP 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' PPKED Other Systems 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CRG 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Medical-VLBERT 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CDGPT2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CGI 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' CCRG Figure 41: Taxonomy of Transformer-based medical report generation approaches based on the mechanism by which they generate clinical reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We reference the papers in ascending order corresponding to their prefix number: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [334], 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [337], 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [332], 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [344], 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [345], 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [346].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [333], 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [330], 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [342], 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [348], 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [349], 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [351], 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [352], 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [353].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Pleural effusion Consolidation Encode GTRi2g Retrieve GTRg2s Paraphrase GTRgs2s GTRg2g Abnormality graph Disease graph Visual feature CNN Templates Report Degenerative changes in the spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' No pleural effusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' There is hyperexpansion of the lungs suggesting underlying emphysema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' No focal airspace consolidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Heart size is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Emphysema Degenerative disease Degenerative change of spine (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='66) Focal airspace consolidation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='01) Hyperexpansion of lungs (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='78) Enlarged heart size (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='04) Tortuous aorta (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='12) Low lung volumes (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='00) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='00 Figure 42: Using an encoder module, KERP creates an abnormality graph from the extracted visual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Then, a retrieval module retrieves a sequence of templates based on detected abnormalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Next, the paraphrase module paraphrases the templates’ terms into a report [330].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Memory-based Systems Concerning the development of systems that rely on a mem- ory mechanism to generate medical reports, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [332] presented a Memory-Driven Transformer (MDT), a model suit- able for the generation of long informative reports and one of the first works on the MIMIC-CXR dataset [343].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' MDT em- ploys a relational memory to exploit characteristics prevailing in reports of similar images, and then the memory is incorpo- rated into the decoder section of the Transformer by implement- ing a memory-driven conditional layer normalization (MCLN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Likewise, Nooralahzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [345] introduced M2 TR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' progressive, a report generation approach that utilizes curricu- lum learning, which is a strategy of training machine learning models by starting with easy samples and gradually increasing the samples’ difficulty [360].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Instead of directly generating full reports from medical images, their work formulates the prob- lem into two steps: first, the Meshed-Memory Transformer (M2 TR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=') [361], as a powerful image captioning model, receives the visual features extracted by a DenseNet [335] backbone and generates high-level global context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Second, BART [362], as a Transformer-based architecture, encodes these contexts with a bidirectional encoder and decodes its output using a left-to- right decoder into coherent reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The overview of the pro- cess is depicted in Figure 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Additionally, You et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [344] proposed AlignTransformer, a framework composed of two modules: Align Hierarchical At- tention (AHA) and Multi-Grained Transformer (MGT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In their approach, first visual features and disease tags are extracted Figure 43: Workflow of the M2 Tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Progressive framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The task is ac- complished in two stages: First, the Meshed-Memory Transformer (M2 TR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=') receives visual features extracted by a DenseNet [335] backbone and generates high-level global context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Second, the BART [362] architecture encodes con- texts with a bidirectional encoder and decodes them with a left-to-right decoder to produce coherent reports [345].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' from the medical image by an image encoder, then they get aligned hierarchically to obtain multi-grained disease-grounded visual features in the AHA module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The obtained grounded fea- tures are capable of tackling the data bias problem by promot- ing a better representation of abnormal sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Next, these grounded visual features are exploited by an adaptive exploit- ing attention (AEA) [361] mechanism in the MGT module for the generation of the medical reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' They also justified their model’s efficiency through the manual evaluation of clinical ra- diologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In MDT-WCL [346], the problem is approached with a weakly supervised contrastive loss, which lends more weight to the reports that are semantically close to the target reports, and a memory-driven Transformer is adopted as the backbone model to store key information in its memory module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To aid the contrastive learning during training, after clustering the re- ports into groups with the K-Means algorithm, each report is assigned a label corresponding to its cluster, and the semanti- cally closed ones are considered to be in the same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Although previous approaches have achieved promising re- sults, they lack the ability to generate mappings between im- ages and texts to align visual-textual information and assist medical diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In order to facilitate visual-textual align- ment, the Cross-modal Memory Network (CMN) [333] ex- tended encoder-decoder methods by utilizing a shared memory for better alignment of information between images and texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 44 High-level Context Report Visual Compared to prior examination, Backbone PosITIVE improved bilateral airspace opaci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' there is a significant improvement POSITIVE minimal streaky opaci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' in aeration bilaterally,with improved UNCERTAIN airspace disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' bilateral airspace opacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Currently, UNCERTAIN atelecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' there are only minimal streaky opacities Visual NEGATIVE large focal pneumothorax effusions in the bilateral midlung,which may Language identified consolidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Language represent mild residual airspace disease, Model NEGATIVE consolidations identified pneumothorax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Model atelectasis, or underlying changes NEGATIVE consolidations definite pleural of chronic lung disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='No large focal identified effusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' consolidations,pneumothorax,or definite NEGATIVE contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' stable mediastinal silhouette pleural effusions identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The mediastinal Visual silhouette is stable and within normal Backbone limits for size and contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='No acute osseous abnormality is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='It uses a pre-trained ResNet [63] as the visual extractor to out- put visual features, then passes them to the cross-modal mem- ory network that utilizes a matrix to store information where each row represents the embedding of information linking im- ages and texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To access the stored information aligning the modalities, memory querying and responding are implemented in a multi-threaded manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Other Systems Other MRG systems focus on solving the problem with dif- ferent ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Lovelace et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [348] proposed a generation frame- work composed of two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the first stage, a Transformer model is adopted to map the input image features extracted by a DenseNet-121 [335] to contextual annotations and learn re- port generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In the second stage, a procedure is introduced to differentiably sample a clinical report from the Transformer decoder and obtain observational clinical information from the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' This differentiability is further employed to fine-tune the model for improving clinical coherence by applying their differentiable CheXpert to the sampled reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Fueled by re- cent progress in explainable artificial intelligence and the in- troduction of algorithms that attempt to provide interpretable prediction in DL-based systems, Likewise, in CDGPT2 [351], the medical image is passed into a Chexnet [363] to provide localizations of 14 types of diseases from the images as vi- sual features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To implement better semantic features, the model was fine-tuned as a multi-label classification problem to extract manual tags from the IU-Xray dataset [326] by replacing the final layer of the model with a layer containing 105 neurons to produce 105 tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The vector representation of the tags is then fed into a pre-trained distilGPT2 [364] as the decoder to gener- ate medical reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Moreover, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [353] presented a confidence-guided report generation (CGRG) approach to sup- port reliability in report generation by quantifying visual and textual uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It’s comprised of an auto-encoder that reconstructs images, a Transformer encoder that encodes the input visual feature extracted by ResNet-101 [63], and a Trans- former decoder for report generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Visual uncertainty is ob- tained by the AutoEncoder, which acts as a guide for the visual feature extractor, and textual uncertainty is quantified based on the introduced Sentence Matched Adjusted Semantic Similar- ity (SMAS) which captures the similarity between the gener- ated reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These uncertainties are further utilized to aid the model optimization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The recent outbreak of COVID-19, one of the deadliest pan- demics, has influenced the research community to alleviate the tedious and time-consuming work of producing medical re- ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' VL-BERT, [365] as an extension of BERT, [36] can be employed as an intelligent medical report generation system to expedite the diagnosis process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Medical-VLBERT [349] in- troduced VL-BERT to the medical report generation domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' It defines the problem as a two-step procedure: First, it uti- lizes two distinct VL-BERTs as terminology encoders to pro- duce terminology-related features (textual and visual), and then these features are fed into a shared language decoder to produce medical textbooks and reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The proposed method takes into account predefined terminology word embeddings that repre- sent medical domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' These embeddings are paired distinctly with two other embeddings as an input to the en- coders: textbook embeddings, which are generated by employ- ing a lookup table, and spatial feature embeddings (termed ”vi- sual context”) that are extracted from medical images by im- plementing DenseNet-121 [335].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The encoders then integrate this pairwise information separately to produce textual and vi- sual terminological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Subsequently, a shared language decoder is trained by utilizing an alternate approach to properly exchange the knowledge captured by the encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Furthermore, in the work of Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' [352], a clas- sification, generation, and interpretation framework (CGI) is proposed to address clinical accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Each term of the frame- work’s name represents a different module to perform the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The classification module learns how to discover diseases and generate their embeddings, which consist of an image and text encoder to extract the global visual features from medi- cal images and obtain text-summarized embeddings from clini- cal documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The generation module is a Transformer model that takes the disease embeddings as input and generates med- ical reports from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The interpretation module then takes these reports for evaluation and fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Discussion and Conclusion This section offers a systematic review of the Trans- former architectures configured for medical report gener- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Compared to previous sections that reviewed ViT- based frameworks to tackle different medical tasks and prob- lems, this section focuses mostly on using standard Trans- formers as the core of a medical report generation sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' A common theme prevailing in these systems is to solve the problem with an encoder-decoder architecture sup- ported by a CNN-based visual backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As mentioned in previous sections, the self-attention mechanism undermines the representation of low-level details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' On the other hand, since medical reports consist of long and multiple sentences, Transformers are of great significance to model long-term dependencies, which assists clinically accurate report gen- eration [366, 352].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To exploit the power of both CNNs and Transformers simultaneously, state-of-the-art MRG sys- tems usually embed CNNs along with Transformers in their frameworks [334, 351, 353].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We have provided information in Table 14 on the reviewed report generation methods con- cerning their architectural type, modality, organ, pre-trained strategy, datasets, metrics, and year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Table 15 contains summarized information about the methodologies, includ- ing their contributions and highlights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In addition, it should be noted that several survey publications have been pub- lished in this field of medicine [327, 355, 367], and the most recent one provided a technical overview of Transformer- based clinical report generation [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We approach our review differently by distinguishing the proposed methods based on the mechanism they used to support the prevailing concerns such as long and coherent text generation, reliabil- ity, and visual-textual biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The ultimate goal of these frameworks is to increase clin- ical accuracy to expedite the diagnosis process and reduce 45 the workloads in radiology professions [348, 352].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Numer- ous works have attempted to facilitate diagnostic decision- making by aligning correlated sections of medical image and textual report that provide valuable information for de- tecting abnormalities [344, 333].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Also, multiple studies em- phasized the importance of universal knowledge, and de- signed a system to incorporate prior information for detect- ing disease [330, 332].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Some research effort was also put into better representation learning by contrasting normal and abnormal samples against each other in representation space by utilizing a contrastive loss as the objective [346].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' One re- cent work was inspired by curriculum learning to imitate the order of the human learning process [345].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Overall, we believe that MRG systems need more research and progression to be robustly incorporated in a practical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Open Challenges and Future Perspectives So far, we discussed the application of Transformers (espe- cially vision Transformers) and reviewed state-of-the-art mod- els in medical image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Even though their effective- ness is exemplified in previous sections by delicately present- ing their ideas and analyzing the significant aspects that were addressed in their proposed methods, there is still room for improvement in many areas to devise a more practical and medically accurate system by leveraging Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Conse- quently, we discuss the challenges and future directions hoping to help researchers gain insight into the limitations and develop more convenient automatic medical systems based on Trans- formers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Explainability Fueled by recent progress in XAI (explainable artificial intel- ligence) and the introduction of algorithms that attempt to pro- vide interpretable prediction in DL-based systems, researchers are putting effort into incorporating XAI methods into con- structing Transformer-based models to promote a more reliable and understandable system in different areas, including medi- cal analysis [368, 369].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Existing approaches usually highlight important regions of the medical image that contribute to the model prediction by employing attention maps [370, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Fur- thermore, Vision Transformers (ViTs) have the ability to pro- vide attention maps that indicate the relevant correlations be- tween the regions of the input and the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, the challenge of numerical instabilities in using propagation- based XAI methods such as LRP [371] and the vagueness of the attention maps, which leads to inaccurate token associations [75, 372], makes interpretable ViTs an open research opportu- nity in computer vision, especially in medical image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We believe that including interpretable vision Transformers, such as ViT-NeT [372], in various medical applications can pro- mote user-friendly predictions and facilitate decision-making in the diagnosis of medical conditions, and is a promising direc- tion in medical research problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Richer Feature Representation An effective and suitable representation space is substantially influential in building medical analysis systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transformers have demonstrated their efficiency in obtaining global informa- tion and capturing long-term dependencies in many areas, such as Natural Language Processing (NLP), Computer Vision, and Speech Recognition [306], and CNNs have proven to be effec- tive in extracting local context from visual data [373].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, this locality usually enables these networks to capture rich local texture representation [374, 375] and lacks model global depen- dency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As a result, many approaches stack Transformers along with CNNs to leverage both local and global information si- multaneously in clinical applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=', medical report gen- eration) [344, 348, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Recent studies stated that the single- scale representation of ViTs hinders improvement in dense pre- diction tasks, so a multi-scaled feature representation is imple- mented which achieves better performance in computer vision tasks, including image classification, object detection, and im- age segmentation [376, 377].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Generalizing this idea to medical applications of ViTs to facilitate devising a clinically suitable system can be considered as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Video-based analysis There has been an increasing interest in the vision commu- nity in extending ViT architectures to video recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Recently, a handful of papers have integrated standard Trans- formers with their models in AI-assisted dynamic clinical tasks [378, 379, 380, 381].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, the scarcity of the proposed ap- proaches puts video-based medical analysis in an infancy stage and open for future investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Another potential research direction is to explore the power of video vision Transformer variants, such as Video Swin Transformer [382], in clinical video understanding and to facilitate automatic robotic surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' High Computational Complexity The robustness of Transformer models in layouts that im- plement large numbers of parameters is one of their strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' While this is a beneficial trait that makes it possible to train models of enormous scale, it leads to the requirement of large resources for training and inferencing [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Particularly dis- advantageous to medical image analysis is that expanding the use of ViTs for pretraining in new tasks and datasets comes with substantial expenses and burdens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Additionally, gathering medical samples can be difficult and the dataset scale is often limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For instance, according to empirical studies in [22], pretraining a ViT-L/16 model on the large-scale dataset of Ima- geNet takes approximately 30 days employing a standard cloud TPUv3 with 8 cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' As a result, a notable number of papers utilized the pre-trained weights of ViT models to exploit the transfer learning strategy to alleviate training load [44, 24, 43], but in some cases, such as dealing with volumetric medical im- ages, where transfer learning doesn’t demonstrate any improve- ments [143, 309], the pretraining process is necessary to cap- ture domain-specific features for generalization and better per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Ultimately, designing effective Transformer systems with fewer parameters while maintaining optimality in terms of clinical accuracy and robustness is a preferable research direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 46 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Transformer-based Registration As reviewed in Section 8, the idea of employing Transform- ers to support efficient medical image registration has become popular in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' The ability of the self-attention mecha- nism assists the learning of long-term visual correlations since their unlimited receptive field promotes a more accurate under- standing of the spatial relationship between moving and fixed images [309, 310].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, registration systems composed of Transformer architectures are still in their infancy and require more research effort to be put into them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Data-Driven Predictions With supervised learning as a popular fashion in building in- telligent systems, the model learns features based on the pro- vided annotations that are suitable to accomplish a specific task, which hinders generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' In other words, super- vised learning modifies the bias-variance trade-off in favor of the strong inductive biases that lead to making assumptions as a means to aid the model in learning a particular task quicker and with higher sample efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' However, these hard assump- tions sacrifice adaptability to other settings and unseen datasets, and the model learns to accomplish its task without having an innate understanding of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' To tackle this issue, unsu- pervised regimes enable the algorithms to act as general de- scriptors and capture features that will assist them in perform- ing efficiently in a wide range of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Similarly, in medical image analysis, adopting Transformer networks with unsuper- vised learning algorithms promotes robustness and generaliz- ability to other datasets and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Medical Software Ecosystems A future direction for advancing in the automatic medical analysis is to provide an open-source environment that contains libraries suitable for solving multiple medical tasks and chal- lenges with Transformer architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Developers can further contribute to the ecosystem by updating and adding additional tasks, bringing novelty, and proposing ideas to enhance perfor- mance and accuracy [132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Companies and organizations can support the system by preparing the necessary computational resources and hardware requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Sample of software pro- totypes in this direction are nnU-Net [383], Ivadomed [384], and preliminary works such as [133], which provides an end- to-end pipeline for implementing deep models on medical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Discussion and Conclusion In this paper, we presented a comprehensive encyclopedic review of the applications of Transformers in medical imag- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' First, we provided preliminary information regarding the Transformer structures and the idea behind the self-attention mechanism in the introduction and background sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Start- ing from Section 3, we reviewed the literature on Transformer architecture in diverse medical imaging tasks, namely, classifi- cation, segmentation, detection, reconstruction, synthesis, reg- istration, and clinical report generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' For each application, we provided a taxonomy and high-level abstraction of the core techniques employed in these models along with the SOTA ap- proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We also provided comparison tables to highlight the pros and cons, network parameters, type of imaging modality they are considering, organ, and the metrics they are using.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Fi- nally, we outlined possible avenues for future research direc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' Acknowledgments This work was funded by the German Re- search Foundation (Deutsche Forschungsgemeinschaft, DFG) under project number 191948804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' We thank Johannes Stegmaier for his contribution to the proofreading of this docu- ment.' 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A med- ical imaging deep learning toolbox, arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content='09984 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} +page_content=' 64' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE1T4oBgHgl3EQf4wXL/content/2301.03505v1.pdf'} diff --git a/1NE0T4oBgHgl3EQfdgAR/content/tmp_files/2301.02376v1.pdf.txt b/1NE0T4oBgHgl3EQfdgAR/content/tmp_files/2301.02376v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9ce342402545848dac18bdc2d5649f8f71d18432 --- /dev/null +++ b/1NE0T4oBgHgl3EQfdgAR/content/tmp_files/2301.02376v1.pdf.txt @@ -0,0 +1,1138 @@ +Improved design and experimental +demonstration of ultrahigh-Q C6-symmetric H1 +hexapole photonic crystal nanocavities +KENTA TAKATA1,2,4, EIICHI KURAMOCHI1,2, AKIHIKO SHINYA1,2 AND +MASAYA NOTOMI1,2,3,5 +1Nanophotonics Center, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan +2NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa +243-0198, Japan +3Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, +Japan +4kenta.takata.ke@hco.ntt.co.jp +5masaya.notomi.mn@hco.ntt.co.jp +Abstract: +An H1 photonic crystal nanocavity is based on a single point defect and has +eigenmodes with a variety of symmetric features. Thus, it is a promising building block for +photonic tight-binding lattice systems that can be used in studies on condensed matter, non- +Hermitian and topological physics. However, improving its radiative quality (𝑄) factor has been +considered challenging. Here, we report the design of a hexapole mode of an H1 nanocavity with +a 𝑄 factor exceeding 108. We achieved such extremely high-𝑄 conditions by designing only four +structural modulation parameters thanks to the C6 symmetry of the mode, despite the need of +more complicated optimizations for many other nanocavities. The fabricated silicon photonic +crystal nanocavities exhibited a systematic change in their resonant wavelengths depending on the +spatial shift of the air holes in units of 1 nm. Out of 26 such samples, we found eight cavities with +loaded 𝑄 factors over one million (1.2 × 106 maximum). We examined the difference between +the theoretical and experimental performances by conducting a simulation of systems with input +and output waveguides and with randomly distributed radii of air holes. Automated optimization +using the same design parameters further increased the theoretical 𝑄 factor by up to 4.5 × 108, +which is two orders of magnitude higher than in the previous studies. Our work elevates the +performance of the H1 nanocavity to the ultrahigh-𝑄 level and paves the way for its large-scale +arrays with unconventional functionalities. +© 2023 Optica Publishing Group +1. +Introduction +Photonic crystal nanocavities (PCNs) in dielectric slabs are a particular series of optical +resonators that exhibit both strong light confinement and small modal volumes [1–12]. These +features enable intense light-matter interactions, which make PCNs very useful for extremely +low-power photonics [13–15], on-chip nonlinear optics [16–18] and quantum optics [19–21]. +Integration of PCNs also opens a route to functional nanophotonic devices, such as slow light +waveguides [22–24], and all-optical switches [25–27], memories [28–30], and transistors [31], +which are potential for information processing. +An H1 PCN comprises a vacancy of a single lattice element [32–35]. Such a point defect +structure takes over the spatial symmetry of its host system. Thus, the eigenmodes of the +Maxwell equations for the H1 nanocavity are also those for the symmetry operations in the +entire point group of the photonic crystal (PhC) [36]. As a result, they are analogous to atomic +orbitals in terms of their symmetric properties, and hence, coupled H1 PCNs work as good +photonic emulators of molecules and tight-binding lattices including basis functions [22,37]. +arXiv:2301.02376v1 [physics.optics] 6 Jan 2023 + +Because their evanescent couplings, resonant frequencies and radiation losses can be controlled by +structural modulation, PCNs can also be combined with unconventional functionalities emerging +in non-Hermitian and topological physics [38–47]. In particular, arrays of H1 PCNs may pave +the way for large-scale two-dimensional crystalline systems [48–53]. This potential is in stark +contrast to most other PCNs based on linear defects, which are less symmetric and thus limited +in their coupling profiles. +However, it is generally more difficult for a smaller PCN to have an ultrahigh 𝑄 factor. Narrower +field distributions in real space result in broader ones in reciprocal space. Parts of such modes +tend to reside in the light cone (LC) and hence turns into radiation fields, namely losses [3]. +We showed two decades ago that a hexapole mode of the H1 nanocavity in a triangular-lattice +PhC slab could have a theoretical 𝑄 factor up to 3 × 106, unlike the other eigenmodes [32,33]. +However, this record was not broken even with an algorithmic optimization [54]. Moreover, the +experimental counterpart was an order of magnitude smaller, namely 3 × 105 [34]. Unfortunately, +there values compared disadvantageously to those of PCNs with larger defect regions [55–59]. +The lack of tightest light confinement seems to be a significant obstacle to using large-scale H1 +nanocavity arrays, for example, to enhance light-matter interactions with bulky coupled modes, +and to make robust optical circuits with topological edge states. +In this article, we design, analyze and experimentally examine the hexapole mode of an H1 +PCN with a theoretical 𝑄 factor (𝑄th) over 108, on the basis of our latest prototype for studying +non-Hermitian physics [44]. Structural modulation in the design maintains the C6v symmetry of +the PCN, which the hexapole mode also respects. As a result, we find that we can dramatically +increase the 𝑄 factor just with four optimization parameters. By elaborating the dependence of +𝑄th on major three parameters in a simulation, we clarify that such extremely high-𝑄 conditions +form a region with some width in the parameter space. Here, we obtained a hexapole mode with +𝑄th = 1.4 × 108 and a modal volume (𝑉) of 0.72(𝜆/𝑛)3. We also compare its field profiles with +those of another H1 PCN based on a previous study in real and reciprocal spaces. +We experimentally investigated a series of silicon (Si) H1 PCNs with different spatial shifts of +air holes. These samples exhibited a systematic variation in their resonant wavelengths, indicating +that undesired variations in the positions of air holes were restricted. We found that eight such +PCNs out of 26 had loaded 𝑄 factors (𝑄exp), which include the effects of the input and output +waveguides, of over one million. The best sample had 𝑄exp = 1.2 × 106, and the cavity’s intrinsic +𝑄 factor (𝑄i) was estimated to be 𝑄i = 1.5 × 106. We also performed a simulation of the system +with randomly varying radii of the air holes to close the gap between 𝑄th and 𝑄exp. +Finally, we performed an automated optimization to further improve 𝑄th. Here, we added the +hole radius of the background PhC as a parameter and found 𝑄th = 4.5 × 108, which is more +than a hundred times those in the previous design. Our results show that the highly symmetric +hexapole mode can achieve both an extremely high 𝑄th and a very small 𝑉 with an inexpensive +optimization. It enables ultrahigh 𝑄exp (> 106) of H1 PCNs and will open up their various +applications. +The remainder of this paper is organized as follows. Section 2 shows the design and modal +properties of our H1 PCN. Section 3 presents experimental results, and numerically analyzes +and discusses them. The automated optimization and resultant impact on the hexapole mode +are summarized in Sec. 4. Section 5 discusses fundamental limitations on the 𝑄 factors of +nanocavities, including ours. Section 6 concludes this study. +2. +Cavity design +2.1. +Structure and scheme +Figure 1(a) depicts the design of our PCN. The system is based on a Si slab with a refractive +index of 𝑛Si = 3.47 and thickness 𝑡. The PhC here is a triangular lattice of circular air holes of +radius 𝑅0 and lattice constant 𝑎. Triangular-lattice PhC slabs are widely used in experiments + +(a) +(b) +R0 +R0 +R1 +s1 +s2 +x +y +z +Fig. 1. (a) Design of H1 PCN based on structural modulation of the innermost and +second innermost layers of air holes with reference to the single point defect (colored +red and orange, respectively). 𝑅0 is the radius of the holes for the background PhC +and the second layer, and 𝑅1 that for the innermost holes. 𝑠1 is a radial shift of the +innermost layer directed outward from the lattice points, and 𝑠2 is that for the second +innermost layer with its regular hexagonal alignment kept. (b) 𝐻𝑧 field distribution of +hexapole mode. +because they have large photonic band gaps for TE-like modes. The lack of a single hole acts as a +point defect and hence forms an H1 nanocavity, which is the simplest structure of PCNs that take +over the C6v symmetry of the PhC. The six holes closest to the defect, which are colored red in +the figure, have a smaller radius 𝑅1 than that of the background PhC (𝑅1 < 𝑅0). This innermost +layer of holes is also shifted radially away from the lattice points by a distance 𝑠1. The second +innermost hole layer comprises the twelve holes located one layer outward from the innermost +ones and is drawn in orange. It is also translated in the radial direction so that it keeps the regular +hexagonal alignment and its half diagonal is increased by a distance 𝑠2. In addition, it’s holes are +of the same radius 𝑅0 as those of the PhC. +We computed the complex eigenfrequencies 𝑓 of the hexapole eigenmode for various cases by +using the finite element method on a commercial solver (COMSOL Multiphysics [60]). With the +defect center defined as the coordinate origin, the system had 11 and 14 layers of holes in the ±𝑥 +and ±𝑦 directions, respectively. A rectangular air region with a height of 3 µm was placed on +each side of the slab. A scattering boundary condition for plane waves is applied to every border +of the computational domain. The 𝑥-𝑦 and 𝑦-𝑧 planes were set as perfect magnetic and electrical +conductors, respectively, for reducing the computational cost. Any changes to these simulation +conditions are noted in what follows. The theoretical 𝑄 factor is given by 𝑄th = Re 𝑓 /(2Im 𝑓 ). +Figure 1(b) shows the 𝑧 component of the magnetic fields (𝐻𝑧) of the hexapole mode along the +𝑥-𝑦 plane. This mode is TE-like and thus characterized by 𝐻𝑧. It is also an eigenmode for the C6 +rotation operator with an eigenvalue of −1. Such an odd parity of a symmetric two-dimensional +multipole contributes to destructive interference in 𝐻𝑧 along the 𝑧 direction corresponding to +Γ point [5, 61]. This feature suppresses radiation loss based on the transverse electric field +components (𝐸𝑥, 𝐸𝑦), as they are linked to 𝐻𝑧 through the Maxwell equations. Thus, structural +modulation maintaining the lattice-matched rotational symmetry is essential to achieving an +ultrahigh 𝑄 factor of the hexapole mode. The other C6-symmetric eigenmode of this cavity is +the monopole mode (not shown). It has an eigenvalue of +1 for the C6 operator and a far lower +𝑄th < 3000 in our simulations. +As illustrated in Fig. 1(a), our design uses only four parameters (𝑅0, 𝑅1, 𝑠1, 𝑠2) to improve + +85 +88 +91 +94 +97 +99 +101 +103 +105 +107 +R1 (nm) +s1 (nm) +105 +106 +107 +108 +Qth +85 +88 +91 +94 +97 +99 +101 +103 +105 +107 +R1 (nm) +s1 (nm) +1.536 +1.547 +1.557 +1.568 +1.578 +λ (µm) +85 +88 +91 +94 +16 +19 +22 +25 +s2 (nm) +s1 (nm) +105 +106 +107 +108 +Qth +85 +88 +91 +94 +16 +19 +22 +25 +s2 (nm) +s1 (nm) +1.548 +1.553 +1.558 +1.563 +1.569 +λ (µm) +(a) +(b) +(d) +(c) +Wavelength +Q factor +Fig. 2. Dependence of (a) resonant wavelength (𝜆) and (b) theoretical 𝑄 factor (𝑄th) of +the hexapole mode on 𝑠1 and 𝑅1 for 𝑠2 = 20.5 nm. (c) 𝜆 and (d) 𝑄th dependent on 𝑠1 +and 𝑠2 for 𝑅1 = 102 nm. Black dots represent sample points in the simulation. The +data among the points are linearly interpolated. A band of parameter conditions for +𝑄th > 108 appears. 𝑅0 = 131 nm, 𝑎 = 426 nm, and 𝑡 = 250 nm. +the 𝑄 factor, unlike recent designs based on costly optimizations of many variables [62–65]. +𝑅0 determines the filling factor of the PhC, which is related to its photonic band gap and thus +the in-plane modal confinement. 𝑅1, 𝑠1 and 𝑠2 affect the local modal properties. The lattice +constant 𝑎 can be varied to adjust the resonant wavelengths of the simulated modes to telecom +ones around 1.55 µm. +2.2. +Resonance properties versus hole shifts +First, let us study the resonance characteristics of the mode for constant 𝑅0 = 131 nm, 𝑎 = 426 nm, +and 𝑡 = 250 nm. Figure 2(a) and (b) are two-dimensional color plots of the resonant wavelength +𝜆 = 𝑐/Re 𝑓 and 𝑄th for isolated (unloaded) H1 PCNs depending on 𝑠1 and 𝑅1. Here, 𝑐 is the +speed of light in vacuum and 𝑠2 = 20.5 nm. The plot of 𝜆 indicates that a small 𝑠1 and large 𝑅1 +squeeze the magnetic poles in Fig. 1(b) and thus yield a short 𝜆, whereas a large 𝑠1 and small 𝑅1 +broaden the magnetic poles and thus increase 𝜆. Remarkably, the 𝑄th plot exhibits a sequence of +optimum points with 𝑄th > 108 forming a linear band. Such a peak distribution indicates that +there is an optimal polar width for every 𝜆 that suppresses local scattering-induced radiation loss. +There is a margin of about ±1.5 nm in 𝑅1 and a wider one in 𝑠1 from each optimum point to have +a 𝑄th > 107. The largest 𝑄 factor here is 𝑄th = 1.43 × 108 for (𝑠1, 𝑅1) = (88.75 nm, 101.75 nm). +In units of 0.5 nm for the parameters, 𝑄th = 1.41 × 108 for (𝑠1, 𝑅1) = (89.5 nm, 102 nm) was +obtained. +Figure 2(c) and (d) depict the dependence of 𝜆 and 𝑄th on 𝑠1 and 𝑠2 for 𝑅1 = 102 nm. There + +is a notable difference between Fig. 2(a) and (c) in the directions of the iso-wavelength contours. +This difference is due to negative correlation between the effect of 𝑅1 and that of 𝑠2; a larger 𝑠2 +results in a longer 𝜆 because of the higher effective index of the cavity region. In contrast, Fig. +2(b) and (d) appear to have more or less similar properties. As the mode wavelength increases +with 𝑠1, the optimal 𝑠2 also becomes larger. 𝑠2 can be used to dramatically improve 𝑄th because +it introduces a gradual variation in the effective potential barrier of the PhC [7,66]. However, the +trace of the extremely high 𝑄 values in Fig. 2(d) is nearly perpendicular to the contour lines in +Fig. 2(c), meaning that the conditions for a much improved 𝑄th are limited for each 𝜆. The peak +value of 𝑄th decreases for large and small 𝑠1 because 𝑅1 is fixed. Overall, a global optimization +for (𝑅1, 𝑠1, 𝑠2) enables us to find the continuous conditions for 𝑄th > 108 in the parameter space. +The best 𝑄th here is 1.46 × 108 for (𝑠1, 𝑠2) = (90.25 nm, 20.75 nm). +2.3. +Modal properties +Next, let us compare the modal shapes in real and reciprocal spaces of the design with +𝑄th > 108 and that in the previous study. Figure 3(a) and (b) show the spatial magnetic intensity +distributions on a common logarithmic scale (log10(|H(r)|2)) along 𝑧 = 0 for hexapole modes +with 𝑄th = 2.0 × 106 and 1.4 × 108, respectively. The PCN shown in (a) is based on Ref. [33] +and does not include 𝑠2 in its design with 𝑅0 = 109 nm, 𝑅1 = 100 nm, 𝑠1 = 78 nm, 𝑎 = 435 nm, +and 𝑡 = 220 nm. The other PCN in (b) corresponds to (𝑠1, 𝑅1) = (89.5 nm, 102 nm) in Fig. 2(a) +and (b). A sizable portion of (a) has evanescent fields with relative intensities of about 10−4, and +visible components with intensities over 10−8 reach the boundaries of the entire geometry. In +comparison, the optimal mode shown in (b) obviously decays faster from the center. This means +that the current design provides stronger in-plane light confinement. +Figure 3(c) and (d) depict the Fourier transforms of the 𝑥 component of the electric fields on a +logarithmic scale (log10(|F (𝐸𝑥(r))|)) along 𝑧 = 0 for the hexapole modes in Fig. 3(a) and (b). +Transverse electric field components lying within the LC measure the magnitude of radiation loss, +because they can directly couple with radiative plane waves [3,67]. As shown in Fig. 3(c), the +previously designed mode has relative Fourier amplitudes of about 10−2.5 distributed in the LC +defined by the black dashed circle. In stark contrast, the radiative field amplitudes are suppressed +over the entire LC for the optimized mode shown in Fig. 3(d). Their maximum value is about +one order of magnitude smaller than that of Fig. 3(c), confirming an improvement in the 𝑄 factor +due to the reduction of the radiation flux. A similar trend is seen in the case of 𝐸𝑦. These modal +properties also support the discussion on Fig. 2(b) and (d). +The standard Purcell mode volume 𝑉 for PCNs is given by [2] +𝑉 = +∫ +𝜖(r)|E(r)|2𝑑3r +max{𝜖(r)|E(r)|2} . +(1) +This definition is accurate in estimating the Purcell effect for high-𝑄 cavities and has been used +for comparison purposes in the literature. Interestingly, the effective volume 𝑉opt = 0.72(𝜆/𝑛)3 +for the mode with 𝑄th = 1.4 × 108 is larger by 9% than that of the previously studied one, +𝑉p = 0.66(𝜆/𝑛)3. The electric energy densities of hexapole modes tend to concentrate mostly on +the sides of the innermost air holes. However, the optimized mode distributes more electric energy +around the point defect than the mode based on Ref. [33] because of the potential modulation by +𝑠2. Thus, it has a reduced maximum energy density or denominator in Eq. (1). +This result shows that we can dramatically improve 𝑄th of the hexapole mode without sacrificing +its small 𝑉. 𝑉opt here is comparable with those of optimized L3 PCNs without hole radius +modulation [64,67], while the hexapole mode has a larger 𝑄th. Thus, our H1 PCNs can be expected +to have 𝑄exp values as high as those ones. In addition, our optimal 𝑄th/𝑉opt = 1.9 × 108(𝑛/𝜆)3 is +slightly better than another L3 nanocavity with 𝑄th/𝑉 = 1.7 × 108(𝑛/𝜆)3 (𝑄th = 1.9 × 108 and +𝑉 = 1.1(𝜆/𝑛)3) designed by the particle-swarm algorithm [65]. + +(a) +(b) +(d) +(c) +a = 435 nm +a = 426 nm +0 +0 +4 +-4 +kx (units of π/a) +-4 +4 +0 +kx (units of π/a) +0 +-4 +4 +4 +-4 +ky (units of π/a) +log10(|H|2) +log10(|(Ex)|) +Fig. 3. (a) Magnetic field intensity distribution in the logarithmic scale (log10(|H(r)|2)) +for the hexpole nanocavity based on the previous work [33] with 𝑎 = 435 nm and +𝑄th = 2.0×106. (b) Same but for the hexpole mode designed in this study with 𝑎 = 435 +nm, 𝑠1 = 89.5 nm, 𝑅1 = 102 nm, and 𝑄th = 1.4 × 108, exhibiting more tightly confined +in-plane evanescent fields than in (a). (c), (d) Absolute Fourier-space distributions of +the 𝑥 components of the electric fields on a logarithmic scale (log10(|F (𝐸𝑥(r))|)) for +the eigenmodes corresponding to (a) and (b), respectively. (d) has significantly reduced +radiative components inside the light line that is marked by the black dashed curve. +In summary, we showed designs of H1 PCNs based on a manual or brute-force search for +extremely high-𝑄 hexapole modes. By focusing on the case for a constant 𝑅0, we found a series +of conditions for 𝑄th > 108 with just three major optimization parameters (𝑅1, 𝑠1, 𝑠2), thanks to +the C6 symmetry of the mode. Introduction of an optical potential modulation with 𝑠2 resulted +in improved light confinement of the optimized mode in both the in-plane and out-of-plane +directions. This point will be examined quantitatively in Sec. 4. +3. +Experimental result and numerical analysis +3.1. +Sample fabrication and measurement +We fabricated Si H1 PCNs of our design for an experimental demonstration. The sample +structures were patterned by electron beam (EB) lithography on a positive EB resist coated +on a silicon-on-insulator (SOI) wafer. The mask pattern was projected to the Si film with a + +0 +-0.5 +-1 +-1.5 +-2 +-2.5 +-3 +-3.5 +-4 +-4.5 +-50 +-1 +-2 +-3 +-4 +-5 +-6 +-7 +-8190.4305 +190.4315 +190.4325 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Normalized power +Frequency (THz) +(b) +(c) +1 μm +Qexp = 1.1×106 +(a) +2 μm +Fig. 4. (a) Laser scope image of a sample with 𝑑 = 5 +√ +3𝑎. The input and output Si +waveguides are broadened and extended to both sides of the sample chip and coupled +with lensed fibers. (b) Close-up SEM image of H1 PCN with 𝑎 = 434 nm. Typical radii +of the small and large air holes are estimated as 𝑅1,s ≈ 106.8 nm and 𝑅0,s ≈ 133.1 +nm. (c) Transmission spectrum of sample with 𝑎 = 434 nm and 𝑠1 = 99.5 nm. The +Lorentzian curve colored red matches the experimental data shown as blue points and +indicates that the cavity has a loaded 𝑄exp of 1.1 × 106. +nominal thickness of 250 nm by inductively coupled plasma etching. The buried oxide (BOX) +layer beneath the PhCs was removed by wet etching with buffered hydrogen fluoride to obtain +air-bridged samples. After the above device processes were completed, the wafer was cleaved so +that the size of each sample chip was 5 mm × 15 mm. +Figure 4(a) is a laser scope image of a PCN sample. The H1 cavity was butt-coupled (loaded) +with two W1 PhC waveguides, each of which had a width of 𝑊0 = +√ +3𝑎 based on the removal +of a single row of air holes. The spatial interval 𝑑 between the cavity and them varied with the +samples, and ones with 𝑑 = 5 +√ +3𝑎 exhibited ultrahigh-𝑄 resonances. Each W1 waveguide was +broadened by 100 nm at either end of the PhC by shifting five pairs of air holes on the sides +outward with a stepwise increment of 20 nm. Consequently, they were efficiently coupled with +air-suspended wire waveguides with a width of 𝑊0. These optical channels were extended farther +and connected to 8 µm-wide slab waveguides that were supported by the BOX layer and led to +the edges of the chip. +A close-up scanning electron microscope (SEM) image of an H1 nanocavity is shown as +Fig. 4(b). Typical radii for the innermost and second innermost hole layers of the resist mask +were estimated as 𝑅1,m ≈ 102.8 nm and 𝑅0,m ≈ 130.4 nm, respectively, which were close to +the condition for 𝑄th > 108 found in Fig. 2. However, the radii of the fabricated samples +became somewhat bigger in the etching process: 𝑅1,s ≈ 106.8 nm and 𝑅0,s ≈ 133.1 nm. We +prepared PCN chips with five distinct lattice constants, 𝑎 = 418, 422, 426, 430, 434 nm. For +the evaluations, we focused on the one with 𝑎 = 434 nm, because it best compensated for the +discrepancies in hole radii between the design and fabrication. +We performed transmission measurements on each sample chip by placing it on a metallic +stage whose temperature was maintained at 25◦C by a Peltier element and a PID controller. +Tapered optical fibers were carefully aligned by using three-axis nano-positioners equipped with +fiber holder stages, so that they were coupled with the slab waveguides at both ends of the chip +and hence formed a measurement channel. The typical coupling loss per such interface was about +10 dB. A coherent transverse electric (TE) polarized light from a tunable laser was injected into +each sample. The output was detected by a power meter synchronized with the wavelength sweep +of the laser. The transverse magnetic (TM) field components of the input and output signals were +filtered out by fiber polarizers. The entire system was based on polarization-maintaining fibers. +We prepared and measured a pair of H1 nanocavity samples with nominally the same structure +for each of 𝑠1; namely the shifts of the innermost holes varied from 89.5 to 101.5 nm in units + +of 1 nm. All these 26 samples had 𝑠2 = 20.5 nm and 𝑑 = 5 +√ +3𝑎. A transmission spectrum of +an H1 nanocavity with 𝑠1 = 99.5 nm is plotted in Fig. 4(c). The experimental data shown as +blue points match the Lorentzian curve (colored red) obtained by a least squares fitting. The +peak frequency (wavelength) was 190.4315 THz (1575.370 nm), and the linewidth of the best-fit +curve was 173.8 MHz. These values give an experimental loaded 𝑄 factor of 𝑄exp = 1.1 × 106. +Here, we have excluded any arbitrariness in determining 𝑄exp of the measured resonance with +discrete data points. The input power was attenuated so that thermal linewidth broadening and +nonlinearity would be avoided. In this case, however, the detection power around resonance tails +tended to be slightly reduced, as indicated by its visible drop near 190.4319 THz. This is because +the power meter had a limited dynamic range with a minimum detectable power of -80 dBm. +We can certainly identify this resonance to be the hexapole mode, because the other cavity +modes typically have 𝑄th < 20000 in our simulations and their wavelength spacing with respect +to the ultrahigh-𝑄 peak is 30 nm or larger. +3.2. +Measured wavelengths and quality factors of H1 PCNs +Figure 5(a) presents the dependence of the measured resonance wavelengths 𝜆 of the hexapole +modes on 𝑠1. To show the correspondence between the data of 𝜆 and 𝑄exp, we divided the +samples into two sets according to their positions, so that each sample in set 1 is closer to the +front edge of the chip than its counterpart in set 2 with the same 𝑠1. It can be clearly seen that +𝜆 is positively correlated with 𝑠1, as predicted in Fig. 2(a) and (c). The variation in 𝜆 within +pairwise samples for each 𝑠1 is so weak that a linear regression of the entire data, shown by the +red line, reproduces their average trend. The slope of the regression line is 1.55 ± 0.032 nm (𝜆) / +nm (𝑠1), and its coefficient of determination is 𝑅2 = 0.990. +Here we define the difference in resonant wavelength between set 1 and 2 as Δ𝜆(𝑠1) = +𝜆1(𝑠1) − 𝜆2(𝑠1), where 𝜆1(𝑠1) and 𝜆2(𝑠1) are the wavelengths of the samples with 𝑠1 in set 1 +and 2, respectively. Δ𝜆 for all 𝑠1 in Fig. 5(a) are calculated, and then their standard deviation is +found to be 𝜎Δ𝜆 = 0.848 nm. Because 𝜆1(𝑠1) and 𝜆2(𝑠1) ideally have the same value and their +variations should stem from numerous independent and random processes during fabrication, +we assume that they have no covariance. Thus, we can estimate the deviation in 𝜆 to be +𝜎𝜆 = [𝜎2 +Δ𝜆/2]1/2 = 0.600 nm. +This result implies that our nanocavities have highly accurate hole positions. Although the +obtained value of 𝜎𝜆 corresponds to a change solely in 𝑠1 of 0.39 nm, in reality, there are other +major factors that affect 𝜎𝜆, such as the hole radii, local Si slab thicknesses and surface roughness. +In addition, the positioning accuracy of the electron beam used in patterning the resist mask is as +small as 0.05 nm. Thus, undesired variations in hole positions, including those in 𝑠1 and 𝑠2, will +be less significant in the actual samples. +The measured loaded 𝑄 factors for the two sample sets are plotted in Fig. 5(b) as a function of +𝑠1. They exhibit a gentle peak centered around 𝑠1 = 94.5 or 95.5 nm; 𝑄exp for these values of 𝑠1 +is significantly larger than that for 𝑠1 = 89.5 and 101.5 nm. The best sample here belongs to set +2 and has 𝑠1 = 96.5 nm and 𝑄exp = 1.2 × 106 with an estimated linewidth of 160.4 MHz. Its +transmission spectrum is shown in the inset of Fig. 5(b). Although the shape of the resonance is +slightly asymmetric, it is still fitted by a Lorentzian function. +Eight samples out of 26 had 𝑄exp > 106. Remarkably, they included ones with 𝑠1 = 90.5 and +99.5 nm, namely off from the peak center. This trend implies that the 𝑄 factors for these PCNs +are much larger in theory but were reduced because of fabrication imperfections. The effect of +disorder is also reflected in the outlier sample with a low 𝑄exp = 3.0 × 105 and 𝑠1 = 96.5 nm in +set 1. + +89.5 +91.5 +93.5 +95.5 +97.5 +99.5 101.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 + Set 1 + Set 2 +Loaded Q factor (×106) +s1 (nm) +89.5 +91.5 +93.5 +95.5 +97.5 +99.5 101.5 +1.558 +1.562 +1.566 +1.570 +1.574 +1.578 +1.582 + Set 1 + Set 2 +Resonant wavelength (µm) +s1 (nm) +89.5 +91.5 +93.5 +95.5 +97.5 +99.5 101.5 +1.558 +1.562 +1.566 +1.570 +1.574 +1.578 +1.582 +Resonant wavelength (µm) +s1 (nm) +89.5 +91.5 +93.5 +95.5 +97.5 +99.5 101.5 +106 +107 +108 + Unloaded + Loaded + WG coupling +Q factor +s1 (nm) +(a) +(b) +(c) +(d) +191.111 191.112 191.113 +0.0 +0.5 +1.0 +Normalized power +Frequency (THz) +Qexp = 1.2×106 +Fig. 5. (a) Dependence of measured 𝜆 on 𝑠1 for two nominally duplicate sets of H1 PCN +samples with 𝑎 = 434 nm, 𝑠2 = 20.5 nm, and 𝑑 = 5 +√ +3𝑎. The grouping of the samples +into sets is based on their positions relative to the front edge of chip (the samples in set +1 are closer to the edge). The red line is a linear regression of the experimental data. +(b) Loaded 𝑄 factor (𝑄exp) as a function of 𝑠1 for the two sample sets. The inset is the +transmission spectrum for the best sample that had 𝑄exp = 1.2 × 106 and 𝑠1 = 96.5 +nm. (c) Simulated 𝜆(𝑠1) for 𝑎 = 434 nm, 𝑡 = 241 nm, 𝑅1 = 106 nm, 𝑅0 = 134 nm, +and 𝑠2 = 20.5 nm, which agrees well with the experimental data. (d) Simulated 𝑄 +factors for the same parameters on a semi-logarithmic scale. Squares show results for +unloaded samples (𝑄th), while dots are for loaded ones (𝑄th,L) including two W1 PhC +waveguides with 𝑑 = 5 +√ +3𝑎 that radiate out the light. Triangles show the 𝑄 factors +𝑄WG due to the losses by the waveguides. +3.3. +Simulation of measured samples +We performed simulations by varying the structural parameters around those estimated from the +SEM image. Figure 5(c) shows the theoretical 𝜆 as a function of 𝑠1 for 𝑎 = 434 nm, 𝑡 = 241 +nm, 𝑅1 = 106 nm, 𝑅0 = 134 nm, and 𝑠2 = 20.5 nm. The theoretical values agree well with the +experimental data. Although the simulation result is slightly convex upward, its average slope +(1.55 nm (𝜆) / nm (𝑠1)) coincides with that of the experimental result. We emphasize that 𝑅1 +and 𝑅0 here are consistent with the measured 𝑅1,s and 𝑅0,s within an error of a few nanometers, +as expected for the current measurement. The value of 𝑡 is smaller than the nominal thickness +250 nm of the Si film, indicating that the PhC slabs were thinned down by the etching processes +and/or that 𝑛Si in the simulation is slightly smaller than that of the actual material. +Moreover, as shown in Fig. 5(d), the corresponding theoretical 𝑄 factors follow the trend seen + +in the experiment. The figure compares 𝑄th for the H1 PCNs with and without two W1 PhC +waveguides with 𝑑 = 5 +√ +3𝑎 extending to the right and left sides of the simulation domain where +the fields are scattered out. The plots are on a semi-logarithmic scale, with the horizontal axis +depicting steps of 1 nm. The loaded 𝑄 factors, 𝑄th,L, are the black dots, and the unloaded ones, 𝑄th, +are the purple squares. Both plots peak at 𝑠1 = 96.5, where 𝑄th,L = 5.9×106 and 𝑄th = 5.9×107. +The loaded hexapole mode for this condition has a theoretical modal volume of 𝑉 = 0.74(𝜆/𝑛)3. +Thus, our best experimental sample is expected to have had 𝑄exp/𝑉 = 1.6 × 106(𝑛/𝜆)3. +The difference between 𝑄th,L and 𝑄th comes from the coupling with the environment via +the waveguides. The impact of this coupling, 𝑄WG, can be derived from the relation 1/𝑄th,L = +1/𝑄th + 1/𝑄WG. The resultant values are plotted as the triangles in Fig. 5(d). They exhibit a +moderate variation with 𝑠1 probably due to the group velocity dispersion of the waveguides +and are about 𝑄WG = 6.6 × 106 around the peak of 𝑄th. As a result, the intrinsic (unloaded) 𝑄 +factor of the optimum sample is estimated to be 𝑄i = [1/𝑄exp − 1/𝑄WG]−1 = 1.5 × 106. The +correspondent 𝑄/𝑉 amounts to 𝑄i/𝑉 = 2.0 × 106(𝑛/𝜆)3, which is comparable with those of +PCNs without having their surface Si passivated with hydrogen [28,56,57,68]. +3.4. +Impact of varying hole radii +We can see that 𝑄exp is still lower than 𝑄th,L and hence it is expected to be affected by reductive +factors other than 𝑄WG. A simple but realistic cause of extra loss is radiative scattering induced +by random variations in the radii and positions of the air holes [55, 69]. The hole radii can +change on the atomic scale order because of stochastic processes in fabrication, such as in the +EB exposure, resist development, and dry and wet etching. On the other hand, the EB shots are +precisely aligned in our lithography process. Thus, the positions of the hole centers are mainly +affected by the small and probabilistic anisotropy of etching or distortion in the shapes of the +holes, part of which is also considered to impact the radii. +Here, we simulated samples with air holes just of varying radii to statistically evaluate the +effect of fabrication imperfections on the 𝑄 factor. The result estimates a dominant portion of the +disorder-induced scatting loss denoted as 1/𝑄scat. We used the parameters that reproduce 𝜆 of +the measured samples and set 𝑠1 = 96.5 nm for 𝑄th = 5.9 × 107 without structural imperfections +or PhC waveguides. The PEC boundary condition of the 𝑦-𝑧 plane was removed so that the +simulation explicitly included all the holes. The small and large holes were assumed to have +random radii sampled from Gaussian distributions with means 𝑅1 and 𝑅0, respectively, and a +common standard deviation (SD) of 𝜎𝑟. The 𝑄 factor obtained in each run is denoted as 𝑄th,F +and satisfies 1/𝑄th,F = 1/𝑄th + 1/𝑄scat. +Figure 6(a) and (b) show 𝜆 and 𝑄th,F for 100 random patterns with 𝜎𝑟 = 1.0 nm. The data +points of both plots look randomly scattered. The mean and SD of the resonant wavelengths are +(𝜇𝜆, 𝜎𝜆) = (1.57084 µm, 1.052 nm) and those of the 𝑄 factors are (𝜇𝑄, 𝜎𝑄) = (2.3 × 106, 1.07 × +106). The wavelengths tend to be distributed symmetrically around 𝜇𝜆, while the 𝑄 factors are +specifically high for some sample points, indicating distinct statistical properties. +We repeated the random simulations for different 𝜎𝑟. The dependence of (𝜇𝜆, 𝜎𝜆) on 𝜎𝑟 and +that of (𝜇𝑄, 𝜎𝑄) are plotted in Fig. 6(c) and (d), respectively. The mean wavelength for each +𝜎𝑟 is mostly convergent at 𝜆 = 1.5710 µm, which is obtained for the case of no disorder. The +deviation in 𝜆 grows proportionally with 𝜎𝑟. The variance of the radii 𝜎2 +𝑟 is directly related to +that of the effective dielectric constant of the PhC slab via the filling fraction of the air holes. +Thus, 𝜎𝑟 affects the deviation of the effective index and has an approximately linear dependence +on 𝜎𝜆. Its slope is estimated as 𝜎𝜆/𝜎𝑟 = 1.11. +In contrast, both 𝜇𝑄 and 𝜎𝑄 tend to be inversely proportional to 𝜎2 +𝑟 . As discussed in Ref. [70], +local variations in the dielectric constant affect the extra scattering rate and hence the loss. By + +20 +40 +60 +80 +100 +1 +0 +2 +4 +6 +8 +10 +Q factor (×106) +Fluctuation pattern index +20 +40 +60 +80 +100 +1 +1.568 +1.569 +1.570 +1.571 +1.572 +1.573 +1.574 +Wavelength (µm) +Fluctuation pattern index +0.0 +0.5 +1.0 +1.5 +2.0 +0 +2 +4 +6 +Mean Q factor µQ (×106) +Deviation of radii σr (nm) +0 +1 +2 +3 +Deviation of Q factor σQ (×106) +0.0 +0.5 +1.0 +1.5 +2.0 +1.568 +1.569 +1.570 +1.571 +1.572 +1.573 +1.574 +Mean wavelength µλ(µm) +Deviation of radii σr (nm) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +Deviation of wavelength σλ (nm) +(a) +(b) +(c) +(d) +Fig. 6. (a) Simulated resonant wavelengths and (b) unloaded 𝑄 factors of H1 PCNs with +100 different random patterns of hole radii for 𝜎𝑟 = 1.0 nm. (c) Mean and standard +deviation of the resonant wavelength (𝜇𝜆, 𝜎𝜆) and (d) those of the 𝑄 factor (𝜇𝑄, 𝜎𝑄) +of the random simulation for different 𝜎𝑟. 𝜇𝜆(𝜎𝑟) converges at the result without +any disorder shown as the black line, while 𝜎𝜆(𝜎𝑟) grows linearly, as indicated by +the regression line in red. Both 𝜇𝑄 and 𝜎𝑄 are inversely proportional to 𝜎2𝑟 . The +approximate statistical properties of the scattering loss are given by Eqs. (2) and (3). +The mean 𝑅0 and 𝑅1 are 134 nm and 106 nm, respectively. The other parameters are +the same as those used for Fig. 5. +subtracting 1/𝑄th from 1/𝑄th,F of the data, the approximate mean and SD of 1/𝑄scat are given by +𝜇[1/𝑄scat] = 6.3 × 10−7𝜎2 +𝑟 , +(2) +𝜎[1/𝑄scat] = 3.3 × 10−7𝜎2 +𝑟 , +(3) +where 𝜎𝑟 is measured in nanometers. +Similar properties have been reported in multi- +heterostructure nanocavities with variations in the positions and radii of the air holes [55,69]. +As mentioned in the discussion of Fig. 5(a), the experimental data suggest 𝜎𝜆 = 0.600 nm. +This value corresponds to 𝜎𝑟 = 0.54 nm via the proportional relation between 𝜎𝜆 and 𝜎𝑟. By +substituting the value of 𝜎𝑟 into Eqs. (2) and (3), we obtain 𝜇[1/𝑄scat] = 1.8 × 10−7 and +𝜎[1/𝑄scat] = 9.6 × 10−8, as the estimated statistical properties of the scattering loss for the +measured samples. The resultant mean 𝑄scat is 5.4 × 106. We should emphasize that we did not +underestimate 𝑄scat by neglecting inaccuracies in the hole positions. The variation in wavelength +in the experiment is attributed solely to 𝜎𝑟, and its entire impact is hence taken into consideration +in obtaining 𝑄scat. +Because the mean 𝑄exp is 𝜇[𝑄exp] ≈ 106 around the optimal condition, this result indicates the +existence of further loss in the experiment with an average 𝑄 factor of (𝜇[1/𝑄exp] − 𝜇[1/𝑄scat] − + +12 +14 +16 +18 +20 +22 +24 +26 +28 +0 +1 +2 +3 +4 +5 +Qth (×108) +s2 (nm) +12 +14 +16 +18 +20 +22 +24 +26 +28 +1.50 +1.52 +1.54 +1.56 +1.58 +1.60 +1.62 +Resonant wavelength (µm) +s2 (nm) +log10(|(Ex)|) +0 +kx (units of π/a) +0 +-4 +4 +4 +-4 +ky (units of π/a) +88 +90 +92 +94 +96 +98 +94.6 +96.8 +99.0 +101.2 +103.4 +105.6 +125.0 +127.5 +130.0 +132.5 +135.0 +R 1 (nm) +R0 (nm) +s1 (nm) +Start +Qth = 9.2×10 +6 +Optimum +Qth = 3.1×10 +8 +Qth = 1.3×10 +8 +Qth = 2.1×10 +8 +(a) +(b) +(c) +(d) +Fig. 7. (a) Evolution of (𝑅0, 𝑅1, 𝑠1) in the Nelder-Mead optimization of 𝑄th for +𝑠2 = 23 nm. Blue arrows indicate the direction of the parameter variation. (b) +log10(|F (𝐸𝑥(r))|) for the optimized hexapole mode for 𝑠2 = 23 nm. The radiative +component lying inside the LC is reduced, compared with Fig. 3. The black dashed +circle denotes the light line. (c) 𝜆 and (d) 𝑄th of the optimized H1 PCNs for different +𝑠2. Both of them tend to be positively correlated with 𝑠2. We obtained 𝑄th = 4.5 × 108 +for the optimized variables (𝑅0, 𝑅1, 𝑠1) ≈ (115.92 nm, 90.258 nm, 85.773 nm) for +𝑠2 = 26 nm. Other fixed parameters are 𝑎 = 426 nm and 𝑡 = 250 nm. +1/𝑄WG)−1 ≈ 1.5 × 106. We attribute part of this loss to a slight amount of EB resist remaining +on the sample. Considering that the laser scope comes into focus twice in scanning the surface, +it is expected to form a very thin layer over the chip. This results in structural asymmetry in +the out-of-plane direction and hence induces extra radiation loss, as is the case with samples +fabricated on sacrificial layers. Its unevenness, which can be seen at the top right of Fig. 4(b) for +example, could also be a source of scattering. We did not try to remove the resist layer from the +chip, because such a process unavoidably thins down the Si layer and thus alters the dependence +of the resonance properties on 𝑠1. The sample quality will be improved in future studies. +4. +Automated optimization +Recent studies have used various automated optimization algorithms to achieve high theoretical +𝑄 factors in PCNs [54, 56, 62–65]. We used the built-in optimization module of COMSOL +Multiphysics and found that the performance of the H1 PCN can further be improved. Here, we +chose the Nelder-Mead method [71], which prepares a symplex in a parameter space and repeats + +0 +-0.5 +-1 +-1.5 +-2 +-2.5 +-3 +-3.5 +-4 +-4.5 +-5its update based on the reflection, expansion, contraction, or shrink process, depending on the +value of the function 𝐹 to be optimized. This scheme does not use any gradient or assume any +approximate form of the function. Thus, it is expected to work regardless of the actual landscape +of 𝐹. We fixed 𝑠2 and obtain a maximal 𝑄th by varying 𝑅0, 𝑅1 and 𝑠1 in each optimization run, +namely 𝐹 = 𝑄th(𝑅0, 𝑅1, 𝑠1). +Figure 7(a) shows the evolution of the parameters in the optimization for 𝑠2 = 23 nm, 𝑎 = 426 +nm, and 𝑡 = 250 nm. Here, the initial point was set as (𝑅0, 𝑅1, 𝑠1) = (128.3 nm, 99.5 nm, 89.4 nm) +with 𝑄th = 9.2 × 106. The variables undergo substantial changes at steps in the early stage +of the operation. The state passes through a condition for 𝑄th > 108 and is then bound in a +region of suboptimal points with 𝑄th < 2 × 108. After a while, however, the algorithm finds +a direction in which 𝑄th is improved beyond 2 × 108. It eventually settles at (𝑅0, 𝑅1, 𝑠1) ≈ +(125.18 nm, 97.421 nm, 89.024 nm) exhibiting the optimum objective, 𝑄th = 3.1 × 108. The +normalized absolute Fourier amplitudes of 𝐸𝑥 for this optimal mode are depicted on a logarithmic +scale in Fig. 7(b). Compared with Fig. 3(d), the domain with the relative amplitudes below 10−5 +in the LC is doubly extended in the 𝑘𝑥 direction. This feature confirms that the light confinement +of this H1 PCN is stronger than that of the manually designed ones shown in Sec. 2. +We repeated the optimization routine with different values of 𝑠2, which is the additional +factor not in the former design examined in Fig. 3(a) and (c). To understand quantitatively +the impact of 𝑠2, we plot the dependences of 𝜆 and 𝑄th of the optimized PCN in Fig. 7(c) +and (d). The resonant wavelength is monotonically red-shifted as 𝑠2 increases. Accordingly, +a larger 𝑠2 results in a higher optimal 𝑄 factor. We find that 𝑄th = 4.5 × 108 for 𝑠2 = 26 nm, +which is more than a hundred-times the values in the previous reports [33, 54]. Remarkably, +the optimized mode also has a small volume of 𝑉opt = 0.71(𝜆/𝑛)3, and thus its 𝑄/𝑉 is as large +as 𝑄th/𝑉opt = 6.3 × 108(𝑛/𝜆)3. This result confirms the striking contribution of the gradual +variation in the optical potential introduced by 𝑠2 to 𝑄th, as mentioned in Sec. 2. +The optimal structural parameters vary greatly with 𝑠2. +We obtained (𝑅0, 𝑅1, 𝑠1) ≈ +(144.23 nm, 111.61 nm, 86.020 nm) and (115.92 nm, 90.258 nm, 85.773 nm) for 𝑠2 = 13 +nm and 26 nm, respectively. 𝑅0 and 𝑅1 tend to be negatively correlated with 𝑠2 and 𝜆, while 𝑠1 +oscillates gently between 82 nm and 92 nm with respect to 𝑠2. Optimization with more parameters +such as (𝑅0, 𝑅1, 𝑠1, 𝑠2, 𝑎) might result in a better 𝑄th. In that case, however, the parameter space +would become larger and contain more local minima of 𝑄th. Thus, the computation would be +much harder in terms of both its convergence and the probability of finding a good solution. We +leave that consideration out of the scope of this study. +5. +Discussion +Experimental 𝑄 factors of PCNs are generally limited by many kinds of defects. Discussing their +impact will allow us to predict how high 𝑄exp could be made in a real PCN device. +A major cause of the reduction of 𝑄 factors is structural imperfections. In our result, the +variations in 𝜆 and 1/𝑄th,F were attributed to those in the hole radii, and 𝜎𝑟 = 0.54 nm and +𝜇[1/𝑄scat] = 1.8 × 10−7 were obtained. A groundbreaking report by Asano et al. on multi- +heterostructure PCNs [72], including one with 𝑄exp = 1.1 × 107, considered the same deviation +𝜎hole in both the positions and radii of the air holes. They estimated 𝜎hole to be 0.25 nm and the +corresponding 𝜇[1/𝑄scat] to be 4.7 × 10−8 for their PCN samples. A monolayer of Si is about +0.135-nm-thick and an air hole has two side walls in the radial direction. Thus, 𝜎hole = 0.25 nm +seems to indicate that the etching process just leaves the uncertainty at the level where a single +atomic layer is removed or not at every Si surface, including the resultant hole displacement. +Both Eq. (2) and the dependence of 𝜇[1/𝑄scat] on 𝜎hole in Ref. [72] are quadratic equations +and have similar coefficients. Even though 𝜎𝑟 and 𝜎hole of the two PCNs can be reduced to the +monolayer level (= 0.135 nm), a dimensionless loss of about 𝜇[1/𝑄scat] ≈ 10−8 remains. This +implies that it is hard to achieve 𝜇[𝑄scat] > 108 for PCNs. + +Another limiting factor is the formation of surface oxidation layers on Si. Every Si/SiO𝑥 +interface has a few kinds of surface states whose spectral densities of states are within the band +gap of Si [73]. They exhibit optical absorption at telecommunication wavelengths (≈ 0.8 eV) and +are known to significantly increase loss in Si photonic devices [74]. This detrimental effect can +be circumvented by passivating Si surfaces with hydrogen via HF etching [75,76]. However, the +Si-H termination is not stable and the surfaces hence suffer from natural oxidation in ambient +conditions. Thus, a combination of this process and subsequent measurement of the samples +in an inert gas-purged chamber seems to be needed in order to achieve 𝑄exp > 107 [72]. For +heterostructure PCNs with oxide layers [77], the inverse of the 𝑄 factor based on absorption +(1/𝑄abs) was estimated to be about 1/(7 × 106) = 1.43 × 10−7, and a large part of it seemed +to stem from the surface states. Although water molecules that adhere to sample surfaces also +induce absorption loss, their impact appears to be an order of magnitude smaller. Repeating the +formation and removal of SiO𝑥 layers can also reduce the surface roughness and hence suppress +extra scattering loss [78,79]. Performing such a process on the bottom surface of Si may also be +helpful in removing dopant contamination that could concentrate around the interface between +the Si and BOX layers [72,80]. +Overall, the 𝑄exp achievable for practical PCNs in air seems to be limited to below 107; with +the hydrogen passivation 𝑄exp may reach on the order of 107. Because PCNs can have such a +high 𝑄/𝑉 coefficient, we should mention that they would also be subject to fluctuations in the +refractive index caused by thermal noise, which induce their linewidth broadening [81]. Although +PCNs are not so affected by ambient temperature, thermal noise may become a problem when +they absorb the injected light. Our experiment showed a symptom of the linewidth broadening, +when the measured transmission power exceeded 1 nW. This feature is attributed to heat, since it +appears as a precursor of bistable transmission based on thermo-optic nonlinearity. A similar +result was seen in a previous report [34]. PCNs with larger 𝑄exp than ours might need a smaller +probe power to avoid it. In that case, a time-resolved ("ring-down") measurement with a pulsed +excitation might be useful [82]. +6. +Conclusion +The theoretical and experimental 𝑄 factors of our hexapole H1 PCNs were 𝑄th > 108 and +𝑄exp > 106. Thanks to the 𝐶6 symmetry of the hexapole mode, our design required optimization +of only four structural modulation parameters. Bands of valid conditions for 𝑄th ⪆ 108 were +found in both the (𝑠1, 𝑅1) and (𝑠1, 𝑠2) parameter spaces. The field distributions of such modes +indicated stronger light confinement in both the in-plane and out-of-plane directions compared +with the previous design that did not use 𝑠2. In the experimental demonstration, the Si H1 PCN +samples exhibited a systematic change in their resonant wavelengths when varying the radial shift +of the innermost holes 𝑠1 in steps of 1 nm. Their maximum loaded 𝑄 factor was 𝑄exp = 1.2 × 106, +and the corresponding cavity’s intrinsic 𝑄 factor was 𝑄i = 1.5 × 106. Repeating an automated +optimization with (𝑅0, 𝑅1, 𝑠1) for different values of the radial shift of the second innermost +holes 𝑠2 resulted in 𝑄th = 4.5 × 108, a more than a hundred-fold improvement compared with +the previous studies. We also discussed some of the major elements that degrade 𝑄exp in reality +and estimated the order of practically obtainable 𝑄exp. Our work spotlights the power of modal +symmetry for improving the performance of nanocavities. It also shows the potential of the H1 +PCN in various applications such as functional photonic devices, quantum information processing, +and large-scale one- and two-dimensional resonator lattices for studying non-Hermitian and +topological photonics and other emergent topics. +Funding. +JSPS KAKENHI Grant Number JP20H05641. +Acknowledgements. +We thank Toshiaki Tamamura, Junichi Asaoka, Osamu Moriwaki, Toshifumi +Watanabe and Mizuki Ikeya for support with the sample fabrication. We are also grateful to Hideaki +Taniyama for support with the complemental FDTD simulation and Shota Kita for fruitful discussion. + +Disclosures. +The authors declare no conflicts of interest. +Data availability. +Data underlying the results presented in this paper are not publicly available at this +time but may be obtained from the authors upon reasonable request. +References +1. +J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='jp 5masaya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='notomi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='mn@hco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='ntt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='jp Abstract: An H1 photonic crystal nanocavity is based on a single point defect and has eigenmodes with a variety of symmetric features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Thus, it is a promising building block for photonic tight-binding lattice systems that can be used in studies on condensed matter, non- Hermitian and topological physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' However, improving its radiative quality (𝑄) factor has been considered challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Here, we report the design of a hexapole mode of an H1 nanocavity with a 𝑄 factor exceeding 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We achieved such extremely high-𝑄 conditions by designing only four structural modulation parameters thanks to the C6 symmetry of the mode, despite the need of more complicated optimizations for many other nanocavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The fabricated silicon photonic crystal nanocavities exhibited a systematic change in their resonant wavelengths depending on the spatial shift of the air holes in units of 1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Out of 26 such samples, we found eight cavities with loaded 𝑄 factors over one million (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='2 × 106 maximum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We examined the difference between the theoretical and experimental performances by conducting a simulation of systems with input and output waveguides and with randomly distributed radii of air holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Automated optimization using the same design parameters further increased the theoretical 𝑄 factor by up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 × 108, which is two orders of magnitude higher than in the previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Our work elevates the performance of the H1 nanocavity to the ultrahigh-𝑄 level and paves the way for its large-scale arrays with unconventional functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' © 2023 Optica Publishing Group 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Introduction Photonic crystal nanocavities (PCNs) in dielectric slabs are a particular series of optical resonators that exhibit both strong light confinement and small modal volumes [1–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' These features enable intense light-matter interactions, which make PCNs very useful for extremely low-power photonics [13–15], on-chip nonlinear optics [16–18] and quantum optics [19–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Integration of PCNs also opens a route to functional nanophotonic devices, such as slow light waveguides [22–24], and all-optical switches [25–27], memories [28–30], and transistors [31], which are potential for information processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' An H1 PCN comprises a vacancy of a single lattice element [32–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Such a point defect structure takes over the spatial symmetry of its host system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Thus, the eigenmodes of the Maxwell equations for the H1 nanocavity are also those for the symmetry operations in the entire point group of the photonic crystal (PhC) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' As a result, they are analogous to atomic orbitals in terms of their symmetric properties, and hence, coupled H1 PCNs work as good photonic emulators of molecules and tight-binding lattices including basis functions [22,37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='02376v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='optics] 6 Jan 2023 Because their evanescent couplings, resonant frequencies and radiation losses can be controlled by structural modulation, PCNs can also be combined with unconventional functionalities emerging in non-Hermitian and topological physics [38–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In particular, arrays of H1 PCNs may pave the way for large-scale two-dimensional crystalline systems [48–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This potential is in stark contrast to most other PCNs based on linear defects, which are less symmetric and thus limited in their coupling profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' However, it is generally more difficult for a smaller PCN to have an ultrahigh 𝑄 factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Narrower field distributions in real space result in broader ones in reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Parts of such modes tend to reside in the light cone (LC) and hence turns into radiation fields, namely losses [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We showed two decades ago that a hexapole mode of the H1 nanocavity in a triangular-lattice PhC slab could have a theoretical 𝑄 factor up to 3 × 106, unlike the other eigenmodes [32,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' However, this record was not broken even with an algorithmic optimization [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Moreover, the experimental counterpart was an order of magnitude smaller, namely 3 × 105 [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Unfortunately, there values compared disadvantageously to those of PCNs with larger defect regions [55–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The lack of tightest light confinement seems to be a significant obstacle to using large-scale H1 nanocavity arrays, for example, to enhance light-matter interactions with bulky coupled modes, and to make robust optical circuits with topological edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In this article, we design, analyze and experimentally examine the hexapole mode of an H1 PCN with a theoretical 𝑄 factor (𝑄th) over 108, on the basis of our latest prototype for studying non-Hermitian physics [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Structural modulation in the design maintains the C6v symmetry of the PCN, which the hexapole mode also respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' As a result, we find that we can dramatically increase the 𝑄 factor just with four optimization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' By elaborating the dependence of 𝑄th on major three parameters in a simulation, we clarify that such extremely high-𝑄 conditions form a region with some width in the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Here, we obtained a hexapole mode with 𝑄th = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4 × 108 and a modal volume (𝑉) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='72(𝜆/𝑛)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We also compare its field profiles with those of another H1 PCN based on a previous study in real and reciprocal spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We experimentally investigated a series of silicon (Si) H1 PCNs with different spatial shifts of air holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' These samples exhibited a systematic variation in their resonant wavelengths, indicating that undesired variations in the positions of air holes were restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We found that eight such PCNs out of 26 had loaded 𝑄 factors (𝑄exp), which include the effects of the input and output waveguides, of over one million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The best sample had 𝑄exp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='2 × 106, and the cavity’s intrinsic 𝑄 factor (𝑄i) was estimated to be 𝑄i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We also performed a simulation of the system with randomly varying radii of the air holes to close the gap between 𝑄th and 𝑄exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Finally, we performed an automated optimization to further improve 𝑄th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Here, we added the hole radius of the background PhC as a parameter and found 𝑄th = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 × 108, which is more than a hundred times those in the previous design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Our results show that the highly symmetric hexapole mode can achieve both an extremely high 𝑄th and a very small 𝑉 with an inexpensive optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' It enables ultrahigh 𝑄exp (> 106) of H1 PCNs and will open up their various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Section 2 shows the design and modal properties of our H1 PCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Section 3 presents experimental results, and numerically analyzes and discusses them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The automated optimization and resultant impact on the hexapole mode are summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Section 5 discusses fundamental limitations on the 𝑄 factors of nanocavities, including ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Section 6 concludes this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Cavity design 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Structure and scheme Figure 1(a) depicts the design of our PCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The system is based on a Si slab with a refractive index of 𝑛Si = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='47 and thickness 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The PhC here is a triangular lattice of circular air holes of radius 𝑅0 and lattice constant 𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Triangular-lattice PhC slabs are widely used in experiments (a) (b) R0 R0 R1 s1 s2 x y z Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (a) Design of H1 PCN based on structural modulation of the innermost and second innermost layers of air holes with reference to the single point defect (colored red and orange, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 𝑅0 is the radius of the holes for the background PhC and the second layer, and 𝑅1 that for the innermost holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 𝑠1 is a radial shift of the innermost layer directed outward from the lattice points, and 𝑠2 is that for the second innermost layer with its regular hexagonal alignment kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (b) 𝐻𝑧 field distribution of hexapole mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' because they have large photonic band gaps for TE-like modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The lack of a single hole acts as a point defect and hence forms an H1 nanocavity, which is the simplest structure of PCNs that take over the C6v symmetry of the PhC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The six holes closest to the defect, which are colored red in the figure, have a smaller radius 𝑅1 than that of the background PhC (𝑅1 < 𝑅0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This innermost layer of holes is also shifted radially away from the lattice points by a distance 𝑠1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The second innermost hole layer comprises the twelve holes located one layer outward from the innermost ones and is drawn in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' It is also translated in the radial direction so that it keeps the regular hexagonal alignment and its half diagonal is increased by a distance 𝑠2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In addition, it’s holes are of the same radius 𝑅0 as those of the PhC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We computed the complex eigenfrequencies 𝑓 of the hexapole eigenmode for various cases by using the finite element method on a commercial solver (COMSOL Multiphysics [60]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' With the defect center defined as the coordinate origin, the system had 11 and 14 layers of holes in the ±𝑥 and ±𝑦 directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' A rectangular air region with a height of 3 µm was placed on each side of the slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' A scattering boundary condition for plane waves is applied to every border of the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The 𝑥-𝑦 and 𝑦-𝑧 planes were set as perfect magnetic and electrical conductors, respectively, for reducing the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Any changes to these simulation conditions are noted in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The theoretical 𝑄 factor is given by 𝑄th = Re 𝑓 /(2Im 𝑓 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Figure 1(b) shows the 𝑧 component of the magnetic fields (𝐻𝑧) of the hexapole mode along the 𝑥-𝑦 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This mode is TE-like and thus characterized by 𝐻𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' It is also an eigenmode for the C6 rotation operator with an eigenvalue of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Such an odd parity of a symmetric two-dimensional multipole contributes to destructive interference in 𝐻𝑧 along the 𝑧 direction corresponding to Γ point [5, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This feature suppresses radiation loss based on the transverse electric field components (𝐸𝑥, 𝐸𝑦), as they are linked to 𝐻𝑧 through the Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Thus, structural modulation maintaining the lattice-matched rotational symmetry is essential to achieving an ultrahigh 𝑄 factor of the hexapole mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The other C6-symmetric eigenmode of this cavity is the monopole mode (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' It has an eigenvalue of +1 for the C6 operator and a far lower 𝑄th < 3000 in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 1(a), our design uses only four parameters (𝑅0, 𝑅1, 𝑠1, 𝑠2) to improve 85 88 91 94 97 99 101 103 105 107 R1 (nm) s1 (nm) 105 106 107 108 Qth 85 88 91 94 97 99 101 103 105 107 R1 (nm) s1 (nm) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='536 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='547 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='557 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='568 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='578 λ (µm) 85 88 91 94 16 19 22 25 s2 (nm) s1 (nm) 105 106 107 108 Qth 85 88 91 94 16 19 22 25 s2 (nm) s1 (nm) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='548 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='553 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='558 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='563 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='569 λ (µm) (a) (b) (d) (c) Wavelength Q factor Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Dependence of (a) resonant wavelength (𝜆) and (b) theoretical 𝑄 factor (𝑄th) of the hexapole mode on 𝑠1 and 𝑅1 for 𝑠2 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (c) 𝜆 and (d) 𝑄th dependent on 𝑠1 and 𝑠2 for 𝑅1 = 102 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Black dots represent sample points in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The data among the points are linearly interpolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' A band of parameter conditions for 𝑄th > 108 appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 𝑅0 = 131 nm, 𝑎 = 426 nm, and 𝑡 = 250 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' the 𝑄 factor, unlike recent designs based on costly optimizations of many variables [62–65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 𝑅0 determines the filling factor of the PhC, which is related to its photonic band gap and thus the in-plane modal confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 𝑅1, 𝑠1 and 𝑠2 affect the local modal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The lattice constant 𝑎 can be varied to adjust the resonant wavelengths of the simulated modes to telecom ones around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='55 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Resonance properties versus hole shifts First, let us study the resonance characteristics of the mode for constant 𝑅0 = 131 nm, 𝑎 = 426 nm, and 𝑡 = 250 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Figure 2(a) and (b) are two-dimensional color plots of the resonant wavelength 𝜆 = 𝑐/Re 𝑓 and 𝑄th for isolated (unloaded) H1 PCNs depending on 𝑠1 and 𝑅1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Here, 𝑐 is the speed of light in vacuum and 𝑠2 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The plot of 𝜆 indicates that a small 𝑠1 and large 𝑅1 squeeze the magnetic poles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 1(b) and thus yield a short 𝜆, whereas a large 𝑠1 and small 𝑅1 broaden the magnetic poles and thus increase 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Remarkably, the 𝑄th plot exhibits a sequence of optimum points with 𝑄th > 108 forming a linear band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Such a peak distribution indicates that there is an optimal polar width for every 𝜆 that suppresses local scattering-induced radiation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' There is a margin of about ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm in 𝑅1 and a wider one in 𝑠1 from each optimum point to have a 𝑄th > 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The largest 𝑄 factor here is 𝑄th = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='43 × 108 for (𝑠1, 𝑅1) = (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='75 nm, 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='75 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In units of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm for the parameters, 𝑄th = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='41 × 108 for (𝑠1, 𝑅1) = (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm, 102 nm) was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Figure 2(c) and (d) depict the dependence of 𝜆 and 𝑄th on 𝑠1 and 𝑠2 for 𝑅1 = 102 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' There is a notable difference between Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 2(a) and (c) in the directions of the iso-wavelength contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This difference is due to negative correlation between the effect of 𝑅1 and that of 𝑠2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' a larger 𝑠2 results in a longer 𝜆 because of the higher effective index of the cavity region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In contrast, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 2(b) and (d) appear to have more or less similar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' As the mode wavelength increases with 𝑠1, the optimal 𝑠2 also becomes larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 𝑠2 can be used to dramatically improve 𝑄th because it introduces a gradual variation in the effective potential barrier of the PhC [7,66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' However, the trace of the extremely high 𝑄 values in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 2(d) is nearly perpendicular to the contour lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 2(c), meaning that the conditions for a much improved 𝑄th are limited for each 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The peak value of 𝑄th decreases for large and small 𝑠1 because 𝑅1 is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Overall, a global optimization for (𝑅1, 𝑠1, 𝑠2) enables us to find the continuous conditions for 𝑄th > 108 in the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The best 𝑄th here is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='46 × 108 for (𝑠1, 𝑠2) = (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='25 nm, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='75 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Modal properties Next, let us compare the modal shapes in real and reciprocal spaces of the design with 𝑄th > 108 and that in the previous study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Figure 3(a) and (b) show the spatial magnetic intensity distributions on a common logarithmic scale (log10(|H(r)|2)) along 𝑧 = 0 for hexapole modes with 𝑄th = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 × 106 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4 × 108, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The PCN shown in (a) is based on Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' [33] and does not include 𝑠2 in its design with 𝑅0 = 109 nm, 𝑅1 = 100 nm, 𝑠1 = 78 nm, 𝑎 = 435 nm, and 𝑡 = 220 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The other PCN in (b) corresponds to (𝑠1, 𝑅1) = (89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm, 102 nm) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 2(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' A sizable portion of (a) has evanescent fields with relative intensities of about 10−4, and visible components with intensities over 10−8 reach the boundaries of the entire geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In comparison, the optimal mode shown in (b) obviously decays faster from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This means that the current design provides stronger in-plane light confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Figure 3(c) and (d) depict the Fourier transforms of the 𝑥 component of the electric fields on a logarithmic scale (log10(|F (𝐸𝑥(r))|)) along 𝑧 = 0 for the hexapole modes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 3(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Transverse electric field components lying within the LC measure the magnitude of radiation loss, because they can directly couple with radiative plane waves [3,67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 3(c), the previously designed mode has relative Fourier amplitudes of about 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 distributed in the LC defined by the black dashed circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In stark contrast, the radiative field amplitudes are suppressed over the entire LC for the optimized mode shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Their maximum value is about one order of magnitude smaller than that of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 3(c), confirming an improvement in the 𝑄 factor due to the reduction of the radiation flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' A similar trend is seen in the case of 𝐸𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' These modal properties also support the discussion on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 2(b) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The standard Purcell mode volume 𝑉 for PCNs is given by [2] 𝑉 = ∫ 𝜖(r)|E(r)|2𝑑3r max{𝜖(r)|E(r)|2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (1) This definition is accurate in estimating the Purcell effect for high-𝑄 cavities and has been used for comparison purposes in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Interestingly, the effective volume 𝑉opt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='72(𝜆/𝑛)3 for the mode with 𝑄th = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4 × 108 is larger by 9% than that of the previously studied one, 𝑉p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='66(𝜆/𝑛)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The electric energy densities of hexapole modes tend to concentrate mostly on the sides of the innermost air holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' However, the optimized mode distributes more electric energy around the point defect than the mode based on Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' [33] because of the potential modulation by 𝑠2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Thus, it has a reduced maximum energy density or denominator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This result shows that we can dramatically improve 𝑄th of the hexapole mode without sacrificing its small 𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 𝑉opt here is comparable with those of optimized L3 PCNs without hole radius modulation [64,67], while the hexapole mode has a larger 𝑄th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Thus, our H1 PCNs can be expected to have 𝑄exp values as high as those ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In addition, our optimal 𝑄th/𝑉opt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='9 × 108(𝑛/𝜆)3 is slightly better than another L3 nanocavity with 𝑄th/𝑉 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='7 × 108(𝑛/𝜆)3 (𝑄th = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='9 × 108 and 𝑉 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='1(𝜆/𝑛)3) designed by the particle-swarm algorithm [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (a) (b) (d) (c) a = 435 nm a = 426 nm 0 0 4 4 kx (units of π/a) 4 4 0 kx (units of π/a) 0 4 4 4 4 ky (units of π/a) log10(|H|2) log10(|\uf046(Ex)|) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (a) Magnetic field intensity distribution in the logarithmic scale (log10(|H(r)|2)) for the hexpole nanocavity based on the previous work [33] with 𝑎 = 435 nm and 𝑄th = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0×106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (b) Same but for the hexpole mode designed in this study with 𝑎 = 435 nm, 𝑠1 = 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm, 𝑅1 = 102 nm, and 𝑄th = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4 × 108, exhibiting more tightly confined in-plane evanescent fields than in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (c), (d) Absolute Fourier-space distributions of the 𝑥 components of the electric fields on a logarithmic scale (log10(|F (𝐸𝑥(r))|)) for the eigenmodes corresponding to (a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (d) has significantly reduced radiative components inside the light line that is marked by the black dashed curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In summary, we showed designs of H1 PCNs based on a manual or brute-force search for extremely high-𝑄 hexapole modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' By focusing on the case for a constant 𝑅0, we found a series of conditions for 𝑄th > 108 with just three major optimization parameters (𝑅1, 𝑠1, 𝑠2), thanks to the C6 symmetry of the mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Introduction of an optical potential modulation with 𝑠2 resulted in improved light confinement of the optimized mode in both the in-plane and out-of-plane directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This point will be examined quantitatively in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Experimental result and numerical analysis 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Sample fabrication and measurement We fabricated Si H1 PCNs of our design for an experimental demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The sample structures were patterned by electron beam (EB) lithography on a positive EB resist coated on a silicon-on-insulator (SOI) wafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The mask pattern was projected to the Si film with a 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 50 1 2 3 4 5 6 7 8190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4305 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4315 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 Normalized power Frequency (THz) (b) (c) 1 μm Qexp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='1×106 (a) 2 μm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (a) Laser scope image of a sample with 𝑑 = 5 √ 3𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The input and output Si waveguides are broadened and extended to both sides of the sample chip and coupled with lensed fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (b) Close-up SEM image of H1 PCN with 𝑎 = 434 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Typical radii of the small and large air holes are estimated as 𝑅1,s ≈ 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='8 nm and 𝑅0,s ≈ 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (c) Transmission spectrum of sample with 𝑎 = 434 nm and 𝑠1 = 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The Lorentzian curve colored red matches the experimental data shown as blue points and indicates that the cavity has a loaded 𝑄exp of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='1 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' nominal thickness of 250 nm by inductively coupled plasma etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The buried oxide (BOX) layer beneath the PhCs was removed by wet etching with buffered hydrogen fluoride to obtain air-bridged samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' After the above device processes were completed, the wafer was cleaved so that the size of each sample chip was 5 mm × 15 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Figure 4(a) is a laser scope image of a PCN sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The H1 cavity was butt-coupled (loaded) with two W1 PhC waveguides, each of which had a width of 𝑊0 = √ 3𝑎 based on the removal of a single row of air holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The spatial interval 𝑑 between the cavity and them varied with the samples, and ones with 𝑑 = 5 √ 3𝑎 exhibited ultrahigh-𝑄 resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Each W1 waveguide was broadened by 100 nm at either end of the PhC by shifting five pairs of air holes on the sides outward with a stepwise increment of 20 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Consequently, they were efficiently coupled with air-suspended wire waveguides with a width of 𝑊0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' These optical channels were extended farther and connected to 8 µm-wide slab waveguides that were supported by the BOX layer and led to the edges of the chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' A close-up scanning electron microscope (SEM) image of an H1 nanocavity is shown as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Typical radii for the innermost and second innermost hole layers of the resist mask were estimated as 𝑅1,m ≈ 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='8 nm and 𝑅0,m ≈ 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4 nm, respectively, which were close to the condition for 𝑄th > 108 found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' However, the radii of the fabricated samples became somewhat bigger in the etching process: 𝑅1,s ≈ 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='8 nm and 𝑅0,s ≈ 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We prepared PCN chips with five distinct lattice constants, 𝑎 = 418, 422, 426, 430, 434 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' For the evaluations, we focused on the one with 𝑎 = 434 nm, because it best compensated for the discrepancies in hole radii between the design and fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We performed transmission measurements on each sample chip by placing it on a metallic stage whose temperature was maintained at 25◦C by a Peltier element and a PID controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Tapered optical fibers were carefully aligned by using three-axis nano-positioners equipped with fiber holder stages, so that they were coupled with the slab waveguides at both ends of the chip and hence formed a measurement channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The typical coupling loss per such interface was about 10 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' A coherent transverse electric (TE) polarized light from a tunable laser was injected into each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The output was detected by a power meter synchronized with the wavelength sweep of the laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The transverse magnetic (TM) field components of the input and output signals were filtered out by fiber polarizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The entire system was based on polarization-maintaining fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We prepared and measured a pair of H1 nanocavity samples with nominally the same structure for each of 𝑠1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' namely the shifts of the innermost holes varied from 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 to 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm in units of 1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' All these 26 samples had 𝑠2 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm and 𝑑 = 5 √ 3𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' A transmission spectrum of an H1 nanocavity with 𝑠1 = 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The experimental data shown as blue points match the Lorentzian curve (colored red) obtained by a least squares fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The peak frequency (wavelength) was 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4315 THz (1575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='370 nm), and the linewidth of the best-fit curve was 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='8 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' These values give an experimental loaded 𝑄 factor of 𝑄exp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='1 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Here, we have excluded any arbitrariness in determining 𝑄exp of the measured resonance with discrete data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The input power was attenuated so that thermal linewidth broadening and nonlinearity would be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In this case, however, the detection power around resonance tails tended to be slightly reduced, as indicated by its visible drop near 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4319 THz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This is because the power meter had a limited dynamic range with a minimum detectable power of -80 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We can certainly identify this resonance to be the hexapole mode, because the other cavity modes typically have 𝑄th < 20000 in our simulations and their wavelength spacing with respect to the ultrahigh-𝑄 peak is 30 nm or larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Measured wavelengths and quality factors of H1 PCNs Figure 5(a) presents the dependence of the measured resonance wavelengths 𝜆 of the hexapole modes on 𝑠1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' To show the correspondence between the data of 𝜆 and 𝑄exp, we divided the samples into two sets according to their positions, so that each sample in set 1 is closer to the front edge of the chip than its counterpart in set 2 with the same 𝑠1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' It can be clearly seen that 𝜆 is positively correlated with 𝑠1, as predicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 2(a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The variation in 𝜆 within pairwise samples for each 𝑠1 is so weak that a linear regression of the entire data, shown by the red line, reproduces their average trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The slope of the regression line is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='032 nm (𝜆) / nm (𝑠1), and its coefficient of determination is 𝑅2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Here we define the difference in resonant wavelength between set 1 and 2 as Δ𝜆(𝑠1) = 𝜆1(𝑠1) − 𝜆2(𝑠1), where 𝜆1(𝑠1) and 𝜆2(𝑠1) are the wavelengths of the samples with 𝑠1 in set 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Δ𝜆 for all 𝑠1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 5(a) are calculated, and then their standard deviation is found to be 𝜎Δ𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='848 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Because 𝜆1(𝑠1) and 𝜆2(𝑠1) ideally have the same value and their variations should stem from numerous independent and random processes during fabrication, we assume that they have no covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Thus, we can estimate the deviation in 𝜆 to be 𝜎𝜆 = [𝜎2 Δ𝜆/2]1/2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='600 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This result implies that our nanocavities have highly accurate hole positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Although the obtained value of 𝜎𝜆 corresponds to a change solely in 𝑠1 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='39 nm, in reality, there are other major factors that affect 𝜎𝜆, such as the hole radii, local Si slab thicknesses and surface roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In addition, the positioning accuracy of the electron beam used in patterning the resist mask is as small as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='05 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Thus, undesired variations in hole positions, including those in 𝑠1 and 𝑠2, will be less significant in the actual samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The measured loaded 𝑄 factors for the two sample sets are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 5(b) as a function of 𝑠1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' They exhibit a gentle peak centered around 𝑠1 = 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 or 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 𝑄exp for these values of 𝑠1 is significantly larger than that for 𝑠1 = 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 and 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The best sample here belongs to set 2 and has 𝑠1 = 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm and 𝑄exp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='2 × 106 with an estimated linewidth of 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Its transmission spectrum is shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Although the shape of the resonance is slightly asymmetric, it is still fitted by a Lorentzian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Eight samples out of 26 had 𝑄exp > 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Remarkably, they included ones with 𝑠1 = 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm, namely off from the peak center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This trend implies that the 𝑄 factors for these PCNs are much larger in theory but were reduced because of fabrication imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The effect of disorder is also reflected in the outlier sample with a low 𝑄exp = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 × 105 and 𝑠1 = 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm in set 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='2 Set 1 Set 2 Loaded Q factor (×106) s1 (nm) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='558 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='562 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='566 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='570 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='574 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='578 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='582 Set 1 Set 2 Resonant wavelength (µm) s1 (nm) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='558 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='562 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='566 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='570 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='574 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='578 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='582 Resonant wavelength (µm) s1 (nm) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 106 107 108 Unloaded Loaded WG coupling Q factor s1 (nm) (a) (b) (c) (d) 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='111 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='112 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 Normalized power Frequency (THz) Qexp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='2×106 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (a) Dependence of measured 𝜆 on 𝑠1 for two nominally duplicate sets of H1 PCN samples with 𝑎 = 434 nm, 𝑠2 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm, and 𝑑 = 5 √ 3𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The grouping of the samples into sets is based on their positions relative to the front edge of chip (the samples in set 1 are closer to the edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The red line is a linear regression of the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (b) Loaded 𝑄 factor (𝑄exp) as a function of 𝑠1 for the two sample sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The inset is the transmission spectrum for the best sample that had 𝑄exp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='2 × 106 and 𝑠1 = 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (c) Simulated 𝜆(𝑠1) for 𝑎 = 434 nm, 𝑡 = 241 nm, 𝑅1 = 106 nm, 𝑅0 = 134 nm, and 𝑠2 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm, which agrees well with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (d) Simulated 𝑄 factors for the same parameters on a semi-logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Squares show results for unloaded samples (𝑄th), while dots are for loaded ones (𝑄th,L) including two W1 PhC waveguides with 𝑑 = 5 √ 3𝑎 that radiate out the light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Triangles show the 𝑄 factors 𝑄WG due to the losses by the waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Simulation of measured samples We performed simulations by varying the structural parameters around those estimated from the SEM image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Figure 5(c) shows the theoretical 𝜆 as a function of 𝑠1 for 𝑎 = 434 nm, 𝑡 = 241 nm, 𝑅1 = 106 nm, 𝑅0 = 134 nm, and 𝑠2 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The theoretical values agree well with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Although the simulation result is slightly convex upward, its average slope (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='55 nm (𝜆) / nm (𝑠1)) coincides with that of the experimental result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We emphasize that 𝑅1 and 𝑅0 here are consistent with the measured 𝑅1,s and 𝑅0,s within an error of a few nanometers, as expected for the current measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The value of 𝑡 is smaller than the nominal thickness 250 nm of the Si film, indicating that the PhC slabs were thinned down by the etching processes and/or that 𝑛Si in the simulation is slightly smaller than that of the actual material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Moreover, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 5(d), the corresponding theoretical 𝑄 factors follow the trend seen in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The figure compares 𝑄th for the H1 PCNs with and without two W1 PhC waveguides with 𝑑 = 5 √ 3𝑎 extending to the right and left sides of the simulation domain where the fields are scattered out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The plots are on a semi-logarithmic scale, with the horizontal axis depicting steps of 1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The loaded 𝑄 factors, 𝑄th,L, are the black dots, and the unloaded ones, 𝑄th, are the purple squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Both plots peak at 𝑠1 = 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5, where 𝑄th,L = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='9×106 and 𝑄th = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='9×107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The loaded hexapole mode for this condition has a theoretical modal volume of 𝑉 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='74(𝜆/𝑛)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Thus, our best experimental sample is expected to have had 𝑄exp/𝑉 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='6 × 106(𝑛/𝜆)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The difference between 𝑄th,L and 𝑄th comes from the coupling with the environment via the waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The impact of this coupling, 𝑄WG, can be derived from the relation 1/𝑄th,L = 1/𝑄th + 1/𝑄WG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The resultant values are plotted as the triangles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' They exhibit a moderate variation with 𝑠1 probably due to the group velocity dispersion of the waveguides and are about 𝑄WG = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='6 × 106 around the peak of 𝑄th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' As a result, the intrinsic (unloaded) 𝑄 factor of the optimum sample is estimated to be 𝑄i = [1/𝑄exp − 1/𝑄WG]−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The correspondent 𝑄/𝑉 amounts to 𝑄i/𝑉 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 × 106(𝑛/𝜆)3, which is comparable with those of PCNs without having their surface Si passivated with hydrogen [28,56,57,68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Impact of varying hole radii We can see that 𝑄exp is still lower than 𝑄th,L and hence it is expected to be affected by reductive factors other than 𝑄WG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' A simple but realistic cause of extra loss is radiative scattering induced by random variations in the radii and positions of the air holes [55, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The hole radii can change on the atomic scale order because of stochastic processes in fabrication, such as in the EB exposure, resist development, and dry and wet etching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' On the other hand, the EB shots are precisely aligned in our lithography process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Thus, the positions of the hole centers are mainly affected by the small and probabilistic anisotropy of etching or distortion in the shapes of the holes, part of which is also considered to impact the radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Here, we simulated samples with air holes just of varying radii to statistically evaluate the effect of fabrication imperfections on the 𝑄 factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The result estimates a dominant portion of the disorder-induced scatting loss denoted as 1/𝑄scat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We used the parameters that reproduce 𝜆 of the measured samples and set 𝑠1 = 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm for 𝑄th = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='9 × 107 without structural imperfections or PhC waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The PEC boundary condition of the 𝑦-𝑧 plane was removed so that the simulation explicitly included all the holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The small and large holes were assumed to have random radii sampled from Gaussian distributions with means 𝑅1 and 𝑅0, respectively, and a common standard deviation (SD) of 𝜎𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The 𝑄 factor obtained in each run is denoted as 𝑄th,F and satisfies 1/𝑄th,F = 1/𝑄th + 1/𝑄scat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Figure 6(a) and (b) show 𝜆 and 𝑄th,F for 100 random patterns with 𝜎𝑟 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The data points of both plots look randomly scattered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The mean and SD of the resonant wavelengths are (𝜇𝜆, 𝜎𝜆) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='57084 µm, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='052 nm) and those of the 𝑄 factors are (𝜇𝑄, 𝜎𝑄) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='3 × 106, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='07 × 106).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The wavelengths tend to be distributed symmetrically around 𝜇𝜆, while the 𝑄 factors are specifically high for some sample points, indicating distinct statistical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We repeated the random simulations for different 𝜎𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The dependence of (𝜇𝜆, 𝜎𝜆) on 𝜎𝑟 and that of (𝜇𝑄, 𝜎𝑄) are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 6(c) and (d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The mean wavelength for each 𝜎𝑟 is mostly convergent at 𝜆 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5710 µm, which is obtained for the case of no disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The deviation in 𝜆 grows proportionally with 𝜎𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The variance of the radii 𝜎2 𝑟 is directly related to that of the effective dielectric constant of the PhC slab via the filling fraction of the air holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Thus, 𝜎𝑟 affects the deviation of the effective index and has an approximately linear dependence on 𝜎𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Its slope is estimated as 𝜎𝜆/𝜎𝑟 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In contrast, both 𝜇𝑄 and 𝜎𝑄 tend to be inversely proportional to 𝜎2 𝑟 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' As discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' [70], local variations in the dielectric constant affect the extra scattering rate and hence the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' By 20 40 60 80 100 1 0 2 4 6 8 10 Q factor (×106) Fluctuation pattern index 20 40 60 80 100 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='568 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='569 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='570 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='571 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='572 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='573 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='574 Wavelength (µm) Fluctuation pattern index 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 0 2 4 6 Mean Q factor µQ (×106) Deviation of radii σr (nm) 0 1 2 3 Deviation of Q factor σQ (×106) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='568 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='569 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='570 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='571 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='572 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='573 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='574 Mean wavelength µλ(µm) Deviation of radii σr (nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 Deviation of wavelength σλ (nm) (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (a) Simulated resonant wavelengths and (b) unloaded 𝑄 factors of H1 PCNs with 100 different random patterns of hole radii for 𝜎𝑟 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (c) Mean and standard deviation of the resonant wavelength (𝜇𝜆, 𝜎𝜆) and (d) those of the 𝑄 factor (𝜇𝑄, 𝜎𝑄) of the random simulation for different 𝜎𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 𝜇𝜆(𝜎𝑟) converges at the result without any disorder shown as the black line, while 𝜎𝜆(𝜎𝑟) grows linearly, as indicated by the regression line in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Both 𝜇𝑄 and 𝜎𝑄 are inversely proportional to 𝜎2𝑟 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The approximate statistical properties of the scattering loss are given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The mean 𝑅0 and 𝑅1 are 134 nm and 106 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The other parameters are the same as those used for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' subtracting 1/𝑄th from 1/𝑄th,F of the data, the approximate mean and SD of 1/𝑄scat are given by 𝜇[1/𝑄scat] = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='3 × 10−7𝜎2 𝑟 , (2) 𝜎[1/𝑄scat] = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='3 × 10−7𝜎2 𝑟 , (3) where 𝜎𝑟 is measured in nanometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Similar properties have been reported in multi- heterostructure nanocavities with variations in the positions and radii of the air holes [55,69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' As mentioned in the discussion of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 5(a), the experimental data suggest 𝜎𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='600 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This value corresponds to 𝜎𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='54 nm via the proportional relation between 𝜎𝜆 and 𝜎𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' By substituting the value of 𝜎𝑟 into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (2) and (3), we obtain 𝜇[1/𝑄scat] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='8 × 10−7 and 𝜎[1/𝑄scat] = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='6 × 10−8, as the estimated statistical properties of the scattering loss for the measured samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The resultant mean 𝑄scat is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We should emphasize that we did not underestimate 𝑄scat by neglecting inaccuracies in the hole positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The variation in wavelength in the experiment is attributed solely to 𝜎𝑟, and its entire impact is hence taken into consideration in obtaining 𝑄scat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Because the mean 𝑄exp is 𝜇[𝑄exp] ≈ 106 around the optimal condition, this result indicates the existence of further loss in the experiment with an average 𝑄 factor of (𝜇[1/𝑄exp] − 𝜇[1/𝑄scat] − 12 14 16 18 20 22 24 26 28 0 1 2 3 4 5 Qth (×108) s2 (nm) 12 14 16 18 20 22 24 26 28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='52 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='54 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='62 Resonant wavelength (µm) s2 (nm) log10(|\uf046(Ex)|) 0 kx (units of π/a) 0 4 4 4 4 ky (units of π/a) 88 90 92 94 96 98 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='6 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='8 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='2 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='6 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='0 R 1 (nm) R0 (nm) s1 (nm) Start Qth = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='2×10 6 Optimum Qth = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='1×10 8 Qth = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='3×10 8 Qth = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='1×10 8 (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (a) Evolution of (𝑅0, 𝑅1, 𝑠1) in the Nelder-Mead optimization of 𝑄th for 𝑠2 = 23 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Blue arrows indicate the direction of the parameter variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (b) log10(|F (𝐸𝑥(r))|) for the optimized hexapole mode for 𝑠2 = 23 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The radiative component lying inside the LC is reduced, compared with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The black dashed circle denotes the light line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (c) 𝜆 and (d) 𝑄th of the optimized H1 PCNs for different 𝑠2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Both of them tend to be positively correlated with 𝑠2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We obtained 𝑄th = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 × 108 for the optimized variables (𝑅0, 𝑅1, 𝑠1) ≈ (115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='92 nm, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='258 nm, 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='773 nm) for 𝑠2 = 26 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Other fixed parameters are 𝑎 = 426 nm and 𝑡 = 250 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 1/𝑄WG)−1 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We attribute part of this loss to a slight amount of EB resist remaining on the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Considering that the laser scope comes into focus twice in scanning the surface, it is expected to form a very thin layer over the chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This results in structural asymmetry in the out-of-plane direction and hence induces extra radiation loss, as is the case with samples fabricated on sacrificial layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Its unevenness, which can be seen at the top right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 4(b) for example, could also be a source of scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We did not try to remove the resist layer from the chip, because such a process unavoidably thins down the Si layer and thus alters the dependence of the resonance properties on 𝑠1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The sample quality will be improved in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Automated optimization Recent studies have used various automated optimization algorithms to achieve high theoretical 𝑄 factors in PCNs [54, 56, 62–65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We used the built-in optimization module of COMSOL Multiphysics and found that the performance of the H1 PCN can further be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Here, we chose the Nelder-Mead method [71], which prepares a symplex in a parameter space and repeats 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 5its update based on the reflection, expansion, contraction, or shrink process, depending on the value of the function 𝐹 to be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This scheme does not use any gradient or assume any approximate form of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Thus, it is expected to work regardless of the actual landscape of 𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We fixed 𝑠2 and obtain a maximal 𝑄th by varying 𝑅0, 𝑅1 and 𝑠1 in each optimization run, namely 𝐹 = 𝑄th(𝑅0, 𝑅1, 𝑠1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Figure 7(a) shows the evolution of the parameters in the optimization for 𝑠2 = 23 nm, 𝑎 = 426 nm, and 𝑡 = 250 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Here, the initial point was set as (𝑅0, 𝑅1, 𝑠1) = (128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='3 nm, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 nm, 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='4 nm) with 𝑄th = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='2 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The variables undergo substantial changes at steps in the early stage of the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The state passes through a condition for 𝑄th > 108 and is then bound in a region of suboptimal points with 𝑄th < 2 × 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' After a while, however, the algorithm finds a direction in which 𝑄th is improved beyond 2 × 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' It eventually settles at (𝑅0, 𝑅1, 𝑠1) ≈ (125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='18 nm, 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='421 nm, 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='024 nm) exhibiting the optimum objective, 𝑄th = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='1 × 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The normalized absolute Fourier amplitudes of 𝐸𝑥 for this optimal mode are depicted on a logarithmic scale in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Compared with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 3(d), the domain with the relative amplitudes below 10−5 in the LC is doubly extended in the 𝑘𝑥 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This feature confirms that the light confinement of this H1 PCN is stronger than that of the manually designed ones shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We repeated the optimization routine with different values of 𝑠2, which is the additional factor not in the former design examined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 3(a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' To understand quantitatively the impact of 𝑠2, we plot the dependences of 𝜆 and 𝑄th of the optimized PCN in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 7(c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The resonant wavelength is monotonically red-shifted as 𝑠2 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Accordingly, a larger 𝑠2 results in a higher optimal 𝑄 factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We find that 𝑄th = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 × 108 for 𝑠2 = 26 nm, which is more than a hundred-times the values in the previous reports [33, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Remarkably, the optimized mode also has a small volume of 𝑉opt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='71(𝜆/𝑛)3, and thus its 𝑄/𝑉 is as large as 𝑄th/𝑉opt = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='3 × 108(𝑛/𝜆)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This result confirms the striking contribution of the gradual variation in the optical potential introduced by 𝑠2 to 𝑄th, as mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The optimal structural parameters vary greatly with 𝑠2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We obtained (𝑅0, 𝑅1, 𝑠1) ≈ (144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='23 nm, 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='61 nm, 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='020 nm) and (115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='92 nm, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='258 nm, 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='773 nm) for 𝑠2 = 13 nm and 26 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 𝑅0 and 𝑅1 tend to be negatively correlated with 𝑠2 and 𝜆, while 𝑠1 oscillates gently between 82 nm and 92 nm with respect to 𝑠2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Optimization with more parameters such as (𝑅0, 𝑅1, 𝑠1, 𝑠2, 𝑎) might result in a better 𝑄th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In that case, however, the parameter space would become larger and contain more local minima of 𝑄th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Thus, the computation would be much harder in terms of both its convergence and the probability of finding a good solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We leave that consideration out of the scope of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Discussion Experimental 𝑄 factors of PCNs are generally limited by many kinds of defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Discussing their impact will allow us to predict how high 𝑄exp could be made in a real PCN device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' A major cause of the reduction of 𝑄 factors is structural imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In our result, the variations in 𝜆 and 1/𝑄th,F were attributed to those in the hole radii, and 𝜎𝑟 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='54 nm and 𝜇[1/𝑄scat] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='8 × 10−7 were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' A groundbreaking report by Asano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' on multi- heterostructure PCNs [72], including one with 𝑄exp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='1 × 107, considered the same deviation 𝜎hole in both the positions and radii of the air holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' They estimated 𝜎hole to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='25 nm and the corresponding 𝜇[1/𝑄scat] to be 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='7 × 10−8 for their PCN samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' A monolayer of Si is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='135-nm-thick and an air hole has two side walls in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Thus, 𝜎hole = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='25 nm seems to indicate that the etching process just leaves the uncertainty at the level where a single atomic layer is removed or not at every Si surface, including the resultant hole displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Both Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' (2) and the dependence of 𝜇[1/𝑄scat] on 𝜎hole in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' [72] are quadratic equations and have similar coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Even though 𝜎𝑟 and 𝜎hole of the two PCNs can be reduced to the monolayer level (= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='135 nm), a dimensionless loss of about 𝜇[1/𝑄scat] ≈ 10−8 remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This implies that it is hard to achieve 𝜇[𝑄scat] > 108 for PCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Another limiting factor is the formation of surface oxidation layers on Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Every Si/SiO𝑥 interface has a few kinds of surface states whose spectral densities of states are within the band gap of Si [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' They exhibit optical absorption at telecommunication wavelengths (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='8 eV) and are known to significantly increase loss in Si photonic devices [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This detrimental effect can be circumvented by passivating Si surfaces with hydrogen via HF etching [75,76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' However, the Si-H termination is not stable and the surfaces hence suffer from natural oxidation in ambient conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Thus, a combination of this process and subsequent measurement of the samples in an inert gas-purged chamber seems to be needed in order to achieve 𝑄exp > 107 [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' For heterostructure PCNs with oxide layers [77], the inverse of the 𝑄 factor based on absorption (1/𝑄abs) was estimated to be about 1/(7 × 106) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='43 × 10−7, and a large part of it seemed to stem from the surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Although water molecules that adhere to sample surfaces also induce absorption loss, their impact appears to be an order of magnitude smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Repeating the formation and removal of SiO𝑥 layers can also reduce the surface roughness and hence suppress extra scattering loss [78,79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Performing such a process on the bottom surface of Si may also be helpful in removing dopant contamination that could concentrate around the interface between the Si and BOX layers [72,80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Overall, the 𝑄exp achievable for practical PCNs in air seems to be limited to below 107;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' with the hydrogen passivation 𝑄exp may reach on the order of 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Because PCNs can have such a high 𝑄/𝑉 coefficient, we should mention that they would also be subject to fluctuations in the refractive index caused by thermal noise, which induce their linewidth broadening [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Although PCNs are not so affected by ambient temperature, thermal noise may become a problem when they absorb the injected light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Our experiment showed a symptom of the linewidth broadening, when the measured transmission power exceeded 1 nW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' This feature is attributed to heat, since it appears as a precursor of bistable transmission based on thermo-optic nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' A similar result was seen in a previous report [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' PCNs with larger 𝑄exp than ours might need a smaller probe power to avoid it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In that case, a time-resolved ("ring-down") measurement with a pulsed excitation might be useful [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Conclusion The theoretical and experimental 𝑄 factors of our hexapole H1 PCNs were 𝑄th > 108 and 𝑄exp > 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Thanks to the 𝐶6 symmetry of the hexapole mode, our design required optimization of only four structural modulation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Bands of valid conditions for 𝑄th ⪆ 108 were found in both the (𝑠1, 𝑅1) and (𝑠1, 𝑠2) parameter spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The field distributions of such modes indicated stronger light confinement in both the in-plane and out-of-plane directions compared with the previous design that did not use 𝑠2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' In the experimental demonstration, the Si H1 PCN samples exhibited a systematic change in their resonant wavelengths when varying the radial shift of the innermost holes 𝑠1 in steps of 1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Their maximum loaded 𝑄 factor was 𝑄exp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='2 × 106, and the corresponding cavity’s intrinsic 𝑄 factor was 𝑄i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Repeating an automated optimization with (𝑅0, 𝑅1, 𝑠1) for different values of the radial shift of the second innermost holes 𝑠2 resulted in 𝑄th = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content='5 × 108, a more than a hundred-fold improvement compared with the previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We also discussed some of the major elements that degrade 𝑄exp in reality and estimated the order of practically obtainable 𝑄exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Our work spotlights the power of modal symmetry for improving the performance of nanocavities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' It also shows the potential of the H1 PCN in various applications such as functional photonic devices, quantum information processing, and large-scale one- and two-dimensional resonator lattices for studying non-Hermitian and topological photonics and other emergent topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' JSPS KAKENHI Grant Number JP20H05641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We thank Toshiaki Tamamura, Junichi Asaoka, Osamu Moriwaki, Toshifumi Watanabe and Mizuki Ikeya for support with the sample fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' We are also grateful to Hideaki Taniyama for support with the complemental FDTD simulation and Shota Kita for fruitful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Disclosures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' References 1.' metadata={'source': 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Matsuo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Suzaki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Segawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Sato, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Kawaguchi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Takahashi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfdgAR/content/2301.02376v1.pdf'} +page_content=' Notomi, “Ultralow-power all-optical RAM based on nanocavities,” Nat.' metadata={'source': 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undirected graph G whose edge weights change over s time slots, the sub-tree scheduling +for wireless sensor networks with partial coverage asks to partition the vertices of G in s non-empty +trees such that the total weight of the trees is minimized. In this note we show that the problem is NP- +hard in both the cases where s (i) is part of the input and (ii) is a fixed instance parameter. In both +our proofs we reduce from the cardinality Steiner tree problem. We additionally give polynomial- +time algorithms for structured inputs of the problem. +Keywords Wireless sensor network, Sub-tree scheduling, Partial coverage, Complexity +1 +Introduction +A central problem in the management of wireless sensor networks is to extend the lifetime of wireless sensors through +operating policies ensuring energy efficiency and/or balancing. Its importance stems from the fact that even a single +failure of a wireless sensor can in principle compromise the effectiveness of the whole network. From the viewpoint +of energy balancing, a general approach to minimize energy consumption is to split the set of sensors into several +non-empty subsets and to subdivide the planning horizon into as many slots, so that the subsets of sensors are operated +sequentially, one at each time slot. +The sub-tree scheduling for wireless sensor networks with partial coverage (STSWSN-PC), introduced by Adasme +(2019), is a particular implementation of such an approach, with the additional requirements that the sensors operated +simultaneously are mutually connected under a tree topology, and each sensor must be active in a unique time slot. +Namely, the STSWSN-PC is defined on an undirected graph G = (V, E) representing the network of sensors, a +number s, 1 ≤ s ≤ |V |, of time slots, and vectors w1, w2, . . . , ws ∈ RE ++ of edge-weights (one for each time slot). The +aim is to find a set T1, T2, . . . , Ts of non-empty vertex-disjoint trees of G covering V and minimizing �s +i=1 wi(Ti). +In the above description, the vertices of G represent the sensors of the network, the edges represent direct links among +sensors, and the weights represent the time slot-dependent power for transmitting information over the corresponding +edges. +The input of the STSWSN-PC is simultaneously defined by the graph G, the number of time slots s, and the values +of the edge weight vectors. The STSWSN-PC may admit efficient optimization algorithms for structured inputs. +For example, when the weights are constant throughout the time slots (i.e., wi ≡ wj for all i, j = 1, 2, . . . , s), the +STSWSN-PC is solvable in polynomial time, e.g., by using Kruskal’s algorithm (Kruskal, 1956) and terminating it at +the first iteration yielding a spanning forest with s trees; when s = 1, the STSWSN-PC boils down to the minimum +spanning tree (MST) problem on general graphs and, as such, is solvable in polynomial time; when s = |V |, the +optimal solution consists of arbitrarily assigning one vertex to each time slot. +However, unstructured instances of the STSWSN-PC have been tackled in Adasme (2019) and Bianchessi (2022) by +means of branch-and-bound and branch-and-cut algorithms, respectively. These approaches implicitly suggest that the +∗Corresponding author +arXiv:2301.00739v1 [cs.CC] 2 Jan 2023 + +On the complexity of STSWSN-PC +1 +3 +6 +7 +2 +4 +5 +8 +(a) +v1 +1 +v1 +2 +v1 +3 +v3 +1 +v3 +2 +v3 +3 +v6 +1 +v6 +2 +v6 +3 +v7 +1 +v7 +2 +v7 +3 +1 +3 +6 +7 +2 +4 +5 +8 +(b) +Figure 1: Example of a CST instance (a), in which terminal vertices are squared-shaped, and of the corresponding +STSWSN-PC instance for k = 5 (b), in which fictitious vertices are diamond-shaped. +problem is theoretically intractable, although its computational complexity is unknown to the best of our knowledge. +The purpose of this note is to fill in this gap. +In Sect. 2 we study the complexity of the STSWSN-PC when s is part of the input, that is, s is not fixed in +{2, 3, . . . , |V | − 1}; in Sect. 3 we study the complexity under the assumption that s is an instance parameter with +a prescribed value ¯s ≥ 2. Through reductions from the (minimum weight) Steiner tree problem (Garey and Johnson, +1990, p. 208), we show that the STSWSN-PC is NP-hard in both cases, thus justifying the usage of implicit enumera- +tion schemes to solve it. Finally, in Sect. 4 we discuss additional structured inputs, other than those mentioned above, +for which the STSWSN-PC is solvable in polynomial time. +2 +NP-hardness when the number of time slots is not fixed +Given an undirected connected graph G = (V, E) with |V | = n vertices and a subset R ⊂ V of terminal vertices, a +Steiner tree is a subtree T of G such that R ⊆ V (T). Given also a weight w(e) ∈ Z+ for each e ∈ E, computing the +Steiner tree of minimum total edge-weight is in general NP-hard, and the problem remains NP-hard if all weights are +equal (Garey and Johnson, 1990, p. 209). In particular, given w(e) = 1 for each e ∈ E, the pair (G, R), and k ∈ Z+ +with |R| − 1 ≤ k ≤ n − 2, the cardinality Steiner tree (CST) problem consisting of determining the existence of a +Steiner tree of G with at most k edges is NP-complete. +We now show that an oracle solving the STSWSN-PC in polynomial time allows to solve the CST in polynomial time, +thus obtaining that the STSWSN-PC is NP-hard. We point out the CST with at most three terminals can be solved +in polynomial time (Arrighi and de Oliveira Oliveira, 2021), therefore we restrict ourselves to CST instances with +|R| ≥ 4. +Given k and (G, R) defining a CST instance as above, we construct a new graph ¯G = ( ¯V , ¯E) obtained from G by +introducing n−k fictitious vertices vr +1, vr +2, . . . , vr +n−k for each terminal vertex r ∈ R and defining ¯E = E ∪ER, where +ER = {(vr +j, r): j = 1, 2, . . . , n − k, r ∈ R}; that is, each fictitious vertex is connected precisely to the corresponding +terminal vertex. An example of such a construction is given in Figure 1. +Next, we define a STSWSN-PC instance I on graph ¯G and s = n−k+1 time slots. Note that, since |R|−1 ≤ k ≤ n−2 +and |R| ≥ 4, then 3 ≤ s ≤ n − 2 < | ¯V |, hence our definition of the number of time slots excludes the polynomially +solvable cases of the STSWSN-PC. +The weights of the time slots are defined as follows: +w1 +e = +�0 +if e ∈ ER +1 +otherwise +(1) +wj +e = +�n +if e ∈ ER +1 +otherwise +∀j = 2, 3, . . . , n − k + 1 +(2) +Lemma 1. Let T ⋆ +1 , T ⋆ +2 , . . . , T ⋆ +n−k+1 be an optimal solution to I. The restriction of T ⋆ +1 to the vertices in V is a Steiner +tree of G. +2 + +On the complexity of STSWSN-PC +Proof. Assume that the restriction of T ⋆ +1 to the vertices of G is not a Steiner tree. Then there is at least a terminal +vertex r⋆ contained in a tree of a time slot after the first one. Since all trees T ⋆ +1 , T ⋆ +2 , . . . , T ⋆ +n−k+1 are connected, at +least one edge of ER belongs to that time slot. By (2) the optimal solution to I has value at least n. Now we show the +existence of a solution with better value. Namely, in the first time slot we consider the tree T1 spanning all vertices +of ¯G except the n − k fictitious vertices linked to r⋆ and we set Tj = {vr⋆ +j−1} for j = 2, 3, . . . , n − k + 1. Then +T1, T2, . . . , Tn−k+1 is a feasible solution whose value is n − 1 by (1). +Now we can prove the main result. In the proof, given S ⊆ ¯V , we denote by δ(S) its cut, namely, the set of edges +having one endpoint in S and the other endpoint outside S. +Proposition 1. There exists a solution to the CST instance given by k and (G, R) if and only if the optimal solution to +I has value at most k. Therefore the STSWSN-PC is NP-hard. +Proof. For the “if” part assume that there exists an optimal solution having value at most k; denoting by T ⋆ the +restriction of its tree of the first time slot to the vertices in V , the nonnegativity of the weights in (1) yields |T ⋆| ≤ k. +Then the result follows from Lemma 1. +Now, let us prove the “only if” part. Assume that there exists a Steiner tree T of G such that |T| ≤ k. We assume, +without loss of generality, that |T| = k: otherwise we repeatedly update T by adding one edge of G belonging to +δ(T), until reaching the required cardinality (this is always possible as G is connected and since the update always +returns a Steiner tree). Then, let ¯v ∈ ¯V \ V be an arbitrary fictitious vertex, define ˆV = ¯V \ {V ∪ {¯v}} as the set of +remaining fictitious vertices, and let V C = V \ V (T) = {v1, v2, . . . , vn−k−1} be the vertices in the complement of +T in G (as |T| = k, T comprises k + 1 vertices). We consider the feasible solution for I given by T1 = T ∪ δ( ˆV ), +T2 = {¯v} and Tj = {vj−2 ∈ V C} for every j = 3, 4, . . . , n − k + 1. By (1)–(2) such a solution has value k. Then the +optimal solution to I has value at most k. +In the above construction, ¯G is obtained from G by appending leaves to its terminal vertices. This is a minor modifi- +cation of the initial graph, hence the STSWSN-PC remains difficult on those classes of graphs which are closed under +such modification and on which the CST is NP-complete. It is the case of chordal bipartite graphs, that is, bipartite +graphs whose cycles C of length at least 6 induce a subgraph with at least |C| + 1 edges. More precisely we have: +Corollary 1. The STSWSN-PC is NP-hard on bipartite chordal graphs. +Proof. Appending leaves to a subset of vertices of a bipartite chordal graph maintains the chordal bipartiteness. Then +the result follows from the NP-completeness of the CST on bipartite chordal graphs proved by M¨uller and Brandst¨adt +(1987). +3 +NP-hardness when the number of time slots is fixed +In this section we consider the complexity of the STSWSN-PC by assuming that we have s = ¯s time slots, with ¯s ≥ 2 +fixed, and we show that the problem remains NP-hard. +We modify the approach of previous section as follows. Let us consider a graph G = (V, E) with |V | = n vertices +and a set R ⊂ V , |R| ≥ 4, of terminal vertices defining an instance of the CST problem. We define a graph +G⋆ = (V ⋆, E⋆) where V ⋆ = V ∪ V R, with V R = {vr +1, vr +2, . . . , vr +¯s−1 : r ∈ R} being a set of fictitious vertices +associated with those in R, and where E⋆ = E ∪ EC ∪ ER, with EC = {(v, w): v, w ∈ V s.t. (v, w) ̸∈ E} and +ER = {(r, vr +j): r ∈ R, j = 1, 2, . . . , ¯s − 1}. That is, G⋆ is obtained by extending G to a complete graph and by +linking each terminal vertex in G to the corresponding ¯s − 1 fictitious vertices (see Figure 2 for an example). +For every e ∈ E⋆ we define the following edge weights: +w1 +e = +� +� +� +0 +if e ∈ ER +1 +if e ∈ E +n +otherwise, +(3) +wj +e = +�n +if e ∈ ER +0 +otherwise. +∀j = 2, 3, . . . , ¯s +(4) +Let I⋆ be the resulting STSWSN-PC instance. +3 + +On the complexity of STSWSN-PC +1 +3 +6 +7 +2 +4 +5 +8 +(a) +v1 +1 +v1 +2 +v3 +1 +v3 +2 +v6 +1 +v6 +2 +v7 +1 +v7 +2 +1 +3 +6 +7 +2 +4 +5 +8 +(b) +Figure 2: Example of a CST instance (a), in which terminal vertices are squared-shaped, and of the corresponding +STSWSN-PC instance for ¯s = 3 (b), in which fictitious vertices are diamond-shaped. +A Steiner tree T of G with k edges corresponds to a solution T1, T2, . . . , T¯s of I⋆ having value k. We distinguish two +cases: +1. if T is not spanning, let r ∈ R be an arbitrary terminal vertex of G and let vr +1, vr +2, . . . , vr +¯s−2 be ¯s − 2 arbitrary +fictitious vertices linked to r. One obtains T1 by extending T with all vertices in V R \ {vr +1, vr +2, . . . , vr +¯s−2} +(whose linking edges in ER have weight 0 in the first time slot, by (3)), by defining T2 as the spanning tree +of the complete graph G⋆ \ V (T1) involving only edges in E ∪ EC (which have weight 0 in the second time +slot, by (4)) and by defining Tj = {vr +j−2} for every j = 3, 4, . . . ¯s; +2. if T is spanning, let r ∈ R be an arbitrary terminal vertex of G and let vr +1, vr +2, . . . , vr +¯s−1 be the ¯s − 1 fictitious +vertices linked to r. One obtains T1 by extending T with all vertices in V R \ {vr +1, vr +2, . . . , vr +¯s−1}, and by +defining Tj = {vr +j−1} for every j = 2, 3, . . . , ¯s. +Note that, since a spanning tree of G is also a Steiner tree, the construction in the above case 2 shows that an optimal +solution to I⋆ has value at most n − 1. Then, as in Lemma 1 and Prop. 1, it is possible to state that if T ⋆ +1 , T ⋆ +2 , . . . , T ⋆ +¯s +is an optimal solution to I⋆, the restriction of T ⋆ +1 to the vertices in V is a Steiner tree of G having the same value. +Indeed, we first observe that T ⋆ +1 has its edges in E ∪ ER, as otherwise (3) would imply that the considered solution +has weight at least n, contradicting its optimality; moreover, if T ⋆ +1 is not a Steiner tree of G, there should be a vertex +r ∈ R belonging to T ⋆ +j with 2 ≤ j ≤ ¯s and, since T ⋆ +1 , T ⋆ +2 , . . . , T ⋆ +¯s are connected, we have that at least one edge of +ER is taken outside the first time slot; then by (4), the considered solution has value at least n, again contradicting its +optimality. +The above arguments prove that the considered CST instance admits a solution if and only if the corresponding +STSWSN-PC instance has value at most k, hence we have: +Proposition 2. The STSWSN-PC with a fixed number ¯s ≥ 2 of time slots is NP-hard. +We remark that the transformation from G to G⋆ used in the above reduction does not allow to state a result similar +to Cor. 1. +4 +Structured polynomially-solvable cases +The results of Prop. 1 and Prop. 2 hold without making any assumption on the structure of the STSWSN-PC instances. +Here we present two polynomially-solvable cases when the input is structured. The first one generalizes the approach +described in the Introduction for the case s = |V |. +Observation 1. When |V | − s is constant the STSWSN-PC is solvable in polynomial time. +Proof. When s = |V | − 1, a feasible solution contains one edge in a time slot and single vertices in all remaining time +slots; then an optimal solution can be determined in O(|V ||E|) time by exhaustively listing all values wj +e for e ∈ E +and 1 ≤ j ≤ |V | − 1 and considering the minimum one. A similar algorithm (of higher time complexity) can be +exhibited for any constant value of |V | − s. +4 + +On the complexity of STSWSN-PC +The second polynomially-solvable case relates to the graph topology: +Observation 2. If G = (V, E) is a tree, the STSWSN-PC with a fixed number ¯s ≥ 2 of time slots is solvable in +polynomial time. +Proof. We can list in O(n¯s−1) all subsets of ¯s − 1 edges whose removal decomposes G into a forest with ¯s trees. For +each such a subset we assign in polynomial time the corresponding trees to the ¯s time slots solving a perfect matching +on the weighted complete bipartite graph B = (T ; S, W) where each vertex in T represents a tree, each vertex of S +represents a time slot and edge eτσ ∈ W linking τ ∈ T to σ ∈ S has weight wσ(τ). +Obs. 2 motivates the following questions that we leave open: (i) When the number of time slots is not fixed, which +is the complexity of the STSWSN-PC defined on trees? (ii) Are there any other graph families (other than trees) for +which the STSWSN-PC is solvable in polynomial time, at least when the number of time slots is fixed? +Acknowledgments +The authors are grateful to Alberto Ceselli and to Emiliano Lancini for their comments on the manuscript. +References +Adasme, P. (2019). Optimal sub-tree scheduling for wireless sensor networks with partial coverage. Computer Stan- +dards & Interfaces, 61, 20–35. +Arrighi, E. and de Oliveira Oliveira, M. (2021). Three Is Enough for Steiner Trees. In D. Coudert and E. Natale, ed- +itors, 19th International Symposium on Experimental Algorithms (SEA 2021), volume 190 of Leibniz International +Proceedings in Informatics (LIPIcs), pages 5:1–5:15, Dagstuhl, Germany. Schloss Dagstuhl – Leibniz-Zentrum f¨ur +Informatik. +Bianchessi, N. (2022). On optimally solving sub-tree scheduling for wireless sensor networks with partial coverage. +Universit`a degli Studi di Milano, http://hdl.handle.net/2434/934107. +Garey, M. R. and Johnson, D. S. (1990). Computers and Intractability; A Guide to the Theory of NP-Completeness. +W. H. Freeman & Co., USA. +Kruskal, J. B. (1956). On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings +of the American Mathematical Society, 7(1), 48–50. +M¨uller, H. and Brandst¨adt, A. (1987). The NP-completeness of Steiner tree and dominating set for chordal bipartite +graphs. Theoretical Computer Science, 53(2-3), 257–265. +5 + diff --git a/2tAyT4oBgHgl3EQf1vk5/content/tmp_files/load_file.txt b/2tAyT4oBgHgl3EQf1vk5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9647782e930995eb16c8cf18a63755bc91e69b89 --- /dev/null +++ b/2tAyT4oBgHgl3EQf1vk5/content/tmp_files/load_file.txt @@ -0,0 +1,248 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf,len=247 +page_content='ON THE COMPLEXITY OF SUB-TREE SCHEDULING FOR WIRELESS SENSOR NETWORKS WITH PARTIAL COVERAGE Michele Barbato∗ Dipartimento di Informatica Universit`a degli Studi di Milano via Celoria 18, 20133 Milano michele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content='barbato@unimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content='it Nicola Bianchessi Dipartimento di Informatica Universit`a degli Studi di Milano via Celoria 18, 20133 Milano nicola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content='bianchessi@unimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content='it ABSTRACT Given an undirected graph G whose edge weights change over s time slots, the sub-tree scheduling for wireless sensor networks with partial coverage asks to partition the vertices of G in s non-empty trees such that the total weight of the trees is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' In this note we show that the problem is NP- hard in both the cases where s (i) is part of the input and (ii) is a fixed instance parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' In both our proofs we reduce from the cardinality Steiner tree problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' We additionally give polynomial- time algorithms for structured inputs of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Keywords Wireless sensor network, Sub-tree scheduling, Partial coverage, Complexity 1 Introduction A central problem in the management of wireless sensor networks is to extend the lifetime of wireless sensors through operating policies ensuring energy efficiency and/or balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Its importance stems from the fact that even a single failure of a wireless sensor can in principle compromise the effectiveness of the whole network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' From the viewpoint of energy balancing, a general approach to minimize energy consumption is to split the set of sensors into several non-empty subsets and to subdivide the planning horizon into as many slots, so that the subsets of sensors are operated sequentially, one at each time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' The sub-tree scheduling for wireless sensor networks with partial coverage (STSWSN-PC), introduced by Adasme (2019), is a particular implementation of such an approach, with the additional requirements that the sensors operated simultaneously are mutually connected under a tree topology, and each sensor must be active in a unique time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Namely, the STSWSN-PC is defined on an undirected graph G = (V, E) representing the network of sensors, a number s, 1 ≤ s ≤ |V |, of time slots, and vectors w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , ws ∈ RE + of edge-weights (one for each time slot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' The aim is to find a set T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , Ts of non-empty vertex-disjoint trees of G covering V and minimizing �s i=1 wi(Ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' In the above description, the vertices of G represent the sensors of the network, the edges represent direct links among sensors, and the weights represent the time slot-dependent power for transmitting information over the corresponding edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' The input of the STSWSN-PC is simultaneously defined by the graph G, the number of time slots s, and the values of the edge weight vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' The STSWSN-PC may admit efficient optimization algorithms for structured inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' For example, when the weights are constant throughout the time slots (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=', wi ≡ wj for all i, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , s), the STSWSN-PC is solvable in polynomial time, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=', by using Kruskal’s algorithm (Kruskal, 1956) and terminating it at the first iteration yielding a spanning forest with s trees;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' when s = 1, the STSWSN-PC boils down to the minimum spanning tree (MST) problem on general graphs and, as such, is solvable in polynomial time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' when s = |V |, the optimal solution consists of arbitrarily assigning one vertex to each time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' However, unstructured instances of the STSWSN-PC have been tackled in Adasme (2019) and Bianchessi (2022) by means of branch-and-bound and branch-and-cut algorithms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' These approaches implicitly suggest that the ∗Corresponding author arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content='00739v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content='CC] 2 Jan 2023 On the complexity of STSWSN-PC 1 3 6 7 2 4 5 8 (a) v1 1 v1 2 v1 3 v3 1 v3 2 v3 3 v6 1 v6 2 v6 3 v7 1 v7 2 v7 3 1 3 6 7 2 4 5 8 (b) Figure 1: Example of a CST instance (a), in which terminal vertices are squared-shaped, and of the corresponding STSWSN-PC instance for k = 5 (b), in which fictitious vertices are diamond-shaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' problem is theoretically intractable, although its computational complexity is unknown to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' The purpose of this note is to fill in this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 2 we study the complexity of the STSWSN-PC when s is part of the input, that is, s is not fixed in {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , |V | − 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 3 we study the complexity under the assumption that s is an instance parameter with a prescribed value ¯s ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Through reductions from the (minimum weight) Steiner tree problem (Garey and Johnson, 1990, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 208), we show that the STSWSN-PC is NP-hard in both cases, thus justifying the usage of implicit enumera- tion schemes to solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Finally, in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 4 we discuss additional structured inputs, other than those mentioned above, for which the STSWSN-PC is solvable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 2 NP-hardness when the number of time slots is not fixed Given an undirected connected graph G = (V, E) with |V | = n vertices and a subset R ⊂ V of terminal vertices, a Steiner tree is a subtree T of G such that R ⊆ V (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Given also a weight w(e) ∈ Z+ for each e ∈ E, computing the Steiner tree of minimum total edge-weight is in general NP-hard, and the problem remains NP-hard if all weights are equal (Garey and Johnson, 1990, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 209).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' In particular, given w(e) = 1 for each e ∈ E, the pair (G, R), and k ∈ Z+ with |R| − 1 ≤ k ≤ n − 2, the cardinality Steiner tree (CST) problem consisting of determining the existence of a Steiner tree of G with at most k edges is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' We now show that an oracle solving the STSWSN-PC in polynomial time allows to solve the CST in polynomial time, thus obtaining that the STSWSN-PC is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' We point out the CST with at most three terminals can be solved in polynomial time (Arrighi and de Oliveira Oliveira, 2021), therefore we restrict ourselves to CST instances with |R| ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Given k and (G, R) defining a CST instance as above, we construct a new graph ¯G = ( ¯V , ¯E) obtained from G by introducing n−k fictitious vertices vr 1, vr 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , vr n−k for each terminal vertex r ∈ R and defining ¯E = E ∪ER, where ER = {(vr j, r): j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , n − k, r ∈ R};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' that is, each fictitious vertex is connected precisely to the corresponding terminal vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' An example of such a construction is given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Next, we define a STSWSN-PC instance I on graph ¯G and s = n−k+1 time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Note that, since |R|−1 ≤ k ≤ n−2 and |R| ≥ 4, then 3 ≤ s ≤ n − 2 < | ¯V |, hence our definition of the number of time slots excludes the polynomially solvable cases of the STSWSN-PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' The weights of the time slots are defined as follows: w1 e = �0 if e ∈ ER 1 otherwise (1) wj e = �n if e ∈ ER 1 otherwise ∀j = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , n − k + 1 (2) Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Let T ⋆ 1 , T ⋆ 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , T ⋆ n−k+1 be an optimal solution to I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' The restriction of T ⋆ 1 to the vertices in V is a Steiner tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 2 On the complexity of STSWSN-PC Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Assume that the restriction of T ⋆ 1 to the vertices of G is not a Steiner tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Then there is at least a terminal vertex r⋆ contained in a tree of a time slot after the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Since all trees T ⋆ 1 , T ⋆ 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , T ⋆ n−k+1 are connected, at least one edge of ER belongs to that time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' By (2) the optimal solution to I has value at least n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Now we show the existence of a solution with better value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Namely, in the first time slot we consider the tree T1 spanning all vertices of ¯G except the n − k fictitious vertices linked to r⋆ and we set Tj = {vr⋆ j−1} for j = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , n − k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Then T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , Tn−k+1 is a feasible solution whose value is n − 1 by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Now we can prove the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' In the proof, given S ⊆ ¯V , we denote by δ(S) its cut, namely, the set of edges having one endpoint in S and the other endpoint outside S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' There exists a solution to the CST instance given by k and (G, R) if and only if the optimal solution to I has value at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Therefore the STSWSN-PC is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' For the “if” part assume that there exists an optimal solution having value at most k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' denoting by T ⋆ the restriction of its tree of the first time slot to the vertices in V , the nonnegativity of the weights in (1) yields |T ⋆| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Then the result follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Now, let us prove the “only if” part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Assume that there exists a Steiner tree T of G such that |T| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' We assume, without loss of generality, that |T| = k: otherwise we repeatedly update T by adding one edge of G belonging to δ(T), until reaching the required cardinality (this is always possible as G is connected and since the update always returns a Steiner tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Then, let ¯v ∈ ¯V \\ V be an arbitrary fictitious vertex, define ˆV = ¯V \\ {V ∪ {¯v}} as the set of remaining fictitious vertices, and let V C = V \\ V (T) = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , vn−k−1} be the vertices in the complement of T in G (as |T| = k, T comprises k + 1 vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' We consider the feasible solution for I given by T1 = T ∪ δ( ˆV ), T2 = {¯v} and Tj = {vj−2 ∈ V C} for every j = 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , n − k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' By (1)–(2) such a solution has value k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Then the optimal solution to I has value at most k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' In the above construction, ¯G is obtained from G by appending leaves to its terminal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' This is a minor modifi- cation of the initial graph, hence the STSWSN-PC remains difficult on those classes of graphs which are closed under such modification and on which the CST is NP-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' It is the case of chordal bipartite graphs, that is, bipartite graphs whose cycles C of length at least 6 induce a subgraph with at least |C| + 1 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' More precisely we have: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' The STSWSN-PC is NP-hard on bipartite chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Appending leaves to a subset of vertices of a bipartite chordal graph maintains the chordal bipartiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Then the result follows from the NP-completeness of the CST on bipartite chordal graphs proved by M¨uller and Brandst¨adt (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 3 NP-hardness when the number of time slots is fixed In this section we consider the complexity of the STSWSN-PC by assuming that we have s = ¯s time slots, with ¯s ≥ 2 fixed, and we show that the problem remains NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' We modify the approach of previous section as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Let us consider a graph G = (V, E) with |V | = n vertices and a set R ⊂ V , |R| ≥ 4, of terminal vertices defining an instance of the CST problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' We define a graph G⋆ = (V ⋆, E⋆) where V ⋆ = V ∪ V R, with V R = {vr 1, vr 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , vr ¯s−1 : r ∈ R} being a set of fictitious vertices associated with those in R, and where E⋆ = E ∪ EC ∪ ER, with EC = {(v, w): v, w ∈ V s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' (v, w) ̸∈ E} and ER = {(r, vr j): r ∈ R, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , ¯s − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' That is, G⋆ is obtained by extending G to a complete graph and by linking each terminal vertex in G to the corresponding ¯s − 1 fictitious vertices (see Figure 2 for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' For every e ∈ E⋆ we define the following edge weights: w1 e = � � � 0 if e ∈ ER 1 if e ∈ E n otherwise, (3) wj e = �n if e ∈ ER 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' ∀j = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , ¯s (4) Let I⋆ be the resulting STSWSN-PC instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 3 On the complexity of STSWSN-PC 1 3 6 7 2 4 5 8 (a) v1 1 v1 2 v3 1 v3 2 v6 1 v6 2 v7 1 v7 2 1 3 6 7 2 4 5 8 (b) Figure 2: Example of a CST instance (a), in which terminal vertices are squared-shaped, and of the corresponding STSWSN-PC instance for ¯s = 3 (b), in which fictitious vertices are diamond-shaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' A Steiner tree T of G with k edges corresponds to a solution T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , T¯s of I⋆ having value k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' We distinguish two cases: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' if T is not spanning, let r ∈ R be an arbitrary terminal vertex of G and let vr 1, vr 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , vr ¯s−2 be ¯s − 2 arbitrary fictitious vertices linked to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' One obtains T1 by extending T with all vertices in V R \\ {vr 1, vr 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , vr ¯s−2} (whose linking edges in ER have weight 0 in the first time slot, by (3)), by defining T2 as the spanning tree of the complete graph G⋆ \\ V (T1) involving only edges in E ∪ EC (which have weight 0 in the second time slot, by (4)) and by defining Tj = {vr j−2} for every j = 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' ¯s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' if T is spanning, let r ∈ R be an arbitrary terminal vertex of G and let vr 1, vr 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , vr ¯s−1 be the ¯s − 1 fictitious vertices linked to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' One obtains T1 by extending T with all vertices in V R \\ {vr 1, vr 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , vr ¯s−1}, and by defining Tj = {vr j−1} for every j = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , ¯s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Note that, since a spanning tree of G is also a Steiner tree, the construction in the above case 2 shows that an optimal solution to I⋆ has value at most n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Then, as in Lemma 1 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 1, it is possible to state that if T ⋆ 1 , T ⋆ 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , T ⋆ ¯s is an optimal solution to I⋆, the restriction of T ⋆ 1 to the vertices in V is a Steiner tree of G having the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Indeed, we first observe that T ⋆ 1 has its edges in E ∪ ER, as otherwise (3) would imply that the considered solution has weight at least n, contradicting its optimality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' moreover, if T ⋆ 1 is not a Steiner tree of G, there should be a vertex r ∈ R belonging to T ⋆ j with 2 ≤ j ≤ ¯s and, since T ⋆ 1 , T ⋆ 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' , T ⋆ ¯s are connected, we have that at least one edge of ER is taken outside the first time slot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' then by (4), the considered solution has value at least n, again contradicting its optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' The above arguments prove that the considered CST instance admits a solution if and only if the corresponding STSWSN-PC instance has value at most k, hence we have: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' The STSWSN-PC with a fixed number ¯s ≥ 2 of time slots is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' We remark that the transformation from G to G⋆ used in the above reduction does not allow to state a result similar to Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 4 Structured polynomially-solvable cases The results of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 1 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 2 hold without making any assumption on the structure of the STSWSN-PC instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Here we present two polynomially-solvable cases when the input is structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' The first one generalizes the approach described in the Introduction for the case s = |V |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Observation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' When |V | − s is constant the STSWSN-PC is solvable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' When s = |V | − 1, a feasible solution contains one edge in a time slot and single vertices in all remaining time slots;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' then an optimal solution can be determined in O(|V ||E|) time by exhaustively listing all values wj e for e ∈ E and 1 ≤ j ≤ |V | − 1 and considering the minimum one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' A similar algorithm (of higher time complexity) can be exhibited for any constant value of |V | − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 4 On the complexity of STSWSN-PC The second polynomially-solvable case relates to the graph topology: Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' If G = (V, E) is a tree, the STSWSN-PC with a fixed number ¯s ≥ 2 of time slots is solvable in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' We can list in O(n¯s−1) all subsets of ¯s − 1 edges whose removal decomposes G into a forest with ¯s trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' For each such a subset we assign in polynomial time the corresponding trees to the ¯s time slots solving a perfect matching on the weighted complete bipartite graph B = (T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' S, W) where each vertex in T represents a tree, each vertex of S represents a time slot and edge eτσ ∈ W linking τ ∈ T to σ ∈ S has weight wσ(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 2 motivates the following questions that we leave open: (i) When the number of time slots is not fixed, which is the complexity of the STSWSN-PC defined on trees?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' (ii) Are there any other graph families (other than trees) for which the STSWSN-PC is solvable in polynomial time, at least when the number of time slots is fixed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Acknowledgments The authors are grateful to Alberto Ceselli and to Emiliano Lancini for their comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' References Adasme, P.' 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+page_content=' The NP-completeness of Steiner tree and dominating set for chordal bipartite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' Theoretical Computer Science, 53(2-3), 257–265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} +page_content=' 5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQf1vk5/content/2301.00739v1.pdf'} diff --git a/6dE0T4oBgHgl3EQfvwHw/content/tmp_files/2301.02624v1.pdf.txt b/6dE0T4oBgHgl3EQfvwHw/content/tmp_files/2301.02624v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..be19e1168e380a8452aa2a4e46fce56f60747ef5 --- /dev/null +++ b/6dE0T4oBgHgl3EQfvwHw/content/tmp_files/2301.02624v1.pdf.txt @@ -0,0 +1,858 @@ +arXiv:2301.02624v1 [math.QA] 6 Jan 2023 +Shapovalov elements of classical and quantum groups +Andrey Mudrov +In memorium of Vladimir Lyachovsky +Moscow Institute of Physics and Technology, +9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russia, +University of Leicester, +University Road, LE1 7RH Leicester, UK, +e-mail: am405@le.ac.uk +Abstract +Shapovalov elements θβ,m of the classical or quantized universal enveloping algebra of a +simple Lie algebra g are parameterized by a positive root β and a positive integer m. They +relate the highest vector of a reducible Verma module with highest vectors of its submodules. +We obtain a factorization of θβ,m to a product of θβ,1 and calculate θβ,1 as a residue of a +matrix element of the inverse Shapovalov form via a generalized Nigel-Moshinsky algorithm. +This way we explicitly express θβ,m of a classical simple Lie algebra through the Cartan-Weyl +basis in g. In the case of quantum groups, we give an analogous formulation through the +entries of the R-matrix (quantum L-operator) in fundamental representations. +Key words: Shapovalov elements, Shapovalov form, Verma modules, singular vectors, Hasse diagrams, +R-matrix +AMS classification codes: 17B10, 17B37 +1 + +1 +Introduction +Category O introduced in [1] for semi-simple Lie algebras and later defined for many other classes +of algebras including quantum groups plays a fundamental role in various fields of mathematics and +mathematical physics. In particular, it accommodates finite-dimensional and numerous important +infinite dimensional representations like parabolic Verma modules and their generalizations [2]. +There are distinguished objects in O called Verma modules that feature a universality property: +all simple modules in O are their quotients. The maximal proper submodule in a Verma module +is generated by extremal vectors [3], which are invariants of the positive triangular subalgebra. +This makes extremal vectors critically important in representation theory. +Extremal vectors in a Verma module Vλ are related with the vacuum vector of highest weight +λ via special elements θβ,m of the (classical or quantum) universal enveloping of the negative Borel +subalgebra that are called Shapovalov elements [4, 5]. They are parameterized with a positive +root β and an integer m ∈ N validating a De Concini-Kac-Kazhdan condition on λ. +In the +classical version, it is 2(λ + ρ, β) − m(β, β) = 0 with ρ being the half-sum of positive roots. This +condition guarantees that the Verma module is reducible. In the special case when the root β is +simple, the element θβ,m is a power f m +β of the simple root vector fβ of weight β. For non-simple +β, the Shapovalov elements are complicated polynomials in negative Chevalley generators with +coefficients in the Cartan subalgebra. It can be viewed as a function θβ,m(λ) of the highest weight +of a generic Verma module Vλ with values in the subalgebra generated by negative root vectors. +A description of extremal vectors in Verma modules over classical Kac-Moody algebras was +obtained in [6] via an interpolation procedure resulting in a calculus of polynomials with complex +exponents. Another approach based on extremal projectors [7] was employed by Zhelobenko in +[8]. He obtained θβ,m for simple Lie algebras as a product of m copies of θβ,1 with shifted weights. +The idea of factorization was also used in a construction of Shapovalov elements for contragredient +Lie superalgebras in [9]. +Factorization of θβ,m into a product of polynomials of lower degree is a great simplification +that is convenient for their analysis. For example it is good for the study of the classical limit in +the case of quantum groups, which is crucial for quantization of conjugacy classes [10]. +With regard to quantum groups, an inductive construction of extremal vectors in Verma mod- +ules was suggested in [11]. Explicit expressions for Shapovalov elements for the A-type appeared +in [12] and recently were obtained in [13] by other methods. It is worthy to note that ordered +PBW-like monomials in θβ,1 deliver an orthogonal basis in generic irreducible Verma modules [12]. +While Zhelobenko’s factorization via extremal projectors simplifies construction of Shapovalov +2 + +elements in the case of classical simple Lie algebras, there remains a problem of explicit descrip- +tion of the factors. +In this paper, we pursue an alternative approach based on the canonical +contravariant bilinear form on Verma modules. It gives expressions for all Shapovalov elements in +a factorized form through root vectors in the classical case and through elements of the R-matrix +in the adjoint representation in the case of quantum groups. +Extremal vectors generate the kernel of the canonical contravariant form on Vλ, which is a +specialization at λ of the ”universal” Shapovalov form on the Borel subalgebra [4] with values in +the Cartan subalgebra. This contravariant form itself is extremely important and has numerous +applications, see e.g. [14, 15, 16, 17]. For generic λ, the module Vλ is irreducible and the form +is non-degenerate. The inverse form gives rise to an element S of extended tensor product of +positive and negative subalgebras of the (quantized) universal enveloping algebra [18]. Sending +the positive leg of S to an auxiliary representation yields a matrix with entries in the negative +subalgebra which we call Shapovalov matrix. +Its explicit description was obtained in [18] by +generalization of Nagel-Moshinski formulas for the lowering operators of sl(n) [19]. They can also +be derived (in the quantum setting) from the ABRR equation [20] on dynamical twist [14]. +Our method relates θβ,m with certain entries of the Shapovalov matrix. This point of view is +quite natural because the kernel of the contravariant form on Vλ results in poles of S. Our approach +not only provides a factorization of θβ,m to a product of θβ,1 but also an efficient description of +θβ,1 in a very elementary way, by a generalized Nagel-Moshinsky rule (3.5) using a technique of +Hasse diagrams. We do it by evaluating residues of matrix elements of S that go singular at a De +Concini-Kac-Kazhdan ”hyperplane”. +Our approach is absolutely parallel for a classical semi-simple Lie algebra g and its quantum +group Uq(g). The classical case can be processed directly or obtained as the limit case q → 1 of +the deformation parameter. Let us describe the method in more detail. +With a module V from the category O and a pair of non-zero vectors vb, va ∈ V we associate +a Shapovalov matrix element, ⟨vb|va⟩, which belongs to the negative Borel (universal enveloping) +subalgebra ˆUq(b−) rationally extended over the Cartan subalgebra. Under certain assumptions +on vb and va, such matrix elements deliver factors in θβ,m. These factors normalize positive root +vectors of a reductive Lie subalgebra l ⊂ g whose negative counterparts annihilate vb. This way +they become lowering operators in the Mickelsson algebras of the pair (g, l), [21]. When λ satisfies +the De Concini-Kac-Kazhdan condition, the factors become θβ,1 shifted by certain weights. +The vector vb should be highest for the minimal simple Lie subalgebra in g that accommodates +the root β and and its weight should satisfy the condition (νb, β) = (β,β) +2 . For finite dimensional +V the latter is equivalent to saying that vb generates a 2-dimensional submodule of the sl(2)- +3 + +subalgebra generated by the root spaces g±β. +The vector va determines a homomorphism Vλ2 → V ⊗ Vλ1, where Vλi are irreducible Verma +modules of highest weights λi and λ2 − λ1 equals the weight of va. Iteration of this construction +yields a chain of homomorphisms +Vλm → V ⊗ Vλm−1 → . . . → V ⊗m ⊗ Vλ0, +where each mapping V ⊗i ⊗ Vλm−i → V ⊗i ⊗ (V ⊗ Vλm−i−1) is identical on the factor V ⊗i. We prove +factorization of ⟨v⊗m +b +|v⊗m +a +⟩ to a product of ⟨vb|va⟩. Then we demonstrate that, under the specified +conditions, that matrix element is proportional to θβ,m(λ) with λ0 = λ. +As a result, we obtain θβ,m(λ) as a product �m−1 +i=0 θβ,1(λi). The factors θβ,1 are calculated by +a general rule (3.5) specialized to the case in Section 5. Viewed as an element of ˆUq(b−), θβ,m +becomes a product of θβ,1 shifted by the integer multiple weights of vb. This shift can be made +trivial by a choice of V if β contains a simple root α with multiplicity 1. Then θβ,m becomes the +m-th power of θβ,1. +It is worthwhile mentioning that θβ,1 can be obtained via an arbitrary auxiliary module V with +a pair of vectors (va, vb = eβva). They all coincide up to a scalar factor on the De Concini-Kac- +Kazhdan ”hyperplane” and generally differ away from it. The problem is to use θβ,1 as a factor +block for constructing θβ,m of higher m. That is why we choose (V, vb, va) in a special way as +described above. On the other hand, since the left tensor leg of S is in the positive subalgebra +Uq(g+) ⊂ Uq(g), it is the structure of Uq(g+)-submodule on V that determines θβ,1. A remarkable +fact is that the cyclic submodule Uq(g+)va in an admissible V turns out to be isomorphic to a +subquotient of the Uq(g+)-module corresponding to g/g+ in the classical limit. This means that +θβ,1 in each case can be calculated via Shapovalov matrix elements from End(g/g+) ⊗ Uq(b−), by +Theorem 5.3. +Except for Section 5, we present only the q-version of the theory. The classical case can be +obtained by sending q to 1. However the final expression for θβ,1 is greatly simplified when q = 1, +so we give a special consideration to it in the Section 5. +2 +Preliminaries +Let g be a simple complex Lie algebra and h ⊂ g its Cartan subalgebra. Fix a triangular de- +composition g = g− ⊕ h ⊕ g+ with maximal nilpotent Lie subalgebras g±. Denote by R ⊂ h∗ the +root system of g, and by R+ the subset of positive roots with basis Π of simple roots. This basis +generates a root lattice Γ ⊂ h∗ with the positive semigroup Γ+ = Z+Π ⊂ Γ. +4 + +For a positive root β ∈ R+ and a simple root α ∈ Π denote by ℓα,β ∈ Z+ the multiplicity with +which α enters β, that is the α-coefficient in the expansion of β over the basis Π. +Choose an ad-invariant form ( . , . ) on g, restrict it to h, and transfer to h∗ by duality. For +every λ ∈ h∗ there is a unique element hλ ∈ h such that µ(hλ) = (µ, λ), for all µ ∈ h∗. For a +non-isotropic µ ∈ h∗ set µ∨ = +2 +(µ,µ)µ and h∨ +µ = +2 +(µ,µ)hµ. +Fundamental weights are denoted by ωα, α ∈ Π. +They are determined by the system of +equations (ωα, β∨) = δα,β, for all α, β ∈ Π. +We assume that q ∈ C is not a root of unity and we understand that when saying ”all q”. By +almost all q we mean all q excepting maybe a finite set of values distinct from q = 1. +The standard Drinfeld-Jimbo quantum group Uq(g) is a complex Hopf algebra with the set of +generators eα, fα, and q±hα labeled by simple roots α and satisfying relations [22, 23] +qhαeβ = q(α,β)eβqhα, +[eα, fβ] = δα,β[hα]q, +qhαfβ = q−(α,β)fβqhα, +α, β ∈ Π. +The symbol [z]q, where z ∈ h + C, stands for +qz−q−z +q−q−1 . +The elements qhα are invertible, with +qhαq−hα = 1, while {eα}α∈Π and {fα}α∈Π also satisfy quantized Serre relations. Their exact form +is not important for this presentation, see [24] for details. +A Hopf algebra structure on Uq(g) is introduced by the comultiplication +∆(fα) = fα ⊗ 1 + q−hα ⊗ fα, +∆(q±hα) = q±hα ⊗ q±hα, +∆(eα) = eα ⊗ qhα + 1 ⊗ eα +set up on the generators and extended as a homomorphism Uq(g) → Uq(g)⊗Uq(g). The antipode is +an algebra and coalgebra anti-automorphism of Uq(g) that acts on the generators by the assignment +γ(fα) = −qhαfα, +γ(q±hα) = q∓hα, +γ(eα) = −eαq−hα. +The counit homomorphism ǫ: Uq(g) → C returns +ǫ(eα) = 0, +ǫ(fα) = 0, +ǫ(qhα) = 1. +We extend the notation fα, eα to all α ∈ R+ meaning the Lusztig root vectors with respect to +some normal ordering of R+, [24]. They are known to generate a Poincare-Birkhoff-Witt (PBW) +basis in Uq(g±). +Denote by Uq(h), Uq(g+), and Uq(g−) subalgebras in Uq(g) generated by {q±hα}α∈Π, {eα}α∈Π, +and {fα}α∈Π, respectively. The quantum Borel subgroups are defined as Uq(b±) = Uq(g±)Uq(h); +they are Hopf subalgebras in Uq(g). We will also need their extended version ˆUq(b±) = Uq(g±) ˆUq(h), +where ˆUq(h) is the ring of fractions of Uq(h) over the multiplicative system generated by [hα − c]q +with α ∈ Γ+ and c ∈ Q. +5 + +Given a Uq(g)-module V , a non-zero vector v is said to be of weight µ if qhαv = q(µ,α)v for all +α ∈ Π. The linear span of such vectors is denoted by V [µ]. A module V is said to be of highest +weight λ if it is generated by a weight vector v ∈ V [λ] that is killed by all eα. Such vector v is +called highest; it is defined up to a non-zero scalar multiplier. +We define an involutive coalgebra anti-automorphism and algebra automorphism σ of Uq(g) +setting it on the generators by the assignment +σ: eα �→ fα, +σ: fα �→ eα, +σ: qhα �→ q−hα. +The involution ω = γ−1 ◦ σ = σ ◦ γ is an algebra anti-automorphism of Uq(g) and preserves the +comultiplication. +A symmetric bilinear form (., .) on a g-module V is called contravariant if +� +xv, w +� += +� +v, ω(x)w +� +for all x ∈ Uq(g), v, w ∈ V . A module of highest weight has a unique C-valued contravariant form +such that squared norm of the highest vector is 1. We call this form canonical and extend this +term to a form on tensor products that is the product of canonical forms on tensor factors. Such +a form is contravariant because ω is a coalgebra map. +Let us recall the definition of Uq(h)-valued Shapovalov form on the Borel subalgebra Uq(b−) +that was introduced for U(g) and studied in [4]. Regard Uq(b−) as a free right Uq(h)-module gen- +erated by Uq(g−). The triangular decomposition Uq(g) = Uq(g−)Uq(h)Uq(g+) facilitates projection +℘: Uq(g) → Uq(h) along the sum g−Uq(g) + Uq(g)g+, where g−Uq(g) and Uq(g)g+ are right and +left ideals generated by positive and negative root vectors, respectively. Set +(x, y) = ℘ +� +ω(x)y +� +, +x, y ∈ Uq(g). +Thus defined the form is Uq(h)-linear and contravariant. It follows that the left ideal Uq(g)g+ is in +the kernel, so the form descends to a Uq(h)-linear form on the quotient Uq(g)/Uq(g)g+ ≃ Uq(b−). +A Verma module Vλ = Uq(g) ⊗Uq(b+) Cλ of highest weight λ ∈ h∗ is induced from the 1- +dimensional Uq(b+)-module Cλ that is trivial on Uq(g+) and returns q(λ,α) on qhα ∈ Uq(h), α ∈ Π. +Its highest vector is denoted by vλ, which is also called vacuum vector. It freely generates Vλ over +Uq(g−). +Specialization of the Shapovalov form at λ ∈ h∗ yields the canonical contravariant C-valued +form (x, y)λ = λ +� +℘ +� +ω(x)y +�� +on Vλ, upon a natural Uq(g−)-module isomorphism Uq(g−) ≃ Vλ +extending the assignment 1 �→ vλ. Conversely, the canonical contravariant form on Vλ regarded as +a function of λ descends to the Shapovalov form if one views Uq(h) as the algebra of polynomial +functions on h∗. By an abuse of terminology, we also mean by Shapovalov form the canonical +contravariant form on Vλ. +6 + +It is known from [25] that the contravariant form on Vλ module goes degenerate if and only if +its highest weight is in the union of +Hβ,m = {λ ∈ h∗ | q2(λ+ρ,β)−m(β,β) = 1} +(2.1) +over β ∈ R+ and m ∈ N, where ρ = 1 +2 +� +α∈R+ α. In the classical case q = 1, Hβ,m becomes a +Kac-Kazhdan hyperplane of weights satisfying 2(λ + ρ, β) = m(β, β). +Recall that a vector v ∈ Vλ of weight λ−µ with µ ∈ Γ+, µ ̸= 0, is called extremal if eαv = 0 for +all α ∈ Π. Extremal vectors are in the kernel of the contravariant form and generate submodules +of the corresponding highest weights. We will be interested in the special case when µ = mβ with +β ∈ R+ and m ∈ N. Then the highest weight λ has to be in Hβ,m. The image θβ,m of v under the +isomorphism Vλ → Uq(g−) is called Shapovalov element of a positive root β and degree m. +For simple β the element θβ,m is just the m-th power of the root vector, θβ,m = f m +β . For +non-simple β, it is a rational trigonometric function Hβ,m → Uq(g−). The goal of this work is to +find explicit expressions for θβ,m with non-simple β. +3 +Shapovalov inverse form and its matrix elements +Define an opposite Verma Uq(g)-module V ′ +λ of lowest weight −λ as follows. The underlying vector +space of V ′ +λ is taken to be Vλ, while the representation homomorphism π′ +λ is twisted by σ, that is +π′ +λ = πλ ◦ σ. The module V ′ +λ is freely generated over Uq(g+) by its lowest vector v′ +λ. +Let σλ : Vλ → V ′ +λ denote the isomorphism of vector spaces, xvλ �→ σ(x)v′ +λ, +x ∈ Uq(g−). It +intertwines the representations homomorphisms π′ +λ◦σ = σλ◦πλ. This map relates the contravariant +form on Vλ with a Uq(g)-invariant pairing Vλ ⊗ V ′ +λ → Vλ ⊗ Vλ → C. +Suppose that the module Vλ is irreducible. Then its invariant pairing is non-degenerate (as well +as the contravariant form on Vλ). The inverse form belongs to a completed tensor product V ′ +λ ˆ⊗Vλ. +Under the isomorphisms Vλ → Uq(g−), V ′ +λ → Uq(g+), it goes to an element that we denote by +S ∈ Uq(g+)ˆ⊗Uq(g−) and call universal Shapovalov matrix. Given a Uq(g+)-locally nilpotent Uq(g)- +module V with representation homomorphism π: Uq(g) → End(V ) the image S = (π ⊗ id)(S) +is a matrix with entries in Uq(g−). It features a rational trigonometric (rational in the classical +case) dependance on λ ∈ h∗. We will assume that V is diagonalizable with finite dimensional +weight spaces. We will also assume that V is endowed with a non-degenerate contravariant form, +for instance, if V is a tensor power of an irreducible module of highest weight. Using terminology +adopted in the quantum inverse scattering theory, we call the module V auxiliary. +7 + +Varying the highest weight λ we get a rational trigonometric dependance of S. As a function of +λ, S is regarded as an element of Uq(g+)ˆ⊗ ˆUq(b−), where ˆUq(b−) is viewed as a right ˆUq(h)-module +freely generated by Uq(g−). This way the weight dependance is accommodated by the right tensor +leg of S. +An explicit expression of S in a weight basis {vi}i∈I ⊂ V , vi ∈ V [νi], can be formulated in +terms of Hasse diagram, H(V ). Such a diagram is associated with any partially ordered sets. In +our case the partial ordering is induced by the Uq(g+)-action on V . Nodes are elements of the +basis {vi}i∈I. Arrows are simple root vectors eα connecting the nodes vi +eα +←− vj whose weight +difference is νi − νj = α. Then a node vi is succeeding a node vj if νi − νj ∈ Γ+\{0}. The matrix +S is triangular: sii = 1 and sij = 0 if νi ̸≻ νj. The entry sij is a rational trigonometric function +h∗ → Uq(g−) taking values in the subspace of weight νj − νi ∈ −Γ+. It is also convenient to +introduce a stronger partial ordering as we will explain below. +Clearly the matrix S depends only on the Uq(b+)-module structure on V . In order to calculate +a particular element sij, we can choose a weight basis that extends a basis in the cyclic submodule +Uq(g+)vj. Then, in particular, sij = 0 if vi ̸∈ Uq(g+)vj. +We define a Hasse sub-diagram H(vi, vj) ⊂ H(V ) that comprises all possible routes from vj to +vi. A node vk ∈ H(V ) is in H(vi, vj) if and only if vi ⪰ vk ⪰ vj. The sub-diagram H(vi, vj) is +associated with a Uq(g+)-module V (vi, vj) that is the quotient of Uq(g+)vj by the sum of cyclic +submodules Uq(g+)vk ⊂ Uq(g+)vj where vk ̸∈ H(vi, vj). It is the module V (vi, vj) that is needed +to calculate a matrix element sij. +We recall a construction of S following [18]. Let {hi}rkg +i=1 ∈ h be an orthonormal basis. The +element q +� +i hi⊗hi belongs to a completion of Uq(h)⊗Uq(h) in the ℏ = ln q-adic topology. Choose an +R-matrix R of Uq(g) such that ˇR = q− � +i hi⊗hiR ∈ Uq(g+)ˆ⊗Uq(g−) and set C = +1 +q−q−1( ˇR − 1 ⊗ 1). +The key identity on C that facilitates the q-version of the theory is [18] +[1 ⊗ eα, C] + (eα ⊗ q−hα)C − C(eα ⊗ qhα) = eα ⊗ [hα]q, +∀α ∈ Π. +(3.2) +In the classical limit, C = � +α∈R+ eα ⊗ fα becomes the polarized split Casimir of g without its +Cartan part. One then recovers an identity +[1 ⊗ eα, C] + [eα ⊗ 1, C] = eα ⊗ hα +(3.3) +for each simple root α. +Let cij ∈ Uq(g−) denote the entries of the matrix (π ⊗ id)(C) ∈ End(V ) ⊗ Uq(g−). We rectify +the partial ordering and the Hasse diagram H(V ) by removing arrows vi ← vj if cij = 0. This will +not affect the formula (3.5) for matrix elements of S. +8 + +For each weight µ ∈ Γ+ put +ηµ = hµ + (µ, ρ) − 1 +2(µ, µ) ∈ h ⊕ C. +(3.4) +Regard ηµ as an affine function on h∗ by the assignment ηµ : ζ �→ (µ, ζ + ρ) − 1 +2(µ, µ), ζ ∈ h∗. +Observe that ηmβ = m +� +hβ + (β, ρ) − m +2 (β, β) +� +. That is, [ηmβ(λ)]q vanishes on Hβ,m (and only on +Hβ,m in the classical case). +For a pair of non-zero vectors v, w ∈ V define a matrix element ⟨w|v⟩ = (w, S1v)S2 ∈ ˆUq(b−), +where S1 ⊗ S2 stands for a Sweedler-like notation for S and the pairing is with respect to a non- +degenerate contravariant form on V . Its specialization at a weight λ is denoted by ⟨w|v⟩λ, which +can be determined from the equality ⟨w|v⟩λvλ = ⟨w|v⟩vλ ∈ Vλ. For each w from V , the map +V → Vλ, v �→ ⟨v|w⟩vλ satisfies: eα⟨v|w⟩vλ = ⟨σ(eα)v|w⟩vλ for all α ∈ Π. This is a consequence of +Uq(g+)-invariance of the tensor S(1 ⊗ vλ) ∈ Uq(g+)ˆ⊗Vλ. +The matrix element ⟨vi|vj⟩ equals sij if (vi, vk) = δik for all k ∈ I. It will be always the case in +what follows. +Fix a ”start” node va and an ”end” node vb such that vb ≻ va. Then a re-scaled matrix element +ˇsab = −sba[ηνb−νa]qq−ηνb−νa can be calculated by the formula +ˇsba = cba + +� +k⩾1 +� +vb≻vk≻...≻v1≻va +cbk . . . c1a +(−1)kqηµk . . . qηµ1 +[ηµk]q . . . [ηµ1]q +∈ ˆUq(b−), +(3.5) +where µl = νl − νa ∈ Γ+, l = 1, . . . , k. Here the summation is performed over all possible routes +(sequences of ordered nodes) from va to vb, see [18] for details. +It is straightforward that Uq(g+)-invariance of the tensor S(va ⊗ vλ) implies +eαˇsba(λ)vλ ∝ [ηνb−νa(λ)]q +� +k +π(eα)bkska(λ)vλ. +(3.6) +The matrix entries ska(λ) carry weight −(νb − νa − α). It follows that ˇsba(λ)vλ is an extremal +vector in Vλ for λ satisfying [ηνb−νa(λ)]q = 0 provided +1. ˇsba(λ) ̸= 0, +2. λ is a regular point for all ska(λ) and all α. +We aim to find an appropriate matrix element for θβ,m that satisfies these conditions. +Let V be a Uq(g)-module with a pair of vectors va, vb ∈ V such that eβva = vb for β ∈ R+. We +call the triple (V, vb, va) a β-representation. +Proposition 3.1. Let (V, vb, va) be a β-representation for β ∈ R+. Then for generic λ ∈ Hβ,1 the +vector ˇsba(λ)vλ ∈ Vλ is extremal. +9 + +Proof. The factors +qηµk +[ηµk ]q in (3.5) go singular on the union of a finite number of the null-sets +{λ ∈ h∗ | [ηµk(λ)]q = 0}. None of µk is collinear to β, hence ˇsba(λ) is regular at generic λ ∈ Hβ,1. +By the same reasoning, all ska(λ) in (3.6) are regular at such λ. Finally, the first term cba (and +only this one) involves the Lusztig root vector fβ, a generator of a PBW basis in Uq(g−). It is +therefore independent of the other terms, and ˇsba(λ) ̸= 0. +Upon identification of ˆUq(b−) with rational Uq(g−)-valued functions on h∗ we conclude that ˇsba +is a Shapovalov element θβ,1 and denote it by θβ. Uniqueness of extremal vector of given weight +implies that all matrix elements ˇsba with vb = eβva deliver the same θβ, up to a scalar factor. +However, they are generally different at λ ̸∈ Hβ,m. When we aim at θβ,m with m > 1, we have to +choose matrix elements for θβ more carefully in order to use them as building blocks. +Note that it was relatively easy to secure the above two conditions in the case of m = 1. For +higher m we will opt a different strategy: we will satisfy the first condition by the very construction +and bypass a proof of the second with different arguments. +4 +Factorization of Shapovalov elements +For a positive root β ∈ Π denote by Πβ ⊂ Π the set of simple roots entering the expansion of +β over the basis Π with positive coefficients. A simple Lie subalgebra, g(β) ⊂ g, generated by +eα, fα with α ∈ Πβ is called support of β. Its universal enveloping algebra is quantized as a Hopf +subalgebra in Uq(g). +Definition 4.1. Let β ∈ R+ be a positive root and (V, vb, va) a β-representation such that eαvb = 0 +for all α ∈ Π, and (νb, β∨) = 1. We call such β-representation admissible. +If a triple is (V, vb, va) is admissible then vb is the highest vector of a Uq(g)-submodule in V . +For finite dimensional dim V < ∞, vb generates a 2-dimensional submodule of the sl(2)-subalgebra +generated by fβ, eβ. The vector vb can be included in an orthonormal basis in V , as required. +Lemma 4.2. Let (V, vb, va) be an admissible β-representation. Set vm +b = v⊗m +b +∈ V ⊗m for m ∈ N. +Pick up λ ∈ h∗ such that all Verma modules Vλk with λk = λ + kνa, k = 0, . . . , m − 1, are +irreducible. Then there is vm +a ∈ V ⊗m of weight mνa such that +⟨vm +b |vm +a ⟩λ0 = ⟨vb|va⟩λm−1 . . . ⟨vb|va⟩λ0. +(4.7) +Proof. Let λ satisfy the required conditions. There is an equivariant map ϕk : Vλk → V ⊗ Vλk−1 +sending the highest vector vλk to an extremal vector S(va ⊗ vλk−1) ∈ V ⊗ Vλk−1. Here S is the +10 + +universal Shapovalov matrix of Vλk−1. Consider a chain of module homomorphisms +Vλm +ϕm +−→ V ⊗ Vλm−1 +id1⊗ϕm−1 +−→ +V ⊗ (V ⊗ Vλm−2) → . . . +idm−1⊗ϕ1 +−→ +V ⊗(m−1) ⊗ (V ⊗ Vλ0), +where idk are the identity operators on V ⊗k. The vector vλm eventually goes over to S(˜vm +a ⊗ vλ0), +where ˜vm +a ∈ V ⊗m is of weight mνa. It is related with v⊗m +a +by an invertible operator from End(V ⊗m), +which is m − 1-fold dynamical twist [14]. +Let us calculate ⟨vm +b |˜vm +a ⟩λ0 by pairing the tensor leg of S(˜vm +a ⊗vλ0) with vm +b = vb ⊗vm−1 +b +. Using +equality S(˜vm +a ⊗ vλ0) = S +� +va ⊗ S(˜vm−1 +a +⊗ vλ0) +� +we reduce ⟨vm +b |˜vm +a ⟩λ0 to +� +vm−1 +b +, ⟨vb|va⟩(1) +λm−1S1˜vm−1 +a +� +⟨vb|va⟩(2) +λm−1 S2(λ0) = ⟨vb|va⟩(2) +λm−1 +� +ω +� +⟨vb|va⟩(1) +λm−1 +� +vm−1 +b +|˜vm−1 +a +� +λ0, +where we use the Sweedler notation ∆(x) = x(1) ⊗ x(2) ∈ Uq(b−) ⊗ Uq(g−) for the coproduct of +x ∈ Uq(g−). Since yqhαvb = ǫ(y)q(α,β)vb for all y ∈ Uq(g+) and α ∈ Γ+, we arrive at +⟨vm +b |˜vm +a ⟩λ0 = q−(β,νb)⟨vb|va⟩λm−1⟨vm−1 +b +|˜vm−1 +a +⟩λ0. +Proceeding by induction on m we conclude that ⟨vm +b |˜vm +a ⟩λ0 equals the right-hand side of (4.7), up +to the factor q−m(β,νb). Finally, set vm +a = qm(β,νb)˜vm +a . This proves the lemma for generic and hence +for all λ where the right-hand side of (4.7) makes sense. +It follows from the above factorization that the least common denominator of the extremal +vector u = S(vm +a ⊗ vλ) ∈ V ⊗m ⊗ Vλ contains +d(λ) = [ηβ(λ + (m − 1)νa)]q = [(λ + ρ, β) − m +2 (β, β)]q. +It comes from the leftmost factor ⟨vb|va⟩λm−1 in the right-hand side of (4.7). Denote by svm +b ,vm +a (λ) +the matrix element ⟨vm +b |vm +a ⟩λ. Since d divides [ηmβ]q, the re-scaled matrix element +ˇsvm +b ,vm +a (λ) = c(λ)d(λ)svm +b ,vm +a (λ) ∝ +m−1 +� +k=0 +θ(λk), +where c(λ) = −q−ηmβ(λm−1) [ηmβ(λ)]q +d(λ) +, is regular and does not vanish at generic λ ∈ Hβ,m because +d(λ) cancels the pole in ⟨vb|va⟩λm−1. Put ˇu = dk(λ)u, where k ⩾ 1 is the maximal degree of this +pole in u. It is an extremal vector in V ⊗m ⊗ Vλ that is regular at generic λ ∈ Hβ,m. +Indeed, let Hµ denote the null set {λ ∈ h∗|[ηµ(λ)]q = 0} for µ ∈ Γ+. Then the Vλ-components +of ˇu may have poles only at λ ∈ ∪µ<βHµ. But each µ is either not collinear to β or µ = lβ with +l < m. In both cases the complement to Hβ,m ∩ Hµ is dense in Hβ,m because q is not a root of +unity. +11 + +Proposition 4.3. For generic λ ∈ Hβ,m, θβ,m(λ) ∝ ˇsvm +b ,vm +a (λ). +Proof. The singular vector ˇu is presentable as +ˇu = vm +a ⊗ dk(λ)vλ + . . . + vm +b ⊗ dk−1(λ)c(λ)ˇsvm +b ,vm +a (λ)vλ. +We argue that ˇu = vm +b ⊗ c(λ)ˇsvm +b ,vm +a (λ)vλ for generic λ in Hβ,m, where d(λ) = 0. Indeed, the +Vλ-components of ˇu span a Uq(g+)-submodule in Vλ. A vector of maximal weight in this span is +extremal and distinct from vλ. But θβ,m(λ)vλ is the only, up to a factor, extremal vector in Vλ, +for generic λ. Therefore k = 1 and θβ,m ∝ ˇsvm +b ,vm +a . +An admissible β-representation can be associated with every simple root α ∈ Πβ if one sets V +to be the irreducible module of highest weight +(β,β) +ℓ(α,α)ωα, where ℓ = ℓα,β is the multiplicity of α with +which it enters β. We denote this module by Vα,β. It is finite dimensional if +(β,β) +ℓ(α,α) ∈ N. Otherwise +it is a parabolic Verma module relative to a Levi subalgebra with the root basis Π\{α}, cf. the +next section. +One can pass to the ”universal form” of θβ regarding it as an element of ˆUq(b−). Then +θβ,m = (τ m−1 +νb +θβ) . . . (τνbθβ) θβ, +(4.8) +where τν is an automorphism of ˆUq(h) generated by the affine shift of h∗ by the weight ν, that is, +(τνϕ)(µ) = ϕ(µ + ν), ϕ ∈ ˆUq(h), µ ∈ h∗. One may ask when the shift is trivial, τνbθβ = θβ, and +θβ,m is just the m-th power of θβ. +Proposition 4.4. Let β be a positive root. Suppose that there is α ∈ Πβ with ℓα,β = 1. Then +θβ,m = θm +β ∈ Uq(b−). +Proof. Let s ⊂ g be a semi-simple subalgebra generated by simple root vectors fµ, eµ with µ ̸= α. +Take for V the module Vα,β with highest weight φ = (β,β) +(α,α)ωα. Put vb to be the highest vector and +va ∝ fβvb. +Both va and vb can be included in an orthonormal basis because they span their weight sub- +spaces in V . Therefore ⟨vb|va⟩ = sba can be calculated by formula (3.5). We write it as +θβ(λ) = cba + +� +vb≻vi≻va +cbisia(λ). +The highest vector vb is killed by s−, therefore the Hasse diagram between va and vb is +vb +eα +←− +fαvb +. . . +va, +12 + +where arrows in the suppressed part are simple root vectors from Uq(s+). But then the only copy +of fα is in cbi while all sia belong to Uq(s−) ˆUq(hs), the extended Borel subalgebra of Uq(s). +Finally, since Πs is orthogonal to νb, we have (µ, νa) = −(µ, β) for all µ ∈ R+ +s . Therefore +θβ(λk) = θβ(λ − kβ), +θβ,m(λ) = +m−1 +� +k=0 +θβ(λ − kβ), +where the product is taken in the descending order from left to right. This proves the plain power +factorization because each θβ carries weight −β. +Conditions of the above proposition are fulfilled for all pairs α, β in the case of sl(n). +5 +Shapovalov elements of degree 1 +In this section we describe the factor θβ entering (4.8), for a particular admissible β-representation +(V, vb, va). We give a complete solution to the problem in the classical case. In the case of q ̸= 1, +we do it up to calculation of the entries of the matrix C in a simple finite dimensional module ˜g +that is a q-deformation of the adjoint module g. Its highest weight is the maximal root, ξ ∈ R+. +To achieve our goals, we need to figure out the Hasse sub-diagram H(vb, va) ⊂ H(V ) that +comprises all possible routes from va to vb. We argue that H(vb, va) can be extracted from a +diagram H(b−) which we introduce below, and the underlying Uq(g+)-modules are isomorphic. +The Uq(g+)-module associated with H(b−) is constructed from ˜g by factoring out the span +of positive weight spaces. In order to distinguish the case of q ̸= 1 from classical and to avoid +confusion with root vectors, we will mark the nodes with tilde. Vectors ˜fη of weights −η ∈ −R+ +are defined uniquely up to a sign if we normalize them by ( ˜fη, ˜fη) = 1. We may assume that they +are deformations of classical root vectors. We take ˜hα = eα ˜fα, α ∈ Π, for basis elements of zero +weight. +For example, the diagram H(b−) in the case of g = g2 is +b− +eα2 +eα1 +eα2 +eα2 +eα1 +˜hα2 +˜fα2 +˜fα1+α2 +˜fα1+2α2 +˜fα1+3α2 +˜f2α1+3α2 +❜ +❜ +❜ +❜ +❜ +❜ +✛ +✟ +✟ +✟ +✟ +✙ +✛ +✛ +✛ +eα1 +eα2 +˜hα1 +˜fα1 +❜ +❜ +✛ +❍ +❍ +❍ +❍ +❨ +From now on we fix V = Vα,β with highest weight φ = +(β,β) +ℓα,β(α,α)ωα and highest vector vb. We +denote by l ⊂ g a reductive Lie subalgebra of maximal rank whose root system is Πl = R\{α} +and by p = l + g+ its parabolic extension. +13 + +In order to construct the start node va ∈ V , we will use the following observation. Recall that +a singular vector � +i wi ⊗ vi in a tensor product W ⊗ V of two irreducible modules of highest +weight defines a Uq(g+)-homomorphism W ∗ → V (and respectively V ∗ → W). +Here W ∗ is +an irreducible Uq(g)-module of lowest weight, which is negative the highest weight of W. The +dual action is defined with the help of antipode γ in the standard way: (xϕ)(w) = ϕ +� +γ(x)w +� +, +for x ∈ Uq(g+), w ∈ W, and ϕ ∈ W ∗. The homomorphism W ∗ → V is implemented via the +assignment ϕ �→ � +i ϕ(wi)vi. We will apply this construction to W = ˜g. +Lemma 5.1. There exists a unique, up to a scalar factor, singular vector u ∈ ˜g ⊗ V of weight φ. +Proof. Let J ⊂ Uq(g−) be the annihilator of the highest vector vb ∈ V . Singular vectors in ˜g ⊗ V +of weight φ are in bijection with vectors ˜h ∈ ˜g of zero weight killed by the left ideal σ(J) ⊂ Uq(g+). +Pick up ˜h ̸= 0 orthogonal to all µ ∈ Πl; it is unique up to a scalar factor. +The ideal J is generated by elements θ ∈ Uq(g+) such that θvb are singular vectors in the +Verma module Vφ covering V . By construction, ˜u is killed by eα ∈ J with α ∈ Πl. If θvφ ∈ Vφ +is a singular vector of weight φ − mη with η ∈ R+\R+ +l , then m > 1. Indeed, since φ = lωα with +positive rational l = +(β,β) +ℓα,β(α,α), we have an inequality l(ωα, η∨) + (ρ, η∨) > 1. Then the condition +(2.1), where λ is replaced with φ and β with η, is fulfilled only if m > 1, since q is not a root of +unity. Then the element σ(θ) kills ˜h because mη with m > 1 is not a weight of ˜g. +Remark that V is finite dimensional if +(β,β) +ℓα,β(α,α) ∈ Z and a parabolic Verma module otherwise +because its highest weight is away from De Concini-Kac-Kazhdan hyperplanes Hη,m with η ∈ +R+\R+ +l . +Now let va ∈ V be the vector of minimal weight in the expansion u = ˜eξ ⊗ va + . . . over the +chosen basis in ˜g (we have omitted the terms of lower weights in the ˜g-factor). Notice that in +the classical case the vector fηvb does not vanish if η ∈ R+\R+ +l because (η, φ) > 0. In particular, +va ∝ fξvb ̸= 0 for the maximal root ξ. For general q, va is killed by the left ideal in Uq(g+) +annihilating the lowest vector ˜fξ ∈ ˜g ≃ ˜g∗, Such va is unique in V up to a scalar factor, because +of Lemma 5.1. +Introduce a partial order on positive roots by writing µ ≺ ν iff fµ ≻ fν in H(b−). This is in +agreement with the partial order on H(g+) ⊂ H(g), which is exactly the Hasse diagram of the root +system R+, [26]. Note that α ≺ β for simple α if and only if α ∈ Πβ. +Proposition 5.2. Let u = ˜eξ ⊗ va + . . . be the singular vector from Lemma 5.1 with va ∈ V of +minimal weight in the expansion over a weight basis in ˜g. Then the Uq(g+)-module generated by +va ∈ V is isomorphic to ˜g(˜hα, ˜fξ), for almost all q. +14 + +Proof. The Uq(g+)-module homomorphism ˜g → V determined by the assignment ˜fξ �→ va factors +through the quotient g(˜hα, ˜fξ) because the kernel includes all ˜fη with η ∈ R+ +l , all ˜hη = eµ ˜fη with +η ∈ Πl, and all negative weight spaces. We are left to prove that it is an isomorphism on g(˜hα, ˜fξ) +for almost all q. It is sufficient to check that it is injective for q = 1 because V rationally depends +on q. But then for each positive root η subject to α ⪯ η ⪯ ξ the vector fηvb is in U(g+)fξvb and +is not zero, because (η, φ) > 0. +It follows that eβva ̸= 0 because eβ ˜fξ ̸= 0. Therefore (V, vb, va) is an admissible β-representation +for almost all q. +Let us consider the classical case in more detail. We choose h∨ +α = +2 +(α,α)hα, α ∈ Π, as a basis +in h ⊂ b−, so that α(h∨ +α) = 2. The root vectors fµ with µ ∈ R+ form a basis in g−. Arrows +labeled by α ∈ Π are h∨ +α +eα +←− fα and fµ +eα +←− fν if µ = ν − α is a positive root. The U(g+)-module +underlying H(b−) is g/g+. +Specialization of the formula (3.5) for θβ requires the knowledge of matrix C = (π ⊗ id)(C) ∈ +End(V ) ⊗ Uq(g−), which is readily available for q = 1. For ν, γ ∈ R+, denote by Cν,γ ∈ C the +scalars such that [eν, fγ] = Cν,γfγ−ν, if γ − ν ∈ R+, Cγ,γ = (β,β) +2 +ℓα,γ +ℓα,β , and Cν,γ = 0 otherwise. Then +(π ⊗ id)(C)(fγvb ⊗ 1) = vb ⊗ Cγ,γfγ + +� +ν≺γ +fγ−νvb ⊗ Cν,γfν, +for all γ satisfying α ⪯ γ ⪯ β. This equality yields all entries of the matrix C needed. The +formula (3.5) becomes +θβ = Cβ,βfβ + +� +k⩾1 +� +ν1+...+νk+1=β +(Cνk+1,γk . . . Cν1,γ0)(fνk+1 . . . fν1) +(−1)k +ηµk . . . ηµ1 +. +(5.9) +The internal summation is performed over all partitions of β to a sum of νi ∈ R+ such that all +γi = γi−1 − νi for i = 1, . . . , k with γ0 = β are in R+ and subject to α ⪯ γi. In particular, +γk = νk+1. The weights µi are defined to be µi = γ0 − γi = ν1 + . . . + νi. Note that in the q ̸= 1 +case the corresponding sum may involve terms with entries of C whose weights are not roots. +Now we summarise the results of this paper. +Theorem 5.3. For each α ≺ β, the rescaled matrix element ⟨˜hα| ˜fβ⟩[ηβ]q with ˜hα, ˜fβ ∈ ˜g, is a +Shapovalov element θβ,1. For general degree m > 1, θβ,m is given by the factorization formula +(4.8) with θβ = θβ,1 and the shift weight νb = +(β,β) +ℓα,β(α,α)ωα. +Proof. Observe that summation formula (3.5) involves only the structure of Uq(g+)-module deter- +mined by the initial and final nodes. That is straightforward with regard to the matrix elements +15 + +of C and also true for the Cartan factors, which depend only on weight differences (mind that +weights in a cyclic Uq(g+)-module generated by a weight vector are fixed up to a constant weight +summand). Furthermore, the nodes of the sub-diagram H(va, vb) can be included in an orthonor- +mal basis whence sba ∝ ⟨vb|va⟩. Now, for almost all q, the theorem follows from Proposition 5.2 +and Proposition 4.3 with Lemma 4.2. Therefore it is true for all q where the factors (4.8) are +defined. +We remark in conclusion that for fixed β ∈ R+ one can pick up α ∈ Πβ delivering the simplest +Hasse diagram H(˜hα, ˜fβ), e.g. with the smallest fundamental group. Such diagrams can be found +amongst subdiagrams in fundamental auxiliary modules of minimal dimension. That also applies +to their associated Uq(g+)-modules. For all non-exceptional types of g, the entries of the matrix C +participating in the route summation formula are calculated in [27], Proposition 2.2. That is also +done for g2 in [28]. This makes the above description of Shapovalov elements for such quantum +groups absolutely explicit. For exceptional g of rank > 2, the problem reduces to calculation of +relevant entries of C. +In the context of quantization of semi-simple conjugacy classes [10], it is crucial to make +sure that θβ,m(λ) tends to f m +β as q → 1. Factorization (4.8) together with the route summation +formula for θβ,1 gives important information about possible singularities of θβ,m(λ) and facilitate +the analysis even without knowing the matrix elements of C. +Acknowledgement +This work is partially supported by the Moscow Institute of Physics and Technology under the +Priority 2030 Strategic Academic Leadership Program and by Russian Science Foundation grant +23-21-00282. The author thanks Vadim Ostapenko and Vladimir Stukopin for stimulating discus- +sions. +References +[1] Bernstein, J. H., Gelfand, I. M., Gelfand, S. I.: On some category of g-modules, Funct. Anal. +Appl. 10 no. 2 (1976), 87–92. +[2] Humphreys, J. Representations of Semisimple Lie Algebras in the BGG Category O, Graduate +Studies in Mathematics 94, AMS, 2008. +16 + +[3] Bernstein, J. H., Gelfand, I. M., Gelfand, S. I.: Structure of representations generated by +highest weight vectors, Funct. Anal. Appl. 5 no. 1 (1971), 1–9. +[4] Shapovalov, N. 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Phys., 56 (2015), 081701. +[28] Baranov, A., Mudrov, A., and Ostapenko, V.: +Quantum exceptional group G2 and its +semisimple conjugacy classes, Alg.& Rep.Theor., 23 (2020) 1827–1848. +18 + diff --git a/6dE0T4oBgHgl3EQfvwHw/content/tmp_files/load_file.txt b/6dE0T4oBgHgl3EQfvwHw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f6214c0f39e2f70dae34a254a1cd3baa2e05d85c --- /dev/null +++ b/6dE0T4oBgHgl3EQfvwHw/content/tmp_files/load_file.txt @@ -0,0 +1,652 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf,len=651 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='02624v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='QA] 6 Jan 2023 Shapovalov elements of classical and quantum groups Andrey Mudrov In memorium of Vladimir Lyachovsky Moscow Institute of Physics and Technology, 9 Institutskiy per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=', Dolgoprudny, Moscow Region, 141701, Russia, University of Leicester, University Road, LE1 7RH Leicester, UK, e-mail: am405@le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='uk Abstract Shapovalov elements θβ,m of the classical or quantized universal enveloping algebra of a simple Lie algebra g are parameterized by a positive root β and a positive integer m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' They relate the highest vector of a reducible Verma module with highest vectors of its submodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We obtain a factorization of θβ,m to a product of θβ,1 and calculate θβ,1 as a residue of a matrix element of the inverse Shapovalov form via a generalized Nigel-Moshinsky algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This way we explicitly express θβ,m of a classical simple Lie algebra through the Cartan-Weyl basis in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' In the case of quantum groups, we give an analogous formulation through the entries of the R-matrix (quantum L-operator) in fundamental representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Key words: Shapovalov elements, Shapovalov form, Verma modules, singular vectors, Hasse diagrams, R-matrix AMS classification codes: 17B10, 17B37 1 1 Introduction Category O introduced in [1] for semi-simple Lie algebras and later defined for many other classes of algebras including quantum groups plays a fundamental role in various fields of mathematics and mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' In particular, it accommodates finite-dimensional and numerous important infinite dimensional representations like parabolic Verma modules and their generalizations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' There are distinguished objects in O called Verma modules that feature a universality property: all simple modules in O are their quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The maximal proper submodule in a Verma module is generated by extremal vectors [3], which are invariants of the positive triangular subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This makes extremal vectors critically important in representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Extremal vectors in a Verma module Vλ are related with the vacuum vector of highest weight λ via special elements θβ,m of the (classical or quantum) universal enveloping of the negative Borel subalgebra that are called Shapovalov elements [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' They are parameterized with a positive root β and an integer m ∈ N validating a De Concini-Kac-Kazhdan condition on λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' In the classical version, it is 2(λ + ρ, β) − m(β, β) = 0 with ρ being the half-sum of positive roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This condition guarantees that the Verma module is reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' In the special case when the root β is simple, the element θβ,m is a power f m β of the simple root vector fβ of weight β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For non-simple β, the Shapovalov elements are complicated polynomials in negative Chevalley generators with coefficients in the Cartan subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It can be viewed as a function θβ,m(λ) of the highest weight of a generic Verma module Vλ with values in the subalgebra generated by negative root vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' A description of extremal vectors in Verma modules over classical Kac-Moody algebras was obtained in [6] via an interpolation procedure resulting in a calculus of polynomials with complex exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Another approach based on extremal projectors [7] was employed by Zhelobenko in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' He obtained θβ,m for simple Lie algebras as a product of m copies of θβ,1 with shifted weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The idea of factorization was also used in a construction of Shapovalov elements for contragredient Lie superalgebras in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Factorization of θβ,m into a product of polynomials of lower degree is a great simplification that is convenient for their analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For example it is good for the study of the classical limit in the case of quantum groups, which is crucial for quantization of conjugacy classes [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' With regard to quantum groups, an inductive construction of extremal vectors in Verma mod- ules was suggested in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Explicit expressions for Shapovalov elements for the A-type appeared in [12] and recently were obtained in [13] by other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It is worthy to note that ordered PBW-like monomials in θβ,1 deliver an orthogonal basis in generic irreducible Verma modules [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' While Zhelobenko’s factorization via extremal projectors simplifies construction of Shapovalov 2 elements in the case of classical simple Lie algebras, there remains a problem of explicit descrip- tion of the factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' In this paper, we pursue an alternative approach based on the canonical contravariant bilinear form on Verma modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It gives expressions for all Shapovalov elements in a factorized form through root vectors in the classical case and through elements of the R-matrix in the adjoint representation in the case of quantum groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Extremal vectors generate the kernel of the canonical contravariant form on Vλ, which is a specialization at λ of the ”universal” Shapovalov form on the Borel subalgebra [4] with values in the Cartan subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This contravariant form itself is extremely important and has numerous applications, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' [14, 15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For generic λ, the module Vλ is irreducible and the form is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The inverse form gives rise to an element S of extended tensor product of positive and negative subalgebras of the (quantized) universal enveloping algebra [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Sending the positive leg of S to an auxiliary representation yields a matrix with entries in the negative subalgebra which we call Shapovalov matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Its explicit description was obtained in [18] by generalization of Nagel-Moshinski formulas for the lowering operators of sl(n) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' They can also be derived (in the quantum setting) from the ABRR equation [20] on dynamical twist [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Our method relates θβ,m with certain entries of the Shapovalov matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This point of view is quite natural because the kernel of the contravariant form on Vλ results in poles of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Our approach not only provides a factorization of θβ,m to a product of θβ,1 but also an efficient description of θβ,1 in a very elementary way, by a generalized Nagel-Moshinsky rule (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='5) using a technique of Hasse diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We do it by evaluating residues of matrix elements of S that go singular at a De Concini-Kac-Kazhdan ”hyperplane”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Our approach is absolutely parallel for a classical semi-simple Lie algebra g and its quantum group Uq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The classical case can be processed directly or obtained as the limit case q → 1 of the deformation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let us describe the method in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' With a module V from the category O and a pair of non-zero vectors vb, va ∈ V we associate a Shapovalov matrix element, ⟨vb|va⟩, which belongs to the negative Borel (universal enveloping) subalgebra ˆUq(b−) rationally extended over the Cartan subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Under certain assumptions on vb and va, such matrix elements deliver factors in θβ,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' These factors normalize positive root vectors of a reductive Lie subalgebra l ⊂ g whose negative counterparts annihilate vb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This way they become lowering operators in the Mickelsson algebras of the pair (g, l), [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' When λ satisfies the De Concini-Kac-Kazhdan condition, the factors become θβ,1 shifted by certain weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The vector vb should be highest for the minimal simple Lie subalgebra in g that accommodates the root β and and its weight should satisfy the condition (νb, β) = (β,β) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For finite dimensional V the latter is equivalent to saying that vb generates a 2-dimensional submodule of the sl(2)- 3 subalgebra generated by the root spaces g±β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The vector va determines a homomorphism Vλ2 → V ⊗ Vλ1, where Vλi are irreducible Verma modules of highest weights λi and λ2 − λ1 equals the weight of va.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Iteration of this construction yields a chain of homomorphisms Vλm → V ⊗ Vλm−1 → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' → V ⊗m ⊗ Vλ0, where each mapping V ⊗i ⊗ Vλm−i → V ⊗i ⊗ (V ⊗ Vλm−i−1) is identical on the factor V ⊗i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We prove factorization of ⟨v⊗m b |v⊗m a ⟩ to a product of ⟨vb|va⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then we demonstrate that, under the specified conditions, that matrix element is proportional to θβ,m(λ) with λ0 = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' As a result, we obtain θβ,m(λ) as a product �m−1 i=0 θβ,1(λi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The factors θβ,1 are calculated by a general rule (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='5) specialized to the case in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Viewed as an element of ˆUq(b−), θβ,m becomes a product of θβ,1 shifted by the integer multiple weights of vb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This shift can be made trivial by a choice of V if β contains a simple root α with multiplicity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then θβ,m becomes the m-th power of θβ,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It is worthwhile mentioning that θβ,1 can be obtained via an arbitrary auxiliary module V with a pair of vectors (va, vb = eβva).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' They all coincide up to a scalar factor on the De Concini-Kac- Kazhdan ”hyperplane” and generally differ away from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The problem is to use θβ,1 as a factor block for constructing θβ,m of higher m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' That is why we choose (V, vb, va) in a special way as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' On the other hand, since the left tensor leg of S is in the positive subalgebra Uq(g+) ⊂ Uq(g), it is the structure of Uq(g+)-submodule on V that determines θβ,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' A remarkable fact is that the cyclic submodule Uq(g+)va in an admissible V turns out to be isomorphic to a subquotient of the Uq(g+)-module corresponding to g/g+ in the classical limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This means that θβ,1 in each case can be calculated via Shapovalov matrix elements from End(g/g+) ⊗ Uq(b−), by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Except for Section 5, we present only the q-version of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The classical case can be obtained by sending q to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' However the final expression for θβ,1 is greatly simplified when q = 1, so we give a special consideration to it in the Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' 2 Preliminaries Let g be a simple complex Lie algebra and h ⊂ g its Cartan subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Fix a triangular de- composition g = g− ⊕ h ⊕ g+ with maximal nilpotent Lie subalgebras g±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Denote by R ⊂ h∗ the root system of g, and by R+ the subset of positive roots with basis Π of simple roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This basis generates a root lattice Γ ⊂ h∗ with the positive semigroup Γ+ = Z+Π ⊂ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' 4 For a positive root β ∈ R+ and a simple root α ∈ Π denote by ℓα,β ∈ Z+ the multiplicity with which α enters β, that is the α-coefficient in the expansion of β over the basis Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Choose an ad-invariant form ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' ) on g, restrict it to h, and transfer to h∗ by duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For every λ ∈ h∗ there is a unique element hλ ∈ h such that µ(hλ) = (µ, λ), for all µ ∈ h∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For a non-isotropic µ ∈ h∗ set µ∨ = 2 (µ,µ)µ and h∨ µ = 2 (µ,µ)hµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Fundamental weights are denoted by ωα, α ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' They are determined by the system of equations (ωα, β∨) = δα,β, for all α, β ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We assume that q ∈ C is not a root of unity and we understand that when saying ”all q”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' By almost all q we mean all q excepting maybe a finite set of values distinct from q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The standard Drinfeld-Jimbo quantum group Uq(g) is a complex Hopf algebra with the set of generators eα, fα, and q±hα labeled by simple roots α and satisfying relations [22, 23] qhαeβ = q(α,β)eβqhα, [eα, fβ] = δα,β[hα]q, qhαfβ = q−(α,β)fβqhα, α, β ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The symbol [z]q, where z ∈ h + C, stands for qz−q−z q−q−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The elements qhα are invertible, with qhαq−hα = 1, while {eα}α∈Π and {fα}α∈Π also satisfy quantized Serre relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Their exact form is not important for this presentation, see [24] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' A Hopf algebra structure on Uq(g) is introduced by the comultiplication ∆(fα) = fα ⊗ 1 + q−hα ⊗ fα, ∆(q±hα) = q±hα ⊗ q±hα, ∆(eα) = eα ⊗ qhα + 1 ⊗ eα set up on the generators and extended as a homomorphism Uq(g) → Uq(g)⊗Uq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The antipode is an algebra and coalgebra anti-automorphism of Uq(g) that acts on the generators by the assignment γ(fα) = −qhαfα, γ(q±hα) = q∓hα, γ(eα) = −eαq−hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The counit homomorphism ǫ: Uq(g) → C returns ǫ(eα) = 0, ǫ(fα) = 0, ǫ(qhα) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We extend the notation fα, eα to all α ∈ R+ meaning the Lusztig root vectors with respect to some normal ordering of R+, [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' They are known to generate a Poincare-Birkhoff-Witt (PBW) basis in Uq(g±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Denote by Uq(h), Uq(g+), and Uq(g−) subalgebras in Uq(g) generated by {q±hα}α∈Π, {eα}α∈Π, and {fα}α∈Π, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The quantum Borel subgroups are defined as Uq(b±) = Uq(g±)Uq(h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' they are Hopf subalgebras in Uq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We will also need their extended version ˆUq(b±) = Uq(g±) ˆUq(h), where ˆUq(h) is the ring of fractions of Uq(h) over the multiplicative system generated by [hα − c]q with α ∈ Γ+ and c ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' 5 Given a Uq(g)-module V , a non-zero vector v is said to be of weight µ if qhαv = q(µ,α)v for all α ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The linear span of such vectors is denoted by V [µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' A module V is said to be of highest weight λ if it is generated by a weight vector v ∈ V [λ] that is killed by all eα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Such vector v is called highest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' it is defined up to a non-zero scalar multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We define an involutive coalgebra anti-automorphism and algebra automorphism σ of Uq(g) setting it on the generators by the assignment σ: eα �→ fα, σ: fα �→ eα, σ: qhα �→ q−hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The involution ω = γ−1 ◦ σ = σ ◦ γ is an algebra anti-automorphism of Uq(g) and preserves the comultiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' A symmetric bilinear form (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=') on a g-module V is called contravariant if � xv, w � = � v, ω(x)w � for all x ∈ Uq(g), v, w ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' A module of highest weight has a unique C-valued contravariant form such that squared norm of the highest vector is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We call this form canonical and extend this term to a form on tensor products that is the product of canonical forms on tensor factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Such a form is contravariant because ω is a coalgebra map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let us recall the definition of Uq(h)-valued Shapovalov form on the Borel subalgebra Uq(b−) that was introduced for U(g) and studied in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Regard Uq(b−) as a free right Uq(h)-module gen- erated by Uq(g−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The triangular decomposition Uq(g) = Uq(g−)Uq(h)Uq(g+) facilitates projection ℘: Uq(g) → Uq(h) along the sum g−Uq(g) + Uq(g)g+, where g−Uq(g) and Uq(g)g+ are right and left ideals generated by positive and negative root vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Set (x, y) = ℘ � ω(x)y � , x, y ∈ Uq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Thus defined the form is Uq(h)-linear and contravariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It follows that the left ideal Uq(g)g+ is in the kernel, so the form descends to a Uq(h)-linear form on the quotient Uq(g)/Uq(g)g+ ≃ Uq(b−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' A Verma module Vλ = Uq(g) ⊗Uq(b+) Cλ of highest weight λ ∈ h∗ is induced from the 1- dimensional Uq(b+)-module Cλ that is trivial on Uq(g+) and returns q(λ,α) on qhα ∈ Uq(h), α ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Its highest vector is denoted by vλ, which is also called vacuum vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It freely generates Vλ over Uq(g−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Specialization of the Shapovalov form at λ ∈ h∗ yields the canonical contravariant C-valued form (x, y)λ = λ � ℘ � ω(x)y �� on Vλ, upon a natural Uq(g−)-module isomorphism Uq(g−) ≃ Vλ extending the assignment 1 �→ vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Conversely, the canonical contravariant form on Vλ regarded as a function of λ descends to the Shapovalov form if one views Uq(h) as the algebra of polynomial functions on h∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' By an abuse of terminology, we also mean by Shapovalov form the canonical contravariant form on Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' 6 It is known from [25] that the contravariant form on Vλ module goes degenerate if and only if its highest weight is in the union of Hβ,m = {λ ∈ h∗ | q2(λ+ρ,β)−m(β,β) = 1} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='1) over β ∈ R+ and m ∈ N, where ρ = 1 2 � α∈R+ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' In the classical case q = 1, Hβ,m becomes a Kac-Kazhdan hyperplane of weights satisfying 2(λ + ρ, β) = m(β, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Recall that a vector v ∈ Vλ of weight λ−µ with µ ∈ Γ+, µ ̸= 0, is called extremal if eαv = 0 for all α ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Extremal vectors are in the kernel of the contravariant form and generate submodules of the corresponding highest weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We will be interested in the special case when µ = mβ with β ∈ R+ and m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then the highest weight λ has to be in Hβ,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The image θβ,m of v under the isomorphism Vλ → Uq(g−) is called Shapovalov element of a positive root β and degree m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For simple β the element θβ,m is just the m-th power of the root vector, θβ,m = f m β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For non-simple β, it is a rational trigonometric function Hβ,m → Uq(g−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The goal of this work is to find explicit expressions for θβ,m with non-simple β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' 3 Shapovalov inverse form and its matrix elements Define an opposite Verma Uq(g)-module V ′ λ of lowest weight −λ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The underlying vector space of V ′ λ is taken to be Vλ, while the representation homomorphism π′ λ is twisted by σ, that is π′ λ = πλ ◦ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The module V ′ λ is freely generated over Uq(g+) by its lowest vector v′ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let σλ : Vλ → V ′ λ denote the isomorphism of vector spaces, xvλ �→ σ(x)v′ λ, x ∈ Uq(g−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It intertwines the representations homomorphisms π′ λ◦σ = σλ◦πλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This map relates the contravariant form on Vλ with a Uq(g)-invariant pairing Vλ ⊗ V ′ λ → Vλ ⊗ Vλ → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Suppose that the module Vλ is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then its invariant pairing is non-degenerate (as well as the contravariant form on Vλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The inverse form belongs to a completed tensor product V ′ λ ˆ⊗Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Under the isomorphisms Vλ → Uq(g−), V ′ λ → Uq(g+), it goes to an element that we denote by S ∈ Uq(g+)ˆ⊗Uq(g−) and call universal Shapovalov matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Given a Uq(g+)-locally nilpotent Uq(g)- module V with representation homomorphism π: Uq(g) → End(V ) the image S = (π ⊗ id)(S) is a matrix with entries in Uq(g−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It features a rational trigonometric (rational in the classical case) dependance on λ ∈ h∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We will assume that V is diagonalizable with finite dimensional weight spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We will also assume that V is endowed with a non-degenerate contravariant form, for instance, if V is a tensor power of an irreducible module of highest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Using terminology adopted in the quantum inverse scattering theory, we call the module V auxiliary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' 7 Varying the highest weight λ we get a rational trigonometric dependance of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' As a function of λ, S is regarded as an element of Uq(g+)ˆ⊗ ˆUq(b−), where ˆUq(b−) is viewed as a right ˆUq(h)-module freely generated by Uq(g−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This way the weight dependance is accommodated by the right tensor leg of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' An explicit expression of S in a weight basis {vi}i∈I ⊂ V , vi ∈ V [νi], can be formulated in terms of Hasse diagram, H(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Such a diagram is associated with any partially ordered sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' In our case the partial ordering is induced by the Uq(g+)-action on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Nodes are elements of the basis {vi}i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Arrows are simple root vectors eα connecting the nodes vi eα ←− vj whose weight difference is νi − νj = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then a node vi is succeeding a node vj if νi − νj ∈ Γ+\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The matrix S is triangular: sii = 1 and sij = 0 if νi ̸≻ νj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The entry sij is a rational trigonometric function h∗ → Uq(g−) taking values in the subspace of weight νj − νi ∈ −Γ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It is also convenient to introduce a stronger partial ordering as we will explain below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Clearly the matrix S depends only on the Uq(b+)-module structure on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' In order to calculate a particular element sij, we can choose a weight basis that extends a basis in the cyclic submodule Uq(g+)vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then, in particular, sij = 0 if vi ̸∈ Uq(g+)vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We define a Hasse sub-diagram H(vi, vj) ⊂ H(V ) that comprises all possible routes from vj to vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' A node vk ∈ H(V ) is in H(vi, vj) if and only if vi ⪰ vk ⪰ vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The sub-diagram H(vi, vj) is associated with a Uq(g+)-module V (vi, vj) that is the quotient of Uq(g+)vj by the sum of cyclic submodules Uq(g+)vk ⊂ Uq(g+)vj where vk ̸∈ H(vi, vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It is the module V (vi, vj) that is needed to calculate a matrix element sij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We recall a construction of S following [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let {hi}rkg i=1 ∈ h be an orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The element q � i hi⊗hi belongs to a completion of Uq(h)⊗Uq(h) in the ℏ = ln q-adic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Choose an R-matrix R of Uq(g) such that ˇR = q− � i hi⊗hiR ∈ Uq(g+)ˆ⊗Uq(g−) and set C = 1 q−q−1( ˇR − 1 ⊗ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The key identity on C that facilitates the q-version of the theory is [18] [1 ⊗ eα, C] + (eα ⊗ q−hα)C − C(eα ⊗ qhα) = eα ⊗ [hα]q, ∀α ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='2) In the classical limit, C = � α∈R+ eα ⊗ fα becomes the polarized split Casimir of g without its Cartan part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' One then recovers an identity [1 ⊗ eα, C] + [eα ⊗ 1, C] = eα ⊗ hα (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='3) for each simple root α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let cij ∈ Uq(g−) denote the entries of the matrix (π ⊗ id)(C) ∈ End(V ) ⊗ Uq(g−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We rectify the partial ordering and the Hasse diagram H(V ) by removing arrows vi ← vj if cij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This will not affect the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='5) for matrix elements of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' 8 For each weight µ ∈ Γ+ put ηµ = hµ + (µ, ρ) − 1 2(µ, µ) ∈ h ⊕ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='4) Regard ηµ as an affine function on h∗ by the assignment ηµ : ζ �→ (µ, ζ + ρ) − 1 2(µ, µ), ζ ∈ h∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Observe that ηmβ = m � hβ + (β, ρ) − m 2 (β, β) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' That is, [ηmβ(λ)]q vanishes on Hβ,m (and only on Hβ,m in the classical case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For a pair of non-zero vectors v, w ∈ V define a matrix element ⟨w|v⟩ = (w, S1v)S2 ∈ ˆUq(b−), where S1 ⊗ S2 stands for a Sweedler-like notation for S and the pairing is with respect to a non- degenerate contravariant form on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Its specialization at a weight λ is denoted by ⟨w|v⟩λ, which can be determined from the equality ⟨w|v⟩λvλ = ⟨w|v⟩vλ ∈ Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For each w from V , the map V → Vλ, v �→ ⟨v|w⟩vλ satisfies: eα⟨v|w⟩vλ = ⟨σ(eα)v|w⟩vλ for all α ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This is a consequence of Uq(g+)-invariance of the tensor S(1 ⊗ vλ) ∈ Uq(g+)ˆ⊗Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The matrix element ⟨vi|vj⟩ equals sij if (vi, vk) = δik for all k ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It will be always the case in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Fix a ”start” node va and an ”end” node vb such that vb ≻ va.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then a re-scaled matrix element ˇsab = −sba[ηνb−νa]qq−ηνb−νa can be calculated by the formula ˇsba = cba + � k⩾1 � vb≻vk≻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='≻v1≻va cbk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' c1a (−1)kqηµk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' qηµ1 [ηµk]q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' [ηµ1]q ∈ ˆUq(b−), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='5) where µl = νl − νa ∈ Γ+, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Here the summation is performed over all possible routes (sequences of ordered nodes) from va to vb, see [18] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It is straightforward that Uq(g+)-invariance of the tensor S(va ⊗ vλ) implies eαˇsba(λ)vλ ∝ [ηνb−νa(λ)]q � k π(eα)bkska(λ)vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='6) The matrix entries ska(λ) carry weight −(νb − νa − α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It follows that ˇsba(λ)vλ is an extremal vector in Vλ for λ satisfying [ηνb−νa(λ)]q = 0 provided 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' ˇsba(λ) ̸= 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' λ is a regular point for all ska(λ) and all α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We aim to find an appropriate matrix element for θβ,m that satisfies these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let V be a Uq(g)-module with a pair of vectors va, vb ∈ V such that eβva = vb for β ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We call the triple (V, vb, va) a β-representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let (V, vb, va) be a β-representation for β ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then for generic λ ∈ Hβ,1 the vector ˇsba(λ)vλ ∈ Vλ is extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The factors qηµk [ηµk ]q in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='5) go singular on the union of a finite number of the null-sets {λ ∈ h∗ | [ηµk(λ)]q = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' None of µk is collinear to β, hence ˇsba(λ) is regular at generic λ ∈ Hβ,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' By the same reasoning, all ska(λ) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='6) are regular at such λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Finally, the first term cba (and only this one) involves the Lusztig root vector fβ, a generator of a PBW basis in Uq(g−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It is therefore independent of the other terms, and ˇsba(λ) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Upon identification of ˆUq(b−) with rational Uq(g−)-valued functions on h∗ we conclude that ˇsba is a Shapovalov element θβ,1 and denote it by θβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Uniqueness of extremal vector of given weight implies that all matrix elements ˇsba with vb = eβva deliver the same θβ, up to a scalar factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' However, they are generally different at λ ̸∈ Hβ,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' When we aim at θβ,m with m > 1, we have to choose matrix elements for θβ more carefully in order to use them as building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Note that it was relatively easy to secure the above two conditions in the case of m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For higher m we will opt a different strategy: we will satisfy the first condition by the very construction and bypass a proof of the second with different arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' 4 Factorization of Shapovalov elements For a positive root β ∈ Π denote by Πβ ⊂ Π the set of simple roots entering the expansion of β over the basis Π with positive coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' A simple Lie subalgebra, g(β) ⊂ g, generated by eα, fα with α ∈ Πβ is called support of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Its universal enveloping algebra is quantized as a Hopf subalgebra in Uq(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let β ∈ R+ be a positive root and (V, vb, va) a β-representation such that eαvb = 0 for all α ∈ Π, and (νb, β∨) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We call such β-representation admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' If a triple is (V, vb, va) is admissible then vb is the highest vector of a Uq(g)-submodule in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For finite dimensional dim V < ∞, vb generates a 2-dimensional submodule of the sl(2)-subalgebra generated by fβ, eβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The vector vb can be included in an orthonormal basis in V , as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let (V, vb, va) be an admissible β-representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Set vm b = v⊗m b ∈ V ⊗m for m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Pick up λ ∈ h∗ such that all Verma modules Vλk with λk = λ + kνa, k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' , m − 1, are irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then there is vm a ∈ V ⊗m of weight mνa such that ⟨vm b |vm a ⟩λ0 = ⟨vb|va⟩λm−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' ⟨vb|va⟩λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let λ satisfy the required conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' There is an equivariant map ϕk : Vλk → V ⊗ Vλk−1 sending the highest vector vλk to an extremal vector S(va ⊗ vλk−1) ∈ V ⊗ Vλk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Here S is the 10 universal Shapovalov matrix of Vλk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Consider a chain of module homomorphisms Vλm ϕm −→ V ⊗ Vλm−1 id1⊗ϕm−1 −→ V ⊗ (V ⊗ Vλm−2) → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' idm−1⊗ϕ1 −→ V ⊗(m−1) ⊗ (V ⊗ Vλ0), where idk are the identity operators on V ⊗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The vector vλm eventually goes over to S(˜vm a ⊗ vλ0), where ˜vm a ∈ V ⊗m is of weight mνa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It is related with v⊗m a by an invertible operator from End(V ⊗m), which is m − 1-fold dynamical twist [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let us calculate ⟨vm b |˜vm a ⟩λ0 by pairing the tensor leg of S(˜vm a ⊗vλ0) with vm b = vb ⊗vm−1 b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Using equality S(˜vm a ⊗ vλ0) = S � va ⊗ S(˜vm−1 a ⊗ vλ0) � we reduce ⟨vm b |˜vm a ⟩λ0 to � vm−1 b , ⟨vb|va⟩(1) λm−1S1˜vm−1 a � ⟨vb|va⟩(2) λm−1 S2(λ0) = ⟨vb|va⟩(2) λm−1 � ω � ⟨vb|va⟩(1) λm−1 � vm−1 b |˜vm−1 a � λ0, where we use the Sweedler notation ∆(x) = x(1) ⊗ x(2) ∈ Uq(b−) ⊗ Uq(g−) for the coproduct of x ∈ Uq(g−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Since yqhαvb = ǫ(y)q(α,β)vb for all y ∈ Uq(g+) and α ∈ Γ+, we arrive at ⟨vm b |˜vm a ⟩λ0 = q−(β,νb)⟨vb|va⟩λm−1⟨vm−1 b |˜vm−1 a ⟩λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Proceeding by induction on m we conclude that ⟨vm b |˜vm a ⟩λ0 equals the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='7), up to the factor q−m(β,νb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Finally, set vm a = qm(β,νb)˜vm a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This proves the lemma for generic and hence for all λ where the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='7) makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It follows from the above factorization that the least common denominator of the extremal vector u = S(vm a ⊗ vλ) ∈ V ⊗m ⊗ Vλ contains d(λ) = [ηβ(λ + (m − 1)νa)]q = [(λ + ρ, β) − m 2 (β, β)]q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It comes from the leftmost factor ⟨vb|va⟩λm−1 in the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Denote by svm b ,vm a (λ) the matrix element ⟨vm b |vm a ⟩λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Since d divides [ηmβ]q, the re-scaled matrix element ˇsvm b ,vm a (λ) = c(λ)d(λ)svm b ,vm a (λ) ∝ m−1 � k=0 θ(λk), where c(λ) = −q−ηmβ(λm−1) [ηmβ(λ)]q d(λ) , is regular and does not vanish at generic λ ∈ Hβ,m because d(λ) cancels the pole in ⟨vb|va⟩λm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Put ˇu = dk(λ)u, where k ⩾ 1 is the maximal degree of this pole in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It is an extremal vector in V ⊗m ⊗ Vλ that is regular at generic λ ∈ Hβ,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Indeed, let Hµ denote the null set {λ ∈ h∗|[ηµ(λ)]q = 0} for µ ∈ Γ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then the Vλ-components of ˇu may have poles only at λ ∈ ∪µ<βHµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' But each µ is either not collinear to β or µ = lβ with l < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' In both cases the complement to Hβ,m ∩ Hµ is dense in Hβ,m because q is not a root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' 11 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For generic λ ∈ Hβ,m, θβ,m(λ) ∝ ˇsvm b ,vm a (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The singular vector ˇu is presentable as ˇu = vm a ⊗ dk(λ)vλ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' + vm b ⊗ dk−1(λ)c(λ)ˇsvm b ,vm a (λ)vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We argue that ˇu = vm b ⊗ c(λ)ˇsvm b ,vm a (λ)vλ for generic λ in Hβ,m, where d(λ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Indeed, the Vλ-components of ˇu span a Uq(g+)-submodule in Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' A vector of maximal weight in this span is extremal and distinct from vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' But θβ,m(λ)vλ is the only, up to a factor, extremal vector in Vλ, for generic λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Therefore k = 1 and θβ,m ∝ ˇsvm b ,vm a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' An admissible β-representation can be associated with every simple root α ∈ Πβ if one sets V to be the irreducible module of highest weight (β,β) ℓ(α,α)ωα, where ℓ = ℓα,β is the multiplicity of α with which it enters β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We denote this module by Vα,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It is finite dimensional if (β,β) ℓ(α,α) ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Otherwise it is a parabolic Verma module relative to a Levi subalgebra with the root basis Π\\{α}, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' One can pass to the ”universal form” of θβ regarding it as an element of ˆUq(b−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then θβ,m = (τ m−1 νb θβ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' (τνbθβ) θβ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='8) where τν is an automorphism of ˆUq(h) generated by the affine shift of h∗ by the weight ν, that is, (τνϕ)(µ) = ϕ(µ + ν), ϕ ∈ ˆUq(h), µ ∈ h∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' One may ask when the shift is trivial, τνbθβ = θβ, and θβ,m is just the m-th power of θβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let β be a positive root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Suppose that there is α ∈ Πβ with ℓα,β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then θβ,m = θm β ∈ Uq(b−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let s ⊂ g be a semi-simple subalgebra generated by simple root vectors fµ, eµ with µ ̸= α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Take for V the module Vα,β with highest weight φ = (β,β) (α,α)ωα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Put vb to be the highest vector and va ∝ fβvb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Both va and vb can be included in an orthonormal basis because they span their weight sub- spaces in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Therefore ⟨vb|va⟩ = sba can be calculated by formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We write it as θβ(λ) = cba + � vb≻vi≻va cbisia(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The highest vector vb is killed by s−, therefore the Hasse diagram between va and vb is vb eα ←− fαvb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' va, 12 where arrows in the suppressed part are simple root vectors from Uq(s+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' But then the only copy of fα is in cbi while all sia belong to Uq(s−) ˆUq(hs), the extended Borel subalgebra of Uq(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Finally, since Πs is orthogonal to νb, we have (µ, νa) = −(µ, β) for all µ ∈ R+ s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Therefore θβ(λk) = θβ(λ − kβ), θβ,m(λ) = m−1 � k=0 θβ(λ − kβ), where the product is taken in the descending order from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This proves the plain power factorization because each θβ carries weight −β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Conditions of the above proposition are fulfilled for all pairs α, β in the case of sl(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' 5 Shapovalov elements of degree 1 In this section we describe the factor θβ entering (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='8), for a particular admissible β-representation (V, vb, va).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We give a complete solution to the problem in the classical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' In the case of q ̸= 1, we do it up to calculation of the entries of the matrix C in a simple finite dimensional module ˜g that is a q-deformation of the adjoint module g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Its highest weight is the maximal root, ξ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' To achieve our goals, we need to figure out the Hasse sub-diagram H(vb, va) ⊂ H(V ) that comprises all possible routes from va to vb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We argue that H(vb, va) can be extracted from a diagram H(b−) which we introduce below, and the underlying Uq(g+)-modules are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The Uq(g+)-module associated with H(b−) is constructed from ˜g by factoring out the span of positive weight spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' In order to distinguish the case of q ̸= 1 from classical and to avoid confusion with root vectors, we will mark the nodes with tilde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Vectors ˜fη of weights −η ∈ −R+ are defined uniquely up to a sign if we normalize them by ( ˜fη, ˜fη) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We may assume that they are deformations of classical root vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We take ˜hα = eα ˜fα, α ∈ Π, for basis elements of zero weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For example, the diagram H(b−) in the case of g = g2 is b− eα2 eα1 eα2 eα2 eα1 ˜hα2 ˜fα2 ˜fα1+α2 ˜fα1+2α2 ˜fα1+3α2 ˜f2α1+3α2 ❜ ❜ ❜ ❜ ❜ ❜ ✛ ✟ ✟ ✟ ✟ ✙ ✛ ✛ ✛ eα1 eα2 ˜hα1 ˜fα1 ❜ ❜ ✛ ❍ ❍ ❍ ❍ ❨ From now on we fix V = Vα,β with highest weight φ = (β,β) ℓα,β(α,α)ωα and highest vector vb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We denote by l ⊂ g a reductive Lie subalgebra of maximal rank whose root system is Πl = R\\{α} and by p = l + g+ its parabolic extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' 13 In order to construct the start node va ∈ V , we will use the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Recall that a singular vector � i wi ⊗ vi in a tensor product W ⊗ V of two irreducible modules of highest weight defines a Uq(g+)-homomorphism W ∗ → V (and respectively V ∗ → W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Here W ∗ is an irreducible Uq(g)-module of lowest weight, which is negative the highest weight of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The dual action is defined with the help of antipode γ in the standard way: (xϕ)(w) = ϕ � γ(x)w � , for x ∈ Uq(g+), w ∈ W, and ϕ ∈ W ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The homomorphism W ∗ → V is implemented via the assignment ϕ �→ � i ϕ(wi)vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We will apply this construction to W = ˜g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' There exists a unique, up to a scalar factor, singular vector u ∈ ˜g ⊗ V of weight φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let J ⊂ Uq(g−) be the annihilator of the highest vector vb ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Singular vectors in ˜g ⊗ V of weight φ are in bijection with vectors ˜h ∈ ˜g of zero weight killed by the left ideal σ(J) ⊂ Uq(g+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Pick up ˜h ̸= 0 orthogonal to all µ ∈ Πl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' it is unique up to a scalar factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The ideal J is generated by elements θ ∈ Uq(g+) such that θvb are singular vectors in the Verma module Vφ covering V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' By construction, ˜u is killed by eα ∈ J with α ∈ Πl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' If θvφ ∈ Vφ is a singular vector of weight φ − mη with η ∈ R+\\R+ l , then m > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Indeed, since φ = lωα with positive rational l = (β,β) ℓα,β(α,α), we have an inequality l(ωα, η∨) + (ρ, η∨) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='1), where λ is replaced with φ and β with η, is fulfilled only if m > 1, since q is not a root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then the element σ(θ) kills ˜h because mη with m > 1 is not a weight of ˜g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Remark that V is finite dimensional if (β,β) ℓα,β(α,α) ∈ Z and a parabolic Verma module otherwise because its highest weight is away from De Concini-Kac-Kazhdan hyperplanes Hη,m with η ∈ R+\\R+ l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Now let va ∈ V be the vector of minimal weight in the expansion u = ˜eξ ⊗ va + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' over the chosen basis in ˜g (we have omitted the terms of lower weights in the ˜g-factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Notice that in the classical case the vector fηvb does not vanish if η ∈ R+\\R+ l because (η, φ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' In particular, va ∝ fξvb ̸= 0 for the maximal root ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For general q, va is killed by the left ideal in Uq(g+) annihilating the lowest vector ˜fξ ∈ ˜g ≃ ˜g∗, Such va is unique in V up to a scalar factor, because of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Introduce a partial order on positive roots by writing µ ≺ ν iff fµ ≻ fν in H(b−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This is in agreement with the partial order on H(g+) ⊂ H(g), which is exactly the Hasse diagram of the root system R+, [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Note that α ≺ β for simple α if and only if α ∈ Πβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let u = ˜eξ ⊗ va + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' be the singular vector from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='1 with va ∈ V of minimal weight in the expansion over a weight basis in ˜g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then the Uq(g+)-module generated by va ∈ V is isomorphic to ˜g(˜hα, ˜fξ), for almost all q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The Uq(g+)-module homomorphism ˜g → V determined by the assignment ˜fξ �→ va factors through the quotient g(˜hα, ˜fξ) because the kernel includes all ˜fη with η ∈ R+ l , all ˜hη = eµ ˜fη with η ∈ Πl, and all negative weight spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We are left to prove that it is an isomorphism on g(˜hα, ˜fξ) for almost all q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It is sufficient to check that it is injective for q = 1 because V rationally depends on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' But then for each positive root η subject to α ⪯ η ⪯ ξ the vector fηvb is in U(g+)fξvb and is not zero, because (η, φ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' It follows that eβva ̸= 0 because eβ ˜fξ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Therefore (V, vb, va) is an admissible β-representation for almost all q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Let us consider the classical case in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We choose h∨ α = 2 (α,α)hα, α ∈ Π, as a basis in h ⊂ b−, so that α(h∨ α) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The root vectors fµ with µ ∈ R+ form a basis in g−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Arrows labeled by α ∈ Π are h∨ α eα ←− fα and fµ eα ←− fν if µ = ν − α is a positive root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The U(g+)-module underlying H(b−) is g/g+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Specialization of the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='5) for θβ requires the knowledge of matrix C = (π ⊗ id)(C) ∈ End(V ) ⊗ Uq(g−), which is readily available for q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For ν, γ ∈ R+, denote by Cν,γ ∈ C the scalars such that [eν, fγ] = Cν,γfγ−ν, if γ − ν ∈ R+, Cγ,γ = (β,β) 2 ℓα,γ ℓα,β , and Cν,γ = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Then (π ⊗ id)(C)(fγvb ⊗ 1) = vb ⊗ Cγ,γfγ + � ν≺γ fγ−νvb ⊗ Cν,γfν, for all γ satisfying α ⪯ γ ⪯ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This equality yields all entries of the matrix C needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='5) becomes θβ = Cβ,βfβ + � k⩾1 � ν1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='+νk+1=β (Cνk+1,γk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Cν1,γ0)(fνk+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' fν1) (−1)k ηµk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' ηµ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='9) The internal summation is performed over all partitions of β to a sum of νi ∈ R+ such that all γi = γi−1 − νi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' , k with γ0 = β are in R+ and subject to α ⪯ γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' In particular, γk = νk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The weights µi are defined to be µi = γ0 − γi = ν1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' + νi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Note that in the q ̸= 1 case the corresponding sum may involve terms with entries of C whose weights are not roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Now we summarise the results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For each α ≺ β, the rescaled matrix element ⟨˜hα| ˜fβ⟩[ηβ]q with ˜hα, ˜fβ ∈ ˜g, is a Shapovalov element θβ,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For general degree m > 1, θβ,m is given by the factorization formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='8) with θβ = θβ,1 and the shift weight νb = (β,β) ℓα,β(α,α)ωα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Observe that summation formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='5) involves only the structure of Uq(g+)-module deter- mined by the initial and final nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' That is straightforward with regard to the matrix elements 15 of C and also true for the Cartan factors, which depend only on weight differences (mind that weights in a cyclic Uq(g+)-module generated by a weight vector are fixed up to a constant weight summand).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Furthermore, the nodes of the sub-diagram H(va, vb) can be included in an orthonor- mal basis whence sba ∝ ⟨vb|va⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Now, for almost all q, the theorem follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='2 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='3 with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Therefore it is true for all q where the factors (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='8) are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' We remark in conclusion that for fixed β ∈ R+ one can pick up α ∈ Πβ delivering the simplest Hasse diagram H(˜hα, ˜fβ), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' with the smallest fundamental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Such diagrams can be found amongst subdiagrams in fundamental auxiliary modules of minimal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' That also applies to their associated Uq(g+)-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For all non-exceptional types of g, the entries of the matrix C participating in the route summation formula are calculated in [27], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' That is also done for g2 in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' This makes the above description of Shapovalov elements for such quantum groups absolutely explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' For exceptional g of rank > 2, the problem reduces to calculation of relevant entries of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' In the context of quantization of semi-simple conjugacy classes [10], it is crucial to make sure that θβ,m(λ) tends to f m β as q → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Factorization (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content='8) together with the route summation formula for θβ,1 gives important information about possible singularities of θβ,m(λ) and facilitate the analysis even without knowing the matrix elements of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' Acknowledgement This work is partially supported by the Moscow Institute of Physics and Technology under the Priority 2030 Strategic Academic Leadership Program and by Russian Science Foundation grant 23-21-00282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' The author thanks Vadim Ostapenko and Vladimir Stukopin for stimulating discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' References [1] Bernstein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dE0T4oBgHgl3EQfvwHw/content/2301.02624v1.pdf'} diff --git a/79AzT4oBgHgl3EQfSPsz/content/tmp_files/2301.01228v1.pdf.txt b/79AzT4oBgHgl3EQfSPsz/content/tmp_files/2301.01228v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..155f1cc5ac44d36d5a9d3046e5621166ee5a19ed --- /dev/null +++ b/79AzT4oBgHgl3EQfSPsz/content/tmp_files/2301.01228v1.pdf.txt @@ -0,0 +1,365 @@ +DMOps: Data Management Operation and Recipes +Eujeong Choi1, Chanjun Park 1 † +1 Upstage +{eujeong, chanjun.park}@upstage.ai +Abstract +Data-centric AI has shed light on the signif- +icance of data within the machine learning +(ML) pipeline. Acknowledging its importance, +various research and policies are suggested +by academia, industry, and government depart- +ments. Although the capability of utilizing ex- +isting data is essential, the capability to build a +dataset has become more important than ever. +In consideration of this trend, we propose a +"Data Management Operation and Recipes" +that will guide the industry regardless of the +task or domain. In other words, this paper +presents the concept of DMOps derived from +real-world experience. By offering a baseline +for building data, we want to help the industry +streamline its data operation optimally. +1 +Introduction +With the emergence of Data-centric AI (Polyzotis +and Zaharia, 2021; Mazumder et al., 2022), various +in-depth research has been introduced in academia +alongside the wide range of policies from indus- +try and government departments (Pencheva et al., +2020). +In the case of academia, there are studies +boosting model performance through large-scale +datasets (Liu et al., 2021; Costa-jussà et al., 2022) +along with the production of benchmark datasets +for objective performance comparison between +models (Wang et al., 2018; Ruder, 2021). Further- +more, there are also benchmark datasets that spe- +cialize in specific tasks (Rajpurkar et al., 2016; Alt +et al., 2020). The government contributes to the +field by implementing public data open policies +and releasing datasets from the National Statistics +department (Panagos et al., 2012). +However, the industry frequently dives into an +untapped and specialized domain, where there is +rarely a ready-to-go dataset. Especially for B2B +companies, there is usually an urgent demand +†Corresponding author. +for data that meets the requirements of their cus- +tomers or their business items (Pustejovsky and +Stubbs, 2012). Since the open source and bench- +mark datasets are normally insufficient to meet +these specific demands, additional data production +is always a necessary step to specialize in a par- +ticular task. As a result, the majority of the AI +businesses started to build their own task-specific +datasets, alongside the emergence of companies +that specialize in operating crowd workers to meet +these demands, and research on efficient data pro- +duction on human-in-the-loop started to make ap- +pearance (Doan, 2018; Wu et al., 2022). +Despite its necessity, there has been a paucity of +studies in the field of data production. To the best of +our knowledge, there has not yet been research that +proposes the entire process starting from analyz- +ing the business standpoint to data annotation and +evaluation. Therefore, we propose a "Data Man- +agement Operation and Recipes" that will assist +in building a dataset efficiently and economically +regardless of task and domain. Specially, we pro- +pose a DMOps that can produce high-quality data +needed in manufacturing deep learning models. +2 +Proposed Data Management Operation +and Recipes (DMOps) +Data management operations involve the integra- +tion of human input and decision-making into a +data management pipeline or system. This involves +tasks such as data annotation, data quality assur- +ance, and other activities that require a human +touch. One way to implement a data management +operation is through the use of recipes. Recipes are +step-by-step instructions for performing a specific +task or set of tasks, and can be used to guide human +workers through the data management process. +Our Data Recipes consists of 12 steps. Through +these steps, we go over the entire process of data op- +eration : from establishing the goal of the project to +delivering the final data to the modeling team. The +arXiv:2301.01228v1 [cs.DB] 2 Jan 2023 + +name and explanation of each step is as follows. +1. Establish the Project Goal: Analyzing the +purpose and requirements of data production +is the first step of the recipes. This step re- +quires collaboration with ML engineer teams +and business operation teams. Through com- +munication, we can decide the input and out- +put format of data that is suitable to the model +of choice, and also set data milestones that fit +the timeline of the business operation team. +2. Secure Raw Data: Researching and collect- +ing raw data is the second step of the recipes. +Three possible cases of collecting raw data are +1) the client providing the raw data, 2) using +open-sourced public data, and 3) purchasing +the raw data from its source platform. The +key issue here is the copyright of each data +source. License information must be checked +thoroughly, and getting a legal review is rec- +ommended before its usage. +3. Data Pre-processing: The third step is im- +proving the quality of the raw data through +pre-processing. Basically the pre-processing +consists of two main tasks: first, adjusting the +format of data regarding its requirements, sec- +ond, filtering non-ethical, privacy invading, +and noisy data (Wiegand et al., 2018; Park +et al., 2020). This step is all about practicing +quality over quantity. +4. Design a Data Schema: Fourth step is de- +signing an annotation system that is efficient +while containing all the information required. +We need to come up with a label system that +can represent human perception by digging +through the collected data with the aid of ML +methods such as unsupervised learning. Also, +figuring out parts that can be somewhat auto- +mated (pseudo-labeling) and parts that need +human intervention (annotating) is essential +in making the process efficient and moreover, +accurate. With few pilot annotation iterations, +the data scheme is expected to reach its opti- +mal design. +5. Prepare a Guideline: Fifth step is the doc- +umentation of the data scheme. Its purpose +is to deliver the designed labeling system to +the expected annotators. The difficulty of the +guideline should be monitored with caution +since the clarity and detailed explanation may +be in a trade-off relationship. +6. Recruit Annotators: Sixth step is recruiting +the annotators. The key is to select workers +that are fit for the task for an efficient and +accurate outcome. The best case would be se- +lecting those who scored high on a test similar +to the actual labeling task. +7. Instruct Annotators: Seventh step is instruct- +ing the annotators with the guideline made +above. In this stage, two-way communication +that draws out questions and debates is the key +whereas one-sided communication is discour- +aged. +8. Data Annotation: This is the step where +data annotators annotate the actual data. It +is the process of transferring the linguis- +tic/cognitive/visual intuition of the construc- +tor into data. Therefore, the data construction +manager must devise a way to unify the differ- +ent intuitions of different builders in a more +general line. When constructing data, it is also +key to continuously respond to the QA of data +builders. +9. Data Inspection: This ninth step is inspecting +the annotated data. During this step, inspec- +tors must identify commonly occurring human +errors and sort out the edge cases through +discussions. Considering the nature of the +Human-in-the-loop process, this step is essen- +tial to ensure the fidelity of the dataset. +10. Data Verification: The tenth step is verifying +the data. When inspecting data, it is necessary +to first determine whether the work has been +completed by observing the given guideline. +Also, 1) data sufficiency, 2) data diversity, 3) +data trustworthiness, 4) data privacy and se- +curity 5) data ethics suitability should be re- +viewed (Roh et al., 2019; Koo et al., 2022). Fi- +nally, data consistency can be identified based +on the inter-annotator agreement (IAA) score. +11. Data Evaluation: Eleventh step is verifying +the quality of data through actual modeling. In +order to quantitatively verify whether the data +is made as planned, various experiments are +conducted such as checking data efficiency by +increasing the amount of data or sectioning + +the data to check the consistency of its qual- +ity (Moon et al., 2021; Park et al., 2021). It is +natural to find artifacts within one’s data; after +identifying the repeated errors, revisiting the +recipes from step 5 is frequently required to +enhance the quality of data. If there are parts +that do not match our purpose while proceed- +ing the steps, we should return to stage 5 and +revise the guideline for another iteration. +12. Data Deliverables: Final step of the recipes +is delivering the final data outcome. In other +words, it is the process of delivering annotated +data to the modeler or customer. When deliv- +ering, the versioning must be adapted to the +protocol, and it is important to reveal the fea- +tures of the data with its sample. Furthermore, +after going through the EDA process, it is rec- +ommended to deliver the data analysis and the +quality evaluation document together. +Figure 1: Process of the Data Management Operation +and Recipes (DMOps) +Why DMOps? +Due to the absence of a standard +data-building process, there are many cases where +the order of steps is mixed up or cannot be applied +task-agnostically. The "DMOps" we propose offers +a fixed process of data production, and at the same +time can be used universally regardless of the task +or domain. Therefore, our recipes can serve as a +baseline for data production. +Data is built through several stages. However the +industry does not have a unified standard of the +order to construct data, so there are many cases +where the stages are scattered or mixed up. How- +ever, when the proposed process is applied, it not +only corrects the scattered order but is also task +agnostic and can be universally applied to any do- +main. In other words, our methodology can serve +as a baseline for data construction. +3 +Conclusion and Future Works +In this paper, we proposed a DMOps that can effi- +ciently produce high-quality data with human an- +notation. The methodology is task agnostic which +allows it to serve as a baseline for any data produc- +tion. In the future, we plan to increase the reliability +of the proposed process through quantitative verifi- +cation at each stage of the process. In addition, we +intend to conduct a study to verify the difference in +data quality depending on whether the data recipes +is applied or not. +References +Christoph Alt, Aleksandra Gabryszak, and Leonhard +Hennig. 2020. Tacred revisited: A thorough evalu- +ation of the tacred relation extraction task. +arXiv +preprint arXiv:2004.14855. +Marta R Costa-jussà, James Cross, Onur Çelebi, Maha +Elbayad, Kenneth Heafield, Kevin Heffernan, Elahe +Kalbassi, Janice Lam, Daniel Licht, Jean Maillard, +et al. 2022. +No language left behind: Scaling +human-centered machine translation. arXiv preprint +arXiv:2207.04672. +AnHai Doan. 2018. Human-in-the-loop data analysis: +a personal perspective. 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Future +Generation Computer Systems. + diff --git a/79AzT4oBgHgl3EQfSPsz/content/tmp_files/load_file.txt b/79AzT4oBgHgl3EQfSPsz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8346252cb07f2974dd6718ab5957cdddba2b95ba --- /dev/null +++ b/79AzT4oBgHgl3EQfSPsz/content/tmp_files/load_file.txt @@ -0,0 +1,218 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf,len=217 +page_content='DMOps: Data Management Operation and Recipes Eujeong Choi1, Chanjun Park 1 † 1 Upstage {eujeong, chanjun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content='park}@upstage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content='ai Abstract Data-centric AI has shed light on the signif- icance of data within the machine learning (ML) pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Acknowledging its importance, various research and policies are suggested by academia, industry, and government depart- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Although the capability of utilizing ex- isting data is essential, the capability to build a dataset has become more important than ever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' In consideration of this trend, we propose a "Data Management Operation and Recipes" that will guide the industry regardless of the task or domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' In other words, this paper presents the concept of DMOps derived from real-world experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' By offering a baseline for building data, we want to help the industry streamline its data operation optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 1 Introduction With the emergence of Data-centric AI (Polyzotis and Zaharia, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Mazumder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=', 2022), various in-depth research has been introduced in academia alongside the wide range of policies from indus- try and government departments (Pencheva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' In the case of academia, there are studies boosting model performance through large-scale datasets (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Costa-jussà et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=', 2022) along with the production of benchmark datasets for objective performance comparison between models (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Ruder, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Further- more, there are also benchmark datasets that spe- cialize in specific tasks (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Alt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' The government contributes to the field by implementing public data open policies and releasing datasets from the National Statistics department (Panagos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' However, the industry frequently dives into an untapped and specialized domain, where there is rarely a ready-to-go dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Especially for B2B companies, there is usually an urgent demand †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' for data that meets the requirements of their cus- tomers or their business items (Pustejovsky and Stubbs, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Since the open source and bench- mark datasets are normally insufficient to meet these specific demands, additional data production is always a necessary step to specialize in a par- ticular task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' As a result, the majority of the AI businesses started to build their own task-specific datasets, alongside the emergence of companies that specialize in operating crowd workers to meet these demands, and research on efficient data pro- duction on human-in-the-loop started to make ap- pearance (Doan, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Despite its necessity, there has been a paucity of studies in the field of data production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' To the best of our knowledge, there has not yet been research that proposes the entire process starting from analyz- ing the business standpoint to data annotation and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Therefore, we propose a "Data Man- agement Operation and Recipes" that will assist in building a dataset efficiently and economically regardless of task and domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Specially, we pro- pose a DMOps that can produce high-quality data needed in manufacturing deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 2 Proposed Data Management Operation and Recipes (DMOps) Data management operations involve the integra- tion of human input and decision-making into a data management pipeline or system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' This involves tasks such as data annotation, data quality assur- ance, and other activities that require a human touch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' One way to implement a data management operation is through the use of recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Recipes are step-by-step instructions for performing a specific task or set of tasks, and can be used to guide human workers through the data management process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Our Data Recipes consists of 12 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Through these steps, we go over the entire process of data op- eration : from establishing the goal of the project to delivering the final data to the modeling team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' The arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content='01228v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content='DB] 2 Jan 2023 name and explanation of each step is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Establish the Project Goal: Analyzing the purpose and requirements of data production is the first step of the recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' This step re- quires collaboration with ML engineer teams and business operation teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Through com- munication, we can decide the input and out- put format of data that is suitable to the model of choice, and also set data milestones that fit the timeline of the business operation team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Secure Raw Data: Researching and collect- ing raw data is the second step of the recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Three possible cases of collecting raw data are 1) the client providing the raw data, 2) using open-sourced public data, and 3) purchasing the raw data from its source platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' The key issue here is the copyright of each data source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' License information must be checked thoroughly, and getting a legal review is rec- ommended before its usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Data Pre-processing: The third step is im- proving the quality of the raw data through pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Basically the pre-processing consists of two main tasks: first, adjusting the format of data regarding its requirements, sec- ond, filtering non-ethical, privacy invading, and noisy data (Wiegand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' This step is all about practicing quality over quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Design a Data Schema: Fourth step is de- signing an annotation system that is efficient while containing all the information required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' We need to come up with a label system that can represent human perception by digging through the collected data with the aid of ML methods such as unsupervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Also, figuring out parts that can be somewhat auto- mated (pseudo-labeling) and parts that need human intervention (annotating) is essential in making the process efficient and moreover, accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' With few pilot annotation iterations, the data scheme is expected to reach its opti- mal design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Prepare a Guideline: Fifth step is the doc- umentation of the data scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Its purpose is to deliver the designed labeling system to the expected annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' The difficulty of the guideline should be monitored with caution since the clarity and detailed explanation may be in a trade-off relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Recruit Annotators: Sixth step is recruiting the annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' The key is to select workers that are fit for the task for an efficient and accurate outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' The best case would be se- lecting those who scored high on a test similar to the actual labeling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Instruct Annotators: Seventh step is instruct- ing the annotators with the guideline made above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' In this stage, two-way communication that draws out questions and debates is the key whereas one-sided communication is discour- aged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Data Annotation: This is the step where data annotators annotate the actual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' It is the process of transferring the linguis- tic/cognitive/visual intuition of the construc- tor into data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Therefore, the data construction manager must devise a way to unify the differ- ent intuitions of different builders in a more general line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' When constructing data, it is also key to continuously respond to the QA of data builders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Data Inspection: This ninth step is inspecting the annotated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' During this step, inspec- tors must identify commonly occurring human errors and sort out the edge cases through discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Considering the nature of the Human-in-the-loop process, this step is essen- tial to ensure the fidelity of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Data Verification: The tenth step is verifying the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' When inspecting data, it is necessary to first determine whether the work has been completed by observing the given guideline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Also, 1) data sufficiency, 2) data diversity, 3) data trustworthiness, 4) data privacy and se- curity 5) data ethics suitability should be re- viewed (Roh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Koo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Fi- nally, data consistency can be identified based on the inter-annotator agreement (IAA) score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Data Evaluation: Eleventh step is verifying the quality of data through actual modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' In order to quantitatively verify whether the data is made as planned, various experiments are conducted such as checking data efficiency by increasing the amount of data or sectioning the data to check the consistency of its qual- ity (Moon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' It is natural to find artifacts within one’s data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' after identifying the repeated errors, revisiting the recipes from step 5 is frequently required to enhance the quality of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' If there are parts that do not match our purpose while proceed- ing the steps, we should return to stage 5 and revise the guideline for another iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Data Deliverables: Final step of the recipes is delivering the final data outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' In other words, it is the process of delivering annotated data to the modeler or customer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' When deliv- ering, the versioning must be adapted to the protocol, and it is important to reveal the fea- tures of the data with its sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Furthermore, after going through the EDA process, it is rec- ommended to deliver the data analysis and the quality evaluation document together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Figure 1: Process of the Data Management Operation and Recipes (DMOps) Why DMOps?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Due to the absence of a standard data-building process, there are many cases where the order of steps is mixed up or cannot be applied task-agnostically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' The "DMOps" we propose offers a fixed process of data production, and at the same time can be used universally regardless of the task or domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Therefore, our recipes can serve as a baseline for data production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Data is built through several stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' However the industry does not have a unified standard of the order to construct data, so there are many cases where the stages are scattered or mixed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' How- ever, when the proposed process is applied, it not only corrects the scattered order but is also task agnostic and can be universally applied to any do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' In other words, our methodology can serve as a baseline for data construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 3 Conclusion and Future Works In this paper, we proposed a DMOps that can effi- ciently produce high-quality data with human an- notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' The methodology is task agnostic which allows it to serve as a baseline for any data produc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' In the future, we plan to increase the reliability of the proposed process through quantitative verifi- cation 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Overview of the germeval 2018 shared task on the identification of offensive language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Xingjiao Wu, Luwei Xiao, Yixuan Sun, Junhang Zhang, Tianlong Ma, and Liang He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' A survey of human-in-the-loop for machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} +page_content=' Future Generation Computer Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79AzT4oBgHgl3EQfSPsz/content/2301.01228v1.pdf'} diff --git a/9dAyT4oBgHgl3EQfdPeW/content/tmp_files/2301.00299v1.pdf.txt b/9dAyT4oBgHgl3EQfdPeW/content/tmp_files/2301.00299v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a876d2e21533bc87c912d2affcb4e71af926bebd --- /dev/null +++ b/9dAyT4oBgHgl3EQfdPeW/content/tmp_files/2301.00299v1.pdf.txt @@ -0,0 +1,1090 @@ +arXiv:2301.00299v1 [stat.AP] 31 Dec 2022 +Definition and clinical validation of Pain Patient +States from high-dimensional mobile data: +application to a chronic pain cohort +1st Jenna M. Reinen +Digital Health +IBM Research +Yorktown Heights, NY +jenna.reinen@ibm.com +2nd Carla Agurto +Digital Health +IBM Research +Yorktown Heights, NY +carla.agurto@ibm.com +3rd Guillermo Cecchi +Digital Health +IBM Research +Yorktown Heights, NY +gcecchi@us.ibm.com +4th Jeffrey L. Rogers +Digital Health +IBM Research +Yorktown Heights, NY +jeffrogers@us.ibm.com +5th NAVITAS and ENVISION Studies Physician Author Group +Clinical Research +Boston Scientific +Valencia, CA +6th Boston Scientific Research Scientists Consortium +Data Research and Engineering +Boston Scientific +Valencia, CA +Abstract—The technical capacity to monitor patients with a +mobile device has drastically expanded, but data produced from +this approach are often difficult to interpret. We present a +solution to produce a meaningful representation of patient status +from large, complex data streams, leveraging both a data-driven +approach, and use clinical knowledge to validate results. Data +were collected from a clinical trial enrolling chronic pain patients, +and included questionnaires, voice recordings, actigraphy, and +standard health assessments. The data were reduced using a +clustering analysis. In an initial exploratory analysis with only +questionnaire data, we found up to 3 stable cluster solutions +that grouped symptoms on a positive to negative spectrum. +Objective features (actigraphy, speech) expanded the cluster +solution granularity. Using a 5 state solution with questionnaire +and actigraphy data, we found significant correlations between +cluster properties and assessments of disability and quality- +of-life. The correlation coefficient values showed an ordinal +distinction, confirming the cluster ranking on a negative to +positive spectrum. This suggests we captured novel, distinct Pain +Patient States with this approach, even when multiple clusters +were equated on pain magnitude. Relative to using complex time +courses of many variables, Pain Patient States holds promise as +an interpretable, useful, and actionable metric for a clinician or +caregiver to simplify and provide timely delivery of care. +Index Terms—chronic pain, digital health, clustering, medical +decision making +I. INTRODUCTION +Recent advances in digital medicine have provided the +opportunity to collect large sets of clinical data to evaluate and +predict critical medical outcomes. For instance, mobile-based +applications, accelerometers, and biosensors are now ubiqui- +tous in phones and watches, enabling one to longitudinally +track variables like mobility and speech, and facilitate patient +symptom self-report. Importantly, these features may associate +with clinical meaning. Large-scale studies have shown that +data from mobile applications tracking daily activity may +predict outcomes relevant to health and illness, such as in +geriatric care and diabetes [1], [2]. Further, language can +assess affective, psycholinguistic, physiological, and cognitive +features can predict physiological and pharmacological [3], +psychiatric [4], and cognitive disease states [5]. These types +of findings have demonstrated the promise of digital health +profiles in understanding patient experience and predicting +important clinical outcomes. +Despite these advances, the size and complexity of the +clinical data generated by mobile applications is nontrivial to +interpret and apply for several reasons. First, digital healthcare +data can exist in multiple formats, creating the need to fuse +vast amount of diverse information [6]. Second, there is a +need for methods that can obtain clear data representations. +These methods should provide interpretation that are manage- +able in size, yet can maintain the characteristics of the raw +information, allowing for patients and healthcare professionals +to interpret and use the output [7]. Attempts to reduce and +understand such data in a biological context have commonly +used data-driven methods, especially those using machine +learning algorithms. This approach offers the advantage of +being able to handle large, multidimensional data sets through +the ability to recognize patterns or joint representations that +are otherwise difficult to identify using standard statistical +approaches, providing knowledge discovery about a particular +Copyright © 2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, +including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or +lists, or reuse of any copyrighted component of this work in other works. + +topic that can span time, location, and scales [8]. In particular, +clustering analysis offers the ability to collapse across oth- +erwise incomprehensible multidimensional data and observe +how features co-occur. In the case of spectrum illnesses that +incorporate a range and variety of symptoms, decomposition +can be helpful, with some outputs having an advantage in +outcomes prediction [9]. But not all results allow for interpre- +tation, and it is particularly susceptible to problems in small, +unvalidated datasets which may result in overfitting and thus +results that are not replicable or generalizable. Further, while +results from unsupervised approaches may reveal meaningful +clinical patterns, few methods exist to formally assign labels, +rank, or identify qualitative aspects from the results of data- +driven approaches through independent validation. +A prime illustration of this problem is in chronic pain, a dis- +ease affecting a substantial percentage of the population [10] +that significantly impacts general function including employ- +ment, mental health, and social interaction. This heterogeneous +condition interacts with well-characterized facets of health, +including mood, sleep, psychosocial function, medication use, +and mobility. However, the current practice for most pain +studies is to evaluate outcomes based on pain magnitude +alone, which does not consider all of the variance shown to +predict treatment success, quality-of-life (QoL), or other mea- +surements of physical, psychological, and social well-being +[11]. But, using all of these features as outcome variables +is nontrivial to compute, conceptualize, and interpret. Few +standard approaches have been developed that incorporate both +the computational methodology required to complete such +a task, and the ability to provide a clinically interpretable +summary of the output. To date, machine learning has been +used to predict pain outcomes, identify clinical subgroups +[12], extract knowledge, and detect structure in biological +and clinical features [13]. In chronic pain, while artificial +intelligence (AI) has been applied to improve diagnoses, fewer +studies apply it to the treatment and management of pain +patients [14], and analyses that use longitudinal data or clinical +validation are extremely limited. +Given the quickly expanding capacity of digital health and +learning algorithms to inform treatment outcomes in complex +illnesses, there is a benefit to developing an approach to +validate health states from multidimensional data. While it +is known that various chronic pain symptoms can co-occur, +it remains currently unknown whether symptom profiles may +be successfully organized into distinct health states. Here, we +propose a method by which we aim to identify clusters from +high-dimensional, longitudinal data in chronic pain patients, +and label them as Pain Patient States that may be operational- +ized for clinical application and decision making [15] [16]. +To this end, we examined data from chronic pain patients in +three subsets of data: 1) with questionnaires only; 2) with +questionnaires plus voice data; and 3) with questionnaires +plus actigraphy data. The dimensionality of each dataset +was reduced into stable clusters using standard unsupervised +clustering algorithms. Next, we quantitatively evaluated the +clusters based on relationships to established health metrics, +using standard assessments as clinical benchmarks in chronic +pain to compare the data-driven results. A clear ordinal rank of +states emerged, allowing us to assign unique qualitative labels +even in clusters that were nearly identical in pain magnitude, +so that they may be used as clinically-informed states. This +system serves as an example of organizing diverse types of +large datasets and anchoring them to known metrics as to +evaluate treatment or assess function. Here, these formerly +convoluted data patterns may now act to contextualize signal, +rank results, track longitudinal health changes, and monitor +meaningful medical outcomes. +II. METHODS +A. Participants and Data Collection +Participants were recruited from pain clinics in on-going, +longitudinal, multi-center, clinical studies (Clinicaltrials.gov +ID: NCT01719055) aimed to understand chronic lower back +and leg pain patients who are candidates for spinal cord +stimulator (SCS) treatment (Boston Scientific, Valencia, CA). +Participants were recruited and enrolled in the NAVITAS +and/or ENVISION studies at multiple United States clinical +sites if they intended to receive or had already received an +SCS trial or implant, were at least 18 years old, and had +been diagnosed with intractable chronic neuropathic pain. +Additionally, subjects may have been previously enrolled in +the RELIEF study (Clinicaltrials.gov ID: NCT01719055). Data +were included in this analysis from each study a subject was +enrolled in. Health-related questionnaires were administered +via an at-home, custom-designed clinical study version of a +digital health ecosystem (Boston Scientific, Valencia, CA) for +up to 36 months. The questions chosen included pain-related +subjective ratings, symptoms hypothesized to contribute to +variability in pain ratings, as well as symptoms hypothesized to +be impacted by pain, specifically pain magnitude, mood, sleep, +alertness, medication use, and activity. Following enrollment, +data were collected in separate in-clinic and at-home data +streams. Mobile data analyzed here included voice recordings, +as well as daily, self-reported symptom monitoring, with the +option to respond more frequently if participants wished. In +addition, subjects were asked to wear a smartwatch to assess +mobility using accelerometer data (Galaxy Watch S2, Samsung +USA, Menlo Park, CA with custom watch application, Boston +Scientific, Valencia, CA). In-clinic assessments were collected +at the baseline (enrollment) visit, and at 1-month, 3-month, +12-month, and optionally 24-month and 36-month visits fol- +lowing enrollment. In the present analysis, we used in-clinic +assessments to evaluate QoL [17] and disability measured by +the Oswestry Disability Index, or ODI [18] questionnaires. +B. Voice data processing +Voice recordings were collected from weekly recordings +based on prompts aimed to understand the participant’s experi- +ence with pain. Speech features for psycholinguistic, sentiment +[19], and acoustic characteristics [20], [21] were extracted +from the audio files using in-house and standard code. Age + +and sex were regressed from all features. Next, to reduce di- +mensionality of these features, a principal components analysis +(PCA) was used (var ≥ 2%) to identify the decomposed +components. These components were later included in a +clustering analysis alongside the 6 features derived from the +questionnaires. +C. Actigraphy data processing +Effective mobility was derived from the watch-based actig- +raphy data. It is a novel metric of physical function and activity +meant to reflect the duration and type of activity a person +experiences beyond steps or activities of daily life. Rates of +activity were calculated into categories for each participant +throughout the day. These categories ranged from Zone 0 +(e.g., resting, using a mobile device while seated) to Zone +4 (e.g., intense or repetitive motion or vigorous exercise) and +were used along with the questionnaire data in the clustering +analysis. +D. Data and Clustering Analysis +For each participant, all available data was downloaded +and selected based on days for which all subjective features +from questionnaires (e.g., overall/leg/back pain, mood, sleep +hours, sleep quality alertness, medication use for opioid/over- +the- counter/non-opioid pain medication, activity interference +due to pain, and activities of daily life), as well as actigraphy +and voice data (where applicable) were present. Patients were +included in the analysis regardless of time point in the study +(e.g., baseline/enrollment, SCS trial period, follow-up), in the +interest of observing a spectrum of pain-related variability and +experience. However, the criteria for removing samples from +the analysis consisted of: 1) any day missing a single data +point, 2) any individual having fewer than 10 total complete +data points, and when applicable 3) individuals who wore the +smartwatch for less than 10 days. All question value responses +were normalized prior to cluster analysis to equate the different +subjective feature values across the individual question, and +data distributions were inspected for abnormalities. Next, each +question categorized to assess pain, sleep, and medication use +were averaged to produce single composite scores for each +modality; for activity, a difference score was taken between +the two questions, in which we include a penalty that account +with pain interferes with any overall activity. If any participant +had answered more than one question on a certain day, the +average of those responses was used to represent the daily +value for that category. Participants were assessed for their +average responses over time in order to determine the extent +to which some participants responded more frequently than +others, and the analysis was rerun without outliers to further +ensure cluster stability. +Cluster definitions were calculated using a k-means cluster- +ing algorithm with Euclidean distance exploring up to cluster +solutions for k = 10. Optimal k was determined using multiple +methods including sum of squares distances and silhouette +values, agglomerative analysis, and consensus clustering. To +ensure clusters were similar across subsamples of participants +exhibiting variability in number of responses included in the +analysis, we repeated the analyses in varying samples of +participants in which highly contributing participants (those +with higher daily average responses) were excluded. Next, we +employed an analogous approach to examine cluster solutions +over the course of time. Generally, we expected the clusters +to remain similar over time with some slight changes (e.g., +higher pain prior to therapy) that would be evident in the +cluster. With this in mind, cluster solution results were then +visually inspected in order to ensure similarities in qualitative +characteristics and are discussed in the results section. +Fig. 1. Conceptual data and methods overview. (A) Data were collected from +a multi-center clinical trial recruiting participant with chronic low back and +leg pain seeking spinal cord stimulator (SCS) treatment. Both in-clinic and at- +home data collection were used to record 1) questionnaire-based daily reports +of pain, mood, activity, medication, alertness, sleep; 2) standard assessments +of QoL (EQ5D) and disability (ODI); 3) voice responses to open-ended +questions about their pain; and 4) actigraphy from a smartwatch. (B) Data from +questionnaires, voice, and actigraphy were subjected to a k-means clustering +analysis and the (C) resulting cluster representation was examined across +features. To validate these clusters, (D) centroid distance to each cluster was +compared to the clinical scores for disability and QoL allowing for (E) an +interpretation and label to be assigned to each cluster. +III. RESULTS +A. Sample demographics and data chronology +In the primary analysis including questionnaires only, 121 +individuals with 11,763 samples of data were used (40.5% +male, mean age 59.4 years old, 17.6 years since pain onset). +In the analysis examining the addition of actigraphy data to +the questionnaire data, 116 individuals with 11,286 samples of +data were used (39.7% male, mean age 59.3 years old, 17.8 +years since pain onset). For the analysis including voice, 2,080 +samples were included. +B. Clustering results and characteristics for questionnaire +data only +Cluster definition was examined for the questionnaire-only +data for k = 2 to 10. Sum of squares distances and silhouette +analyses indicated that a cluster solution of k = 2 or 3 was +stable. Agglomerative hierarchical clustering was repeated to +validate k with cross-methodological clustering, which also +converged on a solution of k = 2 or 3. Given the relative +stability of smaller cluster solutions, we first examined a + +(A) +Clinic Data +Day 5 +ay 55 +Mobile Data +Day 1 +Day 60 +(B) +Decomposed representation +(C) + Examine clusters +(D) +Compare to clinical scores +(E) Interpretation +Mooc +Disability Score = 27 +Medic +Disability Score = 11 +Disability Score = 52 +Mobilitysimple and stable solution of k = 2. Feature characteristics +of the cluster solution for k = 2 were examined by inspecting +mean values for each feature in each cluster (Figure 2A). Re- +sults indicated a clear negative-to-positive grouping of health +features, such that the questionnaire responses of one cluster +appeared to represent a superior health state represented by +better mood, sleep, alertness and activity, and lower ratings +of pain and medication use. The other cluster appeared to +represent an inferior health state, characterized by higher pain +and medication use, with lower ratings for alertness, mood, +sleep, and activity. This analysis was repeated to exclude the +high-responder group in order to ensure that the clusters were +not being driven by the high-responders. Results indicated that +the clusters were very similar both in all participants, and +without the high-responders. Finally, the cluster solutions were +re-examined over the course of time, such that the analysis +was repeated in the baseline period prior to SCS activation, +during the first 6 months of treatment, and the subsequent 6 +months of treatment. Results indicated that the cluster solution +was very similar over time, with some indication of higher +pain prior to treatment. An examination of a 3-cluster solution +revealed a third, intermediate cluster that represented a health +state similar to or in between the two states represented in +the two-cluster solution (Figure 2B). This cluster showed +relatively high ratings of alertness, mood, and sleep, but with +intermediate values for pain, activity and medication use. +A repeated analysis excluding high-responders also showed +an intermediate cluster, with values for each feature with a +magnitude between the previous two clusters. +Fig. 2. Cluster analysis for questionnaire data reveals negative and positive +symptom groups. (A) A two cluster (k = 2) solution resulting from the k-means +analysis of the questionnaire data revealed two clusters of symptoms that +stratified on a negative-to-positive spectrum of pain-related health, in which +one cluster revealed a better health state of better mood, sleep, more reported +activity and alertness, less medication usage, and lower pain. Conversely, the +other cluster depicted a worse health based on the feature means. (B) A +three cluster (k=3) solution also revealed a spectrum of positive to negative +symptom groupings, including superior and inferior states similar to k=2, with +an additional intermediate state showing moderate pain and medication use +but with high mood, sleep, activity, and alertness scores. +C. Decomposing and clustering questionnaire and voice data +Prior to clustering, the results of the principal components +analysis (PCA) of the voice features were inspected. The +results showed that 7 components were present, and character- +ized features such as voiced and unvoiced energy in a speech +signal, negative sentiment, emotional content, and acoustic +voice properties (Table 1). We repeated the clustering analysis +with each of these 7 components included along with the 6 +questionnaire components. With the addition of the 7 compo- +nents, solutions for k of 2 or 5 were possible. For the k = 2 +solution, results showed that in particular, component 4, which +was characterized by high loadings of negative sentiment and +acoustic features associated with emotion, tracked well with +the inferior health cluster (Figure 3A). Further, while not all +components showed the same discrimination between states as +did component 4, there was evidence that the addition of the +voice data expanded the granularity of the state solutions. This +was illustrated by the comparison of a cluster solution with +only questionnaires in which pain was stratified across 3 levels +in all states (Figure 3B). When the cluster solution included +both voice and questionnaires, pain across states expanded to +5 levels (Figure 3C). +TABLE I +DECOMPOSITION OF VOICE FEATURES INTO 7 COMPONENTS +Fig. 3. Adding voice features to cluster analysis improves pain granularity +in state solutions. (A) Clustering analysis was run including 6 questionnaire +components and 7 voice components for a 2-state solution, indicating that +voice features denoting negative sentiment were associated with the poorer +health cluster. (B) A 5-state cluster solution without voice features reveals +three levels of pain magnitude across clusters, while the (C) addition of voice +to a 5-state cluster solution adds further granularity to pain magnitude across +clusters. +D. Clustering results and characteristics for questionnaire +and actigraphy data +Actigraphy data downloaded from the watch were parsed +into mobility Zones 0 - 4 of effective mobility. Inspection of +results indicated that these zones indeed provided granularity +that added description beyond number of steps or self-reported +ADLs (Figure 4). The clustering analysis included the 6 +categories derived from the questionnaires along with the + +(A) +(B) +(C) +MOOD +MOOD +MOOD +ACOUSTIC 3 +ALERTNESS +ACOUSTIC 3 +ALERTNESS +ACOUSTIC 2 +SLEEP +ACOUSTIC 2 +SLEEP + ALERTNESS +8337S +VOICE QUAL +ACTIVITY + VOIGE QUAL +- ACTIVITY + NEGATIVE + PAIN +NEGATIVE +PAIN + EMOT +MEDS USE +ACTIVITY + EMOT +ACOUSTIC +MEDS USE +ACOUSTIC +MEDS USE +ENERGY +ENERGY +MFCC +(ACOUSTIC 1 + PAIN +MECC +ACOUSTIC 1PCA +Largest Loadings +Smallest/Negative Loadings +Component Name +Content: typetoken/speech richness (psycholinguistic), Fisher SWB +Acoustic 1 +Acoustic shape and characteristics of voice spectrum, RASTA #10 +Acoustic: energy, spectral roll off, voicing probability, +Formant 1 (bandwidth) +MFCC +Acoustic: MFcC #2, voicing probability - may be related to how much +Acoustic: MFCC #2, RASTA #2, spectral roll off/voicing probability (how +speech is present (vowel voicing) + much a person is talking) +Acoustic: Formants 1 and 2 (modulation/harmonics of voice), energy, and +Acoustic energy +Acoustic: energy, spectral flux (voice timbre) + RASTA #5 +Content: negative sentiment (VADER), negative emotion (LIWC), +Content: positive sentiment (VADER), positive emotion, tone, reward +Negative emot +"feel" words +(LIWC), compound (positive valence/intensity) +Acoustic: MFCC #3 (frequency band ~250 Hz) +Acoustic voice quality/properties: MFCC #2, spectral entropy, HNR. +Content: tone, compound (positive valence/intensity +Voice quality +unvoiced (% voice not in recording) +Acoustic: jitter features (characteristic of voice time) +Acoustic: formant 1 (modulation in larynx), unvoiced frames (% voice +Acoustic 2 +Acoustic: MFCC 12, 13, 14 (higher frequencies) + not in recording), formant 2 (bandwidth), frequency at max energy +Acoustic: Formant 2 (indicative of emotional content), +Content: tone, compound (positive valence/intensity) +Acoustic 3 +HNR (voice quality) +Acoustic: MFCC #4, slope of LTAS (avg spectrum)(A) +(B) +MOOD +MOOD +ALERTNESS +SLEEP +ALERTNESS +SLEEP +8 0102 0384 0506 07 +0 0.11 0.2: 0.3 +0.6 :0. +MED.USE +-ACTIVITY +MED. USE + ACTIVITY +PAIN +PAINeffective mobility. An analysis for optimal k showed that state +solutions of up to 5 clusters was possible. These clusters +appeared to range from a ”best” state that included low pain +and medication use, and high reports of mood, sleep, alertness, +and effective mobility, to an inferior state that is associated +with high levels of pain and medication use, and low reports of +activity, mood, sleep, alertness, and effective mobility (Figure +5). +Fig. 4. Description of effective mobility zones. Mobility data was parsed into +zones of “effective mobility” based on rates of activity calculated at regular +time window intervals throughout the day. When compared to step counts +and self-reported activities of daily life (ADLs), effective mobility showed +additional computational granularity of participant mobility. +Fig. 5. +Adding mobility features contributes to cluster dimensionality. (A) +A cluster solution including effective mobility identified 5 stable clusters for +which the addition of effective mobility may contribute to additional clusters +relative to the questionnaire-only solutions, still ranging from a negative-to- +positive spectrum and including a best and a worst state. (B) States from the +5-cluster solution show further granularity as it pertains to patient experience +beyond the 2- and 3-state model. +E. Cluster validation and state classification +For the validation analysis, we obtained pairs of metrics +comprised of 1) distances from the cluster centroids on a given +day; and 2) responses to standard assessments (disability, or +ODI, and QoL, or EQ5D measurements focusing on Pain, +Activities, and VAS Health). These two metrics were collected +within one week of each other; any pairs with collection dates +outside of the week window were dropped from analysis. We +first calculated correlations between centroid distances of each +cluster in the two-state solution, and found that the correlations +were statistically significant and consistent in terms of direc- +tion and magnitude for the two states, indicating a clear best +and inferior state (in cluster 1 values were: disability/ODI, r = +0.42, EQ5D Pain, r = 0.47, EQ5D Activities r = 0.37, EQ5D +VAS Health r = −0.32; all p-values <0.001, for cluster 2 +values were: disability/ODI, r = −0.41, EQ5D Pain, r = −0.43, +EQ5D Activities r = −0.38, EQ5D VAS Health r = 0.28; all +p-values < 0.001). This indicated that larger centroid distances +were associated with higher values for the outcomes. Critically, +while most of the validation metric outcomes represented neg- +ative health values with increasing severity including disability, +EQ5D-Pain, EQ5D-Activities, etc., the EQ5D measure of VAS +Health represents health on a positive scale, and as expected +showed an inverse relationship to the findings above. Given +that each cluster was associated with consistent directionality +across all of the standard assessments, we were able to infer +that each of the clusters represented distinct health states, +aligned with what we would have expected to find in patients +across time. +F. Cluster validation with voice data +A similar analysis was repeated using the k = 2 cluster solu- +tion that included voice data. Results indicated that generally +the directionality of the correlations was consistent relative +to prior analyses. However, for several validation metrics, the +magnitude of the r values increased with the addition of voice +features (for disability/ODI, r = 0.47, EQ5D VAS Health r = +−0.49). In particular, assessments that may take into account +negative affect showed an increase in the correlation across +these metrics. Notably, because voice data is collected less +frequently, there was a decrease in sample size relative to +the prior analysis. That said, permutation tests were used to +compare across the two approaches and to ensure that there +were no meaningful differences due to sample size. In all +instances, permutation tests confirmed the significance of prior +findings at p < 0.05. +G. Cluster validation with actigraphy data +Next, we aimed to determine whether correlations between +centroids from a more highly dimensional state solution com- +pared to the standard assessments could provide further ordinal +information about the states. To do this, we ran a similar +analysis using the 5-state solution that was obtained with the +cluster solution including effective mobility. Here, we found +that the correlations across the 5 states also provided evidence +for a consistent ranking of those states from best to worst +(Table 2). +TABLE II +CLUSTER CHARACTERISTICS INCLUDING EFFECTIVE MOBILITY + +e D +State E +31** +r = -0.46** +25** +r = -0.32** +24** +r = -0.35** +19** +r = 0.23** +.2** +r = -0.37**Metric +State A +State B +State C +State +ODI Total +r = 0.46** +r = 0.41** +r = -0.06* +r = -0. +EQ5DActivities +r = 0.28** +r = 0.26** +r = -0.09** +r = -0. +EQ5D Pain +r = 0.42** +r = 0.41** +r = -0.09** +r= -0. +EQ5D HealthVAS +r = -0.18** +r = -0.13** +r = 0.04 ns +r= 0. +EQ5D - Normed Score +r = 0.4** +r = 0.32** +r = -0.12** +r=-0Mood +(A) +(B) +State A + StateB State C State D +StateE + Average + Better +Effective + State A +Pain +mobility. + Sleep +State B +State C +State D +Medication + State E +Activities of +daily living +Mood +70) 0.1 0.2 0.3 0. 4 0.5 0.6 0.7 +Alertness +-Activity. +Sleep +Alertness +Effective +mobility +Worse +Medication +Average +PainZone 0 +Resting, using a mobile phone, remote control +Zone 1 +Dressing, moving around, slowing walking, stretching +Zone 2 +Walking briskly, light exercise +Zone 3 +Running, swimming or exercising +Zone 4 +Intense or repetitive motion or vigorous exercise +Number of Steps +Self reported number of ADLs16.0 +14.3 +3000 +13.8 +13.1 +14.0 +12.9 +2425 +12.0 +2500 +11.4 Worn Hours +11.15 +12.0 +1925 +9.3 1787 +1.2 +2000 +10.0 +8.26 +7.15 Active Hours +7.01 +7.22 +8.0 +6.75 +4.0 +6.77 +1.3 +1500 +803 +1160 +1.2 +1.0 +1117 +1.3 +6.0 +12 +1.2 +Steps +3.3 +1000 +2.5 +2.6 +4.0 +3.3 +574 +2.6 +3.0 +3.3 +1.3 +1.4 +500 +2.0 +1.1 +12 +3.4 +1.3 +1.2 +1.5 +2.2 +2.2 +1.8 +1.3 +1.5 +0.0 +0 +Day 1 +Day 2 +Day 3 +Day 4 +Day 5 +Day 6 +Day7H. Comparison of state timecourse to health events +In an exploratory analysis, we examined the relationship +between state expression change over time relative to known +health events. Here (see Figure 6), we first show that states +represent a more interpretable visualization of health changes +across time relative to examining the timecourse of all vari- +ables at once. Second, several exemplar patients show expected +changes in states before and after implantation of the SCS de- +vice, a procedure that involves surgery and probable eventual +pain relief. +Fig. 6. Examples of patient experiences show that states track with meaningful +clinical events. Top time course for each patient denotes state assignment, +whereas lower time course shows changes in multiple variables. Bar graphs +show the dwell time change before and after a notable event, which here +involves the implantation of a SCS device hypothesized to bring about eventual +pain relief and improvement in QoL. (Here, data in the time courses included +overall, leg, and back pain, sleep hours and quality, number of activities, pain +interference, medication usage for opioid, over-the-counter, and non-opioid +pain medications, alertness, mood, and effective mobility. States are ranked +as A > B > C > D > E, as shown in Table 2.) +IV. DISCUSSION +A. High-dimensional health data can be decomposed mean- +ingfully +Using a unique set of longitudinal questionnaire, mobility, +and speech data, we have developed a novel method to decom- +pose, group, and validate large amounts of chronic pain digital +health data. This study marks one of the only approaches to +create clinically usable pain-related categories from complex +questionnaire, mobility, and speech data across time. This +approach demonstrates that high dimensional, longitudinal +health data from chronic pain patients may be decomposed into +clusters and used to classify patients according to a holistic +status named Pain Patient States. These states have an ordinal +ranking based on clinically-validated standard health assess- +ments. Specifically, we demonstrated that in chronic pain, +we can take multiple streams of information including sleep +hours and quality, mood, pain magnitude at multiple sites, +alertness, multiple types of medication use, ADLs, actigraphy, +and speech in order to represent 3-5 Pain Patient States over +the course of time. The stable solutions that emerged from this +method suggest the discovery of distinct clinical states with +non-obvious properties that may serve as new knowledge that +informs biological mechanisms and clinical care. In addition +to the identification of these Patient Pain States, this improves +upon prior assessments and clinical trials that only use pain +magnitude as an outcome evaluation by considering a much +more comprehensive picture of patient experience in a way +that is clinically interpretable. This approach leverages both +data- and clinically-driven analyses by first using powerful +learning algorithms, and then comparing the output to standard +clinical metrics. Consequently, we are able to transform what +was previously multiple, complex time courses for hundreds +of patients into 3-5 states that are clinically contextualized, +straightforward, and meaningful. +B. The decomposition can be externally validated and ranked +We found that the resulting clusters from our analysis strat- +ified on a negative-to-positive spectrum of health in chronic +pain, and that these clusters were reliable across subsets of +individuals and over time. Importantly, these states provide +valuable, novel information per se, representing new findings +that may define patient experience. Nevertheless, because they +were derived from a purely data-driven analysis, we chose +to compare cluster characteristics to independent standard as- +sessments of disability and QoL. We found not only that good +and bad clusters associate with better and worse disability and +QoL, but that more granular state solutions had a clear ordinal +rank which contextualized the data-driven output (Table 2). +Further, in a 5-state solution (see figure 5A), only 2 levels +of discriminable pain were present for 4 states. This adds +clear dimensionality beyond what pain alone may indicate +about a patient’s well-being. Thus, we were able to assess +5 ordinal steps of health based on multidimensional aspects, +providing evidence that we can offer a more full picture of +patient experience yet preserve interpretability, making these +states meaningful and actionable clinical information. This can +improve precision in outcomes assessments, especially as it +pertains to pain research and clinical trials. +C. Objective data adds granularity to state solutions +In particular, raw objective metrics such as actigraphy +and speech features are too complex to use without some +dimension reduction. However, actigraphy and speech offer +insight into patient experience both because they reflect a novel +behavioral measure and because they involve limited self- +assessment, which is known to be susceptible to psychological +biases. Here we showed that we were able to quantify and se- +lect features from these objective measures in a preprocessing +step, and then incorporate them into a clustering analysis. We +found that one benefit of this approach is that these types of +features indeed add dimensionality to a state solution, and the +preprocessing in this case allowed for the derived features to +add some biological interpretation. Additionally, we identified +speech features that capture negative sentiment, possibly aug- +menting the ability for the states to detect disability versus +wellness as indicated by higher correlation values between +those states and the independent assessments. + +dynamic cnanges in states +State B +tateD +following implant moving +73.3% +0.0% +between multiple states. +State C +State D +13.3% +toteE +6.7% +6.7% +to Implant + Post-Implantt30 Days +Time (Major Ticks Marked every 14 days) +PATIENT 3: DE NOVO SCS PA +A) Longitudinal State Plot for Patient 3 +C) D +State A +State B +State :C +State D +State E +mplar +B) Health Outcomes Plot for Patient 3 +Normalized Values +0.8 +0.6 +0.2 +Time (Major Ticks Markedevery 14 days) +Trial EndTIENT +ellTimeChange +Patient 1 achieves the State A +State A +with SCS therapy during trial and +ateC +10.0% +1.6% +following implant eventually +remaining stable in State B. +StateB +86.7% +A marked reduction of dwell +ateD +time in State C and D is +3.4% +State D +3.3% +observed post-implant. ++ Post-implant defined as days 14 to 44 days after implant to +oImplant +Post-Implantt +account for postsurgical healing. +TIENT +vellTimeChange +State C +Patient 2 achieves cycles +10.0% +State A +State B +13.3% +between State A and State B +36.7% +tateD +following ScS therapy. +80.0% +StateC +20.0% +State E +10.0% +StateD +30.0% +Pre-Implant +Post-Implant+ +TIENT +well TimeChange +tateC +State B +State A +Patient 3 has more +13.3% +10.0% +6.7%PATIENT 1: DE NOVO SCS PA +A) Longitudinal State Plot for Patient 1 +C) Dwe +State A +State B +State C +State D +State E +Trial +Trial +5. +Start +Normalized Values +B) Health Outcomes Plot for Patient 1 +Sta +0.2 +Time (Major Ticks Marked every 14 days) +Trial End to +PATIENT 2: DE NOVO SCS PA +Aj Longitudinal State Plot f +State A. +C) Dw +State B +State C +State D +State E +B) Health Outcomes Plot for Patient 2 +0.8 +0.6 +0.4D. Conclusions +Ultimately, this analysis combined AI and clinical knowl- +edge to successfully reduce complex mobile data into useful +health states that reflect important clinical time points and +changes in patient experience (Figure 6). While all approaches +should be tested and verified broadly across additional popu- +lations and data sets, this approach lays a solid foundation +by which complex datastreams may be reduced into and +authenticated as useful wellness information. We were able to +show that we could successfully use this method in patients +undergoing treatment for chronic pain, with results yielding +new, distinct representations of patient experience. These find- +ings imply it is possible to expand this approach to other +illnesses associated with heterogeneous sets of symptoms. +Finally, while we were able to compare our findings to known +metrics, the health states provide deep insights in and of +themselves that could aid a clinician in medical decision +making and patient care. Given the growing use of digital +health solutions, this approach to define Pain Patient States +holds great promise in harnessing AI-driven solutions to aid +in the care of large groups of chronic pain patients. +ACKNOWLEDGMENT +The NAVITAS and ENVISION Studies Physician Author +Group includes Richard Rauck (The Center for Clinical Re- +search), Eric Loudermilk (PCPMG Clinical Research Unit), +Julio Paez (South Lake Pain Institute), Louis Bojrab (Forest +Health Medical Center), John Noles (River Cities Interven- +tional Pain), Todd Turley (Hope Research Institute), Mohab +Ibrahim (Banner University Medical Center), Amol Patward- +han (Banner University Medical Center), James Scowcroft +(KC Pain Centers), Rene Przkora (University of Florida), +Nathan Miller (Coastal Pain and Spinal Diagnostics), and +Gassan Chaiban (Ochsner Clinic Foundation). +The Boston Scientific Research Scientists Consortium in- +cludes Dat Huynh (Boston Scientific, Data Research and +Engineering), Kristen Lechleiter (Clinical Research, Boston +Scientific), Brad Hershey (Data Research and Engineering, +Boston Scientific), Rex Woon (Data Research and Engineer- +ing, Boston Scientific), and Matt McDonald (Boston Scientific, +Data Research and Engineering). +We wish to acknowledge work by Erhan Bilal (IBM, Digital +Health) for his work on consensus clustering. +REFERENCES +[1] U. 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Wempe, “Praat script to detect syllable nuclei +and measure speech rate automatically,” Behavior Research Methods, +vol. 41, no. 2, pp. 385–390, 2009. + diff --git a/9dAyT4oBgHgl3EQfdPeW/content/tmp_files/load_file.txt b/9dAyT4oBgHgl3EQfdPeW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..442c5c82f8d192055a757e2b98ed34266f14f935 --- /dev/null +++ b/9dAyT4oBgHgl3EQfdPeW/content/tmp_files/load_file.txt @@ -0,0 +1,738 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf,len=737 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='00299v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='AP] 31 Dec 2022 Definition and clinical validation of Pain Patient States from high-dimensional mobile data: application to a chronic pain cohort 1st Jenna M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Reinen Digital Health IBM Research Yorktown Heights, NY jenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='reinen@ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='com 2nd Carla Agurto Digital Health IBM Research Yorktown Heights, NY carla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='agurto@ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='com 3rd Guillermo Cecchi Digital Health IBM Research Yorktown Heights, NY gcecchi@us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='com 4th Jeffrey L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Rogers Digital Health IBM Research Yorktown Heights, NY jeffrogers@us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='com 5th NAVITAS and ENVISION Studies Physician Author Group Clinical Research Boston Scientific Valencia, CA 6th Boston Scientific Research Scientists Consortium Data Research and Engineering Boston Scientific Valencia, CA Abstract—The technical capacity to monitor patients with a mobile device has drastically expanded, but data produced from this approach are often difficult to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' We present a solution to produce a meaningful representation of patient status from large, complex data streams, leveraging both a data-driven approach, and use clinical knowledge to validate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Data were collected from a clinical trial enrolling chronic pain patients, and included questionnaires, voice recordings, actigraphy, and standard health assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' The data were reduced using a clustering analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' In an initial exploratory analysis with only questionnaire data, we found up to 3 stable cluster solutions that grouped symptoms on a positive to negative spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Objective features (actigraphy, speech) expanded the cluster solution granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Using a 5 state solution with questionnaire and actigraphy data, we found significant correlations between cluster properties and assessments of disability and quality- of-life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' The correlation coefficient values showed an ordinal distinction, confirming the cluster ranking on a negative to positive spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' This suggests we captured novel, distinct Pain Patient States with this approach, even when multiple clusters were equated on pain magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Relative to using complex time courses of many variables, Pain Patient States holds promise as an interpretable, useful, and actionable metric for a clinician or caregiver to simplify and provide timely delivery of care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Index Terms—chronic pain, digital health, clustering, medical decision making I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' INTRODUCTION Recent advances in digital medicine have provided the opportunity to collect large sets of clinical data to evaluate and predict critical medical outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' For instance, mobile-based applications, accelerometers, and biosensors are now ubiqui- tous in phones and watches, enabling one to longitudinally track variables like mobility and speech, and facilitate patient symptom self-report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Importantly, these features may associate with clinical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Large-scale studies have shown that data from mobile applications tracking daily activity may predict outcomes relevant to health and illness, such as in geriatric care and diabetes [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Further, language can assess affective, psycholinguistic, physiological, and cognitive features can predict physiological and pharmacological [3], psychiatric [4], and cognitive disease states [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' These types of findings have demonstrated the promise of digital health profiles in understanding patient experience and predicting important clinical outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Despite these advances, the size and complexity of the clinical data generated by mobile applications is nontrivial to interpret and apply for several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' First, digital healthcare data can exist in multiple formats, creating the need to fuse vast amount of diverse information [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Second, there is a need for methods that can obtain clear data representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' These methods should provide interpretation that are manage- able in size, yet can maintain the characteristics of the raw information, allowing for patients and healthcare professionals to interpret and use the output [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Attempts to reduce and understand such data in a biological context have commonly used data-driven methods, especially those using machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' This approach offers the advantage of being able to handle large, multidimensional data sets through the ability to recognize patterns or joint representations that are otherwise difficult to identify using standard statistical approaches, providing knowledge discovery about a particular Copyright © 2022 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' topic that can span time, location, and scales [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' In particular, clustering analysis offers the ability to collapse across oth- erwise incomprehensible multidimensional data and observe how features co-occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' In the case of spectrum illnesses that incorporate a range and variety of symptoms, decomposition can be helpful, with some outputs having an advantage in outcomes prediction [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' But not all results allow for interpre- tation, and it is particularly susceptible to problems in small, unvalidated datasets which may result in overfitting and thus results that are not replicable or generalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Further, while results from unsupervised approaches may reveal meaningful clinical patterns, few methods exist to formally assign labels, rank, or identify qualitative aspects from the results of data- driven approaches through independent validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' A prime illustration of this problem is in chronic pain, a dis- ease affecting a substantial percentage of the population [10] that significantly impacts general function including employ- ment, mental health, and social interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' This heterogeneous condition interacts with well-characterized facets of health, including mood, sleep, psychosocial function, medication use, and mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' However, the current practice for most pain studies is to evaluate outcomes based on pain magnitude alone, which does not consider all of the variance shown to predict treatment success, quality-of-life (QoL), or other mea- surements of physical, psychological, and social well-being [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' But, using all of these features as outcome variables is nontrivial to compute, conceptualize, and interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Few standard approaches have been developed that incorporate both the computational methodology required to complete such a task, and the ability to provide a clinically interpretable summary of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' To date, machine learning has been used to predict pain outcomes, identify clinical subgroups [12], extract knowledge, and detect structure in biological and clinical features [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' In chronic pain, while artificial intelligence (AI) has been applied to improve diagnoses, fewer studies apply it to the treatment and management of pain patients [14], and analyses that use longitudinal data or clinical validation are extremely limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Given the quickly expanding capacity of digital health and learning algorithms to inform treatment outcomes in complex illnesses, there is a benefit to developing an approach to validate health states from multidimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' While it is known that various chronic pain symptoms can co-occur, it remains currently unknown whether symptom profiles may be successfully organized into distinct health states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Here, we propose a method by which we aim to identify clusters from high-dimensional, longitudinal data in chronic pain patients, and label them as Pain Patient States that may be operational- ized for clinical application and decision making [15] [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' To this end, we examined data from chronic pain patients in three subsets of data: 1) with questionnaires only;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' 2) with questionnaires plus voice data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' and 3) with questionnaires plus actigraphy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' The dimensionality of each dataset was reduced into stable clusters using standard unsupervised clustering algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Next, we quantitatively evaluated the clusters based on relationships to established health metrics, using standard assessments as clinical benchmarks in chronic pain to compare the data-driven results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' A clear ordinal rank of states emerged, allowing us to assign unique qualitative labels even in clusters that were nearly identical in pain magnitude, so that they may be used as clinically-informed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' This system serves as an example of organizing diverse types of large datasets and anchoring them to known metrics as to evaluate treatment or assess function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Here, these formerly convoluted data patterns may now act to contextualize signal, rank results, track longitudinal health changes, and monitor meaningful medical outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Participants and Data Collection Participants were recruited from pain clinics in on-going, longitudinal, multi-center, clinical studies (Clinicaltrials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='gov ID: NCT01719055) aimed to understand chronic lower back and leg pain patients who are candidates for spinal cord stimulator (SCS) treatment (Boston Scientific, Valencia, CA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Participants were recruited and enrolled in the NAVITAS and/or ENVISION studies at multiple United States clinical sites if they intended to receive or had already received an SCS trial or implant, were at least 18 years old, and had been diagnosed with intractable chronic neuropathic pain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Additionally, subjects may have been previously enrolled in the RELIEF study (Clinicaltrials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='gov ID: NCT01719055).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Data were included in this analysis from each study a subject was enrolled in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Health-related questionnaires were administered via an at-home, custom-designed clinical study version of a digital health ecosystem (Boston Scientific, Valencia, CA) for up to 36 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' The questions chosen included pain-related subjective ratings, symptoms hypothesized to contribute to variability in pain ratings, as well as symptoms hypothesized to be impacted by pain, specifically pain magnitude, mood, sleep, alertness, medication use, and activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Following enrollment, data were collected in separate in-clinic and at-home data streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Mobile data analyzed here included voice recordings, as well as daily, self-reported symptom monitoring, with the option to respond more frequently if participants wished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' In addition, subjects were asked to wear a smartwatch to assess mobility using accelerometer data (Galaxy Watch S2, Samsung USA, Menlo Park, CA with custom watch application, Boston Scientific, Valencia, CA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' In-clinic assessments were collected at the baseline (enrollment) visit, and at 1-month, 3-month, 12-month, and optionally 24-month and 36-month visits fol- lowing enrollment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' In the present analysis, we used in-clinic assessments to evaluate QoL [17] and disability measured by the Oswestry Disability Index, or ODI [18] questionnaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Voice data processing Voice recordings were collected from weekly recordings based on prompts aimed to understand the participant’s experi- ence with pain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Speech features for psycholinguistic, sentiment [19], and acoustic characteristics [20], [21] were extracted from the audio files using in-house and standard code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Age and sex were regressed from all features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Next, to reduce di- mensionality of these features, a principal components analysis (PCA) was used (var ≥ 2%) to identify the decomposed components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' These components were later included in a clustering analysis alongside the 6 features derived from the questionnaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Actigraphy data processing Effective mobility was derived from the watch-based actig- raphy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' It is a novel metric of physical function and activity meant to reflect the duration and type of activity a person experiences beyond steps or activities of daily life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Rates of activity were calculated into categories for each participant throughout the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' These categories ranged from Zone 0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=', resting, using a mobile device while seated) to Zone 4 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=', intense or repetitive motion or vigorous exercise) and were used along with the questionnaire data in the clustering analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Data and Clustering Analysis For each participant, all available data was downloaded and selected based on days for which all subjective features from questionnaires (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=', overall/leg/back pain, mood, sleep hours, sleep quality alertness, medication use for opioid/over- the- counter/non-opioid pain medication, activity interference due to pain, and activities of daily life), as well as actigraphy and voice data (where applicable) were present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Patients were included in the analysis regardless of time point in the study (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=', baseline/enrollment, SCS trial period, follow-up), in the interest of observing a spectrum of pain-related variability and experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' However, the criteria for removing samples from the analysis consisted of: 1) any day missing a single data point, 2) any individual having fewer than 10 total complete data points, and when applicable 3) individuals who wore the smartwatch for less than 10 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' All question value responses were normalized prior to cluster analysis to equate the different subjective feature values across the individual question, and data distributions were inspected for abnormalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Next, each question categorized to assess pain, sleep, and medication use were averaged to produce single composite scores for each modality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' for activity, a difference score was taken between the two questions, in which we include a penalty that account with pain interferes with any overall activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' If any participant had answered more than one question on a certain day, the average of those responses was used to represent the daily value for that category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Participants were assessed for their average responses over time in order to determine the extent to which some participants responded more frequently than others, and the analysis was rerun without outliers to further ensure cluster stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Cluster definitions were calculated using a k-means cluster- ing algorithm with Euclidean distance exploring up to cluster solutions for k = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Optimal k was determined using multiple methods including sum of squares distances and silhouette values, agglomerative analysis, and consensus clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' To ensure clusters were similar across subsamples of participants exhibiting variability in number of responses included in the analysis, we repeated the analyses in varying samples of participants in which highly contributing participants (those with higher daily average responses) were excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Next, we employed an analogous approach to examine cluster solutions over the course of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Generally, we expected the clusters to remain similar over time with some slight changes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=', higher pain prior to therapy) that would be evident in the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' With this in mind, cluster solution results were then visually inspected in order to ensure similarities in qualitative characteristics and are discussed in the results section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Conceptual data and methods overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' (A) Data were collected from a multi-center clinical trial recruiting participant with chronic low back and leg pain seeking spinal cord stimulator (SCS) treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Both in-clinic and at- home data collection were used to record 1) questionnaire-based daily reports of pain, mood, activity, medication, alertness, sleep;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' 2) standard assessments of QoL (EQ5D) and disability (ODI);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' 3) voice responses to open-ended questions about their pain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' and 4) actigraphy from a smartwatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' (B) Data from questionnaires, voice, and actigraphy were subjected to a k-means clustering analysis and the (C) resulting cluster representation was examined across features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' To validate these clusters, (D) centroid distance to each cluster was compared to the clinical scores for disability and QoL allowing for (E) an interpretation and label to be assigned to each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Sample demographics and data chronology In the primary analysis including questionnaires only, 121 individuals with 11,763 samples of data were used (40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='5% male, mean age 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='4 years old, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='6 years since pain onset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' In the analysis examining the addition of actigraphy data to the questionnaire data, 116 individuals with 11,286 samples of data were used (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='7% male, mean age 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='3 years old, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='8 years since pain onset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' For the analysis including voice, 2,080 samples were included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Clustering results and characteristics for questionnaire data only Cluster definition was examined for the questionnaire-only data for k = 2 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Sum of squares distances and silhouette analyses indicated that a cluster solution of k = 2 or 3 was stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Agglomerative hierarchical clustering was repeated to validate k with cross-methodological clustering, which also converged on a solution of k = 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Given the relative stability of smaller cluster solutions, we first examined a (A) Clinic Data Day 5 ay 55 Mobile Data Day 1 Day 60 (B) Decomposed representation (C) Examine clusters (D) Compare to clinical scores (E) Interpretation Mooc Disability Score = 27 Medic Disability Score = 11 Disability Score = 52 Mobilitysimple and stable solution of k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Feature characteristics of the cluster solution for k = 2 were examined by inspecting mean values for each feature in each cluster (Figure 2A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Re- sults indicated a clear negative-to-positive grouping of health features, such that the questionnaire responses of one cluster appeared to represent a superior health state represented by better mood, sleep, alertness and activity, and lower ratings of pain and medication use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' The other cluster appeared to represent an inferior health state, characterized by higher pain and medication use, with lower ratings for alertness, mood, sleep, and activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' This analysis was repeated to exclude the high-responder group in order to ensure that the clusters were not being driven by the high-responders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Results indicated that the clusters were very similar both in all participants, and without the high-responders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Finally, the cluster solutions were re-examined over the course of time, such that the analysis was repeated in the baseline period prior to SCS activation, during the first 6 months of treatment, and the subsequent 6 months of treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Results indicated that the cluster solution was very similar over time, with some indication of higher pain prior to treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' An examination of a 3-cluster solution revealed a third, intermediate cluster that represented a health state similar to or in between the two states represented in the two-cluster solution (Figure 2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' This cluster showed relatively high ratings of alertness, mood, and sleep, but with intermediate values for pain, activity and medication use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' A repeated analysis excluding high-responders also showed an intermediate cluster, with values for each feature with a magnitude between the previous two clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Cluster analysis for questionnaire data reveals negative and positive symptom groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' (A) A two cluster (k = 2) solution resulting from the k-means analysis of the questionnaire data revealed two clusters of symptoms that stratified on a negative-to-positive spectrum of pain-related health, in which one cluster revealed a better health state of better mood, sleep, more reported activity and alertness, less medication usage, and lower pain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Conversely, the other cluster depicted a worse health based on the feature means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' (B) A three cluster (k=3) solution also revealed a spectrum of positive to negative symptom groupings, including superior and inferior states similar to k=2, with an additional intermediate state showing moderate pain and medication use but with high mood, sleep, activity, and alertness scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Decomposing and clustering questionnaire and voice data Prior to clustering, the results of the principal components analysis (PCA) of the voice features were inspected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' The results showed that 7 components were present, and character- ized features such as voiced and unvoiced energy in a speech signal, negative sentiment, emotional content, and acoustic voice properties (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' We repeated the clustering analysis with each of these 7 components included along with the 6 questionnaire components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' With the addition of the 7 compo- nents, solutions for k of 2 or 5 were possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' For the k = 2 solution, results showed that in particular, component 4, which was characterized by high loadings of negative sentiment and acoustic features associated with emotion, tracked well with the inferior health cluster (Figure 3A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Further, while not all components showed the same discrimination between states as did component 4, there was evidence that the addition of the voice data expanded the granularity of the state solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' This was illustrated by the comparison of a cluster solution with only questionnaires in which pain was stratified across 3 levels in all states (Figure 3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' When the cluster solution included both voice and questionnaires, pain across states expanded to 5 levels (Figure 3C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' TABLE I DECOMPOSITION OF VOICE FEATURES INTO 7 COMPONENTS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Adding voice features to cluster analysis improves pain granularity in state solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' (A) Clustering analysis was run including 6 questionnaire components and 7 voice components for a 2-state solution, indicating that voice features denoting negative sentiment were associated with the poorer health cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' (B) A 5-state cluster solution without voice features reveals three levels of pain magnitude across clusters, while the (C) addition of voice to a 5-state cluster solution adds further granularity to pain magnitude across clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Clustering results and characteristics for questionnaire and actigraphy data Actigraphy data downloaded from the watch were parsed into mobility Zones 0 - 4 of effective mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Inspection of results indicated that these zones indeed provided granularity that added description beyond number of steps or self-reported ADLs (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' The clustering analysis included the 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='categories derived from the questionnaires along with the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='(A) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='(B) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='(C) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='MOOD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='MOOD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='MOOD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ACOUSTIC 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ALERTNESS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ACOUSTIC 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ALERTNESS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ACOUSTIC 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='SLEEP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ACOUSTIC 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='SLEEP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ALERTNESS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='8337S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='VOICE QUAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ACTIVITY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='VOIGE QUAL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ACTIVITY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='NEGATIVE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='PAIN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='NEGATIVE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='PAIN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='EMOT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='MEDS USE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ACTIVITY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='EMOT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ACOUSTIC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='MEDS USE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ACOUSTIC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='MEDS USE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ENERGY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ENERGY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='MFCC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='(ACOUSTIC 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='PAIN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='MECC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='ACOUSTIC 1PCA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='Largest Loadings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='Smallest/Negative Loadings ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='Component Name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='Content: typetoken/speech richness (psycholinguistic),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Fisher SWB Acoustic 1 Acoustic shape and characteristics of voice spectrum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' RASTA #10 Acoustic: energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' spectral roll off,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' voicing probability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Formant 1 (bandwidth) MFCC Acoustic: MFcC #2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' voicing probability - may be related to how much Acoustic: MFCC #2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' RASTA #2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' spectral roll off/voicing probability (how speech is present (vowel voicing) much a person is talking) Acoustic: Formants 1 and 2 (modulation/harmonics of voice),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' and Acoustic energy Acoustic: energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' spectral flux (voice timbre) RASTA #5 Content: negative sentiment (VADER),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' negative emotion (LIWC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Content: positive sentiment (VADER),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' positive emotion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' tone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' reward Negative emot "feel" words (LIWC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' compound (positive valence/intensity) Acoustic: MFCC #3 (frequency band ~250 Hz) Acoustic voice quality/properties: MFCC #2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' spectral entropy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' HNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Content: tone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' compound (positive valence/intensity Voice quality unvoiced (% voice not in recording) Acoustic: jitter features (characteristic of voice time) Acoustic: formant 1 (modulation in larynx),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' unvoiced frames (% voice Acoustic 2 Acoustic: MFCC 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' 14 (higher frequencies) not in recording),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' formant 2 (bandwidth),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' frequency at max energy Acoustic: Formant 2 (indicative of emotional content),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Content: tone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' compound (positive valence/intensity) Acoustic 3 HNR (voice quality) Acoustic: MFCC #4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' slope of LTAS (avg spectrum)(A) (B) MOOD MOOD ALERTNESS SLEEP ALERTNESS SLEEP 8 0102 0384 0506 07 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='6 :0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' MED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='USE ACTIVITY MED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' USE ACTIVITY PAIN PAINeffective mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' An analysis for optimal k showed that state solutions of up to 5 clusters was possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' These clusters appeared to range from a ”best” state that included low pain and medication use, and high reports of mood, sleep, alertness, and effective mobility, to an inferior state that is associated with high levels of pain and medication use, and low reports of activity, mood, sleep, alertness, and effective mobility (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Description of effective mobility zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Mobility data was parsed into zones of “effective mobility” based on rates of activity calculated at regular time window intervals throughout the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' When compared to step counts and self-reported activities of daily life (ADLs), effective mobility showed additional computational granularity of participant mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Adding mobility features contributes to cluster dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' (A) A cluster solution including effective mobility identified 5 stable clusters for which the addition of effective mobility may contribute to additional clusters relative to the questionnaire-only solutions, still ranging from a negative-to- positive spectrum and including a best and a worst state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' (B) States from the 5-cluster solution show further granularity as it pertains to patient experience beyond the 2- and 3-state model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Cluster validation and state classification For the validation analysis, we obtained pairs of metrics comprised of 1) distances from the cluster centroids on a given day;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' and 2) responses to standard assessments (disability, or ODI, and QoL, or EQ5D measurements focusing on Pain, Activities, and VAS Health).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' These two metrics were collected within one week of each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' any pairs with collection dates outside of the week window were dropped from analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' We first calculated correlations between centroid distances of each cluster in the two-state solution, and found that the correlations were statistically significant and consistent in terms of direc- tion and magnitude for the two states, indicating a clear best and inferior state (in cluster 1 values were: disability/ODI, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='42, EQ5D Pain, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='47, EQ5D Activities r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='37, EQ5D VAS Health r = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' all p-values <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='001, for cluster 2 values were: disability/ODI, r = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='41, EQ5D Pain, r = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='43, EQ5D Activities r = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='38, EQ5D VAS Health r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='28;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' all p-values < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' This indicated that larger centroid distances were associated with higher values for the outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Critically, while most of the validation metric outcomes represented neg- ative health values with increasing severity including disability, EQ5D-Pain, EQ5D-Activities, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=', the EQ5D measure of VAS Health represents health on a positive scale, and as expected showed an inverse relationship to the findings above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Given that each cluster was associated with consistent directionality across all of the standard assessments, we were able to infer that each of the clusters represented distinct health states, aligned with what we would have expected to find in patients across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Cluster validation with voice data A similar analysis was repeated using the k = 2 cluster solu- tion that included voice data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Results indicated that generally the directionality of the correlations was consistent relative to prior analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' However, for several validation metrics, the magnitude of the r values increased with the addition of voice features (for disability/ODI, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='47, EQ5D VAS Health r = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' In particular, assessments that may take into account negative affect showed an increase in the correlation across these metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Notably, because voice data is collected less frequently, there was a decrease in sample size relative to the prior analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' That said, permutation tests were used to compare across the two approaches and to ensure that there were no meaningful differences due to sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' In all instances, permutation tests confirmed the significance of prior findings at p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Cluster validation with actigraphy data Next, we aimed to determine whether correlations between centroids from a more highly dimensional state solution com- pared to the standard assessments could provide further ordinal information about the states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' To do this, we ran a similar analysis using the 5-state solution that was obtained with the cluster solution including effective mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Here, we found that the correlations across the 5 states also provided evidence for a consistent ranking of those states from best to worst (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' TABLE II CLUSTER CHARACTERISTICS INCLUDING EFFECTIVE MOBILITY e D State E 31** r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='46** 25** r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='32** 24** r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='35** 19** r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='23** .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='2** r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='37**Metric State A State B State C State ODI Total r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='46** r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='41** r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='06* r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' EQ5DActivities r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='28** r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='26** r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='09** r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' EQ5D Pain r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='42** r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='41** r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='09** r= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' EQ5D HealthVAS r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='18** r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='13** r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='04 ns r= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' EQ5D - Normed Score r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='4** r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='32** r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='12** r=-0Mood (A) (B) State A StateB State C State D StateE Average Better Effective State A Pain mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Sleep State B State C State D Medication State E Activities of daily living Mood 70) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='3 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vigorous exercise Number of Steps Self reported number of ADLs16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='3 3000 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='9 2425 12.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='0 0 Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day7H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Comparison of state timecourse to health events In an exploratory analysis, we examined the relationship between state expression change over time relative to known health events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Here (see Figure 6), we first show that states represent a more interpretable visualization of health changes across time relative to examining the timecourse of all vari- ables at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Second, several exemplar patients show expected changes in states before and after implantation of the SCS de- vice, a procedure that involves surgery and probable eventual pain relief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Examples of patient experiences show that states track with meaningful clinical events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Top time course for each patient denotes state assignment, whereas lower time course shows changes in multiple variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Bar graphs show the dwell time change before and after a notable event, which here involves the implantation of a SCS device hypothesized to bring about eventual pain relief and improvement in QoL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' (Here, data in the time courses included overall, leg, and back pain, sleep hours and quality, number of activities, pain interference, medication usage for opioid, over-the-counter, and non-opioid pain medications, alertness, mood, and effective mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' States are ranked as A > B > C > D > E, as shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=') IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' High-dimensional health data can be decomposed mean- ingfully Using a unique set of longitudinal questionnaire, mobility, and speech data, we have developed a novel method to decom- pose, group, and validate large amounts of chronic pain digital health data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' This study marks one of the only approaches to create clinically usable pain-related categories from complex questionnaire, mobility, and speech data across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' This approach demonstrates that high dimensional, longitudinal health data from chronic pain patients may be decomposed into clusters and used to classify patients according to a holistic status named Pain Patient States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' These states have an ordinal ranking based on clinically-validated standard health assess- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Specifically, we demonstrated that in chronic pain, we can take multiple streams of information including sleep hours and quality, mood, pain magnitude at multiple sites, alertness, multiple types of medication use, ADLs, actigraphy, and speech in order to represent 3-5 Pain Patient States over the course of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' The stable solutions that emerged from this method suggest the discovery of distinct clinical states with non-obvious properties that may serve as new knowledge that informs biological mechanisms and clinical care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' In addition to the identification of these Patient Pain States, this improves upon prior assessments and clinical trials that only use pain magnitude as an outcome evaluation by considering a much more comprehensive picture of patient experience in a way that is clinically interpretable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' This approach leverages both data- and clinically-driven analyses by first using powerful learning algorithms, and then comparing the output to standard clinical metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Consequently, we are able to transform what was previously multiple, complex time courses for hundreds of patients into 3-5 states that are clinically contextualized, straightforward, and meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' The decomposition can be externally validated and ranked We found that the resulting clusters from our analysis strat- ified on a negative-to-positive spectrum of health in chronic pain, and that these clusters were reliable across subsets of individuals and over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Importantly, these states provide valuable, novel information per se, representing new findings that may define patient experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Nevertheless, because they were derived from a purely data-driven analysis, we chose to compare cluster characteristics to independent standard as- sessments of disability and QoL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' We found not only that good and bad clusters associate with better and worse disability and QoL, but that more granular state solutions had a clear ordinal rank which contextualized the data-driven output (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Further, in a 5-state solution (see figure 5A), only 2 levels of discriminable pain were present for 4 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' This adds clear dimensionality beyond what pain alone may indicate about a patient’s well-being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Thus, we were able to assess 5 ordinal steps of health based on multidimensional aspects, providing evidence that we can offer a more full picture of patient experience yet preserve interpretability, making these states meaningful and actionable clinical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' This can improve precision in outcomes assessments, especially as it pertains to pain research and clinical trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Objective data adds granularity to state solutions In particular, raw objective metrics such as actigraphy and speech features are too complex to use without some dimension reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' However, actigraphy and speech offer insight into patient experience both because they reflect a novel behavioral measure and because they involve limited self- assessment, which is known to be susceptible to psychological biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Here we showed that we were able to quantify and se- lect features from these objective measures in a preprocessing step, and then incorporate them into a clustering analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' We found that one benefit of this approach is that these types of features indeed add dimensionality to a state solution, and the preprocessing in this case allowed for the derived features to add some biological interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Additionally, we identified speech features that capture negative sentiment, possibly aug- menting the ability for the states to detect disability versus wellness as indicated by higher correlation values between those states and the independent assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' dynamic cnanges in states State B tateD following implant moving 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='3% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='0% between multiple states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' State C State D 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='3% toteE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='7% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='7% to Implant Post-Implantt30 Days Time (Major Ticks Marked every 14 days) PATIENT 3: DE NOVO SCS PA A) Longitudinal State Plot for Patient 3 C) D State A State B State :C State D State E mplar B) Health Outcomes Plot for Patient 3 Normalized Values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='2 Time (Major Ticks Markedevery 14 days) Trial EndTIENT ellTimeChange Patient 1 achieves the State A State A with SCS therapy during trial and ateC 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='0% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='6% following implant eventually remaining stable in State B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' StateB 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='7% A marked reduction of dwell ateD time in State C and D is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='4% State D 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='3% observed post-implant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' + Post-implant defined as days 14 to 44 days after implant to oImplant Post-Implantt account for postsurgical healing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' TIENT vellTimeChange State C Patient 2 achieves cycles 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='0% State A State B 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='3% between State A and State B 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='7% tateD following ScS therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='0% StateC 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='0% State E 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='0% StateD 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='0% Pre-Implant Post-Implant+ TIENT well TimeChange tateC State B State A Patient 3 has more 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='3% 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='0% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='7%PATIENT 1: DE NOVO SCS PA A) Longitudinal State Plot for Patient 1 C) Dwe State A State B State C State D State E Trial Trial 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Start Normalized Values B) Health Outcomes Plot for Patient 1 Sta 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='2 Time (Major Ticks Marked every 14 days) Trial End to PATIENT 2: DE NOVO SCS PA Aj Longitudinal State Plot f State A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' C) Dw State B State C State D State E B) Health Outcomes Plot for Patient 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content='4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Conclusions Ultimately, this analysis combined AI and clinical knowl- edge to successfully reduce complex mobile data into useful health states that reflect important clinical time points and changes in patient experience (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' While all approaches should be tested and verified broadly across additional popu- lations and data sets, this approach lays a solid foundation by which complex datastreams may be reduced into and authenticated as useful wellness information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' We were able to show that we could successfully use this method in patients undergoing treatment for chronic pain, with results yielding new, distinct representations of patient experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' These find- ings imply it is possible to expand this approach to other illnesses associated with heterogeneous sets of symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Finally, while we were able to compare our findings to known metrics, the health states provide deep insights in and of themselves that could aid a clinician in medical decision making and patient care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Given the growing use of digital health solutions, this approach to define Pain Patient States holds great promise in harnessing AI-driven solutions to aid in the care of large groups of chronic pain patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' ACKNOWLEDGMENT The NAVITAS and ENVISION Studies Physician Author Group includes Richard Rauck (The Center for Clinical Re- search),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Eric Loudermilk (PCPMG Clinical Research Unit),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Julio Paez (South Lake Pain Institute),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Louis Bojrab (Forest Health Medical Center),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' John Noles (River Cities Interven- tional Pain),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Todd Turley (Hope Research Institute),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Mohab Ibrahim (Banner University Medical Center),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Amol Patward- han (Banner University Medical Center),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' James Scowcroft (KC Pain Centers),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Rene Przkora (University of Florida),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Nathan Miller (Coastal Pain and Spinal Diagnostics),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' and Gassan Chaiban (Ochsner Clinic Foundation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' The Boston Scientific Research Scientists Consortium in- cludes Dat Huynh (Boston Scientific, Data Research and Engineering), Kristen Lechleiter (Clinical Research, Boston Scientific), Brad Hershey (Data Research and Engineering, Boston Scientific), Rex Woon (Data Research and Engineer- ing, Boston Scientific), and Matt McDonald (Boston Scientific, Data Research and Engineering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' We wish to acknowledge work by Erhan Bilal (IBM, Digital Health) for his work on consensus clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' REFERENCES [1] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Ekelund, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} +page_content=' Brage, S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dAyT4oBgHgl3EQfdPeW/content/2301.00299v1.pdf'} diff --git a/A9AzT4oBgHgl3EQfv_5R/content/tmp_files/2301.01715v1.pdf.txt b/A9AzT4oBgHgl3EQfv_5R/content/tmp_files/2301.01715v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a180799841c4f0a8be586e1e7ab988575129e682 --- /dev/null +++ b/A9AzT4oBgHgl3EQfv_5R/content/tmp_files/2301.01715v1.pdf.txt @@ -0,0 +1,808 @@ +A Comparison of Fundamental Methods for Iso-surface +Extraction + +JAN PATERA1, VÁCLAV SKALA2 +Department of Computer Science and Engineering +Faculty of Applied Sciences, University of West Bohemia +Univerzitní 22, Plzeň +CZECH REPUBLIC +hopatera@kiv.zcu.cz, skala@kiv.zcu.cz +http://herakles.zcu.cz + +Abstract: In this paper four fundamental methods for an iso-surface extraction are compared, +based on cell decomposition to tetrahedra. The methods are compared both on mathematically +generated data sets as well as on real data sets. The comparison using mathematical data is +made from different points of view such as area approximation, volume approximation. On +the other hand, the Hausdorff distance and root mean square are used to compare methods on +real data sets. The presented comparison can be helpful when deciding among tested methods +which one to choose, as well as when we need to compare a newly developed method with +other existing approaches. + +Key-Words: Comparison, Iso-surface extraction, Error, Hausdorff distance, Volume data, +Computer graphics. + +1 Introduction + +In the recent period of time volume data have started to play a significant role in many +scientific areas and are spread across many professions. In medical field, various devices, +such as Computed Tomography (CT) scanners, Magnetic Resonance Imaging (MRI) scanners +produce volume data. The volume data are also produced as a result of mathematical or +physical simulations and experiments and researchers need to visualize such data. + +1 Supported by the Ministry of Education of Czech Republic; project number MSM +235200005 (Information Systems and Technologies) +2 Supported by project NoE – 3DTV PLT 511568 +Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004 + +There are two main techniques for the volume data visualization. The first approach is +based on volume rendering (ray-tracing-like methods), the second one on surface rendering +(iso-surface-extraction-like methods). The volume rendering methods are complex and work +with the whole volume data. This paper is concentrated on surface rendering methods that +visualizes surfaces stored in the volume data (so called iso-surfaces). The extracted +iso-surface is determined by a threshold value. All the points on the iso-surface have their +value equal to the threshold. +The field of the iso-surface extraction is quite large. There are many approaches used +for the iso-surface extraction such as view-dependent techniques, parallel or distributed +approaches, external memory (or sometimes called I/O) techniques, multiresolution (LOD) +based extractions and others. In general, we can describe the iso-surface generation and +visualization process with the following steps: +1. Search for all active cells (cells that are intersected by the iso-surface) +2. The iso-surface and normal vectors approximation within these cells (e.g. by a triangle +set) +3. Iso-surfaces visualization (visualization of a set of triangles; different iso-surfaces can +be visualized with different colors depending on a selected threshold value, alpha +blending, etc.) +The first phase of the iso-surface extraction can be accelerated using a wide set of speed up +algorithms [7], [9], [10], [11], [17] or [18]. However, we are interested not that much in speed +of the extraction process but in properties of the output set of triangles. +As there are many various methods for the iso-surface generation and each such a +method generates generally different approximation of a searched iso-surface for a given +threshold, there is no way how to compare the resulting iso-surfaces to each other unless we +know how the iso-surface should look like. We try to compare generated iso-surfaces +produced by different methods. + +Such a comparison can be made with respect to the volume data. When we generate +the volume data using some mathematical or physical model, we are able to gain some +additional information concerning the object that is utilized to make a comparison more +informative and objective. As additional information, we assume e.g. possibility to compute +area or volume of such an object. For real data sets, when we do not have any additional +information concerning the scanned object, we can just use general approaches for +comparison, such as Hausdorff distance or root mean square (RMS) distance. +Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004 + + +This paper is organized in the following way. At first, compared methods are +described. Afterwards, we will explain used approaches for the comparison and how the data +are generated. The last two sections are devoted to the error analysis, methods comparisons +and conclusion. + +2 Method Description + +2.1 Marching Cubes + +There are many kinds of volume data. From simulations, we often get unstructured volume +data. In the other hand from medical imaging the output data is structured one. We aimed at +comparison of iso-surface generation methods that are used for structured data, especially for +regular grids. Compared methods do not differ in the kind of used interpolation but only in the +way they divide a cell into tetrahedra. The well-known method is Marching Cubes (MC) +method that was firstly published by Lorensen and Cline [12]. + +The input volume data consist of samples organized into a regular 3D Cartesian grid. +From such a grid, we can easily obtain a set of cells. The cell has in this case a cube shape and +consists of eight corresponding samples from two adjacent sample planes. Four samples are +from the first plane and four samples are from the second plane. MC method processes +sequentially all the cells that can be found in volume data. The iso-surface, which we are +looking for, is specified by a threshold value. + +Each cell is processed separately. Firstly, the cell index is computed. The cell has eight +vertices, let us name them from A to H, and each vertex has its data value. Depending on a +selected threshold the vertex is assigned a binary value index = ABCDEFGHB. Each bit of the +index is 0 when the data value in the corresponding vertex is less than the threshold and 1 +otherwise. + +Based on the index, we are able to distinguish 256 cases how the iso-surface can +intersect the cell, because each vertex can be inside or outside of the iso-surface. When the +index is 0 or 255 the cell is not intersected by the iso-surface, otherwise such a cell is called +an active cell. The purpose of the index will be described later. For an active cell, normal +vectors are computed in all its vertices using symmetric or asymmetric difference of data +samples. +Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004 + + +Each index represents a different case how the iso-surface can intersect the cell. All +these cases can be tabularized and easily triangulated using linear interpolation. The triangles +vertices lay on the cell edges. Note, that triangles vertices are interpolated only on the cell +edges, this will not be true for other methods. Maximum of four triangles per the cell is +needed to approximate the iso-surface. For each triangle vertex a normal vector is computed +from normal vectors in the cell vertices, using linear interpolation as well. + +Each cell face is shared by another cell. Due to such sharing, the iso-surface is +continuous among adjacent cells. Note that there can be ambiguous faces at which the +triangulation proposed by [12] will produce holes. There are few approaches how to avoid the +holes. Ambiguous cases can be detected and a special triangulation can be applied [16]. The +cells can be divided into tetrahedra and resulting simplices triangulated in a little bit different +way as we will describe in the next section. Other approaches are out of the scope of this +paper, see [2], [3], [6], [13], [14], [15]. + +The algorithm complexity of MC method is O(N), where N is the number of all cells. + +2.1 Marching Tetrahedra + +Marching Tetrahedra (MT) method is based on the same principle as MC method. The +significant difference is that the cube cell is furthermore split into tetrahedra. There are two +main splitting schemes. The cell is divided into five tetrahedra (MT5) [8], [15] or the cell is +divided into six tetrahedra (MT6) [15]. There are several ways how the cube cell can be +divided into five (e.g. Fig. 1) or six tetrahedra (e.g. Fig. 2). + +For the five tetrahedra scheme, it is necessary to alternate two different splitting +schemes. Otherwise, the continuity of the extracted iso-surface will not be maintained +properly. + +Fig. 1 - MT5 tetrahedra division of the cell + +Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004 + + +Fig. 2 - Three tetrahedra from a half of the cube, the second half is divided in similar way + +After the cell is split into tetrahedra (four vertices), the index=ABCDB for each tetrahedron is +computed separately and tetrahedron is processed separately in the similar way as the cube +cell in the MC method. There are only 16 possibilities how the tetrahedron can be intersected +with iso-surface. These methods generate at most two triangles per tetrahedron. + +Five or six tetrahedra decomposition introduces new edges at which the triangles +vertices are to be interpolated. For five tetrahedra the interpolation will be held on face +diagonals of the cube cell, for six tetrahedra both face and internal diagonals are used. + +If we look at five tetrahedra division, there is one tetrahedron with different shape and +size. For six tetrahedra splitting, all the tetrahedra are the same. + +2.3 Centered Cubic Lattice + +The last method that will be compared is Centered Cubic Lattice (CCL) method, see [5]. This +method is little bit different, because it splits the cube cell into 24 tetrahedra. + +The difference is that the resulting tetrahedra are partially shared between adjacent +cells and a new data value is introduced to the center of gravity of the processed cell, Fig. 3. +There are several ways how to compute the value of the central sample, e.g. the arithmetic +mean or weighted mean. + +Each tetrahedron is then processed separately in the same way as in MT5 or MT6 +methods. + +As well as in previous methods this kind of splitting introduces new edges at which +the interpolation will be made. These are edges among adjacent central points. + +In this division scheme, all the 24 tetrahedra are the same as to the dimensions +(similarly to MT6 method). +Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004 + + +There are also other possible decompositions of the cube cell, e.g. [19] that +decomposes parallelepiped cell into two tetrahedra and one octahedron. These techniques +were not included into our study. + + +Fig. 3 - Centered Cubic Lattice division for one cell face + +3 Comparison Approaches + +3.1 Hausdorff Distance + +As mentioned before, we use Hausdorff distance [20] for comparisons mainly for iso-surfaces +that are extracted from real data sets. At first, we define a distance between a point p (from +surface S) and a surface S’ (with points p’) as +d(p, S’)=min||p-p’||, +for all p’ from S’. Now we can define Hausdorff distance between two surfaces S and S’ as +dH(S,S’)=max d(p,S’), +for all p from S. Note important thing that Hausdorff distance is not symmetrical +d(S,S’)≠d(S’,S). When we call d(S,S’) a forward and d(S’,S) a backward distance, we can +define a symmetrical Hausdorff distance [1] as +dSH(S,S’)=max(d(S,S’), d(S’,S)). +The symmetrical difference provides better error measurement for two surfaces. We utilized a +METRO software tool (described in [4]) for accurate computation of Hausdorff distance of +two discrete surfaces (triangle meshes). The METRO tool was mainly used to compare +original mesh with its simplified (e.g. decimated) version. We use it for comparison of two +iso-surfaces, each generated with different method. + +Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004 + +3.2 Root Mean Square Distance + +We use also the Root Mean Square (RMS) of computed distances. RMS distance in discrete +case is defined as [20] +n +x +x +S +S +RMS +n +2 +2 +1 +... +)' +, +( ++ ++ += +, +where n is a number of points of a mesh S’, xi (where i=1.. n) represents the distance of +corresponding point pi’ from S xi=d(pi’, S). We compare S’ to S. + +Note that RMS is not symmetrical as well as Hausdorff distance. We do not use +symmetrical RMS distance in our tests, thus it is not defined here. This measurement is +computed with METRO tool as well. + +Both the Hausdorff distance and the RMS distance are calculated according to some +source mesh using METRO tool. As such a mesh, we use a mesh generated with MC method. + +3.3 Mathematical Data + +At first, we should mention how the testing data are generated from basic mathematical +objects. For such objects we need to know an equation. Let us consider for example a sphere. +Each vertex of a regular grid has its coordinates and we have to assign it a value. The vertex +value is computed as a distance of the grid vertex (known coordinates) from the object surface +(known equation). The zero threshold then represents the object surface in volume data. + +As we know the object equation and its dimensions, we are able to compute some +additional information concerning the object, such as surface area, object volume, triangles +position difference from the object surface, etc. We believe that these properties are worth to +compute, because they can help us to differentiate among the quality of methods. + +Surface area – the iso-surface is generated by an extraction method in a form of a set +of triangles. We compute the total area as a sum of all triangles area. Than we can compute +the area of mathematical object and compare it with iso-surface area obtained. For special +objects such as sphere, we are able to track the error behaviour dependency on the sphere +radius. + +Volume enclosed with the iso-surface – for basic objects the volume is computed +using appropriate formula. The volume enclosed with the iso-surface is computed in the +following manner (for tetrahedra only). There are three cases for a tetrahedron: +Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004 + +1. The whole tetrahedron is outside of the iso-surface – does not affect the total volume +computation. +2. The whole tetrahedron is inside – the whole tetrahedron contributes to the total +volume. The tetrahedron volume is computed easily. +3. The tetrahedron is intersected with the iso-surface – we have to compute the part of the +tetrahedron which is inside of the iso-surface. As there are at most two triangles +generated per tetrahedron, these triangles form two small tetrahedra with appropriate +tetrahedron vertex and we are able to compute the volume of the tetrahedron part +which contributes to the total volume. +Triangles position difference – we measure the difference between triangle center of gravity +and object surface. This gives us information about triangles position difference compared to +the object surface. + +The three mentioned geometric properties are the main aspects that we used for +extraction methods output comparison. The obtained results are showed in the next section. + +4 Results + +At first, we should describe the data sets used for our comparisons and give the reasons why +we chose them. The main part of the used data set is a set of mathematically generated +objects, Fig. 4. A real data set was used to show how the Hausdorff distance is dependent on +applied iso-surface extraction method. The brief description of used data sets follows in +upcoming paragraphs. + + +Fig. 4 - Objects (csph, torus, sombrero, cube, sphere and noisedsph) + +Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004 + +4.1 Used Objects + +Sphere – sphere is an example of an object that we use to follow the error behaviour +dependency on sphere radius. The sphere equation used for data generation is a modified +implicit equation +r +s +z +s +y +s +x +z +y +x +F +Z +Y +X +− +− ++ +− ++ +− += +2 +2 +2 +) +( +) +( +) +( +) +, +, +( + +where x, y and z are samples coordinates, sx, sy and sz are the sphere centre coordinates, r is +sphere radius and F(x,y,z) is a corresponding sample value. This equation assigns data value +to all the volume data samples. The sphere is then represented with a zero threshold +iso-surface. The samples that are inside of the sphere have negative value, on the sphere zero +value and samples placed out of the sphere have positive value. The sample value represents +the distance of the sample from the sphere surface. The radius was 25 in our experiments. + +Cell edge has a length 1 for our purposes. The object dimensions (e.g. radius, edge +length) are then related to a cell edge length. + +Noised sphere – (noisedsph) to study the influence of the noise to the shape of the +output set of triangles we generate a noised sphere. The random noise is introduced (added) to +all samples of the volume data. The size of the noise is given in percentage from the sphere +radius size. We used radius 25 and 10% noise. + +Cube – this kind of an object we use to follow the behaviour and properties of the +iso-surface on edges. We will show the iso-surface difference mainly visually. Data are +generated similarly as in the previous case using the distance of sample from the closest face, +edge or vertex. The inner, on surface and outer samples have the negative, zero and positive +value respectively. Cube was generated using a=b=c=42. + +Cube minus sphere – (csph) such an object was constructed to combine both +properties of the sphere (r=25) and cube (a=b=c=42). The generation of it is a little bit +complicated. At first, the cube is generated in the volume data. Afterwards, the values of all +samples that are closer to the sphere than to the cube are modified to the new distance. + +Torus – is the typical mathematically generated object. Torus is defined with the +following equation [20] +a +z +y +x +c +z +y +x +F +− ++ ++ +− += +2 +2 +2 +2 +) +( +) +, +, +( + +where x, y and z are samples coordinates, c is a torus main radius, a is a torus secondary +radius and F(x,y,z) is a corresponding sample value. The samples value are negative, zero or +positive as well. Torus dimensions are c=20 and a=42 in our case. +Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004 + + +Sombrero – is the last mathematically generated object we use. It is a surface defined +with the mathematical equation (taken from Derive mathematical program) + +2 +2 +2 +2 +)) +( +cos( +) +, +, +( +z +x +c +z +x +b +a +y +z +y +x +F ++ ++ ++ +⋅ +⋅ +− += + +where x, y and z are sample coordinates and F(x,y,z) is a corresponding sample value and a, b +and c are constants modifying the shape of the function. Sombrero parameters we used are +a=12, b=0.25 and c=3. + +Real data sets – Samples of real data set have only positive values that represent a +density of the space in the sample position (we used engine.vol, ctmayo.vol and hplogo.vol +sets). + +4.2 Tests and Results + +For all our mathematically generated objects, we are able to compute the triangles position +difference compared to the mathematical object. Firstly, a triangle center of gravity is +computed. As we have the routines for point to object distance computation, we can compute +the distance of the center of gravity of the triangle from the appropriate object. The overall +position difference PERR is computed as + +n +objDist +P +n +i +ERR +∑ += += +1 +|) +, +( +| +iT +O + +where Ti (i goes from 1 to n) is the center of gravity of the i-th triangle, n is the number of +triangles and objDist(O, X) is the distance of point X from an object O surface. + +Triangles Position Difference +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +cube +csph +sphere +torus +TPD +MC +MT5 +MT6 +CCL + +Fig. 5 - Triangles position difference comparison (edge vs. smooth object) + +Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004 + +The position difference for a sombrero object was slightly smaller and similar to the results +obtained for a sphere. For a cube the CCL method gives the worst results, see Fig. 5. This is +probably due to different interpolation of the cube edges (Fig. 6). A csph object has more +edges than a cube itself. The more tetrahedra we create the worse results we get. Surprisingly +for a torus the MT6 method gives the greatest position difference. We think this is because of +the interpolation at a cell interior edge (the longest one). + + +Fig. 6 - Iso-surface on edges (MT5, MC, MT6, CCL) + +Note that RMS distance is related to the MC method. For a sphere and a torus the obtained +results were slightly less than results for a sombrero. Again, when the object has edges the +CCL method is the worst from the view of RMS distance, see Fig. 7. For noisedsph object the +CCL method gives the best results. We suppose that the central cell sample value computation +(using arithmetic mean) filters data a little bit as well. + +RMS Comparison +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +cube +csph +noisedsph +sombrero +RMS +MT5 +MT6 +CCL + +Fig. 7 - RMS distance histogram + +Again, a sphere and sombrero give approximately similar results compared to torus. From the +view of Hausdorff distance the MT6 method gives the worst results for all tested objects, see +Fig. 8. As you can see for noisedsph the CCL method is the best choice. The best choice in +Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004 + +this case is probably MT5 method because it does not generate as much triangles as CCL +method. + +Hausdorff Distance Comparison +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +cube +csph +noisedsph +torus +Hausdorff dist. +MT5 +MT6 +CCL + +Fig. 8 - Hausdorff distance histogram + +The more tetrahedra is used the larger area is extracted for all tested objects that have edges, +see Fig. 9. The results in Fig. 9 and Fig. 10 are relative due to mathematical results. For +objects like torus (does not have edges) the results were approximately the same as for a +sphere. We think that for the area approximation purposes the best choice is MC method. + +Area Comparison +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +cube +csph +sphere +Area +Math +MC +MT5 +MT6 +CCL + +Fig. 9 - Area comparison (relative to mathematical volume) + +The MT5 method is in most cases slightly better than MT6 method and both methods are +approaching to the original volume from below, see Fig. 10. The CCL method in the other +hand is in most cases approaching mathematically computed volume from above. MC method +is not included because it is hard to compute the volume enclosed with the iso-surface (due to +256 cases). + +Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004 + +Volume Comparison +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +cube +csph +sphere +torus +Volume +Math +MT5 +MT6 +CCL + +Fig. 10 - Volume comparison (relative to mathematical volume) + +4.3 Sphere Additional Test + +A relative volume error is defined in a following way +V +V +V +Error +TR − += + +where VTR is a volume enclosed with iso-surface triangles, V is mathematically computed +volume of the sphere. + +The CCL method is the best choice for the volume approximation, see Fig. 11. We +assume that it is due to high number of tetrahedra. The CCL method error oscillates about +zero value. MT5 gives slightly better results than MT6 method. The progress of error is +similar. Both methods are approaching the zero error from below. Another thing we compare +is a number of extracted triangles. + +Error of Volume Approximation (Sphere, r=10 to 100) +-0.06 +-0.05 +-0.04 +-0.03 +-0.02 +-0.01 +0.00 +0.01 +0.02 +0.03 +0 +20 +40 +60 +80 +100 +120 +Radius +Error[%] +MT5 +MT6 +CCL +Fig. 11 - Sphere volume error graph + +Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004 + +It is a known fact that a number of generated triangles is mainly dependent on the type of the +cell division, see Fig. 12. MC works with a cube cell (at most four triangles per cell) and it +does not divide it into tetrahedral (at most two triangles per tetrahedron). MT5 divides the +cube cell into 5 tetrahedra, MT6 into 6 tetrahedra. In fact, CCL divides the cube cell into 24 +tetrahedra, but these tetrahedra also contain parts of adjacent cube cells. When we sum the +volume of all 24 tetrahedra, we obtain two cube cells volume, so on average 12 tetrahedra per +cube cell. + +Number of Extracted Triangles +0 +200000 +400000 +600000 +800000 +1000000 +1200000 +1400000 +1600000 +1800000 +0 +20 +40 +60 +80 +100 +Radius +Triangles +MC +MT5 +MT6 +CCL + +Fig. 12 - Number of extracted triangles + +4 Conclusions + +We compared fundamental methods for the iso-surface extraction evaluating Hausdorff +distance, RMS distance, triangles position difference and iso-surface area and volume. + +Hausdorff distance is in fact the biggest distance between two compared surfaces +(extreme distance). In general, we are more interested in average distance between two +surfaces (the RMS distance). In this case, the CCL method generally gives worse results +compared to other methods. If we look at a position difference, the MC method seems to be +generally the best one. The quality of the extracted set of triangles for noised sphere was in +general bad. Interesting is that a volume of objects is approximated with the similar difference +no matter of method used except for csph object. + +It is important to realize that for real data we do not know the exact area or volume of +the object. Hence, the speculations such that the Hausdorff distance is bigger or lower are not +completely correct. + +Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004 + +Acknowledgements + +We want to thank to Dr. Ivana Kolingerová for her help and support during preparation of this +paper. + +References + +[1] +Aspert,N., Santa-Cruz,D., Ebrahimi,T.: Mesh Measuring Errors Between Surfaces +Using The Hausdorff Distance, In Proceedings of the IEEE International Conference in +Multimedia and Expo (ICME) 2002, Vol. 1, pages 705-708, Lausanne, Switzerland, +August 26-29, 2002 +[2] +Bonnel,K.S., Duchaineau,M.A., Schikore,D.R., Hamann,B., Joy,K.I.: Material Interface +Reconstruction, IEEE Transactions on Visualization and Computer Graphics, Vol. 9, +No. 4, pages 500-511, 2003 +[3] +Cignoni,P., Ganovelli,F., Montani,C., Scopigno,R.: Reconstruction of Topologically +Correct and Adaptive Trilinear Isosurfaces, Computers & Graphics, Vol. 24, No. 3, +pages 399-418, 2000 +[4] +Cignoni,P., Rocchini,C., Scopigno,R.: Metro: Measuring Error on Simplified Surfaces, +Computer Graphics Forum, Blackwell Publishers, Vol. 17, No. 2, pages 167-174, June +1998 +[5] +Chan,S.L., Purisima,E.O.: A New Tetrahedral Scheme for Iso-surface Generation, +Computers & Graphics, Vol. 22, No. 1, pages 82-90, Elsevier Science Limited, 1998 +[6] +Chernyaev,E.V.: Marching Cubes 33: Construction of Topologically Correct +Isosurfaces, Institute for High Energy Physics, Moscow, Russia, Report CN/95-17, +1995 +[7] +Giles,M., Haimes,R.: Advanced Interactive Visualization for CFD, Computing Systems +in Engineering, Vol. 1, No.1, pages 51-62, 1990 +[8] +Hall,M. 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Jan Patera (http://zcu.cz/~hopatera) is a PhD student and a part-time +tutor at the Department of Computer Sciences at the University of West +Bohemia in Plzeň in Czech Republic. He graduated in the field of computer +graphics at the University of West Bohemia in 2002. He is a member of the +Center of Computer Graphics and Data Visualization (CGDV). His research +activities concern volume data, iso-surface extraction, algorithms and data +visualization. + + +Vaclav Skala is a full professor of Computer Science at the Faculty of +Applied Sciences at the University of West Bohemia in Plzen, Czech +Republic. He is responsible for courses on Computer Graphics, Algorithms +for Computer Graphics, Visualization, Multimedia Systems, Programming +in Windows, .NET Technologies at the Department of Computer Science. +He is a member of The Visual Computer and Computers&Graphics +editorial boards, Eurographics Executive Committee and member of +program committees of established international conferences. He has been a +research fellow or lecturing at the Brunel University (London, U.K.), +Moscow Technical University (Russia), Gavle University (Sweden) and +others institutions in Europe. He organizes the WSCG International +Conferences in Central Europe on Computer Graphics, Visualization and +Computer Vision (http://wscg.zcu.cz) held annually since 1992 and .NET +Technologies conferences (http://dotnet.zcu.cz). He is interested in +algorithms, data structures, mathematics, computer graphics, computer +vision and visualization. He has been responsible for several research +projects as well. Currently he is a director of the Center of Computer +Graphics and Visualization (http://herakles.zcu.cz). + + +Machine Graphics and Vision, Polish Academy of Sciences, Vol.13, No.4., pp.329-344, ISSN 1230-0535, 2004 + diff --git a/A9AzT4oBgHgl3EQfv_5R/content/tmp_files/load_file.txt b/A9AzT4oBgHgl3EQfv_5R/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a0f2651b4d1447d72d4748507bcc97d7f75a0503 --- /dev/null +++ b/A9AzT4oBgHgl3EQfv_5R/content/tmp_files/load_file.txt @@ -0,0 +1,520 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf,len=519 +page_content='A Comparison of Fundamental Methods for Iso-surface Extraction JAN PATERA1, VÁCLAV SKALA2 Department of Computer Science and Engineering Faculty of Applied Sciences, University of West Bohemia Univerzitní 22, Plzeň CZECH REPUBLIC hopatera@kiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='zcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='cz, skala@kiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='zcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='cz http://herakles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='zcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='cz Abstract: In this paper four fundamental methods for an iso-surface extraction are compared, based on cell decomposition to tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The methods are compared both on mathematically generated data sets as well as on real data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The comparison using mathematical data is made from different points of view such as area approximation, volume approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' On the other hand, the Hausdorff distance and root mean square are used to compare methods on real data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The presented comparison can be helpful when deciding among tested methods which one to choose, as well as when we need to compare a newly developed method with other existing approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Key-Words: Comparison, Iso-surface extraction, Error, Hausdorff distance, Volume data, Computer graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 1 Introduction In the recent period of time volume data have started to play a significant role in many scientific areas and are spread across many professions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' In medical field, various devices, such as Computed Tomography (CT) scanners, Magnetic Resonance Imaging (MRI) scanners produce volume data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The volume data are also produced as a result of mathematical or physical simulations and experiments and researchers need to visualize such data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 1 Supported by the Ministry of Education of Czech Republic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' project number MSM 235200005 (Information Systems and Technologies) 2 Supported by project NoE – 3DTV PLT 511568 Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004 There are two main techniques for the volume data visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The first approach is based on volume rendering (ray-tracing-like methods), the second one on surface rendering (iso-surface-extraction-like methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The volume rendering methods are complex and work with the whole volume data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' This paper is concentrated on surface rendering methods that visualizes surfaces stored in the volume data (so called iso-surfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The extracted iso-surface is determined by a threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' All the points on the iso-surface have their value equal to the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The field of the iso-surface extraction is quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' There are many approaches used for the iso-surface extraction such as view-dependent techniques, parallel or distributed approaches, external memory (or sometimes called I/O) techniques, multiresolution (LOD) based extractions and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' In general, we can describe the iso-surface generation and visualization process with the following steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Search for all active cells (cells that are intersected by the iso-surface) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The iso-surface and normal vectors approximation within these cells (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' by a triangle set) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Iso-surfaces visualization (visualization of a set of triangles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' different iso-surfaces can be visualized with different colors depending on a selected threshold value, alpha blending, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=') The first phase of the iso-surface extraction can be accelerated using a wide set of speed up algorithms [7], [9], [10], [11], [17] or [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' However, we are interested not that much in speed of the extraction process but in properties of the output set of triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' As there are many various methods for the iso-surface generation and each such a method generates generally different approximation of a searched iso-surface for a given threshold, there is no way how to compare the resulting iso-surfaces to each other unless we know how the iso-surface should look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' We try to compare generated iso-surfaces produced by different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Such a comparison can be made with respect to the volume data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' When we generate the volume data using some mathematical or physical model, we are able to gain some additional information concerning the object that is utilized to make a comparison more informative and objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' As additional information, we assume e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' possibility to compute area or volume of such an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' For real data sets, when we do not have any additional information concerning the scanned object, we can just use general approaches for comparison, such as Hausdorff distance or root mean square (RMS) distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004 This paper is organized in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' At first, compared methods are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Afterwards, we will explain used approaches for the comparison and how the data are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The last two sections are devoted to the error analysis, methods comparisons and conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 2 Method Description 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='1 Marching Cubes There are many kinds of volume data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' From simulations, we often get unstructured volume data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' In the other hand from medical imaging the output data is structured one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' We aimed at comparison of iso-surface generation methods that are used for structured data, especially for regular grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Compared methods do not differ in the kind of used interpolation but only in the way they divide a cell into tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The well-known method is Marching Cubes (MC) method that was firstly published by Lorensen and Cline [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The input volume data consist of samples organized into a regular 3D Cartesian grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' From such a grid, we can easily obtain a set of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The cell has in this case a cube shape and consists of eight corresponding samples from two adjacent sample planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Four samples are from the first plane and four samples are from the second plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' MC method processes sequentially all the cells that can be found in volume data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The iso-surface, which we are looking for, is specified by a threshold value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Each cell is processed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Firstly, the cell index is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The cell has eight vertices, let us name them from A to H, and each vertex has its data value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Depending on a selected threshold the vertex is assigned a binary value index = ABCDEFGHB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Each bit of the index is 0 when the data value in the corresponding vertex is less than the threshold and 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Based on the index, we are able to distinguish 256 cases how the iso-surface can intersect the cell, because each vertex can be inside or outside of the iso-surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' When the index is 0 or 255 the cell is not intersected by the iso-surface, otherwise such a cell is called an active cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The purpose of the index will be described later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' For an active cell, normal vectors are computed in all its vertices using symmetric or asymmetric difference of data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004 Each index represents a different case how the iso-surface can intersect the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' All these cases can be tabularized and easily triangulated using linear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The triangles vertices lay on the cell edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Note, that triangles vertices are interpolated only on the cell edges, this will not be true for other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Maximum of four triangles per the cell is needed to approximate the iso-surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' For each triangle vertex a normal vector is computed from normal vectors in the cell vertices, using linear interpolation as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Each cell face is shared by another cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Due to such sharing, the iso-surface is continuous among adjacent cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Note that there can be ambiguous faces at which the triangulation proposed by [12] will produce holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' There are few approaches how to avoid the holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Ambiguous cases can be detected and a special triangulation can be applied [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The cells can be divided into tetrahedra and resulting simplices triangulated in a little bit different way as we will describe in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Other approaches are out of the scope of this paper, see [2], [3], [6], [13], [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The algorithm complexity of MC method is O(N), where N is the number of all cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='1 Marching Tetrahedra Marching Tetrahedra (MT) method is based on the same principle as MC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The significant difference is that the cube cell is furthermore split into tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' There are two main splitting schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The cell is divided into five tetrahedra (MT5) [8], [15] or the cell is divided into six tetrahedra (MT6) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' There are several ways how the cube cell can be divided into five (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 1) or six tetrahedra (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' For the five tetrahedra scheme, it is necessary to alternate two different splitting schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Otherwise, the continuity of the extracted iso-surface will not be maintained properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 1 - MT5 tetrahedra division of the cell Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 2 - Three tetrahedra from a half of the cube, the second half is divided in similar way After the cell is split into tetrahedra (four vertices), the index=ABCDB for each tetrahedron is computed separately and tetrahedron is processed separately in the similar way as the cube cell in the MC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' There are only 16 possibilities how the tetrahedron can be intersected with iso-surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' These methods generate at most two triangles per tetrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Five or six tetrahedra decomposition introduces new edges at which the triangles vertices are to be interpolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' For five tetrahedra the interpolation will be held on face diagonals of the cube cell, for six tetrahedra both face and internal diagonals are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' If we look at five tetrahedra division, there is one tetrahedron with different shape and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' For six tetrahedra splitting, all the tetrahedra are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='3 Centered Cubic Lattice The last method that will be compared is Centered Cubic Lattice (CCL) method, see [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' This method is little bit different, because it splits the cube cell into 24 tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The difference is that the resulting tetrahedra are partially shared between adjacent cells and a new data value is introduced to the center of gravity of the processed cell, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' There are several ways how to compute the value of the central sample, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' the arithmetic mean or weighted mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Each tetrahedron is then processed separately in the same way as in MT5 or MT6 methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' As well as in previous methods this kind of splitting introduces new edges at which the interpolation will be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' These are edges among adjacent central points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' In this division scheme, all the 24 tetrahedra are the same as to the dimensions (similarly to MT6 method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004 There are also other possible decompositions of the cube cell, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' [19] that decomposes parallelepiped cell into two tetrahedra and one octahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' These techniques were not included into our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 3 - Centered Cubic Lattice division for one cell face 3 Comparison Approaches 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='1 Hausdorff Distance As mentioned before, we use Hausdorff distance [20] for comparisons mainly for iso-surfaces that are extracted from real data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' At first, we define a distance between a point p (from surface S) and a surface S’ (with points p’) as d(p, S’)=min||p-p’||, for all p’ from S’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Now we can define Hausdorff distance between two surfaces S and S’ as dH(S,S’)=max d(p,S’), for all p from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Note important thing that Hausdorff distance is not symmetrical d(S,S’)≠d(S’,S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' When we call d(S,S’) a forward and d(S’,S) a backward distance, we can define a symmetrical Hausdorff distance [1] as dSH(S,S’)=max(d(S,S’), d(S’,S)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The symmetrical difference provides better error measurement for two surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' We utilized a METRO software tool (described in [4]) for accurate computation of Hausdorff distance of two discrete surfaces (triangle meshes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The METRO tool was mainly used to compare original mesh with its simplified (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' decimated) version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' We use it for comparison of two iso-surfaces, each generated with different method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='2 Root Mean Square Distance We use also the Root Mean Square (RMS) of computed distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' RMS distance in discrete case is defined as [20] n x x S S RMS n 2 2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=" )' , ( + + = , where n is a number of points of a mesh S’, xi (where i=1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='. n) represents the distance of corresponding point pi’ from S xi=d(pi’, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' We compare S’ to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Note that RMS is not symmetrical as well as Hausdorff distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' We do not use symmetrical RMS distance in our tests, thus it is not defined here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' This measurement is computed with METRO tool as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Both the Hausdorff distance and the RMS distance are calculated according to some source mesh using METRO tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' As such a mesh, we use a mesh generated with MC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='3 Mathematical Data At first, we should mention how the testing data are generated from basic mathematical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' For such objects we need to know an equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Let us consider for example a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Each vertex of a regular grid has its coordinates and we have to assign it a value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The vertex value is computed as a distance of the grid vertex (known coordinates) from the object surface (known equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The zero threshold then represents the object surface in volume data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' As we know the object equation and its dimensions, we are able to compute some additional information concerning the object, such as surface area, object volume, triangles position difference from the object surface, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' We believe that these properties are worth to compute, because they can help us to differentiate among the quality of methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Surface area – the iso-surface is generated by an extraction method in a form of a set of triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' We compute the total area as a sum of all triangles area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Than we can compute the area of mathematical object and compare it with iso-surface area obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' For special objects such as sphere, we are able to track the error behaviour dependency on the sphere radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Volume enclosed with the iso-surface – for basic objects the volume is computed using appropriate formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The volume enclosed with the iso-surface is computed in the following manner (for tetrahedra only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' There are three cases for a tetrahedron: Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The whole tetrahedron is outside of the iso-surface – does not affect the total volume computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The whole tetrahedron is inside – the whole tetrahedron contributes to the total volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The tetrahedron volume is computed easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The tetrahedron is intersected with the iso-surface – we have to compute the part of the tetrahedron which is inside of the iso-surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' As there are at most two triangles generated per tetrahedron, these triangles form two small tetrahedra with appropriate tetrahedron vertex and we are able to compute the volume of the tetrahedron part which contributes to the total volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Triangles position difference – we measure the difference between triangle center of gravity and object surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' This gives us information about triangles position difference compared to the object surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The three mentioned geometric properties are the main aspects that we used for extraction methods output comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The obtained results are showed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 4 Results At first, we should describe the data sets used for our comparisons and give the reasons why we chose them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The main part of the used data set is a set of mathematically generated objects, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' A real data set was used to show how the Hausdorff distance is dependent on applied iso-surface extraction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The brief description of used data sets follows in upcoming paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 4 - Objects (csph, torus, sombrero, cube, sphere and noisedsph) Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='1 Used Objects Sphere – sphere is an example of an object that we use to follow the error behaviour dependency on sphere radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The sphere equation used for data generation is a modified implicit equation r s z s y s x z y x F Z Y X − − + − + − = 2 2 2 ) ( ) ( ) ( ) , , ( where x, y and z are samples coordinates, sx, sy and sz are the sphere centre coordinates, r is sphere radius and F(x,y,z) is a corresponding sample value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' This equation assigns data value to all the volume data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The sphere is then represented with a zero threshold iso-surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The samples that are inside of the sphere have negative value, on the sphere zero value and samples placed out of the sphere have positive value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The sample value represents the distance of the sample from the sphere surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The radius was 25 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Cell edge has a length 1 for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The object dimensions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' radius, edge length) are then related to a cell edge length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Noised sphere – (noisedsph) to study the influence of the noise to the shape of the output set of triangles we generate a noised sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The random noise is introduced (added) to all samples of the volume data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The size of the noise is given in percentage from the sphere radius size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' We used radius 25 and 10% noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Cube – this kind of an object we use to follow the behaviour and properties of the iso-surface on edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' We will show the iso-surface difference mainly visually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Data are generated similarly as in the previous case using the distance of sample from the closest face, edge or vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The inner, on surface and outer samples have the negative, zero and positive value respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Cube was generated using a=b=c=42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Cube minus sphere – (csph) such an object was constructed to combine both properties of the sphere (r=25) and cube (a=b=c=42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The generation of it is a little bit complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' At first, the cube is generated in the volume data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Afterwards, the values of all samples that are closer to the sphere than to the cube are modified to the new distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Torus – is the typical mathematically generated object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Torus is defined with the following equation [20] a z y x c z y x F − + + − = 2 2 2 2 ) ( ) , , ( where x, y and z are samples coordinates, c is a torus main radius, a is a torus secondary radius and F(x,y,z) is a corresponding sample value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The samples value are negative, zero or positive as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Torus dimensions are c=20 and a=42 in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004 Sombrero – is the last mathematically generated object we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' It is a surface defined with the mathematical equation (taken from Derive mathematical program) 2 2 2 2 )) ( cos( ) , , ( z x c z x b a y z y x F + + + ⋅ ⋅ − = where x, y and z are sample coordinates and F(x,y,z) is a corresponding sample value and a, b and c are constants modifying the shape of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Sombrero parameters we used are a=12, b=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='25 and c=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Real data sets – Samples of real data set have only positive values that represent a density of the space in the sample position (we used engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='vol, ctmayo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='vol and hplogo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='vol sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='2 Tests and Results For all our mathematically generated objects, we are able to compute the triangles position difference compared to the mathematical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Firstly, a triangle center of gravity is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' As we have the routines for point to object distance computation, we can compute the distance of the center of gravity of the triangle from the appropriate object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The overall position difference PERR is computed as n objDist P n i ERR ∑ = = 1 |) , ( | iT O where Ti (i goes from 1 to n) is the center of gravity of the i-th triangle, n is the number of triangles and objDist(O, X) is the distance of point X from an object O surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Triangles Position Difference 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='6 cube csph sphere torus TPD MC MT5 MT6 CCL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 5 - Triangles position difference comparison (edge vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' smooth object) Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004 The position difference for a sombrero object was slightly smaller and similar to the results obtained for a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' For a cube the CCL method gives the worst results, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' This is probably due to different interpolation of the cube edges (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' A csph object has more edges than a cube itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The more tetrahedra we create the worse results we get.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Surprisingly for a torus the MT6 method gives the greatest position difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' We think this is because of the interpolation at a cell interior edge (the longest one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 6 - Iso-surface on edges (MT5, MC, MT6, CCL) Note that RMS distance is related to the MC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' For a sphere and a torus the obtained results were slightly less than results for a sombrero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Again, when the object has edges the CCL method is the worst from the view of RMS distance, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' For noisedsph object the CCL method gives the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' We suppose that the central cell sample value computation (using arithmetic mean) filters data a little bit as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' RMS Comparison 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='25 cube csph noisedsph sombrero RMS MT5 MT6 CCL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 7 RMS distance histogram Again, a sphere and sombrero give approximately similar results compared to torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' From the view of Hausdorff distance the MT6 method gives the worst results for all tested objects, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' As you can see for noisedsph the CCL method is the best choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The best choice in Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004 this case is probably MT5 method because it does not generate as much triangles as CCL method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Hausdorff Distance Comparison 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4 cube csph noisedsph torus Hausdorff dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' MT5 MT6 CCL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 8 Hausdorff distance histogram The more tetrahedra is used the larger area is extracted for all tested objects that have edges, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 9 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 10 are relative due to mathematical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' For objects like torus (does not have edges) the results were approximately the same as for a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' We think that for the area approximation purposes the best choice is MC method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Area Comparison 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='8 cube csph sphere Area Math MC MT5 MT6 CCL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 9 - Area comparison (relative to mathematical volume) The MT5 method is in most cases slightly better than MT6 method and both methods are approaching to the original volume from below, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The CCL method in the other hand is in most cases approaching mathematically computed volume from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' MC method is not included because it is hard to compute the volume enclosed with the iso-surface (due to 256 cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004 Volume Comparison 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='2 cube csph sphere torus Volume Math MT5 MT6 CCL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 10 - Volume comparison (relative to mathematical volume) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='3 Sphere Additional Test A relative volume error is defined in a following way V V V Error TR − = where VTR is a volume enclosed with iso-surface triangles, V is mathematically computed volume of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The CCL method is the best choice for the volume approximation, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' We assume that it is due to high number of tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The CCL method error oscillates about zero value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' MT5 gives slightly better results than MT6 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The progress of error is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Both methods are approaching the zero error from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Another thing we compare is a number of extracted triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Error of Volume Approximation (Sphere, r=10 to 100) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='06 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='05 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='04 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='03 0 20 40 60 80 100 120 Radius Error[%] MT5 MT6 CCL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 11 - Sphere volume error graph Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004 It is a known fact that a number of generated triangles is mainly dependent on the type of the cell division, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' MC works with a cube cell (at most four triangles per cell) and it does not divide it into tetrahedral (at most two triangles per tetrahedron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' MT5 divides the cube cell into 5 tetrahedra, MT6 into 6 tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' In fact, CCL divides the cube cell into 24 tetrahedra, but these tetrahedra also contain parts of adjacent cube cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' When we sum the volume of all 24 tetrahedra, we obtain two cube cells volume, so on average 12 tetrahedra per cube cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Number of Extracted Triangles 0 200000 400000 600000 800000 1000000 1200000 1400000 1600000 1800000 0 20 40 60 80 100 Radius Triangles MC MT5 MT6 CCL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 12 Number of extracted triangles 4 Conclusions We compared fundamental methods for the iso-surface extraction evaluating Hausdorff distance, RMS distance, triangles position difference and iso-surface area and volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Hausdorff distance is in fact the biggest distance between two compared surfaces (extreme distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' In general, we are more interested in average distance between two surfaces (the RMS distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' In this case, the CCL method generally gives worse results compared to other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' If we look at a position difference, the MC method seems to be generally the best one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' The quality of the extracted set of triangles for noised sphere was in general bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Interesting is that a volume of objects is approximated with the similar difference no matter of method used except for csph object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' It is important to realize that for real data we do not know the exact area or volume of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Hence, the speculations such that the Hausdorff distance is bigger or lower are not completely correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004 Acknowledgements We want to thank to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Ivana Kolingerová for her help and support during preparation of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' References [1] Aspert,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', Santa-Cruz,D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', Ebrahimi,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' : Mesh Measuring Errors Between Surfaces Using The Hausdorff Distance, In Proceedings of the IEEE International Conference in Multimedia and Expo (ICME) 2002, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' 1, pages 705-708, Lausanne, Switzerland, August 26-29, 2002 [2] Bonnel,K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='S.' metadata={'source': 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[20] Weisstein,E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' : MathWorld, A Wolfram Web Resource, http://mathworld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='wolfram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='com Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004 Ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Jan Patera (http://zcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='cz/~hopatera) is a PhD student and a part-time tutor at the Department of Computer Sciences at the University of West Bohemia in Plzeň in Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' He graduated in the field of computer graphics at the University of West Bohemia in 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' He is a member of the Center of Computer Graphics and Data Visualization (CGDV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' His research activities concern volume data, iso-surface extraction, algorithms and data visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Vaclav Skala is a full professor of Computer Science at the Faculty of Applied Sciences at the University of West Bohemia in Plzen, Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' He is responsible for courses on Computer Graphics, Algorithms for Computer Graphics, Visualization, Multimedia Systems, Programming in Windows, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='NET Technologies at the Department of Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' He is a member of The Visual Computer and Computers&Graphics editorial boards, Eurographics Executive Committee and member of program committees of established international conferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' He has been a research fellow or lecturing at the Brunel University (London, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='), Moscow Technical University (Russia), Gavle University (Sweden) and others institutions in Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' He organizes the WSCG International Conferences in Central Europe on Computer Graphics, Visualization and Computer Vision (http://wscg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='zcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='cz) held annually since 1992 and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='NET Technologies conferences (http://dotnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='zcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='cz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' He is interested in algorithms, data structures, mathematics, computer graphics, computer vision and visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' He has been responsible for several research projects as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Currently he is a director of the Center of Computer Graphics and Visualization (http://herakles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='zcu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='cz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=' Machine Graphics and Vision, Polish Academy of Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='13, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} +page_content='329-344, ISSN 1230-0535, 2004' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AzT4oBgHgl3EQfv_5R/content/2301.01715v1.pdf'} diff --git a/AtE1T4oBgHgl3EQfDQOp/content/tmp_files/2301.02875v1.pdf.txt b/AtE1T4oBgHgl3EQfDQOp/content/tmp_files/2301.02875v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..131348b4aa33b4004211f90e798fcfea0056bde6 --- /dev/null +++ b/AtE1T4oBgHgl3EQfDQOp/content/tmp_files/2301.02875v1.pdf.txt @@ -0,0 +1,1449 @@ +arXiv:2301.02875v1 [math.NA] 7 Jan 2023 +SCIENCE CHINA Mathematics +1 +XXXX Vol. XX No. XX XX–XX +www.SciChina.com +www.springerlink.com +An iterative two-grid method for strongly non- +linear elliptic boundary value problems +Jiajun Zhan1, Lei Yang1, Xiaoqing Xing2,†, Liuqiang Zhong2 +1 School of Computer Science and Engineering, Faculty of Innovation Engineering, Macau University of Science +and Technology, Macao SAR 999078, China; +2 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China +Email: 2109853gii30011@student.must.edu.mo, leiyang@must.edu.mo, xingxq@scnu.edu.cn, zhong@scnu.edu.cn +Abstract +We design and analyze an iterative two-grid algorithm for the finite element discretizations +of strongly nonlinear elliptic boundary value problems in this paper. We propose an iterative two-grid +algorithm, in which a nonlinear problem is first solved on the coarse space, and then a symmetric positive +definite problem is solved on the fine space. The innovation of this paper lies in the establishment +of a first convergence analysis, which requires simultaneous estimation of four interconnected error +estimates. +We also present some numerical experiments to confirm the efficiency of the proposed +algorithm. +Keywords: +iterative two-grid method, convergence, strongly nonlinear elliptic problems. +MSC(2020): +65N30, 65M12, 35J60 +1 +Introduction +The two-grid methods are first proposed for nonselfadjoint problems and indefinite elliptic +problems [6, 10]. Then, the two-grid methods are extended to solve semiliinear elliptic problems +[7], quasi-linear and nonlinear elliptic problems [8, 9], respectively. Especially, for nonlinear +elliptic problems, the basic idea of two-grid methods is to first obtain a rough solution by +solving the original problem in a “coarse mesh” with mesh size H, and then correct the rough +solution by solving a symmetric positive definite (SPD) system in a “fine mesh” with mesh size +h. Noticing the mesh size of “coarse mesh” is much smaller than that of “fine mesh”, it is not +difficult to solve an original problem in “coarse mesh”. Therefore, two-grid methods reduce +the computational complexity of solving the original problem to solving a SPD problem and +dramatically improve the computational speed. Recently, Bi, Wang and Lin [1] presented a +two-grid algorithm to solve the strongly nonlinear elliptic problems and provided a posteriori +error estimator for the two-grid methods. It’s noted that the literature mentioned above is all +about non-iterative two-grid methods. +As is well-known, the mesh size H of “coarse mesh” and h of “fine mesh” should satisfy a +certain relationship for the optimal convergence order in non-iterative two-grid methods. The +iterative two-grid methods have the advantage over the non-iterative two-grid methods in that, +the distance between the mesh sizes H and h can be enlarged by increasing the iteration counts +† Corresponding author + +2 +Jiajun Zhan & et al. +with the same accuracy. However, there is only a small amount of literature on iterative two-grid +methods of conforming finite element discretization for elliptic problems. Xu [9] first proposed +and analyzed an iterative two-grid method for non-symmetric positive definite elliptic problems. +Zhang, Fan and Zhong [11] designed some iterative two-grid algorithms for semilinear elliptic +problems and provided the corresponding convergence analysis. To our knowledge, there is +not any published literature on the iterative two-grid algorithm of conforming finite element +discretization for strongly nonlinear elliptic boundary value problems. +In this paper, an iterative two-grid algorithm for solving strongly nonlinear elliptic problems +is studied. The discrete system of strongly nonlinear elliptic problems is presented at first. And +then, an iterative two-grid algorithm is proposed for the discrete system, which is obtained by +applying a non-iterative two-grid algorithm of [8] in a successive fashion. Finally, a challenging +convergence analysis of the proposed algorithm is provided. Despite the fact that our algorithm +is simply obtained by [8], the convergence analysis of the non-iterative two-grid algorithm could +not be directly applied to the iterative two-grid algorithm. Here we complete this challenging +convergence analysis by mathematical induction which can also be used in solving semilinear +elliptic problems by iterative two-grid algorithms in [11]. However, we must emphasize that the +convergence analysis of our algorithm is significantly different from the one of [11]. Compared +with the current work [11], our convergence analysis is far more difficult and complex, and +specific challenges could be reflected in: (1) the higher order derivative component of our model +problem is still nonlinear; (2) the interconnectedness of the error estimates causes formidable +obstacle for the convergence analysis (See the proof of Lemma 4.7). +To avoid the repeated use of generic but unspecified constants, x ≲ y is used to denote x ⩽ +Cy, where C are some positive constants which do not depend on the mesh size. Furthermore +the constants C may denote different values under different circumstances. For some specific +constants, we use the constant C with some subscript to denote. +2 +Model problems and discrete systems +In this section, we present the continuous and discrete variational problems of strongly nonlinear +elliptic problems, and provide the corresponding well-posedness and priori error estimates. +Given a bounded convex polygonal domain Ω ⊂ R2 with the boundary ∂Ω. We denote +W m,p(Ω) as the standard Sobolev space with norm ∥ · ∥m,p,Ω and seminorm | · |m,p,Ω, where the +integers m ⩾ 0 and p ⩾ 1. For convenience, we also denote Hm(Ω) = W m,2(Ω), ∥·∥m = ∥·∥m,2,Ω +and H1 +0(Ω) := {u ∈ H1(Ω) : u|∂Ω = 0}. +We consider the following strongly nonlinear elliptic problems: +� +−∇ · a(x, u, ∇u) + f(x, u, ∇u) = 0, in Ω, +u = 0, on ∂Ω, +(2.1) +where a(x, y, z) : ¯Ω × R × R2 → R2 and f(x, y, z) : ¯Ω × R × R2 → R. When a(x, u, ∇u) and +f(x, u, ∇u) take different functions, different problems are available, such as mean curvature +flow, Bratu’s problem and so on(See [3]). +We assume that a(x, y, z) and f(x, y, z) are second order continuous differentiable functions. +For simplicity, we denote that ay(w) = Dya(x, w, ∇w), az(w) = Dza(x, w, ∇w), fy(w) = +Dyf(x, w, ∇w) and fz(w) = Dzf(x, w, ∇w), and similar notations are applied to the second +order derivatives of a(x, y, z) and f(x, y, z). + +An iterative two-grid method for strongly nonlinear elliptic boundary value problems +3 +Remark 2.1 +Since a(x, y, z) and f(x, y, z) are second order continuous differentiable func- +tions, there exists a positive constant ˜C as upper bound with respect to all the first and second +order derivatives of a(·, ·, ·) and f(·, ·, ·). +We denote +A(v, ϕ) = (a(x, v, ∇v), ∇ϕ) + (f(x, v, ∇v), ϕ), +∀v, ϕ ∈ H1 +0(Ω). +(2.2) +By Green formula, it’s easy to see that the solution u ∈ H1 +0(Ω) of (2.1) satisfies +A(u, v) = 0, +∀v ∈ H1 +0(Ω). +(2.3) +The Fr´echet derivative L′ of (2.1) at w is given by +L′(w)v = −∇ · (ay(w)v + az(w)∇v) + fy(w)v + fz(w)∇v. +In the following, we gives some of our basic assumptions (Similar assumptions also could be +found in [9] or [3]). Firstly, the problem (2.3) has a solution u ∈ H1 +0(Ω)∩Hr+1(Ω)∩W 2,2+ε(Ω) +(ε > 0 and integer r ⩾ 1). Secondly, for the solution u of (2.3), there exists a positive constant +α0 such that +ξT az(u)ξ ⩾ α0|ξ|2, +∀ξ ∈ R2, x ∈ ¯Ω. +(2.4) +Finally, L′(u) : H1 +0(Ω) → H−1(Ω) is an isomorphism. These assumptions guarantee that u is +an isolated solution of (2.3). +Let Th be a conforming quasi-uniform triangulation on Ω, where the mesh size h denotes +the maximum of the circumscribed circle diameters of element K ∈ Th. By this, any element +K ∈ Th is contained in (contains) a circle of radius ˆC1h (respectively, ˆC2h), where the constant +ˆC1 and ˆC2 do not depend on mesh size h, and there is no hanging node on Th. +The finite element space Vh on Th is defined as +Vh = {vh ∈ H1 +0(Ω) : vh|K ∈ Pr(K), ∀ K ∈ Th}, +where Pr(K) is the set of polynomials of degree at most integer r on K. +Here is the discrete system of (2.3): Find uh ∈ Vh such that +A (uh, vh) = 0, +∀vh ∈ Vh. +(2.5) +The following lemma presents the well-posedness of the variational problem (2.5) and its +priori error estimates, which can be found in Lemma 3.2 and Theorem 3.4 of [9], respectively. +Lemma 2.2 +Assume u is the solution of problem (2.3), then when h is small enough, the +discrete variational problem (2.5) exists a unique solution uh ∈ Vh, and the following priori +error estimate holds +∥u − uh∥1,p ≲ hr, +if u ∈ W r+1,p(Ω), 2 ⩽ p ⩽ ∞. +(2.6) +3 +Iterative two-grid algorithms + +4 +Jiajun Zhan & et al. +In this section, we present an iterative two-grid algorithm for the variational problems (2.3). +Let Th and TH be two quasi-uniform, conforming and nested mesh in Ω. Furthermore the +mesh size h of Th and H of TH satisfy, for some 0 < λ < 1, +H = O(hλ) +and +h < H < 1. +For present the iterative two-grid algorithm, we introduce the form B(w; v, χ) (induced by +L′) by , for a fixed w and any v, χ ∈ H1 +0(Ω), +B(w; v, χ) = (ay(w)v, ∇χ) + (az(w)∇v, ∇χ) + (fy(w)v, χ) + (fz(w)∇v, χ). +(3.1) +Remark 3.1 +The form B(w; ·, ·) is a bilinear form for fixed w. +To our knowledge, the two-grid algorithms of strongly nonlinear problems are firstly pro- +posed in [8]. Here one of two-grid algorithms from Algorithm 3.3 of [8] is given. +Algorithm 3.1 +1. Find uH ∈ VH, such that +A(uH, vH) = 0, +∀vH ∈ VH. +2. Find uh ∈ Vh, such that +B(uH; uh, vh) = B(uH; uH, vh) − A(uH, vh), +∀vh ∈ Vh. +Remark 3.2 +In the Algorithm 3.1, we first solve a nonlinear problem in a coarse space +VH. However, because dim(VH) is relatively small, the calculated amount of solving a nonlinear +problem in VH is not excessive. As for the second step of Algorithm 3.1, noticing that B(uH; ·, ·) +is a bilinear form with given uH, we simply need to solve a linear problem in Vh, for which there +are numerous concerning fast algorithms. +In [8], Xu had showed that the solution uh of Algorithm 3.1 could be a good approximation +with respect to finite element solution uh at a low cost, namely, +∥uh − uh∥1 ≲ H2. +(3.2) +Using triangle inequality, (2.6) with r = 1 and (3.2), we obtain the error estimate of Algorithm +3.1, +∥u − uh∥1 ⩽ ∥u − uh∥1 + ∥uh − uh∥1 ≲ h + H2. +(3.3) +Next, putting the Algorithm 3.1 into a successive fashion, we obtain our iterative two-grid +algorithm. +Algorithm 3.2 +Let u0 +h = uH be the solution of (2.5) in VH. Assume that uk +h ∈ Vh has been +obtained, then uk+1 +h +∈ Vh can be obtained by the following two steps. +Step 1. Find ek +H ∈ VH such that, for any vH ∈ VH, +A(uk +h + ek +H, vH) = 0. +(3.4) +Step 2. Find uk+1 +h +∈ Vh such that, for any vh ∈ Vh, +B(uk +h + ek +H; uk+1 +h +, vh) = B(uk +h + ek +H; uk +h + ek +H, vh) − A(uk +h + ek +H, vh). +(3.5) + +An iterative two-grid method for strongly nonlinear elliptic boundary value problems +5 +Remark 3.3 +Noticing the uniqueness of finite element solution (See Lemma 2.2), (2.5), +u0 +h = uH and (3.4) with k = 0, we can see that e0 +H = 0, which means u0 +h + e0 +H = uH. By +observing the the Step 2 of Algorithm 3.2 and the second step of Algorithm 3.1, the conclusion +is that Algorithm 3.2 is same with Algorithm 3.1 when k = 0. +In comparison to [8], our method is still valid for high order conforming finite elements, +whereas [8] only considered piecewise linear finite element space. Here gives the error estimate +of our algorithm (See Theorem 4.9), +∥u − uk +h∥1 ≲ hr + Hr+k. +(3.6) +Specially, if we choose finite element space Vh as piecewise linear finite element space, i.e. r = 1, +the error estimate (3.6) of Algorithm 3.2 could be written as +∥u − uk +h∥1 ≲ h + H1+k. +To achieve the optimal convergence order, the relationship h = H2 should be satisfied in +Algorithm 3.1 (See (3.3)). But in Algorithm 3.2, we could expand the distance between the +mesh size H and h by increasing the iteration counts k. +4 +Convergence analysis +In this section, we provide the corresponding convergence analysis of Algorithm 3.2. To this +end, we need to introduce some preliminaries based on form B(w; v, χ) at first. +4.1 +Some preliminaries based on form B(w; v, χ) +In this subsection, we present some properties of form B(w; v, χ) and introduce two discrete +Green function. +Firstly, with fixed w, by Remark 2.1 and Cauchy-Schwarz inequality, it’s easy to obtain +that the form B(w; ·, ·) is continuous, i.e., +B(w; v, χ) ≲ ∥v∥1∥χ∥1, +∀ v, χ ∈ H1 +0(Ω). +(4.1) +Secondly, we present a lemma which provides the Babuˇska-Brezzi(BB) conditions of form +B(·; ·, ·) in Vh. And this lemma can be proved using similar arguments in Lemma 2.2 of [9]. +Lemma 4.1 +Assume u is the solution of problem (2.3), then when h is small enough, we +have, for any wh ∈ Vh, +∥wh∥1 ≲ sup +vh∈Vh +B (u; wh, vh) +∥vh∥1 +and +∥wh∥1 ≲ sup +vh∈Vh +B (u; vh, wh) +∥vh∥1 +. +(4.2) +Proof. +For the solution u of (2.3), a projection operator Ph : H1 +0(Ω) → Vh is defined by +(az(u)∇Phv, ∇χh) = (az(u)∇v, ∇χh), +∀v ∈ H1 +0(Ω), χh ∈ Vh. +(4.3) +By (2.4), we can know that the projection operator Ph is well-defined. Taking v = vh ∈ Vh ⊂ +H1 +0(Ω) and χh = Phvh − vh, and using (2.4), we could prove that the projection operator Ph +is identity operator for space Vh. Substituting χh = Phv into (4.3), and using (2.4), Poincar´e +inequality, Remark 2.1 and Cauchy–Schwarz inequality, it holds that +∥Phv∥1 ≲ ∥v∥1, +∀v ∈ H1 +0(Ω). +(4.4) + +6 +Jiajun Zhan & et al. +By (2.4), duality argument and (4.4), we can obtain (See Theorem 3.2.5 in [2]) +∥v − Phv∥0 ≲ h∥v∥1, +∀v ∈ H1 +0(Ω). +(4.5) +For any wh ∈ Vh, v ∈ H1 +0(Ω), by (3.1), Green formula, (4.3), Remark 2.1, Cauchy-Schwarz +inequality and (4.5), we have +B(u; wh, v − Phv) += +(ay(u)wh, ∇(v − Phv)) + (az(u)∇wh, ∇(v − Phv)) ++(fy(u)wh, v − Phv) + (fz(u)∇wh, v − Phv) += +((∇ · ay(u))wh, v − Phv) + (ay(u) · ∇wh, v − Phv) ++(fy(u)wh, v − Phv) + (fz(u)∇wh, v − Phv) +≲ +∥wh∥1∥v − Phv∥0 +≲ +h∥wh∥1∥v∥1. +(4.6) +Noticing that L′(u) : H1 +0(Ω) → H−1(Ω) is an isomorphism, using (4.6) and (4.4), we obtain +that +∥wh∥1 +≲ +sup +v∈H1 +0 (Ω) +B(u; wh, v) +∥v∥1 +≲ +sup +v∈H1 +0 (Ω) +B(u; wh, v − Phv) +∥v∥1 ++ sup +v∈H1 +0 Ω +B(u; wh, Phv) +∥v∥1 +≲ +h∥wh∥1 + +sup +v∈H1 +0 (Ω) +B(u; wh, Phv) +∥Phv∥1 +. +Taking h sufficiently small in the above inequality with projection operator Ph being identity +operator for Vh, we could obtain the first estimate of (4.2). The proof of the second estimate +of (4.2) is similar. +Next, we provide another BB condition of the form B(·; ·, ·). +Lemma 4.2 +Assume u is the solution of (2.3) and Ψ satisfying ∥u − Ψ∥1,∞ ≲ H, then when +H is small enough, for any wh ∈ Vh, it holds that +∥wh∥1 ≲ sup +vh∈Vh +B (Ψ; wh, vh) +∥vh∥1 +and +∥wh∥1 ≲ sup +vh∈Vh +B (Ψ; vh, wh) +∥vh∥1 +. +(4.7) +Proof. +Using the definition (3.1) of form B, Taylor expansion h(y, z) = h(y0, z0)+∂yh(˜θ1, ˜θ2)(y− +y0) + ∂zh(˜θ1, ˜θ2)(z − z0), where ˜θ1 is between y and y0 and ˜θ2 is between z and z0, Remark 2.1, +and H¨older inequality, we obtain +B(u; wh, vh) − B(Ψ; wh, vh) += +((ay(u) − ay(Ψ))wh, ∇vh) + ((az(u) − az(Ψ))∇wh, ∇vh) ++((fy(u) − fy(Ψ))wh, vh) + ((fz(u) − fz(Ψ))∇wh, vh) += +(ayy(θ1)(u − Ψ)wh, ∇vh) + (ayz(θ1)∇(u − Ψ)wh, ∇vh) ++(azy(θ2)(u − Ψ)∇wh, ∇vh) + (∇(u − Ψ)T azz(θ2)∇wh, ∇vh) ++(fyy(θ3)(u − Ψ)wh, vh) + (fyz(θ3) · ∇(u − Ψ)wh, vh) ++(fzy(θ4) · ∇wh(u − Ψ), vh) + (∇(u − Ψ)T fzz(θ4)∇wh, vh) +≲ +∥u − Ψ∥1,∞∥wh∥1∥vh∥1, +(4.8) + +An iterative two-grid method for strongly nonlinear elliptic boundary value problems +7 +where θi (i = 1, 2, 3, 4) are between u and Ψ. +By Lemma 4.1, (4.8) and ∥u − Ψ∥1,∞ ≲ H, it is obtained that +∥wh∥1 +≲ +sup +vh∈Vh +B(u; wh, vh) − B (Ψ; wh, vh) +∥vh∥1 ++ sup +vh∈Vh +B (Ψ; wh, vh) +∥vh∥1 +≲ +∥u − Ψ∥1,∞∥wh∥1 + sup +vh∈Vh +B (Ψ; wh, vh) +∥vh∥1 +≲ +H∥wh∥1 + sup +vh∈Vh +B (Ψ; wh, vh) +∥vh∥1 +. +Taking H sufficiently small into the above inequality, we can derive the first estimate of (4.7). +The proof of the second estimate of (4.7) is similar. +Remark 4.3 +According to (2.6), Lemma 4.2 still holds with replacing Ψ by the finite element +solution uh of (2.5). +For more concise notations and the subsequent analysis, we denote +Ek = uh − uk +h, +(4.9) +uk,1 +h += uk +h + ek +H, +(4.10) +where uh is the solution of problem (2.5) and, uk +h and ek +H are given by Algorithm 3.2. It’s noted +that these notation will be used frequently in the rest of this paper. +Remark 4.4 +For k ⩾ 0, assume that Ek, uk,1 +h +and ek +H are given by (4.9), (4.10) and Algorithm +3.2, respectively. If both ∥Ek∥1,∞ ≲ H and ∥ek +H∥1,∞ ≲ H are provided, the Lemma 4.2 still +holds with replacing Ψ by uk,1 +h . In fact, using (4.10), (4.9), triangle inequality, (2.6) with r ⩾ 1 +and h < H, ∥Ek∥1,∞ ≲ H and ∥ek +H∥1,∞ ≲ H, we derive that +∥u − uk,1 +h ∥1,∞ ⩽ ∥u − uh∥1,∞ + ∥Ek∥1,∞ + ∥ek +H∥1,∞ ≲ H. +Therefore, the Lemma 4.2 still holds with Ψ = uk,1 +h . +And then, for the finite element solution uh of (2.5) and any fixed x ∈ Ω, we introduce the +Green functions gx +H ∈ VH, which be defined by +B(uh; vH, gx +H) = ∂vH(x), +∀vH ∈ VH, +(4.11) +where ∂ denotes either +∂ +∂x1 or +∂ +∂x2 . It’s easy to see that the Green function gx +H is well-defined +by Remark 4.3. +Assume uk,1 +h +is given by (4.10), similarly, for any fixed x ∈ Ω, we introduce the Green +functions gk,x +h +∈ Vh by +B(uk,1 +h ; vh, gk,x +h +) = ∂vh(x), +∀vh ∈ Vh. +(4.12) +By Remark 4.4, we also can see that Green function gk,x +h +is well-defined. +Here give some estimates of the above two Green functions gx +H and gk,x +h +(See Lemma 3.3 of +[4], or (2.10) and (2.11) of [9]) +∥gx +H∥1,1 ≲ | log H| +and +∥gk,x +h +∥1,1 ≲ | log h|. +(4.13) + +8 +Jiajun Zhan & et al. +At last, for any v ∈ H1 +0(Ω) ∩ W 1,∞(Ω), using (3.1.11) of [2], it could be obtained that +∥v∥1,∞ ≲ |v|1,∞. +(4.14) +4.2 +Error estimate +In this subsection, we present the convergence analysis of Algorithm 3.2 by a series of lemmas. +Lemma 4.5 +Assume uk,1 +h , Ek and ek +H are given by (4.10), (4.9) and Algorithm 3.2, respec- +tively, then we have, for any vh ∈ Vh, +B(uk,1 +h ; Ek+1, vh) ≲ (∥Ek∥1,∞ + ∥ek +H∥1,∞)(∥Ek∥1 + ∥ek +H∥1)∥vh∥1, +(4.15) +B(uk,1 +h ; Ek+1, vh) ≲ (∥Ek∥2 +1,∞ + ∥ek +H∥2 +1,∞)∥vh∥1,1. +(4.16) +Proof. +Using (4.9), Remark 3.1, (3.5), (2.5) and (2.2), it is obtained that +B(uk,1 +h ; Ek+1, vh) += +B(uk,1 +h ; uh, vh) − B(uk,1 +h ; uk+1 +h +, vh) += +B(uk,1 +h ; uh, vh) − B(uk,1 +h ; uk +h + ek +H, vh) ++A(uk,1 +h , vh) − A(uh, vh). += +B(uk,1 +h ; Ek − ek +H, vh) + (a(uk,1 +h , ∇uk,1 +h ), vh) + (f(uk,1 +h , ∇uk,1 +h ), vh) +−(a(uh, ∇uh), vh) − (f(uh, ∇uh), vh) +:= +A1 − A2 − A3, +(4.17) +where +A1 += +B(uk,1 +h ; Ek − ek +H, vh), +A2 += +(a(uh, ∇uh), vh) − (a(uk,1 +h , ∇uk,1 +h ), vh), +A3 += +(f(uh, ∇uh), vh) − (f(uk,1 +h , ∇uk,1 +h ), vh). +For A1, using the definition (3.1) of B, we have +A1 += +(ay(uk,1 +h )(Ek − ek +H), ∇vh) + (az(uk,1 +h )∇(Ek − ek +H), ∇vh) ++(fy(uk,1 +h )(Ek − ek +H), vh) + (fz(uk,1 +h )∇(Ek − ek +H), vh). +(4.18) +For A2, using second order Taylor expansion, (4.10) and (4.9), we obtain +A2 += +(ay(uk,1 +h )(Ek − ek +H), ∇vh) + (az(uk,1 +h )∇(Ek − ek +H), ∇vh) ++(ayy(θ5)(Ek − ek +H)2, ∇vh) + 2(ayz(θ5)∇(Ek − ek +H)(Ek − ek +H), ∇vh) ++(∇(Ek − ek +H)T azz(θ5)∇(Ek − ek +H), ∇vh), +(4.19) +where θ5 is between uh and uk,1 +h . +Similarly for A3, using second order Taylor expansion, (4.10) and (4.9), it is obtained that +A3 += +(fy(uk,1 +h )(Ek − ek +H), vh) + (fz(uk,1 +h )∇(Ek − ek +H), vh) ++(fyy(θ6)(Ek − ek +H)2, vh) + 2(fyz(θ6) · ∇(Ek − ek +H)(Ek − ek +H), vh) ++(∇(Ek − ek +H)T fzz(θ6)∇(Ek − ek +H), vh), +(4.20) + +An iterative two-grid method for strongly nonlinear elliptic boundary value problems +9 +where θ6 is between uh and uk,1 +h . +Noticing the sum of the first order derivative items about a(·, ·, ·) and f(·, ·, ·) in (4.19) and +(4.20) exactly equal A1. Substituting (4.18), (4.19) and (4.20) into (4.17), it’s obtained that +B(uk,1 +h ; Ek+1, vh) += +−(ayy(θ5)(Ek − ek +H)2, ∇vh) − 2(ayz(θ5)∇(Ek − ek +H)(Ek − ek +H), ∇vh) +−(∇(Ek − ek +H)T azz(θ5)∇(Ek − ek +H), ∇vh) − (fyy(θ6)(Ek − ek +H)2, vh) +−2(fyz(θ6) · ∇(Ek − ek +H)(Ek − ek +H), vh) +−(∇(Ek − ek +H)T fzz(θ6)∇(Ek − ek +H), vh). +(4.21) +Applying Remark 2.1, H¨older inequality and triangle inequality into (4.21), we could obtain +B(uk,1 +h ; Ek+1, vh) +≲ +∥Ek − ek +H∥1,∞∥Ek − ek +H∥1∥vh∥1 +⩽ +(∥Ek∥1,∞ + ∥ek +H∥1,∞)(∥Ek∥1 + ∥ek +H∥1)∥vh∥1, +which completes the proof of (4.15). Similarly, we could obtain (4.16) by (4.21). +Lemma 4.6 +Assume that uk,1 +h , ek +H and Ek are defined by (4.10), Algorithm 3.2 and (4.9), +respectively, then we have +B(uk,1 +h ; ek +H, vH) ≲ (∥Ek∥1 + ∥Ek+1∥1)∥vH∥1, +∀vH ∈ VH. +(4.22) +Proof. +Taking vh = vH into (3.5) and using (3.4), we obtain +B(uk,1 +h ; uk+1 +h +, vH) = B(uk,1 +h ; uk +h + ek +H, vH). +Rewriting the the above equation with Remark 3.1, and then using (4.9), (4.1) and triangle +inequality, we have +B(uk,1 +h ; ek +H, vH) += +B(uk,1 +h ; uk+1 +h +− uk +h, vH) += +B(uk,1 +h ; uk+1 +h +− uh + uh − uk +h, vH) += +B(uk,1 +h ; Ek − Ek+1, vH) +≲ +∥Ek − Ek+1∥1∥vH∥1 +⩽ +(∥Ek∥1 + ∥Ek+1∥1) ∥vH∥1, +which completes the proof. +Lemma 4.7 +Assume that Ek and ek +H are given by (4.9) and Algorithm 3.2, respectively, and +r ⩾ 1, when h is small enough, then for any integer k ⩾ 1, +∥Ek∥1 ≲ Hr+k, +∥Ek∥1,∞ ≲ | log h|H2, +∥ek +H∥1,∞ ≲ H, +∥ek +H∥1 ≲ Hr+k. +(4.23) +Proof. +Here we use mathematical induction to prove that (4.23) is true. +By (3.4), u0 +h = uH, (2.5) and the uniqueness of finite element solution (See Lemma 2.2), it +could be seen that e0 +H = 0. + +10 +Jiajun Zhan & et al. +Making use of triangle inequality, (2.6) and h ⩽ H, we have +∥E0∥1 ⩽ ∥u − uh∥1 + ∥u − uH∥1 ≲ hr + Hr ⩽ Hr, +(4.24) +∥E0∥1,∞ ⩽ ∥u − uh∥1,∞ + ∥u − uH∥1,∞ ≲ hr + Hr ⩽ Hr. +(4.25) +Next, we will prove (4.23) is true when k = 1. +(i) For ∥E1∥1 ≲ Hr+1. +Noticing that r ⩾ 1 and e0 +H = 0, and using (4.25), we have +∥E0∥1,∞ ≲ H and ∥e0 +H∥1,∞ ≲ H , which could derive the BB condition of form B(u0,1 +h ; ·, ·) (See +Remark 4.4). Using the BB condition of form B(u0,1 +h ; ·, ·), (4.15), (4.25), ek +H = 0, (4.24), r ⩾ 1 +and H < 1, it’s obtained that +∥E1∥1 +≲ +sup +vh∈Vh +B(u0,1 +h ; E1, vh) +∥vh∥1 +≲ +(∥E0∥1,∞ + ∥e0 +H∥1,∞)(∥E0∥1 + ∥e0 +H∥1) +≲ +(Hr + 0)(Hr + 0) +≲ +Hr+1. +(4.26) +(ii) For ∥E1∥1,∞ ≲ | log h|H2. For k = 0 and any fixed x ∈ Ω, taking vh = E1 into (4.12), +using (4.16), (4.25), e0 +H = 0, (4.13), r ⩾ 1 and H < 1, we obtain +∂E1(x) += +B(u0,1 +h ; E1, g0,x +h ) +≲ +(∥E0∥2 +1,∞ + ∥e0 +H∥2 +1,∞)∥g0,x +h ∥1,1 +≲ +(H2r + 0)| log h| +≲ +| log h|H2. +Further using the arbitrariness of x and (4.14), we derive that +∥E1∥1,∞ ≲ | log h|H2. +(iii) For ∥e1 +H∥1,∞ ≲ H. Using ∥E1∥1,∞ ≲ | log h|H2 and Lemma A.1 (The specific content +of lemma and proof are referred to Appendix), we obtain +∥e1 +H∥1,∞ ≲ H. +(iv) For ∥e1 +H∥1 ≲ Hr+1. Noticing that ∥E1∥1,∞ ≲ | log h|H2 and ∥e1 +H∥1,∞ ≲ H are satisfied, +therefore the BB condition of form B(u1,1 +h ; ·, ·) holds (See Remark 4.4). Using the BB condition +of B(u1,1 +h ; ·, ·) and (4.15), it’s obtained that +∥E2∥1 +≲ +sup +vh +B(u1,1 +h ; E2, vh) +∥vh∥1 +≲ +(∥E1∥1,∞ + ∥e1 +H∥1,∞)(∥E1∥1 + ∥e1 +H∥1). +(4.27) +Using the BB condition of form B(u1,1 +h ; ·, ·), (4.22) with k = 1, (4.27), ∥E1∥1,∞ ≲ | log h|H2 +and ∥e1 +H∥1,∞ ≲ H, we have +∥e1 +H∥1 +≲ +sup +vH∈VH +B(u1,1 +h ; e1 +H, vH) +∥vH∥1 +≲ +∥E1∥1 + ∥E2∥1 +≲ +∥E1∥1 + (∥E1∥1,∞ + ∥e1 +H∥1,∞)(∥E1∥1 + ∥e1 +H∥1) +≲ +∥E1∥1 + (| log h|H2 + H)(∥E1∥1 + ∥e1 +H∥1). + +An iterative two-grid method for strongly nonlinear elliptic boundary value problems +11 +Taking H be small enough in the above inequality, and using (4.26), it’s obtained that +∥e1 +H∥1 ≲ ∥E1∥1 ≲ Hr+1. +We assume (4.23) is true when k = l, i.e., +∥El∥1 ≲ Hr+l, +∥El∥1,∞ ≲ | log h|H2, +∥el +H∥1,∞ ≲ H, +∥el +H∥1 ≲ Hr+l. +(4.28) +Next, we will prove (4.23) also holding when k = l + 1. +(i) For ∥El+1∥1 ≲ Hr+l+1. +Noticing that ∥El∥1,∞ ≲ | log h|H2 and ∥el +H∥1,∞ ≲ H are +satisfied, therefore the BB condition of form B(ul,1 +h ; ·, ·) holds (See Remark 4.4). Using the BB +condition of form B(ul,1 +h ; ·, ·), (4.15), (4.28) and H < 1, we obtain +∥El+1∥1 +≲ +sup +vh∈Vh +B(ul,1 +h ; El+1, vh) +∥vh∥1 +≲ +(∥El∥1,∞ + ∥el +H∥1,∞)(∥El∥1 + ∥el +H∥1) +≲ +(| log h|H2 + H)(Hr+l + Hr+l) +≲ +Hr+l+1. +(4.29) +(ii) For ∥El+1∥1,∞ ≲ | log h|H2. Taking vh = El+1 into (4.12) with k = l, using (4.16), +(4.28) and (4.13), we obtain +∂El+1(x) += +B(ul,1 +h ; El+1, gl,x +h ) +≲ +(∥El∥2 +1,∞ + ∥el +H∥2 +1,∞)∥gl,x +h ∥1,1 +≲ +(| log h|2H4 + H2)| log h| +≲ +| log h|H2, +which combining the arbitrariness of x and (4.14), it could be derived that +∥El+1∥1,∞ ≲ | log h|H2. +(iii) For ∥el+1 +H ∥1,∞ ≲ H. Using ∥El+1∥1,∞ ≲ | log h|H2 and Lemma A.1, we obtain +∥el+1 +H ∥1,∞ ≲ H. +(4.30) +(iv) For ∥el+1 +H ∥1 ≲ Hr+l+1. Noticing that ∥El+1∥1,∞ ≲ | log h|H2 and ∥el+1 +H ∥1,∞ ≲ H are +satisfied, therefore the BB condition of form B(ul+1,1 +h +; ·, ·) holds (See Remark 4.4). Using the +BB condition of form B(ul+1,1 +h +; ·, ·), (4.15), it’s obtained that +∥El+2∥1 +≲ +sup +vh +B(ul+1,1 +h +; El+2, vh) +∥vh∥1 +≲ +(∥El+1∥1,∞ + ∥el+1 +H ∥1,∞)(∥El+1∥1 + ∥el+1 +H ∥1). +(4.31) +Using the BB condition of form B(ul+1,1 +h +; ·, ·), (4.22) with k = l + 1, (4.31), ∥El+1∥1,∞ ≲ +| log h|H2 and ∥el+1 +H ∥1,∞ ≲ H, we have +∥el+1 +H ∥1 +≲ +sup +vH∈VH +B(ul+1,1 +h +; el+1 +H , vH) +∥vH∥1 +≲ +∥El+1∥1 + ∥El+2∥1 +≲ +∥El+1∥1 + (∥El+1∥1,∞ + ∥el+1 +H ∥1,∞)(∥El+1∥1 + ∥el+1 +H ∥1) +≲ +∥El+1∥1 + (| log h|H2 + H)(∥El+1∥1 + ∥el+1 +H ∥1). + +12 +Jiajun Zhan & et al. +Taking H be small enough in the above inequality and using (4.29), it’s obtained that +∥el+1 +H ∥1 ≲ ∥El+1∥1 ≲ Hr+l+1. +By mathematical induction, the conclusion is obtained. +Remark 4.8 +Although we just use the estimation ∥Ek+1∥1 ≲ Hr+k+1 in our main result +(See Theorem 4.9), the availability of ∥Ek+1∥1 ≲ Hr+k+1 requires the support of ∥Ek∥1,∞ ≲ +| log h|H2, ∥ek +H∥1,∞ ≲ H and ∥ek +H∥1 ≲ Hr+k. +Here gives the main result of this paper. +Theorem 4.9 +Assume that u is the solution of (2.3) and uk +h is given by Algorithm 3.2, then +we have +∥u − uk +h∥1 ≲ hr + Hr+k. +(4.32) +Proof. +Using triangle inequality, (4.9), (2.6) and Lemma 4.7, we could obtain that +∥u − uk +h∥1 ⩽ ∥u − uh∥1 + ∥Ek∥1 ≲ hr + Hr+k, +which completes the proof. +5 +Numerical experiments +In this section, we present some numerical experiments to show the efficiency of the pro- +posed iterative two-grid algorithm. We implemented these experiments by the software package +FEALPy of programming language Python [5]. Specially in the Step 1 of Algorithm 3.2, we +solve the nonlinear systems by Newton iteration methods with relative residual 10−8. +We adopt the following mean curvature flow problem as our model problem: +−∇ · +� +∇u +(1 + |∇u|2)1/2 +� += g in Ω, +u = 0 on ∂Ω, +where the computational domain Ω = (0, 1)× (0, 1), the exact solution u = x(1 − x)2y(1 − y)ex, +and g is so chosen according to the exact solution. +Firstly, we choose conforming piecewise linear finite element space as Vh, namely choose +r = 1. According to Theorem 4.9, we should keep hr = Hr+k hold in order to achieve the +optimal convergence order. Therefore in Table 1, we present some numerical results in different +mesh size with h = H2 for k = 1. In this case, our algorithm is same with Algorithm 3.1 (See +Remark 3.3). Furthermore, we could observe that ∥u−u1 +h∥1 ∗max{H2, h}−1 are stable in Table +1, which agrees with (4.32) in Theorem 4.9. +Table 1: k = 1, r = 1 +H +h +∥u − u1 +h∥1 +∥u − u1 +h∥1 ∗ max{H2, h}−1 +1/9 +1/81 +2.74E-03 +0.221953 +1/10 +1/100 +2.22E-03 +0.221967 +1/11 +1/121 +1.83E-03 +0.221977 +1/12 +1/144 +1.54E-03 +0.221983 + +An iterative two-grid method for strongly nonlinear elliptic boundary value problems +13 +And then, we increase the iterative counts k to expand the distance between H and h, which +is shown in Tables 2 and 3. We also observe that ∥u − u1 +h∥1 ∗ max{H1+k, h}−1 are stable. +Table 2: k = 2, r = 1 +H +h +∥u − u2 +h∥1 +∥u − u2 +h∥1 ∗ max{H3, h}−1 +1/3 +1/27 +8.20E-03 +0.221524 +1/4 +1/64 +3.47E-03 +0.221784 +1/5 +1/125 +1.77E-03 +0.221826 +1/6 +1/216 +1.03E-03 +0.221836 +Table 3: k = 3, r = 1 +H +h +∥u − u3 +h∥1 +∥u − u3 +h∥1 ∗ max{H4, h}−1 +1/2 +1/16 +1.38E-02 +0.220944 +1/3 +1/81 +2.74E-03 +0.221805 +1/4 +1/256 +8.67E-04 +0.221837 +At last, we implement similar numerical experiments for high order finite element space +with r = 2 and r = 3 in Tables 4-9. By observation, all these numerical experiments are in +support of (4.32) in Theorem 4.9. +Table 4: k = 1, r = 2 +H +h +∥u − u1 +h∥1 +∥u − u1 +h∥1 ∗ max{H3, h2}−1 +1/4 +1/8 +2.57E-03 +0.164578 +1/9 +1/27 +2.30E-04 +0.167320 +1/16 +1/64 +4.09E-05 +0.167553 +1/25 +1/125 +1.07E-05 +0.167590 +1/36 +1/216 +3.59E-06 +0.167599 +Acknowledgements +The work of the first and second authors were partially funded by the +Science and Technology Development Fund, Macau SAR (Nos. 0070/2019/A2, 0031/2022/A1). +The third author was supported by the National Natural Science Foundation of China (Grant +No. +11901212). The third and fourth authors are also supported by the National Natural +Science Foundation of China (Grant No. 12071160). +References +[1] +Bi C J, Wang C, Lin Y P. A posteriori error estimates of two-grid finite element methods for nonlinear +elliptic problems. J Sci Comput, 2018, 74: 23–48 +[2] +Ciarlet P G. The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, No. 40. +SIAM, Philadelphia, 2002 +[3] +Gudi T, Nataraj N, Pani A. hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary +value problems. Numer Math, 2008, 109: 233–268 +[4] +Thom´ee V, Xu J C, Zhang N Y. Superconvergence of the gradient in piecewise linear finite-element approx- +imation to a parabolic problem. SIAM J Numer Anal, 1989, 26: 553–573 +[5] +Wei H Y, Huang Y Q. Fealpy: Finite element analysis library in python. https://github.com/weihuayi/ +fealpy, Xiangtan University, 2017-2021 +[6] +Xu J C. Iterative methods by SPD and small subspace solvers for nonsymmetric or indefinite problems. In: +Proceedings of the 5th International Symposium on Domain Decomposition Methods for Partial Differential + +14 +Jiajun Zhan & et al. +Table 5: k = 2, r = 2 +H +h +∥u − u2 +h∥1 +∥u − u2 +h∥1 ∗ max{H4, h2}−1 +1/8 +1/64 +4.09E-05 +0.167552 +1/9 +1/81 +2.55E-05 +0.167571 +1/10 +1/100 +1.68E-05 +0.167582 +1/11 +1/121 +1.14E-05 +0.167589 +1/12 +1/144 +8.08E-06 +0.167593 +Table 6: k = 3, r = 2 +H +h +∥u − u3 +h∥1 +∥u − u3 +h∥1 ∗ max{H5, h2}−1 +1/5 +1/55 +5.54E-05 +0.167534 +1/6 +1/90 +2.07E-05 +0.160874 +1/7 +1/126 +1.06E-05 +0.167590 +1/8 +1/184 +4.95E-06 +0.162211 +1/9 +1/243 +2.84E-06 +0.167600 +Table 7: k = 1, r = 3 +H +h +∥u − u1 +h∥1 +∥u − u1 +h∥1 ∗ max{H4, h3}−1 +1/8 +1/16 +1.83E-05 +0.075054 +1/27 +1/81 +1.40E-07 +0.074662 +1/64 +1/256 +4.44E-09 +0.074543 +Table 8: k = 2, r = 3 +H +h +∥u − u2 +h∥1 +∥u − u2 +h∥1 ∗ max{H5, h3}−1 +1/8 +1/32 +2.28E-06 +0.074870 +1/9 +1/36 +1.60E-06 +0.074838 +1/10 +1/40 +1.17E-06 +0.074810 +1/11 +1/55 +4.49E-07 +0.072343 +1/12 +1/60 +3.46E-07 +0.074716 +Table 9: k = 3, r = 3 +H +h +∥u − u3 +h∥1 +∥u − u3 +h∥1 ∗ max{H6, h3}−1 +1/8 +1/64 +2.85E-07 +0.074703 +1/9 +1/81 +1.40E-07 +0.074662 +1/10 +1/100 +7.46E-08 +0.074630 +1/11 +1/121 +4.21E-08 +0.074606 +1/12 +1/144 +2.50E-08 +0.074588 + +An iterative two-grid method for strongly nonlinear elliptic boundary value problems +15 +Equations. Siam, Philadelphia, 1992, 106–118 +[7] +Xu J C. A novel two-grid method for semilinear elliptic equations. SIAM J Sci Comput, 1994, 15: 231–237 +[8] +Xu J C. Some two-grid finite element methods. In: Domain Decomposition Methods in Science and Engi- +neering (Quarteroni, Alfio and P´eriaux, Jacques and Kuznetsov, Yuri A and Widlund, Olof B eds). Contemp +Math, vol. 157, Amer Math Soc, 1994, 79–87 +[9] +Xu J C. Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J Numer Anal, 1996, 33: +1759–1777 +[10] +Xu J C, Cai X C. A preconditioned GMRES method for nonsymmetric or indefinite problems. Math Comp, +1992, 59: 311–319 +[11] +Zhang W F, Fan R H, Zhong L Q. Iterative two-grid methods for semilinear elliptic equations. Comput +Math Appl, 2020, 80: 522–530 +Appendix A +The purpose of this appendix is to provide the proof of Lemma A.1. +Lemma A.1 +Assume ek +H is given in (3.4) and ∥Ek∥1,∞ ≲ | log h|H2, when H is small enough, it holds that +∥ek +H∥1,∞ ≲ H. +(A.1) +Before we present the proof of Lemma A.1, we need to introduce some preliminaries and lemmas. +For the finite element solution uh of (2.5), we introduce a projection operator ˆPH : H1 +0(Ω) → VH, which be +defined by, +B(uh; ˆPHw, vH) = B(uh; w, vH), +∀w ∈ H1 +0(Ω), vH ∈ VH. +(A.2) +It’s easy to derive that ˆPH is well-defined by the BB-conditions of form B(uh; ·, ·) which could be obtained by +Remark 4.3. Furthermore, the projection operator ˆPH satisfies the following estimate +∥ ˆPHw∥1,∞ ≲ | log H|∥w∥1,∞, +∀w ∈ W 1,∞(Ω). +(A.3) +In fact, taking vH = ˆPHw in (4.11), and using (A.2), (3.1), Remark 2.1, H¨older inequality and (4.13), we obtain +∂ ˆPHw(x) += +B(uh; ˆPHw, gx +H) += +B(uh; w, gx +H) += +(ay(uh)w, ∇gx +H) + (az(uh)∇w, ∇gx +H) + (fy(uh)w, gx +H) + (fz(uh)∇w, gx +H) +≲ +∥w∥1,∞∥gx +H∥1,1 +≲ +| log H|∥w∥1,∞. +Finally using of the arbitrariness of x and (4.14), we could obtain (A.3). +By Taylor expansion, we have (the detailed proof can be found in Lemma 3.1 of [9]) +A(w, χ) = A(v, χ) + B(v; w − v, χ) + R(η; v, w, χ), +∀w, v, χ ∈ H1 +0(Ω), +(A.4) +where the forms A(·, ·) and B(·; ·, ·) are given by (2.2) and (3.1), respectively, η = v + t(w − v) and +R(η; v, w, χ) += +� 1 +0 +� +(ayy(η)(v − w)2, ∇χ) + 2(ayz(η)∇(v − w)(v − w), ∇χ) ++(∇(v − w)T azz(η)∇(v − w), ∇χ) + (fyy(η)(v − w)2, χ) ++2(fyz(η) · ∇(v − w)(v − w), χ) + (∇(v − w)T fzz(η)∇(v − w), χ) +� +(1 − t)dt. +For the proof of Lemma A.1, we introduce a operator Φ as follow. Assume uh is the solution of (2.5), Ek, +R, uk +h are given in (4.9), (A.4) and Algorithm 3.2, respectively, we defined operator Φ : VH → VH by, for any +wH ∈ VH, +B(uh; Φ(wH), vH) = B(uh; Ek, vH) − R(uh + t(wH − Ek); uh, uk +h + wH, vH), +∀vH ∈ VH. +(A.5) +By the BB-conditions of form B(uh; ·, ·) (See Remark 4.3), it’s easy to prove that operator Φ is well-defined. +We define a space +QH = {vH ∈ VH : ∥vH − ˆPHEk∥1,∞ ⩽ H}, +(A.6) +where ˆPH is a projection operator defined by (A.2). Since QH is a finite dimensional space, it’s easy to see that +QH is a non-empty compact convex subset. +Next, we will use Brouwer fixed point theorem to prove that (A.5) has a fixed point ¯wH in QH. +Lemma A.2 +Assume ∥Ek∥1,∞ ≲ | log h|H2, then when H is small enough, we have Φ(QH) ⊂ QH. + +16 +Jiajun Zhan & et al. +Proof. +For any wH ∈ QH, vH ∈ VH, rewriting (A.5) with (A.2), we have +B(uh; Φ(wH) − ˆPHEk, vH) = −R(uh + t(wH − Ek); uh, uk +h + wH, vH). +(A.7) +Substituting vH = Φ(wH) − ˆPHEk into (4.11) and using (A.7), Remark 2.1, H¨older inequality, (4.9), triangle +inequality, (4.13), (A.3), (A.6) and ∥Ek∥1,∞ ≲ | log h|H2, it is obtained that +∂(Φ(wH) − ˆPHEk)(x) += +B(uh; Φ(wH) − ˆPHEk, gx +H) += +−R(uh + t(wH − Ek); uh, uk +h + wH, gx +H) +≲ +∥Ek − wH∥2 +1,∞∥gx +H∥1,1 +≲ +(∥Ek − ˆPHEk∥2 +1,∞ + ∥ ˆPHEk − wH∥2 +1,∞)| log H| +≲ +((1 + | log H|)2∥Ek∥2 +1,∞ + H2)| log H| +≲ +((1 + | log H|)2| log h|2H4 + H2)| log H|. +Further using the arbitrariness of x and (4.14), the proof is finished. +Lemma A.3 +Assume ∥Ek∥1,∞ ≲ | log h|H2, then the operator Φ is continuous in VH. +Proof. +For any w1, w2 ∈ QH, by (A.5), we have +B(uh; Φ(w1) − Φ(w2), vH) += +R(uh + t(w2 − Ek); uh, uk +h + w2, vH) +−R(uh + t(w1 − Ek); uh, uk +h + w1, vH). +(A.8) +Noticing that the definition of R in (A.4), for the terms concerning ayy on the right hand side of (A.8), we can +use Remark 2.1 and H¨older inequality to obtain that +(ayy(uh + t(w2 − Ek))(Ek − w2)2, ∇vH) − (ayy(uh + t(w1 − Ek))(Ek − w1)2, ∇vH) += +(ayy(uh + t(w2 − Ek))(Ek − w2)2, ∇vH) +−(ayy(uh + t(w1 − Ek))(Ek − w2)2, ∇vH) ++(ayy(uh + t(w1 − Ek))(Ek − w2)2, ∇vH) +−(ayy(uh + t(w1 − Ek))(Ek − w1)2, ∇vH) += +([ayy(uh + t(w2 − Ek)) − ayy(uh + t(w1 − Ek))] (Ek − w2)2, ∇vH) ++(ayy(uh + t(w1 − Ek)) +� +−2Ekw2 + w2 +2 + 2Ekw1 − w2 +1 +� +, ∇vH) += +([ayy(uh + t(w2 − Ek)) − ayy(uh + t(w1 − Ek))] (Ek − w2)2, ∇vH) ++(ayy(uh + t(w1 − Ek)) (2Ek − w1 − w2) (w1 − w2), ∇vH) +≲ +∥ayy(uh + t(w2 − Ek)) − ayy(uh + t(w1 − Ek))∥0,∞∥(Ek − w2)2∥0∥vH∥1 ++∥2Ek − w1 − w2∥0∥w1 − w2∥0,∞∥vH∥1. +(A.9) +For ∥(Ek − w2)2∥0, we use triangle inequality, (A.3), (A.6) and ∥Ek∥1,∞ ≲ | log h|H2, it’s obtained that +∥(Ek − w2)2∥0 +⩽ +∥Ek − w2∥2 +1,∞ +≲ +∥Ek∥2 +1,∞ + ∥ ˆPHEk∥2 +1,∞ + ∥ ˆPHEk − w2∥2 +1,∞ +≲ +∥Ek∥2 +1,∞ + | log H|2∥Ek∥2 +1,∞ + H2 +≲ +| log h|2H4 + | log H|2| log h|2H4 + H2 +:= +C1(H), +(A.10) +where C1(H) is a constant depending on H. +Similarly, for ∥2Ek − w1 − w2∥0, there also exists a constant C2(H) such that +∥2Ek − w1 − w2∥0 ≲ C2(H). +(A.11) +Substituting (A.10) and (A.11) into (A.9), it’s could be obtained that +(ayy(uh + t(w2 − Ek))(Ek − w2)2, ∇vH) − (ayy(uh + t(w1 − Ek))(Ek − w1)2, ∇vH) +≲ +C(H) +� +∥ayy(uh + t(w2 − Ek)) − ayy(uh + t(w1 − Ek))∥0,∞ ++∥w1 − w2∥0,∞ +� +∥vH∥1, +where C(H) = max{C1(H), C2(H)} . +The rest of the items on the right hand side of (A.8) have similar +results, and here is omitted. +The conclusion follows from the above discussion, (A.8), the BB-conditions of + +An iterative two-grid method for strongly nonlinear elliptic boundary value problems +17 +form B(uh; ·, ·) (See Remark 4.3) and the continuity of second order derivatives of a(·, ·, ·) and f(·, ·, ·) (See the +assumptions about a(·, ·, ·) and f(·, ·, ·) in Section 2). +At last, we present the proof of Lemma A.1 by Brouwer fixed point theorem. +Proof of Lemma A.1. +Making use of Lemmas A.2 and A.3 and Brouwer fixed point theorem, we know that +(A.5) exists a fixed point ¯wH in QH. +Taking w = uk +h + ¯wH, v = uh and χ = vH into (A.4), and then using (2.5) with VH ⊂ Vh, Remark 3.1, +(4.9), ¯wH = Φ( ¯wH) and (A.5), we obtain that +A(uk +h + ¯wH, vH) += +A(uh, vH) + B(uh; uk +h + ¯wH − uh, vH) + R(η; uh, uk +h + ¯wH, vH) += +B(uh; ¯wH, vH) − B(uh; Ek, vH) + R(η; uh, uk +h + ¯wH, vH) += +B(uh; Φ( ¯wH), vH) − B(uh; Ek, vH) + R(η; uh, uk +h + ¯wH, vH) += +0, +(A.12) +where η = uh + t( ¯wH − Ek). By the uniqueness of finite element solution (See Lemma 2.2), (3.4) and (A.12), +we can see that ¯wH = ek +H, which implies ek +H ∈ QH. +At last, using triangle inequality, (A.6), (A.3) and ∥Ek∥1,∞ ≲ | log h|H2, we obtain +∥ek +H∥1,∞ +⩽ +∥ek +H − ˆPHEk∥1,∞ + ∥ ˆPHEk∥1,∞ +≲ +H + | log H|∥Ek∥1,∞ +≲ +H + | log H|| log h|H2, +which completes the proof. + diff --git a/AtE1T4oBgHgl3EQfDQOp/content/tmp_files/load_file.txt b/AtE1T4oBgHgl3EQfDQOp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1b5bd8daff87e979623a0cc7553d49785b7ac20a --- /dev/null +++ b/AtE1T4oBgHgl3EQfDQOp/content/tmp_files/load_file.txt @@ -0,0 +1,788 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf,len=787 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='02875v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='NA] 7 Jan 2023 SCIENCE CHINA Mathematics 1 XXXX Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' XX No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' XX XX–XX www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='SciChina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='com www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='springerlink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='com An iterative two-grid method for strongly non- linear elliptic boundary value problems Jiajun Zhan1, Lei Yang1, Xiaoqing Xing2,†, Liuqiang Zhong2 1 School of Computer Science and Engineering, Faculty of Innovation Engineering, Macau University of Science and Technology, Macao SAR 999078, China;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' 2 School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China Email: 2109853gii30011@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='must.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='mo, leiyang@must.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='mo, xingxq@scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='cn, zhong@scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='cn Abstract We design and analyze an iterative two-grid algorithm for the finite element discretizations of strongly nonlinear elliptic boundary value problems in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' We propose an iterative two-grid algorithm, in which a nonlinear problem is first solved on the coarse space, and then a symmetric positive definite problem is solved on the fine space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' The innovation of this paper lies in the establishment of a first convergence analysis, which requires simultaneous estimation of four interconnected error estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' We also present some numerical experiments to confirm the efficiency of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Keywords: iterative two-grid method, convergence, strongly nonlinear elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' MSC(2020): 65N30, 65M12, 35J60 1 Introduction The two-grid methods are first proposed for nonselfadjoint problems and indefinite elliptic problems [6, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Then, the two-grid methods are extended to solve semiliinear elliptic problems [7], quasi-linear and nonlinear elliptic problems [8, 9], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Especially, for nonlinear elliptic problems, the basic idea of two-grid methods is to first obtain a rough solution by solving the original problem in a “coarse mesh” with mesh size H, and then correct the rough solution by solving a symmetric positive definite (SPD) system in a “fine mesh” with mesh size h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Noticing the mesh size of “coarse mesh” is much smaller than that of “fine mesh”, it is not difficult to solve an original problem in “coarse mesh”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Therefore, two-grid methods reduce the computational complexity of solving the original problem to solving a SPD problem and dramatically improve the computational speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Recently, Bi, Wang and Lin [1] presented a two-grid algorithm to solve the strongly nonlinear elliptic problems and provided a posteriori error estimator for the two-grid methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' It’s noted that the literature mentioned above is all about non-iterative two-grid methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' As is well-known, the mesh size H of “coarse mesh” and h of “fine mesh” should satisfy a certain relationship for the optimal convergence order in non-iterative two-grid methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' The iterative two-grid methods have the advantage over the non-iterative two-grid methods in that, the distance between the mesh sizes H and h can be enlarged by increasing the iteration counts † Corresponding author 2 Jiajun Zhan & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' with the same accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' However, there is only a small amount of literature on iterative two-grid methods of conforming finite element discretization for elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Xu [9] first proposed and analyzed an iterative two-grid method for non-symmetric positive definite elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Zhang, Fan and Zhong [11] designed some iterative two-grid algorithms for semilinear elliptic problems and provided the corresponding convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' To our knowledge, there is not any published literature on the iterative two-grid algorithm of conforming finite element discretization for strongly nonlinear elliptic boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' In this paper, an iterative two-grid algorithm for solving strongly nonlinear elliptic problems is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' The discrete system of strongly nonlinear elliptic problems is presented at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' And then, an iterative two-grid algorithm is proposed for the discrete system, which is obtained by applying a non-iterative two-grid algorithm of [8] in a successive fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Finally, a challenging convergence analysis of the proposed algorithm is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Despite the fact that our algorithm is simply obtained by [8], the convergence analysis of the non-iterative two-grid algorithm could not be directly applied to the iterative two-grid algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Here we complete this challenging convergence analysis by mathematical induction which can also be used in solving semilinear elliptic problems by iterative two-grid algorithms in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' However, we must emphasize that the convergence analysis of our algorithm is significantly different from the one of [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Compared with the current work [11], our convergence analysis is far more difficult and complex, and specific challenges could be reflected in: (1) the higher order derivative component of our model problem is still nonlinear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (2) the interconnectedness of the error estimates causes formidable obstacle for the convergence analysis (See the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' To avoid the repeated use of generic but unspecified constants, x ≲ y is used to denote x ⩽ Cy, where C are some positive constants which do not depend on the mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Furthermore the constants C may denote different values under different circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' For some specific constants, we use the constant C with some subscript to denote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' 2 Model problems and discrete systems In this section, we present the continuous and discrete variational problems of strongly nonlinear elliptic problems, and provide the corresponding well-posedness and priori error estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Given a bounded convex polygonal domain Ω ⊂ R2 with the boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' We denote W m,p(Ω) as the standard Sobolev space with norm ∥ · ∥m,p,Ω and seminorm | · |m,p,Ω, where the integers m ⩾ 0 and p ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' For convenience, we also denote Hm(Ω) = W m,2(Ω), ∥·∥m = ∥·∥m,2,Ω and H1 0(Ω) := {u ∈ H1(Ω) : u|∂Ω = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' We consider the following strongly nonlinear elliptic problems: � −∇ · a(x, u, ∇u) + f(x, u, ∇u) = 0, in Ω, u = 0, on ∂Ω, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1) where a(x, y, z) : ¯Ω × R × R2 → R2 and f(x, y, z) : ¯Ω × R × R2 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' When a(x, u, ∇u) and f(x, u, ∇u) take different functions, different problems are available, such as mean curvature flow, Bratu’s problem and so on(See [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' We assume that a(x, y, z) and f(x, y, z) are second order continuous differentiable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' For simplicity, we denote that ay(w) = Dya(x, w, ∇w), az(w) = Dza(x, w, ∇w), fy(w) = Dyf(x, w, ∇w) and fz(w) = Dzf(x, w, ∇w), and similar notations are applied to the second order derivatives of a(x, y, z) and f(x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' An iterative two-grid method for strongly nonlinear elliptic boundary value problems 3 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 Since a(x, y, z) and f(x, y, z) are second order continuous differentiable func- tions, there exists a positive constant ˜C as upper bound with respect to all the first and second order derivatives of a(·, ·, ·) and f(·, ·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' We denote A(v, ϕ) = (a(x, v, ∇v), ∇ϕ) + (f(x, v, ∇v), ϕ), ∀v, ϕ ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2) By Green formula, it’s easy to see that the solution u ∈ H1 0(Ω) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1) satisfies A(u, v) = 0, ∀v ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3) The Fr´echet derivative L′ of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1) at w is given by L′(w)v = −∇ · (ay(w)v + az(w)∇v) + fy(w)v + fz(w)∇v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' In the following, we gives some of our basic assumptions (Similar assumptions also could be found in [9] or [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Firstly, the problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3) has a solution u ∈ H1 0(Ω)∩Hr+1(Ω)∩W 2,2+ε(Ω) (ε > 0 and integer r ⩾ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Secondly, for the solution u of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3), there exists a positive constant α0 such that ξT az(u)ξ ⩾ α0|ξ|2, ∀ξ ∈ R2, x ∈ ¯Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4) Finally, L′(u) : H1 0(Ω) → H−1(Ω) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' These assumptions guarantee that u is an isolated solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Let Th be a conforming quasi-uniform triangulation on Ω, where the mesh size h denotes the maximum of the circumscribed circle diameters of element K ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' By this, any element K ∈ Th is contained in (contains) a circle of radius ˆC1h (respectively, ˆC2h), where the constant ˆC1 and ˆC2 do not depend on mesh size h, and there is no hanging node on Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' The finite element space Vh on Th is defined as Vh = {vh ∈ H1 0(Ω) : vh|K ∈ Pr(K), ∀ K ∈ Th}, where Pr(K) is the set of polynomials of degree at most integer r on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Here is the discrete system of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3): Find uh ∈ Vh such that A (uh, vh) = 0, ∀vh ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) The following lemma presents the well-posedness of the variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) and its priori error estimates, which can be found in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4 of [9], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 Assume u is the solution of problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3), then when h is small enough, the discrete variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) exists a unique solution uh ∈ Vh, and the following priori error estimate holds ∥u − uh∥1,p ≲ hr, if u ∈ W r+1,p(Ω), 2 ⩽ p ⩽ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='6) 3 Iterative two-grid algorithms 4 Jiajun Zhan & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' In this section, we present an iterative two-grid algorithm for the variational problems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Let Th and TH be two quasi-uniform, conforming and nested mesh in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Furthermore the mesh size h of Th and H of TH satisfy, for some 0 < λ < 1, H = O(hλ) and h < H < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' For present the iterative two-grid algorithm, we introduce the form B(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' v, χ) (induced by L′) by , for a fixed w and any v, χ ∈ H1 0(Ω), B(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' v, χ) = (ay(w)v, ∇χ) + (az(w)∇v, ∇χ) + (fy(w)v, χ) + (fz(w)∇v, χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 The form B(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·) is a bilinear form for fixed w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' To our knowledge, the two-grid algorithms of strongly nonlinear problems are firstly pro- posed in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Here one of two-grid algorithms from Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3 of [8] is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Find uH ∈ VH, such that A(uH, vH) = 0, ∀vH ∈ VH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Find uh ∈ Vh, such that B(uH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uh, vh) = B(uH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uH, vh) − A(uH, vh), ∀vh ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 In the Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, we first solve a nonlinear problem in a coarse space VH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' However, because dim(VH) is relatively small, the calculated amount of solving a nonlinear problem in VH is not excessive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' As for the second step of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, noticing that B(uH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·) is a bilinear form with given uH, we simply need to solve a linear problem in Vh, for which there are numerous concerning fast algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' In [8], Xu had showed that the solution uh of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 could be a good approximation with respect to finite element solution uh at a low cost, namely, ∥uh − uh∥1 ≲ H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2) Using triangle inequality, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='6) with r = 1 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2), we obtain the error estimate of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, ∥u − uh∥1 ⩽ ∥u − uh∥1 + ∥uh − uh∥1 ≲ h + H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3) Next, putting the Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 into a successive fashion, we obtain our iterative two-grid algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 Let u0 h = uH be the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) in VH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Assume that uk h ∈ Vh has been obtained, then uk+1 h ∈ Vh can be obtained by the following two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Find ek H ∈ VH such that, for any vH ∈ VH, A(uk h + ek H, vH) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4) Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Find uk+1 h ∈ Vh such that, for any vh ∈ Vh, B(uk h + ek H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uk+1 h , vh) = B(uk h + ek H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uk h + ek H, vh) − A(uk h + ek H, vh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) An iterative two-grid method for strongly nonlinear elliptic boundary value problems 5 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3 Noticing the uniqueness of finite element solution (See Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5), u0 h = uH and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4) with k = 0, we can see that e0 H = 0, which means u0 h + e0 H = uH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' By observing the the Step 2 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 and the second step of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, the conclusion is that Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 is same with Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 when k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' In comparison to [8], our method is still valid for high order conforming finite elements, whereas [8] only considered piecewise linear finite element space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Here gives the error estimate of our algorithm (See Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9), ∥u − uk h∥1 ≲ hr + Hr+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='6) Specially, if we choose finite element space Vh as piecewise linear finite element space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' r = 1, the error estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='6) of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 could be written as ∥u − uk h∥1 ≲ h + H1+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' To achieve the optimal convergence order, the relationship h = H2 should be satisfied in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 (See (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' But in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2, we could expand the distance between the mesh size H and h by increasing the iteration counts k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' 4 Convergence analysis In this section, we provide the corresponding convergence analysis of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' To this end, we need to introduce some preliminaries based on form B(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' v, χ) at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 Some preliminaries based on form B(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' v, χ) In this subsection, we present some properties of form B(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' v, χ) and introduce two discrete Green function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Firstly, with fixed w, by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 and Cauchy-Schwarz inequality, it’s easy to obtain that the form B(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·) is continuous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=', B(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' v, χ) ≲ ∥v∥1∥χ∥1, ∀ v, χ ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1) Secondly, we present a lemma which provides the Babuˇska-Brezzi(BB) conditions of form B(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·) in Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' And this lemma can be proved using similar arguments in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 of [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 Assume u is the solution of problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3), then when h is small enough, we have, for any wh ∈ Vh, ∥wh∥1 ≲ sup vh∈Vh B (u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' wh, vh) ∥vh∥1 and ∥wh∥1 ≲ sup vh∈Vh B (u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' vh, wh) ∥vh∥1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' For the solution u of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3), a projection operator Ph : H1 0(Ω) → Vh is defined by (az(u)∇Phv, ∇χh) = (az(u)∇v, ∇χh), ∀v ∈ H1 0(Ω), χh ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3) By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4), we can know that the projection operator Ph is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Taking v = vh ∈ Vh ⊂ H1 0(Ω) and χh = Phvh − vh, and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4), we could prove that the projection operator Ph is identity operator for space Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Substituting χh = Phv into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3), and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4), Poincar´e inequality, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 and Cauchy–Schwarz inequality, it holds that ∥Phv∥1 ≲ ∥v∥1, ∀v ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4) 6 Jiajun Zhan & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4), duality argument and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4), we can obtain (See Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5 in [2]) ∥v − Phv∥0 ≲ h∥v∥1, ∀v ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) For any wh ∈ Vh, v ∈ H1 0(Ω), by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1), Green formula, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3), Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, Cauchy-Schwarz inequality and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5), we have B(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' wh, v − Phv) = (ay(u)wh, ∇(v − Phv)) + (az(u)∇wh, ∇(v − Phv)) +(fy(u)wh, v − Phv) + (fz(u)∇wh, v − Phv) = ((∇ · ay(u))wh, v − Phv) + (ay(u) · ∇wh, v − Phv) +(fy(u)wh, v − Phv) + (fz(u)∇wh, v − Phv) ≲ ∥wh∥1∥v − Phv∥0 ≲ h∥wh∥1∥v∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='6) Noticing that L′(u) : H1 0(Ω) → H−1(Ω) is an isomorphism, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4), we obtain that ∥wh∥1 ≲ sup v∈H1 0 (Ω) B(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' wh, v) ∥v∥1 ≲ sup v∈H1 0 (Ω) B(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' wh, v − Phv) ∥v∥1 + sup v∈H1 0 Ω B(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' wh, Phv) ∥v∥1 ≲ h∥wh∥1 + sup v∈H1 0 (Ω) B(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' wh, Phv) ∥Phv∥1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Taking h sufficiently small in the above inequality with projection operator Ph being identity operator for Vh, we could obtain the first estimate of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' The proof of the second estimate of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Next, we provide another BB condition of the form B(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 Assume u is the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3) and Ψ satisfying ∥u − Ψ∥1,∞ ≲ H, then when H is small enough, for any wh ∈ Vh, it holds that ∥wh∥1 ≲ sup vh∈Vh B (Ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' wh, vh) ∥vh∥1 and ∥wh∥1 ≲ sup vh∈Vh B (Ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' vh, wh) ∥vh∥1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Using the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1) of form B, Taylor expansion h(y, z) = h(y0, z0)+∂yh(˜θ1, ˜θ2)(y− y0) + ∂zh(˜θ1, ˜θ2)(z − z0), where ˜θ1 is between y and y0 and ˜θ2 is between z and z0, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, and H¨older inequality, we obtain B(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' wh, vh) − B(Ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' wh, vh) = ((ay(u) − ay(Ψ))wh, ∇vh) + ((az(u) − az(Ψ))∇wh, ∇vh) +((fy(u) − fy(Ψ))wh, vh) + ((fz(u) − fz(Ψ))∇wh, vh) = (ayy(θ1)(u − Ψ)wh, ∇vh) + (ayz(θ1)∇(u − Ψ)wh, ∇vh) +(azy(θ2)(u − Ψ)∇wh, ∇vh) + (∇(u − Ψ)T azz(θ2)∇wh, ∇vh) +(fyy(θ3)(u − Ψ)wh, vh) + (fyz(θ3) · ∇(u − Ψ)wh, vh) +(fzy(θ4) · ∇wh(u − Ψ), vh) + (∇(u − Ψ)T fzz(θ4)∇wh, vh) ≲ ∥u − Ψ∥1,∞∥wh∥1∥vh∥1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='8) An iterative two-grid method for strongly nonlinear elliptic boundary value problems 7 where θi (i = 1, 2, 3, 4) are between u and Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='8) and ∥u − Ψ∥1,∞ ≲ H, it is obtained that ∥wh∥1 ≲ sup vh∈Vh B(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' wh, vh) − B (Ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' wh, vh) ∥vh∥1 + sup vh∈Vh B (Ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' wh, vh) ∥vh∥1 ≲ ∥u − Ψ∥1,∞∥wh∥1 + sup vh∈Vh B (Ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' wh, vh) ∥vh∥1 ≲ H∥wh∥1 + sup vh∈Vh B (Ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' wh, vh) ∥vh∥1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Taking H sufficiently small into the above inequality, we can derive the first estimate of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' The proof of the second estimate of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='7) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3 According to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='6), Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 still holds with replacing Ψ by the finite element solution uh of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' For more concise notations and the subsequent analysis, we denote Ek = uh − uk h, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9) uk,1 h = uk h + ek H, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='10) where uh is the solution of problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) and, uk h and ek H are given by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' It’s noted that these notation will be used frequently in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4 For k ⩾ 0, assume that Ek, uk,1 h and ek H are given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='10) and Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' If both ∥Ek∥1,∞ ≲ H and ∥ek H∥1,∞ ≲ H are provided, the Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 still holds with replacing Ψ by uk,1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' In fact, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9), triangle inequality, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='6) with r ⩾ 1 and h < H, ∥Ek∥1,∞ ≲ H and ∥ek H∥1,∞ ≲ H, we derive that ∥u − uk,1 h ∥1,∞ ⩽ ∥u − uh∥1,∞ + ∥Ek∥1,∞ + ∥ek H∥1,∞ ≲ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Therefore, the Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 still holds with Ψ = uk,1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' And then, for the finite element solution uh of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) and any fixed x ∈ Ω, we introduce the Green functions gx H ∈ VH, which be defined by B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' vH, gx H) = ∂vH(x), ∀vH ∈ VH, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='11) where ∂ denotes either ∂ ∂x1 or ∂ ∂x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' It’s easy to see that the Green function gx H is well-defined by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Assume uk,1 h is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='10), similarly, for any fixed x ∈ Ω, we introduce the Green functions gk,x h ∈ Vh by B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' vh, gk,x h ) = ∂vh(x), ∀vh ∈ Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='12) By Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4, we also can see that Green function gk,x h is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Here give some estimates of the above two Green functions gx H and gk,x h (See Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3 of [4], or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='11) of [9]) ∥gx H∥1,1 ≲ | log H| and ∥gk,x h ∥1,1 ≲ | log h|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='13) 8 Jiajun Zhan & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' At last, for any v ∈ H1 0(Ω) ∩ W 1,∞(Ω), using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='11) of [2], it could be obtained that ∥v∥1,∞ ≲ |v|1,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='14) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 Error estimate In this subsection, we present the convergence analysis of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 by a series of lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5 Assume uk,1 h , Ek and ek H are given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9) and Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2, respec- tively, then we have, for any vh ∈ Vh, B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Ek+1, vh) ≲ (∥Ek∥1,∞ + ∥ek H∥1,∞)(∥Ek∥1 + ∥ek H∥1)∥vh∥1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='15) B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Ek+1, vh) ≲ (∥Ek∥2 1,∞ + ∥ek H∥2 1,∞)∥vh∥1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9), Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2), it is obtained that B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Ek+1, vh) = B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uh, vh) − B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uk+1 h , vh) = B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uh, vh) − B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uk h + ek H, vh) +A(uk,1 h , vh) − A(uh, vh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' = B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Ek − ek H, vh) + (a(uk,1 h , ∇uk,1 h ), vh) + (f(uk,1 h , ∇uk,1 h ), vh) −(a(uh, ∇uh), vh) − (f(uh, ∇uh), vh) := A1 − A2 − A3, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='17) where A1 = B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Ek − ek H, vh), A2 = (a(uh, ∇uh), vh) − (a(uk,1 h , ∇uk,1 h ), vh), A3 = (f(uh, ∇uh), vh) − (f(uk,1 h , ∇uk,1 h ), vh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' For A1, using the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1) of B, we have A1 = (ay(uk,1 h )(Ek − ek H), ∇vh) + (az(uk,1 h )∇(Ek − ek H), ∇vh) +(fy(uk,1 h )(Ek − ek H), vh) + (fz(uk,1 h )∇(Ek − ek H), vh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='18) For A2, using second order Taylor expansion, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9), we obtain A2 = (ay(uk,1 h )(Ek − ek H), ∇vh) + (az(uk,1 h )∇(Ek − ek H), ∇vh) +(ayy(θ5)(Ek − ek H)2, ∇vh) + 2(ayz(θ5)∇(Ek − ek H)(Ek − ek H), ∇vh) +(∇(Ek − ek H)T azz(θ5)∇(Ek − ek H), ∇vh), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='19) where θ5 is between uh and uk,1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Similarly for A3, using second order Taylor expansion, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9), it is obtained that A3 = (fy(uk,1 h )(Ek − ek H), vh) + (fz(uk,1 h )∇(Ek − ek H), vh) +(fyy(θ6)(Ek − ek H)2, vh) + 2(fyz(θ6) · ∇(Ek − ek H)(Ek − ek H), vh) +(∇(Ek − ek H)T fzz(θ6)∇(Ek − ek H), vh), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='20) An iterative two-grid method for strongly nonlinear elliptic boundary value problems 9 where θ6 is between uh and uk,1 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Noticing the sum of the first order derivative items about a(·, ·, ·) and f(·, ·, ·) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='19) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='20) exactly equal A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='18), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='19) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='20) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='17), it’s obtained that B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Ek+1, vh) = −(ayy(θ5)(Ek − ek H)2, ∇vh) − 2(ayz(θ5)∇(Ek − ek H)(Ek − ek H), ∇vh) −(∇(Ek − ek H)T azz(θ5)∇(Ek − ek H), ∇vh) − (fyy(θ6)(Ek − ek H)2, vh) −2(fyz(θ6) · ∇(Ek − ek H)(Ek − ek H), vh) −(∇(Ek − ek H)T fzz(θ6)∇(Ek − ek H), vh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='21) Applying Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, H¨older inequality and triangle inequality into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='21), we could obtain B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Ek+1, vh) ≲ ∥Ek − ek H∥1,∞∥Ek − ek H∥1∥vh∥1 ⩽ (∥Ek∥1,∞ + ∥ek H∥1,∞)(∥Ek∥1 + ∥ek H∥1)∥vh∥1, which completes the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Similarly, we could obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='16) by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='6 Assume that uk,1 h , ek H and Ek are defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='10), Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9), respectively, then we have B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ek H, vH) ≲ (∥Ek∥1 + ∥Ek+1∥1)∥vH∥1, ∀vH ∈ VH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='22) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Taking vh = vH into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4), we obtain B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uk+1 h , vH) = B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uk h + ek H, vH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Rewriting the the above equation with Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, and then using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1) and triangle inequality, we have B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ek H, vH) = B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uk+1 h − uk h, vH) = B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uk+1 h − uh + uh − uk h, vH) = B(uk,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Ek − Ek+1, vH) ≲ ∥Ek − Ek+1∥1∥vH∥1 ⩽ (∥Ek∥1 + ∥Ek+1∥1) ∥vH∥1, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='7 Assume that Ek and ek H are given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9) and Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2, respectively, and r ⩾ 1, when h is small enough, then for any integer k ⩾ 1, ∥Ek∥1 ≲ Hr+k, ∥Ek∥1,∞ ≲ | log h|H2, ∥ek H∥1,∞ ≲ H, ∥ek H∥1 ≲ Hr+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='23) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Here we use mathematical induction to prove that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='23) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4), u0 h = uH, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) and the uniqueness of finite element solution (See Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2), it could be seen that e0 H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' 10 Jiajun Zhan & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Making use of triangle inequality, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='6) and h ⩽ H, we have ∥E0∥1 ⩽ ∥u − uh∥1 + ∥u − uH∥1 ≲ hr + Hr ⩽ Hr, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='24) ∥E0∥1,∞ ⩽ ∥u − uh∥1,∞ + ∥u − uH∥1,∞ ≲ hr + Hr ⩽ Hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='25) Next, we will prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='23) is true when k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (i) For ∥E1∥1 ≲ Hr+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Noticing that r ⩾ 1 and e0 H = 0, and using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='25), we have ∥E0∥1,∞ ≲ H and ∥e0 H∥1,∞ ≲ H , which could derive the BB condition of form B(u0,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·) (See Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Using the BB condition of form B(u0,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='15), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='25), ek H = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='24), r ⩾ 1 and H < 1, it’s obtained that ∥E1∥1 ≲ sup vh∈Vh B(u0,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' E1, vh) ∥vh∥1 ≲ (∥E0∥1,∞ + ∥e0 H∥1,∞)(∥E0∥1 + ∥e0 H∥1) ≲ (Hr + 0)(Hr + 0) ≲ Hr+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='26) (ii) For ∥E1∥1,∞ ≲ | log h|H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' For k = 0 and any fixed x ∈ Ω, taking vh = E1 into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='12), using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='16), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='25), e0 H = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='13), r ⩾ 1 and H < 1, we obtain ∂E1(x) = B(u0,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' E1, g0,x h ) ≲ (∥E0∥2 1,∞ + ∥e0 H∥2 1,∞)∥g0,x h ∥1,1 ≲ (H2r + 0)| log h| ≲ | log h|H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Further using the arbitrariness of x and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='14), we derive that ∥E1∥1,∞ ≲ | log h|H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (iii) For ∥e1 H∥1,∞ ≲ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Using ∥E1∥1,∞ ≲ | log h|H2 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 (The specific content of lemma and proof are referred to Appendix), we obtain ∥e1 H∥1,∞ ≲ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (iv) For ∥e1 H∥1 ≲ Hr+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Noticing that ∥E1∥1,∞ ≲ | log h|H2 and ∥e1 H∥1,∞ ≲ H are satisfied, therefore the BB condition of form B(u1,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·) holds (See Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Using the BB condition of B(u1,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='15), it’s obtained that ∥E2∥1 ≲ sup vh B(u1,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' E2, vh) ∥vh∥1 ≲ (∥E1∥1,∞ + ∥e1 H∥1,∞)(∥E1∥1 + ∥e1 H∥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='27) Using the BB condition of form B(u1,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='22) with k = 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='27), ∥E1∥1,∞ ≲ | log h|H2 and ∥e1 H∥1,∞ ≲ H, we have ∥e1 H∥1 ≲ sup vH∈VH B(u1,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' e1 H, vH) ∥vH∥1 ≲ ∥E1∥1 + ∥E2∥1 ≲ ∥E1∥1 + (∥E1∥1,∞ + ∥e1 H∥1,∞)(∥E1∥1 + ∥e1 H∥1) ≲ ∥E1∥1 + (| log h|H2 + H)(∥E1∥1 + ∥e1 H∥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' An iterative two-grid method for strongly nonlinear elliptic boundary value problems 11 Taking H be small enough in the above inequality, and using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='26), it’s obtained that ∥e1 H∥1 ≲ ∥E1∥1 ≲ Hr+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' We assume (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='23) is true when k = l, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=', ∥El∥1 ≲ Hr+l, ∥El∥1,∞ ≲ | log h|H2, ∥el H∥1,∞ ≲ H, ∥el H∥1 ≲ Hr+l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='28) Next, we will prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='23) also holding when k = l + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (i) For ∥El+1∥1 ≲ Hr+l+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Noticing that ∥El∥1,∞ ≲ | log h|H2 and ∥el H∥1,∞ ≲ H are satisfied, therefore the BB condition of form B(ul,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·) holds (See Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Using the BB condition of form B(ul,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='15), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='28) and H < 1, we obtain ∥El+1∥1 ≲ sup vh∈Vh B(ul,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' El+1, vh) ∥vh∥1 ≲ (∥El∥1,∞ + ∥el H∥1,∞)(∥El∥1 + ∥el H∥1) ≲ (| log h|H2 + H)(Hr+l + Hr+l) ≲ Hr+l+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='29) (ii) For ∥El+1∥1,∞ ≲ | log h|H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Taking vh = El+1 into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='12) with k = l, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='16), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='28) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='13), we obtain ∂El+1(x) = B(ul,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' El+1, gl,x h ) ≲ (∥El∥2 1,∞ + ∥el H∥2 1,∞)∥gl,x h ∥1,1 ≲ (| log h|2H4 + H2)| log h| ≲ | log h|H2, which combining the arbitrariness of x and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='14), it could be derived that ∥El+1∥1,∞ ≲ | log h|H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (iii) For ∥el+1 H ∥1,∞ ≲ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Using ∥El+1∥1,∞ ≲ | log h|H2 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, we obtain ∥el+1 H ∥1,∞ ≲ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='30) (iv) For ∥el+1 H ∥1 ≲ Hr+l+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Noticing that ∥El+1∥1,∞ ≲ | log h|H2 and ∥el+1 H ∥1,∞ ≲ H are satisfied, therefore the BB condition of form B(ul+1,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·) holds (See Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Using the BB condition of form B(ul+1,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='15), it’s obtained that ∥El+2∥1 ≲ sup vh B(ul+1,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' El+2, vh) ∥vh∥1 ≲ (∥El+1∥1,∞ + ∥el+1 H ∥1,∞)(∥El+1∥1 + ∥el+1 H ∥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='31) Using the BB condition of form B(ul+1,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='22) with k = l + 1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='31), ∥El+1∥1,∞ ≲ | log h|H2 and ∥el+1 H ∥1,∞ ≲ H, we have ∥el+1 H ∥1 ≲ sup vH∈VH B(ul+1,1 h ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' el+1 H , vH) ∥vH∥1 ≲ ∥El+1∥1 + ∥El+2∥1 ≲ ∥El+1∥1 + (∥El+1∥1,∞ + ∥el+1 H ∥1,∞)(∥El+1∥1 + ∥el+1 H ∥1) ≲ ∥El+1∥1 + (| log h|H2 + H)(∥El+1∥1 + ∥el+1 H ∥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' 12 Jiajun Zhan & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Taking H be small enough in the above inequality and using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='29), it’s obtained that ∥el+1 H ∥1 ≲ ∥El+1∥1 ≲ Hr+l+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' By mathematical induction, the conclusion is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='8 Although we just use the estimation ∥Ek+1∥1 ≲ Hr+k+1 in our main result (See Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9), the availability of ∥Ek+1∥1 ≲ Hr+k+1 requires the support of ∥Ek∥1,∞ ≲ | log h|H2, ∥ek H∥1,∞ ≲ H and ∥ek H∥1 ≲ Hr+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Here gives the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9 Assume that u is the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3) and uk h is given by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2, then we have ∥u − uk h∥1 ≲ hr + Hr+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='32) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Using triangle inequality, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='6) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='7, we could obtain that ∥u − uk h∥1 ⩽ ∥u − uh∥1 + ∥Ek∥1 ≲ hr + Hr+k, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' 5 Numerical experiments In this section, we present some numerical experiments to show the efficiency of the pro- posed iterative two-grid algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' We implemented these experiments by the software package FEALPy of programming language Python [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Specially in the Step 1 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2, we solve the nonlinear systems by Newton iteration methods with relative residual 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' We adopt the following mean curvature flow problem as our model problem: −∇ · � ∇u (1 + |∇u|2)1/2 � = g in Ω, u = 0 on ∂Ω, where the computational domain Ω = (0, 1)× (0, 1), the exact solution u = x(1 − x)2y(1 − y)ex, and g is so chosen according to the exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Firstly, we choose conforming piecewise linear finite element space as Vh, namely choose r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' According to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9, we should keep hr = Hr+k hold in order to achieve the optimal convergence order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Therefore in Table 1, we present some numerical results in different mesh size with h = H2 for k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' In this case, our algorithm is same with Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 (See Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Furthermore, we could observe that ∥u−u1 h∥1 ∗max{H2, h}−1 are stable in Table 1, which agrees with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='32) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Table 1: k = 1, r = 1 H h ∥u − u1 h∥1 ∥u − u1 h∥1 ∗ max{H2, h}−1 1/9 1/81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='74E-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='221953 1/10 1/100 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='22E-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='221967 1/11 1/121 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='83E-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='221977 1/12 1/144 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='54E-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='221983 An iterative two-grid method for strongly nonlinear elliptic boundary value problems 13 And then, we increase the iterative counts k to expand the distance between H and h, which is shown in Tables 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' We also observe that ∥u − u1 h∥1 ∗ max{H1+k, h}−1 are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Table 2: k = 2, r = 1 H h ∥u − u2 h∥1 ∥u − u2 h∥1 ∗ max{H3, h}−1 1/3 1/27 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='20E-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='221524 1/4 1/64 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='47E-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='221784 1/5 1/125 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='77E-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='221826 1/6 1/216 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='03E-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='221836 Table 3: k = 3, r = 1 H h ∥u − u3 h∥1 ∥u − u3 h∥1 ∗ max{H4, h}−1 1/2 1/16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='38E-02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='220944 1/3 1/81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='74E-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='221805 1/4 1/256 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='67E-04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='221837 At last, we implement similar numerical experiments for high order finite element space with r = 2 and r = 3 in Tables 4-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' By observation, all these numerical experiments are in support of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='32) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Table 4: k = 1, r = 2 H h ∥u − u1 h∥1 ∥u − u1 h∥1 ∗ max{H3, h2}−1 1/4 1/8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='57E-03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='164578 1/9 1/27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='30E-04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='167320 1/16 1/64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='09E-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='167553 1/25 1/125 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='07E-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='167590 1/36 1/216 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='59E-06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='167599 Acknowledgements The work of the first and second authors were partially funded by the Science and Technology Development Fund, Macau SAR (Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' 0070/2019/A2, 0031/2022/A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' The third author was supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' 11901212).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' The third and fourth authors are also supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' 12071160).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' References [1] Bi C J, Wang C, Lin Y P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' A posteriori error estimates of two-grid finite element methods for nonlinear elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' J Sci Comput, 2018, 74: 23–48 [2] Ciarlet P G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' The Finite Element Method for Elliptic Problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Classics in Applied Mathematics, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' SIAM, Philadelphia, 2002 [3] Gudi T, Nataraj N, Pani A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Numer Math, 2008, 109: 233–268 [4] Thom´ee V, Xu J C, Zhang N Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Superconvergence of the gradient in piecewise linear finite-element approx- imation to a parabolic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' SIAM J Numer Anal, 1989, 26: 553–573 [5] Wei H Y, Huang Y Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Fealpy: Finite element analysis library in python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='com/weihuayi/ fealpy, Xiangtan University, 2017-2021 [6] Xu J C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Iterative methods by SPD and small subspace solvers for nonsymmetric or indefinite problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' In: Proceedings of the 5th International Symposium on Domain Decomposition Methods for Partial Differential 14 Jiajun Zhan & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Table 5: k = 2, r = 2 H h ∥u − u2 h∥1 ∥u − u2 h∥1 ∗ max{H4, h2}−1 1/8 1/64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='09E-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='167552 1/9 1/81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='55E-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='167571 1/10 1/100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='68E-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='167582 1/11 1/121 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='14E-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='167589 1/12 1/144 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='08E-06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='167593 Table 6: k = 3, r = 2 H h ∥u − u3 h∥1 ∥u − u3 h∥1 ∗ max{H5, h2}−1 1/5 1/55 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='54E-05 0.' metadata={'source': 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+page_content='162211 1/9 1/243 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='84E-06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='167600 Table 7: k = 1, r = 3 H h ∥u − u1 h∥1 ∥u − u1 h∥1 ∗ max{H4, h3}−1 1/8 1/16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='83E-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='075054 1/27 1/81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='40E-07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='074662 1/64 1/256 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='44E-09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='074543 Table 8: k = 2, r = 3 H h ∥u − u2 h∥1 ∥u − u2 h∥1 ∗ max{H5, h3}−1 1/8 1/32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='28E-06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='074870 1/9 1/36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='60E-06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='074838 1/10 1/40 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='85E-07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='074703 1/9 1/81 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='40E-07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='074662 1/10 1/100 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='46E-08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='074630 1/11 1/121 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='21E-08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='074606 1/12 1/144 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='50E-08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='074588 An iterative two-grid method for strongly nonlinear elliptic boundary value problems 15 Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Siam, Philadelphia, 1992, 106–118 [7] Xu J C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' A novel two-grid method for semilinear elliptic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' SIAM J Sci Comput, 1994, 15: 231–237 [8] Xu J C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Some two-grid finite element methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' In: Domain Decomposition Methods in Science and Engi- neering (Quarteroni, Alfio and P´eriaux, Jacques and Kuznetsov, Yuri A and Widlund, Olof B eds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Contemp Math, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' 157, Amer Math Soc, 1994, 79–87 [9] Xu J C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Two-grid discretization techniques for linear and nonlinear PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' SIAM J Numer Anal, 1996, 33: 1759–1777 [10] Xu J C, Cai X C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' A preconditioned GMRES method for nonsymmetric or indefinite problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Math Comp, 1992, 59: 311–319 [11] Zhang W F, Fan R H, Zhong L Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Iterative two-grid methods for semilinear elliptic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Comput Math Appl, 2020, 80: 522–530 Appendix A The purpose of this appendix is to provide the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 Assume ek H is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4) and ∥Ek∥1,∞ ≲ | log h|H2, when H is small enough, it holds that ∥ek H∥1,∞ ≲ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1) Before we present the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, we need to introduce some preliminaries and lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' For the finite element solution uh of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5), we introduce a projection operator ˆPH : H1 0(Ω) → VH, which be defined by, B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ˆPHw, vH) = B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' w, vH), ∀w ∈ H1 0(Ω), vH ∈ VH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2) It’s easy to derive that ˆPH is well-defined by the BB-conditions of form B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·) which could be obtained by Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Furthermore, the projection operator ˆPH satisfies the following estimate ∥ ˆPHw∥1,∞ ≲ | log H|∥w∥1,∞, ∀w ∈ W 1,∞(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3) In fact, taking vH = ˆPHw in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='11), and using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1), Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, H¨older inequality and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='13), we obtain ∂ ˆPHw(x) = B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ˆPHw, gx H) = B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' w, gx H) = (ay(uh)w, ∇gx H) + (az(uh)∇w, ∇gx H) + (fy(uh)w, gx H) + (fz(uh)∇w, gx H) ≲ ∥w∥1,∞∥gx H∥1,1 ≲ | log H|∥w∥1,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Finally using of the arbitrariness of x and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='14), we could obtain (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' By Taylor expansion, we have (the detailed proof can be found in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 of [9]) A(w, χ) = A(v, χ) + B(v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' w − v, χ) + R(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' v, w, χ), ∀w, v, χ ∈ H1 0(Ω), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4) where the forms A(·, ·) and B(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·) are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1), respectively, η = v + t(w − v) and R(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' v, w, χ) = � 1 0 � (ayy(η)(v − w)2, ∇χ) + 2(ayz(η)∇(v − w)(v − w), ∇χ) +(∇(v − w)T azz(η)∇(v − w), ∇χ) + (fyy(η)(v − w)2, χ) +2(fyz(η) · ∇(v − w)(v − w), χ) + (∇(v − w)T fzz(η)∇(v − w), χ) � (1 − t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' For the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, we introduce a operator Φ as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Assume uh is the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5), Ek, R, uk h are given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4) and Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2, respectively, we defined operator Φ : VH → VH by, for any wH ∈ VH, B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Φ(wH), vH) = B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Ek, vH) − R(uh + t(wH − Ek);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uh, uk h + wH, vH), ∀vH ∈ VH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) By the BB-conditions of form B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·) (See Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3), it’s easy to prove that operator Φ is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' We define a space QH = {vH ∈ VH : ∥vH − ˆPHEk∥1,∞ ⩽ H}, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='6) where ˆPH is a projection operator defined by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Since QH is a finite dimensional space, it’s easy to see that QH is a non-empty compact convex subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Next, we will use Brouwer fixed point theorem to prove that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) has a fixed point ¯wH in QH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 Assume ∥Ek∥1,∞ ≲ | log h|H2, then when H is small enough, we have Φ(QH) ⊂ QH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' 16 Jiajun Zhan & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' For any wH ∈ QH, vH ∈ VH, rewriting (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2), we have B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Φ(wH) − ˆPHEk, vH) = −R(uh + t(wH − Ek);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uh, uk h + wH, vH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='7) Substituting vH = Φ(wH) − ˆPHEk into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='11) and using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='7), Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, H¨older inequality, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9), triangle inequality, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='13), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='6) and ∥Ek∥1,∞ ≲ | log h|H2, it is obtained that ∂(Φ(wH) − ˆPHEk)(x) = B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Φ(wH) − ˆPHEk, gx H) = −R(uh + t(wH − Ek);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uh, uk h + wH, gx H) ≲ ∥Ek − wH∥2 1,∞∥gx H∥1,1 ≲ (∥Ek − ˆPHEk∥2 1,∞ + ∥ ˆPHEk − wH∥2 1,∞)| log H| ≲ ((1 + | log H|)2∥Ek∥2 1,∞ + H2)| log H| ≲ ((1 + | log H|)2| log h|2H4 + H2)| log H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Further using the arbitrariness of x and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='14), the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3 Assume ∥Ek∥1,∞ ≲ | log h|H2, then the operator Φ is continuous in VH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' For any w1, w2 ∈ QH, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5), we have B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Φ(w1) − Φ(w2), vH) = R(uh + t(w2 − Ek);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uh, uk h + w2, vH) −R(uh + t(w1 − Ek);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uh, uk h + w1, vH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='8) Noticing that the definition of R in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4), for the terms concerning ayy on the right hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='8), we can use Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 and H¨older inequality to obtain that (ayy(uh + t(w2 − Ek))(Ek − w2)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ∇vH) − (ayy(uh + t(w1 − Ek))(Ek − w1)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ∇vH) = (ayy(uh + t(w2 − Ek))(Ek − w2)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ∇vH) −(ayy(uh + t(w1 − Ek))(Ek − w2)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ∇vH) +(ayy(uh + t(w1 − Ek))(Ek − w2)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ∇vH) −(ayy(uh + t(w1 − Ek))(Ek − w1)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ∇vH) = ([ayy(uh + t(w2 − Ek)) − ayy(uh + t(w1 − Ek))] (Ek − w2)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ∇vH) +(ayy(uh + t(w1 − Ek)) � −2Ekw2 + w2 2 + 2Ekw1 − w2 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ∇vH) = ([ayy(uh + t(w2 − Ek)) − ayy(uh + t(w1 − Ek))] (Ek − w2)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ∇vH) +(ayy(uh + t(w1 − Ek)) (2Ek − w1 − w2) (w1 − w2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ∇vH) ≲ ∥ayy(uh + t(w2 − Ek)) − ayy(uh + t(w1 − Ek))∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='∞∥(Ek − w2)2∥0∥vH∥1 +∥2Ek − w1 − w2∥0∥w1 − w2∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='∞∥vH∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9) For ∥(Ek − w2)2∥0, we use triangle inequality, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='6) and ∥Ek∥1,∞ ≲ | log h|H2, it’s obtained that ∥(Ek − w2)2∥0 ⩽ ∥Ek − w2∥2 1,∞ ≲ ∥Ek∥2 1,∞ + ∥ ˆPHEk∥2 1,∞ + ∥ ˆPHEk − w2∥2 1,∞ ≲ ∥Ek∥2 1,∞ + | log H|2∥Ek∥2 1,∞ + H2 ≲ | log h|2H4 + | log H|2| log h|2H4 + H2 := C1(H), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='10) where C1(H) is a constant depending on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Similarly, for ∥2Ek − w1 − w2∥0, there also exists a constant C2(H) such that ∥2Ek − w1 − w2∥0 ≲ C2(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='11) Substituting (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='10) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='11) into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9), it’s could be obtained that (ayy(uh + t(w2 − Ek))(Ek − w2)2, ∇vH) − (ayy(uh + t(w1 − Ek))(Ek − w1)2, ∇vH) ≲ C(H) � ∥ayy(uh + t(w2 − Ek)) − ayy(uh + t(w1 − Ek))∥0,∞ +∥w1 − w2∥0,∞ � ∥vH∥1, where C(H) = max{C1(H), C2(H)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' The rest of the items on the right hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='8) have similar results, and here is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' The conclusion follows from the above discussion, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='8), the BB-conditions of An iterative two-grid method for strongly nonlinear elliptic boundary value problems 17 form B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ·, ·) (See Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3) and the continuity of second order derivatives of a(·, ·, ·) and f(·, ·, ·) (See the assumptions about a(·, ·, ·) and f(·, ·, ·) in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' At last, we present the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1 by Brouwer fixed point theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Making use of Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3 and Brouwer fixed point theorem, we know that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) exists a fixed point ¯wH in QH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Taking w = uk h + ¯wH, v = uh and χ = vH into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4), and then using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5) with VH ⊂ Vh, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='9), ¯wH = Φ( ¯wH) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='5), we obtain that A(uk h + ¯wH, vH) = A(uh, vH) + B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uk h + ¯wH − uh, vH) + R(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uh, uk h + ¯wH, vH) = B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' ¯wH, vH) − B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Ek, vH) + R(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uh, uk h + ¯wH, vH) = B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Φ( ¯wH), vH) − B(uh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' Ek, vH) + R(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' uh, uk h + ¯wH, vH) = 0, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='12) where η = uh + t( ¯wH − Ek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' By the uniqueness of finite element solution (See Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='4) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='12), we can see that ¯wH = ek H, which implies ek H ∈ QH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content=' At last, using triangle inequality, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='6), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} +page_content='3) and ∥Ek∥1,∞ ≲ | log h|H2, we obtain ∥ek H∥1,∞ ⩽ ∥ek H − ˆPHEk∥1,∞ + ∥ ˆPHEk∥1,∞ ≲ H + | log H|∥Ek∥1,∞ ≲ H + | log H|| log h|H2, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE1T4oBgHgl3EQfDQOp/content/2301.02875v1.pdf'} diff --git a/DdAzT4oBgHgl3EQfif0c/vector_store/index.pkl b/DdAzT4oBgHgl3EQfif0c/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..07ecb8ee2b77fc8ec1979f74f5598f4e0bb6eb28 --- /dev/null +++ b/DdAzT4oBgHgl3EQfif0c/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c11ae16beb401dfd6c2dd54589e229d3e5be68e94f672cfbdb9460089c6eefb2 +size 199820 diff --git a/F9AzT4oBgHgl3EQfi_2P/content/tmp_files/2301.01509v1.pdf.txt b/F9AzT4oBgHgl3EQfi_2P/content/tmp_files/2301.01509v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e8b8666acd15390ddc817de72f0acbfbe6b7507c --- /dev/null +++ b/F9AzT4oBgHgl3EQfi_2P/content/tmp_files/2301.01509v1.pdf.txt @@ -0,0 +1,1201 @@ +Magnetization dynamics with time-dependent spin-density functional theory: +significance of exchange-correlation torques +Daniel Hill, Justin Shotton, and Carsten A. Ullrich∗ +Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA +(Dated: January 5, 2023) +In spin-density-functional theory (SDFT) for noncollinear magnetic materials, the Kohn-Sham +system features exchange-correlation (xc) scalar potentials and magnetic fields. The significance of +the xc magnetic fields is not very well explored; in particular, they can give rise to local torques +on the magnetization, which are absent in standard local and semilocal approximations. +Exact +benchmark solutions for a five-site extended Hubbard lattice at half filling and in the presence of +spin-orbit coupling are compared with SDFT results obtained using orbital-dependent exchange- +only approximations. The magnetization dynamics following short-pulse excitations is found to be +reasonably well described in the exchange-only approximation for weak to moderate interactions. +For stronger interactions and near transitions between magnetically ordered and frustrated phases, +exchange and correlation torques tend to compensate each other and must both be accounted for. +I. +INTRODUCTION +Spin dynamics in magnetic systems is a research area +of much current activity. Spintronics [1], which is con- +cerned with the manipulation of electronic spins, spin +currents, spin textures, and spin excitations, has created +a wealth of scientific knowledge and many avenues for +new technologies. +Prominent examples are spin waves +for encoding and transmitting information (magnonics) +[2, 3], skyrmions for magnetic information storage [4– +8], and single-spin qubits for quantum computation [9]. +Another related area of much interest is ultrafast demag- +netization induced by femtosecond laser pulses [10–14]. +Computational approaches to simulate magnetization +dynamics in a wide variety of systems are typically based +on the Landau-Lifshitz-Gilbert (LLG) equation of motion +[15, 16]. The LLG equation provides a classical descrip- +tion of the time evolution of the magnetization vector +m(t) in response to a time-dependent perturbation (typ- +ically, a short pulse or a periodic driving field) or evolving +from a nonequilibrium initial state. +Materials proper- +ties such as anisotropy, deformations, strain, and various +forms of damping can be built into the LLG approach +via phenomenological or “second-principles” parameters. +In this paper, we are less concerned with these spe- +cific materials properties; instead of LLG we will use a +fully quantum mechanical description of the electronic +charge and spin degrees of freedom, and our focus will +be specifically on the impact of electron-electron interac- +tions on the magnetization dynamics. To be more clear, +we consider a system of N interacting electrons under +the influence of a time-dependent scalar potential V (r, t) +and a time-dependent magnetic field B(r, t) which cou- +ples only to the electron spin (and not to orbital motion). +∗ ullrichc@missouri.edu +The associated many-body Hamiltonian is given by +ˆH = +N +� +j +� +−∇2 +j +2 + V (rj, t) + σj · B(rj, t) +� ++ 1 +2 +N +� +j̸=k +1 +|rj − rk| , +(1) +where σj is the vector of Pauli matrices acting on the +spin of the jth electron, and we define the magnetic field +strength such that the Bohr magneton, µB = e¯h/2m, +does not explicitly appear in the Hamiltonian ˆH. +We +use atomic units (e = m = ¯h = 4πϵ0 = 1) throughout. +From the Heisenberg equation of motion for ˆH, Capelle +et al. showed that the magnetization has the following +time evolution [17]: +dm(r, t) +dt ++ ˆ∇ · J(r, t) = m(r, t) × B(r, t) , +(2) +where J(r, t) is the spin-current tensor. Equation (2) is +exact but not very helpful in practice since J(r, t) re- +quires the many-body wave function associated with ˆH. +A more practical (but still in principle exact) alternative +is time-dependent spin-density functional theory (TD- +SDFT). The idea of TD-SDFT is to consider an auxiliary +system of noninteracting fermions, acted upon by an “ef- +fective” scalar potential and magnetic field, Veff(r, t) and +Beff(r, t), such that the same density n(r, t) and magne- +tization m(r, t) are produced as in the physical system. +The resulting equation of motion, the TD-SDFT coun- +terpart to Eq. (2), is [17] +dm(r, t) +dt ++ ˆ∇ · JKS(r, t) = m(r, t) × Beff(r, t) . +(3) +Here, JKS(r, t) is the Kohn-Sham spin-current tensor, +which is easily determined from the noninteracting wave +function, and the effective magnetic field is defined as +Beff(r, t) = B(r, t) + Bxc(r, t), where the exchange- +correlation (xc) magnetic field Bxc is a functional of the +arXiv:2301.01509v1 [cond-mat.str-el] 4 Jan 2023 + +2 +density and magnetization. Formally, m(r, t) is the same +in Eqs. (2) and (3), but J and JKS are in general dif- +ferent (the difference lies in the transverse component). +Thus, the so-called xc torque, +τxc(r, t) = m(r, t) × Bxc(r, t) , +(4) +ensures that TD-SDFT produces the correct magnetiza- +tion dynamics [17]. +While all of this is clear at the formal level, the ex- +act form of Bxc is unknown and must be approximated +in practice. This immediately raises several questions: +which approximations of Bxc are available, and do they +produce xc torques? +And, how important are the xc +torques for the magnetization dynamics? +A number of approximations for Bxc have been derived +within ground-state SDFT for noncollinear magnetism +[18–20]; via the adiabatic approximation, they immedi- +ately carry over to TD-SDFT. The most widely used ap- +proach, pioneered by K¨ubler et al. [21, 22] and imple- +mented in many popular electronic structure codes, is +to use standard local or semilocal xc functionals such as +the local spin-density approximation (LSDA) or general- +ized gradient approximations (GGAs), and assume a lo- +cal spin quantization axis which is aligned with the local +magnetization vector m(r, t); this produces a Bxc(r, t) +that is parallel to m(r, t) everywhere. We see right away +from Eq. (4) that this class of approximations does not +produce any xc torques. +Approximations for Bxc that do include xc torque ef- +fects can be constructed in several ways. +Existing lo- +cal and semilocal functionals (LSDA and GGAs) have +been modified [23–26] or used in a source-free construc- +tion [27], and new gradient-corrected functionals were +constructed based using the spin-spiral state of the elec- +tron gas as reference system [28–30]. +More consistent +derivations of xc meta-GGAs, starting from noncollinear +generalizations of the exchange hole and the two-body +density matrix, were recently presented [31, 32]. Vari- +ous orbital-dependent functionals were generalized to the +case of noncollinear magnetization [33–35]. +Existing applications of ground-state SDFT to non- +collinear magnetic materials [25, 26, 33] and model sys- +tems [36] seem to suggest that xc torques are of relatively +minor importance for magnetic structure and energetics, +although the torques themselves may not be insignificant +[32]. On the other hand, there are good reasons to expect +that xc torques will be more impactful for magnetization +dynamics: they explicitly appear in the equation of mo- +tion, Eq. (3), and even if τxc(r, t) is relatively small at +a given r and t, its effect can accumulate over time. So +far, however, there has been no systematic attempt to +assess this hypothesis. We are only aware of one study +in the literature, where Dewhurst et al. [37] used their +source-free Bxc functional to simulate laser-induced spin +dynamics in bulk Co and Ni and Co-Pt and Ni-Pt inter- +faces. They found that xc torques were significant only +if they are not overshadowed by magnetic anisotropy ef- +fects (i.e., in bulk, and not at interfaces), and that they +FIG. 1. +Geometry of the 5-site Hubbard cluster used in +this work. The arrows indicate the ordering of the nearest- +neighbor sum in Eq. (6), accounting for the directional hop- +ping due to SOC. +give rise to rather slow spin rotation compared to other +forms of spin dynamics, induced optically or via spin- +orbit coupling (SOC). +In this paper, our goal is to assess the importance of +xc torques in frustrated magnetic systems. +Exchange- +frustrated solids such as spin glasses and kagome antifer- +romagnetic lattices are characterized by many competing +noncollinear spin configurations and quantum spin liquid +phases [38–40], and may therefore exhibit an enhanced +sensitivity to subtle xc torque effects. Needless to say, ex- +tended spin frustrated solids are challenging to describe, +and exact or quasi-exact benchmark results are hard to +come by. We will therefore limit ourselves to small model +systems which capture the spirit of spin frustration and +yet are computationally manageable. +Here, we will consider small Hubbard-type model +systems along similar lines as in our earlier studies +[35, 36, 41]; by including SOC we can generate intrin- +sically noncollinear ground states. In particular, we will +focus on a five-site half-filled Hubbard bowtie as a mini- +mal model for studying xc torque effects in the presence +of magnetic frustration. We will generate both exact and +SDFT phase diagrams of spin configurations for this sys- +tem and explore the spin dynamics for different config- +urations in the phase diagram. +The TD-SDFT treat- +ment will be based on orbital-dependent exchange-only +functionals, and we will compare with exact solutions +of the many-body time-dependent Schr¨odinger equation. +Focusing on a few representative case studies, we will +gain insight into the significance of xc torques in differ- +ent regimes. +The paper is organized as follows. In Sec. II the ex- +tended Hubbard model and the SDFT framework are in- +troduced and the exact and SDFT magnetic phase dia- +grams are discussed. In Sec. III we describe some techni- +cal aspects of the TD-SDFT modeling such as the choice +of initial state. +In Sec. IV the results of exact diago- +nalization and SDFT models are compared for the cases +with moderate to strong correlations and non-local inter- +actions. Conclusions are given in Sec. V. + +2 +4 +3 +53 +II. +EXACT AND SDFT MAGNETIC +STRUCTURE OF HUBBARD CLUSTERS +A. +Definition of the model +In this paper we limit ourselves to (TD-)SDFT in the +exchange-only approximation. As discussed earlier [36], +the standard Hubbard model with on-site interactions +does not give rise to any exchange torques. If one wishes +to study exchange torque effects it is necessary to work +with an extended Hubbard model instead. We will con- +sider, in the following, a half-filled 5-site Hubbard cluster +in a bowtie shape, as shown in Fig. 1. Here, we go be- +yond Ref. [36] and include SOC through a modification +of the kinetic-energy operator, where the hopping term +becomes complex and the hopping acquires a directional- +ity [42, 43]. Thus, our inhomogeneous extended Hubbard +model with SOC is described by the Hamiltonian +ˆHmodel = ˆHT + ˆHU + ˆHext . +(5) +The first term is a hopping term with SOC absorbed into +a spin dependent phase factor, +ˆHT = −th +� +⟨j,j′⟩ +� +σ +e−iσθc† +jσcj′σ + h.c., +(6) +where h.c. stands for Hermitian conjugate. Here, th = +√ +T 2 + C2 is the generalized hopping strength parameter +which depends on nearest neighbor hopping strength T +and spin orbit coupling C, j is the site index for the +geometry shown in Fig. 1, cjσ is the annihilation operator +for an electron of spin σ at site j, the brackets ⟨. . .⟩ denote +an ordered sum over nearest neighbors with the order +indicated by the arrows in Fig. 1, and σ = ±1 labels +spin-up and -down. +Furthermore, θ is the SOC angle +which parameterizes the strength of the SOC parameter +C relative to the conventional hopping term T [44–46]. +The second term in the model Hamiltonian (5) com- +prises the on-site and nearest-neighbor interaction terms, +ˆHU = U0 +� +j +nj↑nj↓ + U1 +� +⟨j,j′⟩ +� +σ,σ′ +njσnj′σ′ , +(7) +where njσ = c† +jσcjσ is the spin σ particle number density +at site j, and U0 and U1 are the on-site and nearest- +neighbor repulsion strengths, respectively. For the pur- +poses of this paper, we set U1 = +1 +2U0, a fairly typical +choice for modeling real materials [47], and we restrict +the hopping parameter th and on-site interaction param- +eter U0 to be of similar orders of magnitude. Finite non- +local interactions are necessary for nontrivial exchange +torques, but we avoid the much stronger interactions +regime because the charge degrees of freedom tend to +freeze out as U0 and U1 become large, resulting in the +dynamics being dominated by a simpler pure-spin low- +energy effective model. +Lastly, ˆHext contains the couplings to the external po- +tential and external magnetic field, +ˆHext = +� +j +(Vjnj + Bj · mj) , +(8) +where Vj is the scalar potential and Bj is the magnetic +field on site j, the total density is nj = nj↑ + nj↓, and +the magnetization is given by mj = � +σ,σ′ c† +jσ⃗σσσ′cjσ′ +with ⃗σ = (σx, σy, σz) denoting a vector composed of +the Pauli matrices. We keep the external field param- +eters each less than the on-site interaction and hopping, +Vj, |Bj| < U0, th. +These external field parameters are +not strictly set to zero because they can be used to break +degeneracy in order to fix a symmetry breaking state, +and because, as discussed in Section III, small variation +of these parameters in the exact model is found to be +useful in matching the SDFT initial state and the exact +initial state more accurately. +B. +Magnetic phase diagram of the Hubbard bowtie +We use exact diagonalization of ˆHmodel to construct +benchmark solutions with which to compare our SDFT +results. Figure 2a shows the exact phase diagram of the +half-filled Hubbard bowtie in a plane whose x − y axes +are defined by C = (th/U0) sin θ and T = (th/U0) cos θ; +the SOC angle θ is here measured with respect to the +kinetic energy axis. Similar phase diagrams for the half- +filled Hubbard trimer were obtained by Tabrizi et al. +[43]. Within the above specified regime the model has +a phase transition at θc = nπ/3 for any integer n. For +the case of zero external fields, the ground state of the +5-site model at half filling is degenerate and magnetically +ordered with a nontrivial noncollinear spin structure (ex- +cept at isolated points in the phase diagram where the +spins are ferromagnetically aligned) indicating magnetic +frustration. +On the phase boundary, θc, the ground state exhibits +a symmetry breaking charge density wave (CDW) in the +form of a spontaneous charge polarization along the x- +axis of Fig. 1. In Fig. 2a the states shown outside the +phase diagram image are the states at the critical angles +θc. A specific choice of charge polarization is depicted +in order to show the corresponding spin state. The sites +with no spin indicated do not necessarily have zero mag- +netic moment, but it tends to be orders of magnitude +smaller. The states shown inside the shaded segments of +the phase diagram are those of the midpoint angles be- +tween the phase boundaries, e.g. θ = 30◦. As θ changes, +the relative angles of the spins change as well, with the +fastest changes occurring in the vicinity of the phase tran- +sitions. Thus, the phase transitions at θc are not discon- +tinuous, rather they appear to be a zero temperature, +finite model analog of a second order phase transition, +although the continuous transition occurs over a rather +narrow range of θ. + +4 +T +C +60∘ +T +60∘ +C +a +b +CDW +CDW +CDW +CDW +CDW +CDW +FIG. 2. +(a) Magnetic phase diagram of the half-filled 5-site Hubbard model, obtained using exact diagonalization. +The +red arrows indicate the relative in-plane spin direction of the state depicted (taken at the midpoint angle between the phase +boundaries). The blue pluses and minuses indicate the direction of electric polarization for the CDW critical angle states +for the specific spin arrangement shown. (b) Corresponding magnetic phase diagram using exchange-only SDFT, showing the +broadening of the phase boundary states. The phase diagram has approximately the same states as for the exact diagonalization +phase diagram, but the critical angles, where a CDW occurs, acquire a width of a few degrees. +The complete phase diagram of the ground state of +our 5-site Hubbard bowtie and other finite and extended +triangular lattice systems is of interest in and by itself, +especially with respect to their symmetries. +A more +complete formal analysis of the phase boundaries and +other symmetry-related properties will be the subject of +a forthcoming study. +C. +Exchange-only SDFT +Exact exchange in noncollinear SDFT has been defined +in Ref. [35]. Starting point is the exchange energy +Ex = −1 +2 +� � +drdr′ +|r − r′|Tr +� +γ(r, r′)γ(r′, r) +� +. +(9) +Here, γ denotes the one-particle spin-density matrix, a +2 × 2 matrix in spin space whose elements are given +by γσξ(r, r′) = �N +j ψjσ(r)ψ∗ +jξ(r′), constructed from two- +component spinor Kohn-Sham orbitals, where σ =↑, ↓ +and likewise for ξ; Tr is the trace over spin indices. The +exact noncollinear exchange potential then follows by +minimizing Ex with respect to the orbitals, under the +constraint that the orbitals come from a single-particle +equation with a local potential—this is the so-called op- +timized effective potential (OEP) approach [48]. +This +approach is system-independent, i.e., it can be defined in +real space and for lattice models alike. +The exact-exchange OEP requires solving an integral +equation; we use here instead a simplification known as +the Krieger-Li-Iafrate (KLI) approximation [49]. +The +construction and numerical solution of the noncollinear +KLI approximation have been discussed in detail in Refs. +[32, 35]. KLI directly yields a scalar exchange potential +and an exchange magnetic field with moderate numeri- +cal effort and with very little loss of accuracy compared +to the full OEP. In time-dependent SDFT, the exact- +exchange OEP formally carries a memory [50]. The time- +dependent KLI, on the other hand, is an adiabatic ap- +proximation. +KLI +for +noncollinear +systems +produces +exchange +torques in extended Hubbard systems [36]. For the pur- +poses of the present study, we also define a projected KLI +(KLIp) in which the exchange magnetic field Bx on each +lattice site is projected along the local magnetization di- +rection, and which therefore has no exchange torques. +D. +SDFT phase diagram +In the SDFT modeling of the Hamiltonian (5), a simi- +lar magnetic phase diagram is obtained as the exact one +shown in Fig. 2a. The main difference is that the phase + +5 +boundaries at the critical angles θc are not as sharp as +in the exact case but quite diffuse, as schematically de- +picted in Fig. 2b. This is mainly due to the well-known +tendency of SDFT to prefer symmetry breaking, unless +highly accurate correlation functionals are used. +The broadened phase boundary region has a tendency +to exhibit “charge sloshing” [51] in the Kohn-Sham self- +consistency iterations. Charge sloshing spoils the con- +vergence behavior and must be overcome with special +measures, e.g. charge preconditioning or imaginary time +propagation [52]. A sufficiently strong external potential +Vj can also be applied to one side of the model in or- +der to prevent charge sloshing. A fairly strong external +potential in the exchange-only SDFT modeling is also +necessary in the vicinity of θc in order to match to the +exact initial state because correlation effects tend to be +stronger close to the phase boundaries (see Sec. III). +For the simulations of section IV D, where the SDFT +calculations are not tethered to an exact initial solu- +tion, charge sloshing can arise in the stronger interaction +regime, even far from the critical angle θc. +We found +that replacing the Kohn-Sham self-consistency loop with +an imaginary time propagation algorithm [52] for com- +puting the SDFT ground state was useful in mitigating +charge sloshing. +III. +TIME PROPAGATION AND CHOICE OF +INITIAL STATE +In order to compare the dynamics of the exact and +TD-SDFT solutions, we excite the system with a small, +localized magnetic field burst along the y direction dur- +ing a brief number of time steps. To propagate the full +time-dependent many-body Schr¨odinger equation for our +Hubbard bowtie we use a standard Crank-Nicolson algo- +rithm. +The time-dependent Kohn-Sham equations are +also propagated using Crank-Nicolson, including a pre- +dictor-corrector scheme (one corrector step suffices) [53]. +Since our interest is predominantly in the dynamical +effects comparing KLI and KLIp, we start in both cases +from the same ground state. This means that the ex- +change torques must be included in the calculation of +the KLIp initial state, as this is required in order to +have KLIp start with the same initial conditions as the +full KLI simulations; however, these torques are frozen +in, effectively in the form of an external magnetic field. +By contrast, in full KLI the exchange torques are time- +dependent as the system evolves. +Compared to the differences between exchange-only +SDFT and exact many-body benchmarks, the differences +between KLI and KLIp are small and can easily be over- +shadowed. Since we are here interested in relatively sub- +tle dynamical exchange torque effects, it is desirable to +start from a KLI initial state with external scalar poten- +tial Vj and magnetic field Bj chosen to reproduce the +exact density and magnetization. With some effort, Vj +and Bj can be numerically constructed by minimizing +TABLE I. SOC angle θ and interaction strength U0 for the +three ground states considered in Sec. IV, the total magnitude +of the exact xc torque and the exchange-only torque, and the +correlation and exchange energies. +θ +U0 +Σj|τxc| +Σj|τ KLI +x +| +Ec +Ex +30◦ +1 +4.2 × 10−2 +2.6 × 10−2 +-0.214 +-1.84 +30◦ +3 +4.8 × 10−2 +1.2 × 10−1 +-0.236 +-5.82 +60◦ +1 +1.3 × 10−4 +2.0 × 10−3 +-0.448 +-1.92 +the functional +F(Vj, Bj) = +� +j +� +(nj − n(0) +j )2 + |mj − m(0) +j |2� +, +(10) +where n(0) +j +and m(0) +j +are the target density and mag- +netization, respectively. +For each simulation matched +to an exact initial state, we minimize F to an accu- +racy of at least F = 10−25. The minimization is done +via a conjugate gradient method with randomized resets +when a local minimum of insufficient accuracy is reached. +Searching over Vj and Bj of only the SDFT simulations +to find the minimum of F is extremely computationally +expensive due to the high dimensionality of the parame- +ter space. In order to overcome this issue, we switch to +minimizing F with respect to the external fields of the +exact solution once F <∼ 10−4. Minimizing with respect +to exact solution parameters is less computationally ex- +pensive due to the much smoother response of the exact +solution to small changes in the external fields. +IV. +RESULTS AND DISCUSSION +The model system shown in Fig. 1 is simple yet ex- +hibits quite a rich range of structural and dynamical be- +havior. The parameter space to be explored comprises +the hopping strength th, the SOC angle θ, and the inter- +action strength U0 (fixing U1 = U0/2). In the following +we set th = 1 and limit ourselves to three representative +choices of (θ, U0) in the magnetic phase diagram. This +will already be sufficient to gain insight into the signifi- +cance of the xc torques. +Table I gives an overview of the three parameter sets, +the ground-state exchange and correlation energies Ex +and Ec, and the magnitude of the exact xc torque τxc +and of the exchange-only torque τx. These will be further +discussed below. +A. +θ = 30◦, U0 = 1 +We first consider the case θ = 30◦, which is in the +middle of the spin-frustrated region shown in yellow in +the phase diagrams of Fig. 2, and for weak interaction +strength U0 = 1. The magnetization dynamics compari- +son of exact, KLI, and KLIp is shown in Fig. 3a, which + +6 +exact +KLI +KLIproj + b +a +Frequency +Y-axis Magentization (x10−5) +Spectral Amplitude (a.u.) +FIG. 3. +Comparison of exact, KLI, and KLIp modeling for +the case of θ = 30◦ and U0 = 1. +(a) Dynamics of the y- +component of the magnetization of a corner site exited by a +small, short, local burst of magnetic field in the y direction. +(b) Associated spectral amplitude (in arbitrary units), calcu- +lated via Fourier transform of the data shown in part (a). +depicts the magnetization along the y-direction of a cor- +ner site. By construction (see Sec. III), all three methods +start from the same initial value. +KLI and KLIp stay fairly close to one another for +much of the run time due to the relative smallness of the +Hubbard interaction, which indicates that the exchange +torques are not very important in the chosen regime. For +the first few cycles of the precessional motion triggered by +the short pulse, exchange-only SDFT is quite close to the +exact result. In spite of that, both KLI and KLIp start +to diverge significantly from the exact solution around +t = 15, which shows that the correlation effects, although +relatively small, eventually start playing a nonnegligible +role in the time evolution of the system. +To gain further insight, we perform a spectral analysis +of the time-dependent data via Fourier transformation +of the amplitude of the magnetization oscillations, which +reveals the spectrum of magnetic excitations. As shown +in Fig. 3b, KLI and KLIp agree well with the exact spec- +trum at low frequencies (up to about a frequency ω = 3). +At higher frequencies, the SDFT spectra differ from the + b +a +Frequency +Y-axis Magentization (x10−5) +Spectral Amplitude (a.u.) +FIG. 4. +Same as Fig. 3 but for U0 = 3. +exact spectra, which may be due to the fact that we are +using here an adiabatic approximation which does not +produce double or higher excitations [53] and hence does +not capture all peaks. However, KLI and KLIp remain +very close to each other throughout, illustrating again +that exchange torques are insignificant here. +B. +θ = 30◦, U0 = 3 +For the second case, we remain at θ = 30◦, away +from the phase boundaries, but increase the interaction +strength into the moderately strongly interacting regime, +at U0 = 3. The real time magnetization dynamics and +amplitude spectrum are shown in Fig. 4. Clearly, KLI +and KLIp start to differ from each other almost right +away, which points to the more important role of the +exchange torques. +At first glance, it is surprising to see that the projected +KLI, which has no torques, agrees better with the exact +magnetization oscillations, at least for the first few cycles. +To explain this, it is helpful to consider the magnitudes +of the initial τxc and τx given in Table I. For U0 = 1, the +sum of the exchange torques is comparable to the sum of +the xc torques (within a factor 1.6); at U0 = 3, on the +other hand, the exchange torques are much larger than + +my[0] +0.12826 +KS +0.12828 +KSproj +0.12830 +0.12832 +0.12834 +SIxe +0.12836 +-0.12838 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +Timemy[0] +-0.12826 +exact +KS +-0.12828 +KSproj +-0.12830 +-0.12832 +-0.12834 +-0.12836 +-0.12838 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +Timemy[O] FFT +10-1 +exact +KLI +10-2 +KLIproj +FFT amplitude (a.u.) +10-4 +10-5 +10-6. +10-7, +0 +2 +4 +5 +6 +Excitation energy (in units of tg)my[o] +le-5-1.567e. +exact +-5.2 +KLI +KLIproj +5.4 +5.6 +5.8 +6.0 +6.2 +-6.4 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +Timemy[O] FFT +10-2 +amplitude (a.u.) +10 +-4 +10-5 +FT +10-6 +exact +KLI +KLIproj +10-7 +0 +2 +3 +1 +4 +5 +6 +Excitation energy (in units of tg)my[0] +1e-5-3.28e-1 +6.0 +6 +6.4 +6.6 +6.8 +7.0 +-7.2 +exact +-7.4 +KLI +KLIproj +-7.6 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +Time7 + b +a +Frequency +Y-axis Magentization (x10−5) +Spectral Amplitude (a.u.) +FIG. 5. +Same as Fig. 3 but for θ = 60◦. +the xc torques, which suggests that the correlation con- +tribution to the torques becomes relatively much more +important. In other words, exchange-only overestimates +the torques, and correlation compensates for it. KLIp +avoids this overestimation (better no exchange torque at +all, than too much of it), and brings the dynamics closer +to the exact case. Notice that this could have not been +anticipated just from looking at the exchange and cor- +relation energies Ex and Ec of the initial state, which +would have suggested that the exchange is dominant. +The Fourier spectrum in Fig. 4b is less clear: while +both KLI and KLIp seem to reproduce the rough trends +of the exact spectrum, it is difficult to say which one of +them agrees better. Neither of them captures the details +of the exact spectrum particularly well. +C. +θ = 60◦, U0 = 1 +Lastly, we consider the case of θ = 60◦ and U0 = 1, see +Fig. 5. This state is at a critical angle of the magnetic +phase diagram where artificial charge density symmetry +breaking in exchange-only SDFT is prevalent, indicat- +ing that strong correlations are needed to reproduce the +exact results. As shown in Table I, Ec is significantly en- +hanced relative to Ex, compared to the case of θ = 30◦. +∑j |τKLI +x +| +0 +1 +2 +3 +4 +Fave +U0 +FIG. 6. +Red (right axis): Comparison between KLI and +KLIp solutions as a function of interaction strength for the +case of θ = 30◦ and U0, quantified by the time-averaged dis- +tance measure Fave, Eq. (11). Blue (left axis): Σj|τ KLI +x +| of +the Hubbard bowtie ground state versus U0. +Correspondingly, the exchange torques are lower, due to +the localization of the magnetization to one side of the +system. The strong correlation effects at the transition +angle result in both KLI and KLIp diverging from the +exact solution fairly quickly. The magnetization oscilla- +tions calculated with KLI and KLIp match each other +fairly well, at least for the first few cycles, but then dif- +ferences start to accumulate. +The Fourier spectrum, see Fig. 5b, has well defined ex- +citations, which are fairly well captured by both KLI and +KLIp, but some inaccuracies are noticeable at both high +and low frequencies. +Notably, KLIp performs slightly +better at estimating the gaps in the spectrum for mid- +range frequency excitations. The better performance of +KLIp occurs, similarly to Section IV B, due to the KLI +exchange-only approximation substantially overestimat- +ing the xc torques, with no correlation to compensate +(see Table I). +D. +Distance between KLI and KLIp versus U0 +The effect of the exchange torques can be further quan- +tified by introducing the time-averaged distance measure +Fave = 1 +t +� t +0 +dt′ � +j +� � +nKLI +j +− nKLIp +j +�2 ++ +���mKLI +j +− mKLIp +j +��� +2 � +, +(11) +where we calculate the time average over a short time +(t = 2) after initial excitation. +This provides an esti- +mate of the degree of divergence between the solutions +which can be compared with interaction strength and the +magnitude of ground state KLI exchange torques. +Figure 6 shows the time-averaged distance measure +(11) between KLI and KLIp as a function of U0 at +θ = 30◦, and, for the sake of comparison, the sum of + +my[O] FFT +10-1 +exact +KLI +10-2 +KLIproj +FFT amplitude (a.u.) +10-4 +10-5 +10-6. +10-7_ +0 +2 +3 +4 +5 +7 +6 +Excitation energy (in units of tg)my[o] +1e-5-2.521e-1 +3.2 +Y-axis Magnetization Magnitude +3.4 +3.6 +3.8 +4.0 +4.2 +exact +KLI +-4.4 +KLIproj +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +20.0 +Time0.25 +0.12 +0.10 +0.20 +0.08 +0.15 +0.06 +0.10 +0.04 +0.05 +0.02 +0.00 +0.00 +0.0 +0.5 +1.0 +1.5 +2.08 +the magnitudes of the KLI exchange torques of the corre- +sponding initial states. Both Fave and Σj|τ KLI +x +| start out +linearly for small interaction strengths U0 and keep in- +creasing well into the moderate interaction regime, where +Fave appears to start leveling off around U0 = 2. +A comparison with exact time-dependent xc torques +is, unfortunately, not possible; even the construction of +the exact Σj|τxc| over the whole range of U0 is numeri- +cally too demanding, except for the three cases in Table +I. Nevertheless, we can infer from the results presented +in Fig. +6 that both exchange and correlation torques +must be accounted for even for relatively low interaction +strengths in order to accurately describe the dynamics. +V. +CONCLUSION +We have performed exact and approximate, exchange- +only (TD)-SDFT calculations on a half-filled 5-site Hub- +bard cluster with varying interaction and SOC strengths. +The purpose of this study was to assess the significance of +many-body magnetic torques for the description of spin +dynamics. We considered three scenarios with weak and +moderate interactions and close to and away from a tran- +sition between different magnetic phases. While this is +clearly not an exhaustive exploration of the parameter +space, the examples studied here are good representa- +tives and allow us to draw meaningful conclusions. +We find that exchange torques become increasingly im- +portant as non-local interactions become stronger, with +an approximately linear dependence at low interactions +(see Fig. 6), but the relationship becomes nonlinear for +more general interaction strengths. Strong correlations in +the vicinity of phase boundaries reduce the importance of +exchange torques due to localization. When correlations +are particularly strong, they appear to counteract the ex- +change torques, leading to a net reduction of the total xc +torques. This suggests that when lacking a sufficiently ac- +curate correlation functional, completely projecting out +the xc torques may improve the overall accuracy of TD- +SDFT magnetic dynamics, at least for short times. +The challenge for future work is clearly to construct +correlation functionals that produce accurate torques, +and test these against benchmarks. A good starting point +will be to do this for similar finite Hubbard models, fol- +lowed by tests for the magnetization dynamics in real +magnetic materials in the linear and nonlinear regime. +ACKNOWLEDGMENTS +This work was supported by DOE Grant No. +DE- +SC0019109. The authors wish to thank Aurora Pribram- +Jones for helpful discussion. +[1] I. ˇZuti´c, J. Fabian, and S. Das Sarma, Spintronics: Fun- +damentals and applications, Rev. Mod. Phys. 76, 323 +(2004). +[2] S. M. Rezende, Fundamentals of Magnonics, Lecture +Notes in Physics, Vol. 969 (Springer, Heidelberg, 2020). +[3] A. Barman, G. Gubbiotti, S. Ladak, A. O. 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Ullrich, Time-dependent density-functional theory: +concepts and applications (Oxford University Press, Ox- +ford, 2012). + diff --git a/F9AzT4oBgHgl3EQfi_2P/content/tmp_files/load_file.txt b/F9AzT4oBgHgl3EQfi_2P/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..823122d3b9b791b62e71abc133e62bc8e8dcf9e9 --- /dev/null +++ b/F9AzT4oBgHgl3EQfi_2P/content/tmp_files/load_file.txt @@ -0,0 +1,870 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf,len=869 +page_content='Magnetization dynamics with time-dependent spin-density functional theory: significance of exchange-correlation torques Daniel Hill, Justin Shotton, and Carsten A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Ullrich∗ Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA (Dated: January 5, 2023) In spin-density-functional theory (SDFT) for noncollinear magnetic materials, the Kohn-Sham system features exchange-correlation (xc) scalar potentials and magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The significance of the xc magnetic fields is not very well explored;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' in particular, they can give rise to local torques on the magnetization, which are absent in standard local and semilocal approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Exact benchmark solutions for a five-site extended Hubbard lattice at half filling and in the presence of spin-orbit coupling are compared with SDFT results obtained using orbital-dependent exchange- only approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The magnetization dynamics following short-pulse excitations is found to be reasonably well described in the exchange-only approximation for weak to moderate interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' For stronger interactions and near transitions between magnetically ordered and frustrated phases, exchange and correlation torques tend to compensate each other and must both be accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' INTRODUCTION Spin dynamics in magnetic systems is a research area of much current activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Spintronics [1], which is con- cerned with the manipulation of electronic spins, spin currents, spin textures, and spin excitations, has created a wealth of scientific knowledge and many avenues for new technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Prominent examples are spin waves for encoding and transmitting information (magnonics) [2, 3], skyrmions for magnetic information storage [4– 8], and single-spin qubits for quantum computation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Another related area of much interest is ultrafast demag- netization induced by femtosecond laser pulses [10–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Computational approaches to simulate magnetization dynamics in a wide variety of systems are typically based on the Landau-Lifshitz-Gilbert (LLG) equation of motion [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The LLG equation provides a classical descrip- tion of the time evolution of the magnetization vector m(t) in response to a time-dependent perturbation (typ- ically, a short pulse or a periodic driving field) or evolving from a nonequilibrium initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Materials proper- ties such as anisotropy, deformations, strain, and various forms of damping can be built into the LLG approach via phenomenological or “second-principles” parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' In this paper, we are less concerned with these spe- cific materials properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' instead of LLG we will use a fully quantum mechanical description of the electronic charge and spin degrees of freedom, and our focus will be specifically on the impact of electron-electron interac- tions on the magnetization dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' To be more clear, we consider a system of N interacting electrons under the influence of a time-dependent scalar potential V (r, t) and a time-dependent magnetic field B(r, t) which cou- ples only to the electron spin (and not to orbital motion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' ∗ ullrichc@missouri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='edu The associated many-body Hamiltonian is given by ˆH = N � j � −∇2 j 2 + V (rj, t) + σj · B(rj, t) � + 1 2 N � j̸=k 1 |rj − rk| , (1) where σj is the vector of Pauli matrices acting on the spin of the jth electron, and we define the magnetic field strength such that the Bohr magneton, µB = e¯h/2m, does not explicitly appear in the Hamiltonian ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' We use atomic units (e = m = ¯h = 4πϵ0 = 1) throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' From the Heisenberg equation of motion for ˆH, Capelle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' showed that the magnetization has the following time evolution [17]: dm(r, t) dt + ˆ∇ · J(r, t) = m(r, t) × B(r, t) , (2) where J(r, t) is the spin-current tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Equation (2) is exact but not very helpful in practice since J(r, t) re- quires the many-body wave function associated with ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' A more practical (but still in principle exact) alternative is time-dependent spin-density functional theory (TD- SDFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The idea of TD-SDFT is to consider an auxiliary system of noninteracting fermions, acted upon by an “ef- fective” scalar potential and magnetic field, Veff(r, t) and Beff(r, t), such that the same density n(r, t) and magne- tization m(r, t) are produced as in the physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The resulting equation of motion, the TD-SDFT coun- terpart to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' (2), is [17] dm(r, t) dt + ˆ∇ · JKS(r, t) = m(r, t) × Beff(r, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' (3) Here, JKS(r, t) is the Kohn-Sham spin-current tensor, which is easily determined from the noninteracting wave function, and the effective magnetic field is defined as Beff(r, t) = B(r, t) + Bxc(r, t), where the exchange- correlation (xc) magnetic field Bxc is a functional of the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='01509v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='str-el] 4 Jan 2023 2 density and magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Formally, m(r, t) is the same in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' (2) and (3), but J and JKS are in general dif- ferent (the difference lies in the transverse component).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Thus, the so-called xc torque, τxc(r, t) = m(r, t) × Bxc(r, t) , (4) ensures that TD-SDFT produces the correct magnetiza- tion dynamics [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' While all of this is clear at the formal level, the ex- act form of Bxc is unknown and must be approximated in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' This immediately raises several questions: which approximations of Bxc are available, and do they produce xc torques?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' And, how important are the xc torques for the magnetization dynamics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' A number of approximations for Bxc have been derived within ground-state SDFT for noncollinear magnetism [18–20];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' via the adiabatic approximation, they immedi- ately carry over to TD-SDFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The most widely used ap- proach, pioneered by K¨ubler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' [21, 22] and imple- mented in many popular electronic structure codes, is to use standard local or semilocal xc functionals such as the local spin-density approximation (LSDA) or general- ized gradient approximations (GGAs), and assume a lo- cal spin quantization axis which is aligned with the local magnetization vector m(r, t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' this produces a Bxc(r, t) that is parallel to m(r, t) everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' We see right away from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' (4) that this class of approximations does not produce any xc torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Approximations for Bxc that do include xc torque ef- fects can be constructed in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Existing lo- cal and semilocal functionals (LSDA and GGAs) have been modified [23–26] or used in a source-free construc- tion [27], and new gradient-corrected functionals were constructed based using the spin-spiral state of the elec- tron gas as reference system [28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' More consistent derivations of xc meta-GGAs, starting from noncollinear generalizations of the exchange hole and the two-body density matrix, were recently presented [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Vari- ous orbital-dependent functionals were generalized to the case of noncollinear magnetization [33–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Existing applications of ground-state SDFT to non- collinear magnetic materials [25, 26, 33] and model sys- tems [36] seem to suggest that xc torques are of relatively minor importance for magnetic structure and energetics, although the torques themselves may not be insignificant [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' On the other hand, there are good reasons to expect that xc torques will be more impactful for magnetization dynamics: they explicitly appear in the equation of mo- tion, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' (3), and even if τxc(r, t) is relatively small at a given r and t, its effect can accumulate over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' So far, however, there has been no systematic attempt to assess this hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' We are only aware of one study in the literature, where Dewhurst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' [37] used their source-free Bxc functional to simulate laser-induced spin dynamics in bulk Co and Ni and Co-Pt and Ni-Pt inter- faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' They found that xc torques were significant only if they are not overshadowed by magnetic anisotropy ef- fects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=', in bulk, and not at interfaces), and that they FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Geometry of the 5-site Hubbard cluster used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The arrows indicate the ordering of the nearest- neighbor sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' (6), accounting for the directional hop- ping due to SOC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' give rise to rather slow spin rotation compared to other forms of spin dynamics, induced optically or via spin- orbit coupling (SOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' In this paper, our goal is to assess the importance of xc torques in frustrated magnetic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Exchange- frustrated solids such as spin glasses and kagome antifer- romagnetic lattices are characterized by many competing noncollinear spin configurations and quantum spin liquid phases [38–40], and may therefore exhibit an enhanced sensitivity to subtle xc torque effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Needless to say, ex- tended spin frustrated solids are challenging to describe, and exact or quasi-exact benchmark results are hard to come by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' We will therefore limit ourselves to small model systems which capture the spirit of spin frustration and yet are computationally manageable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Here, we will consider small Hubbard-type model systems along similar lines as in our earlier studies [35, 36, 41];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' by including SOC we can generate intrin- sically noncollinear ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' In particular, we will focus on a five-site half-filled Hubbard bowtie as a mini- mal model for studying xc torque effects in the presence of magnetic frustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' We will generate both exact and SDFT phase diagrams of spin configurations for this sys- tem and explore the spin dynamics for different config- urations in the phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The TD-SDFT treat- ment will be based on orbital-dependent exchange-only functionals, and we will compare with exact solutions of the many-body time-dependent Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Focusing on a few representative case studies, we will gain insight into the significance of xc torques in differ- ent regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' II the ex- tended Hubbard model and the SDFT framework are in- troduced and the exact and SDFT magnetic phase dia- grams are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' III we describe some techni- cal aspects of the TD-SDFT modeling such as the choice of initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' IV the results of exact diago- nalization and SDFT models are compared for the cases with moderate to strong correlations and non-local inter- actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Conclusions are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 2 4 3 53 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' EXACT AND SDFT MAGNETIC STRUCTURE OF HUBBARD CLUSTERS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Definition of the model In this paper we limit ourselves to (TD-)SDFT in the exchange-only approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' As discussed earlier [36], the standard Hubbard model with on-site interactions does not give rise to any exchange torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' If one wishes to study exchange torque effects it is necessary to work with an extended Hubbard model instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' We will con- sider, in the following, a half-filled 5-site Hubbard cluster in a bowtie shape, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Here, we go be- yond Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' [36] and include SOC through a modification of the kinetic-energy operator, where the hopping term becomes complex and the hopping acquires a directional- ity [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Thus, our inhomogeneous extended Hubbard model with SOC is described by the Hamiltonian ˆHmodel = ˆHT + ˆHU + ˆHext .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' (5) The first term is a hopping term with SOC absorbed into a spin dependent phase factor, ˆHT = −th � ⟨j,j′⟩ � σ e−iσθc† jσcj′σ + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=', (6) where h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' stands for Hermitian conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Here, th = √ T 2 + C2 is the generalized hopping strength parameter which depends on nearest neighbor hopping strength T and spin orbit coupling C, j is the site index for the geometry shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 1, cjσ is the annihilation operator for an electron of spin σ at site j, the brackets ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='⟩ denote an ordered sum over nearest neighbors with the order indicated by the arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 1, and σ = ±1 labels spin-up and -down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Furthermore, θ is the SOC angle which parameterizes the strength of the SOC parameter C relative to the conventional hopping term T [44–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The second term in the model Hamiltonian (5) com- prises the on-site and nearest-neighbor interaction terms, ˆHU = U0 � j nj↑nj↓ + U1 � ⟨j,j′⟩ � σ,σ′ njσnj′σ′ , (7) where njσ = c† jσcjσ is the spin σ particle number density at site j, and U0 and U1 are the on-site and nearest- neighbor repulsion strengths, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' For the pur- poses of this paper, we set U1 = 1 2U0, a fairly typical choice for modeling real materials [47], and we restrict the hopping parameter th and on-site interaction param- eter U0 to be of similar orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Finite non- local interactions are necessary for nontrivial exchange torques, but we avoid the much stronger interactions regime because the charge degrees of freedom tend to freeze out as U0 and U1 become large, resulting in the dynamics being dominated by a simpler pure-spin low- energy effective model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Lastly, ˆHext contains the couplings to the external po- tential and external magnetic field, ˆHext = � j (Vjnj + Bj · mj) , (8) where Vj is the scalar potential and Bj is the magnetic field on site j, the total density is nj = nj↑ + nj↓, and the magnetization is given by mj = � σ,σ′ c† jσ⃗σσσ′cjσ′ with ⃗σ = (σx, σy, σz) denoting a vector composed of the Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' We keep the external field param- eters each less than the on-site interaction and hopping, Vj, |Bj| < U0, th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' These external field parameters are not strictly set to zero because they can be used to break degeneracy in order to fix a symmetry breaking state, and because, as discussed in Section III, small variation of these parameters in the exact model is found to be useful in matching the SDFT initial state and the exact initial state more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Magnetic phase diagram of the Hubbard bowtie We use exact diagonalization of ˆHmodel to construct benchmark solutions with which to compare our SDFT results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Figure 2a shows the exact phase diagram of the half-filled Hubbard bowtie in a plane whose x − y axes are defined by C = (th/U0) sin θ and T = (th/U0) cos θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' the SOC angle θ is here measured with respect to the kinetic energy axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Similar phase diagrams for the half- filled Hubbard trimer were obtained by Tabrizi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Within the above specified regime the model has a phase transition at θc = nπ/3 for any integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' For the case of zero external fields, the ground state of the 5-site model at half filling is degenerate and magnetically ordered with a nontrivial noncollinear spin structure (ex- cept at isolated points in the phase diagram where the spins are ferromagnetically aligned) indicating magnetic frustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' On the phase boundary, θc, the ground state exhibits a symmetry breaking charge density wave (CDW) in the form of a spontaneous charge polarization along the x- axis of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 2a the states shown outside the phase diagram image are the states at the critical angles θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' A specific choice of charge polarization is depicted in order to show the corresponding spin state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The sites with no spin indicated do not necessarily have zero mag- netic moment, but it tends to be orders of magnitude smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The states shown inside the shaded segments of the phase diagram are those of the midpoint angles be- tween the phase boundaries, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' θ = 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' As θ changes, the relative angles of the spins change as well, with the fastest changes occurring in the vicinity of the phase tran- sitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Thus, the phase transitions at θc are not discon- tinuous, rather they appear to be a zero temperature, finite model analog of a second order phase transition, although the continuous transition occurs over a rather narrow range of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 4 T C 60∘ T 60∘ C a b CDW CDW CDW CDW CDW CDW FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' (a) Magnetic phase diagram of the half-filled 5-site Hubbard model, obtained using exact diagonalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The red arrows indicate the relative in-plane spin direction of the state depicted (taken at the midpoint angle between the phase boundaries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The blue pluses and minuses indicate the direction of electric polarization for the CDW critical angle states for the specific spin arrangement shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' (b) Corresponding magnetic phase diagram using exchange-only SDFT, showing the broadening of the phase boundary states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The phase diagram has approximately the same states as for the exact diagonalization phase diagram, but the critical angles, where a CDW occurs, acquire a width of a few degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The complete phase diagram of the ground state of our 5-site Hubbard bowtie and other finite and extended triangular lattice systems is of interest in and by itself, especially with respect to their symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' A more complete formal analysis of the phase boundaries and other symmetry-related properties will be the subject of a forthcoming study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Exchange-only SDFT Exact exchange in noncollinear SDFT has been defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Starting point is the exchange energy Ex = −1 2 � � drdr′ |r − r′|Tr � γ(r, r′)γ(r′, r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' (9) Here, γ denotes the one-particle spin-density matrix, a 2 × 2 matrix in spin space whose elements are given by γσξ(r, r′) = �N j ψjσ(r)ψ∗ jξ(r′), constructed from two- component spinor Kohn-Sham orbitals, where σ =↑, ↓ and likewise for ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Tr is the trace over spin indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The exact noncollinear exchange potential then follows by minimizing Ex with respect to the orbitals, under the constraint that the orbitals come from a single-particle equation with a local potential—this is the so-called op- timized effective potential (OEP) approach [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' This approach is system-independent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=', it can be defined in real space and for lattice models alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The exact-exchange OEP requires solving an integral equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' we use here instead a simplification known as the Krieger-Li-Iafrate (KLI) approximation [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The construction and numerical solution of the noncollinear KLI approximation have been discussed in detail in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' [32, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' KLI directly yields a scalar exchange potential and an exchange magnetic field with moderate numeri- cal effort and with very little loss of accuracy compared to the full OEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' In time-dependent SDFT, the exact- exchange OEP formally carries a memory [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The time- dependent KLI, on the other hand, is an adiabatic ap- proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' KLI for noncollinear systems produces exchange torques in extended Hubbard systems [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' For the pur- poses of the present study, we also define a projected KLI (KLIp) in which the exchange magnetic field Bx on each lattice site is projected along the local magnetization di- rection, and which therefore has no exchange torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' SDFT phase diagram In the SDFT modeling of the Hamiltonian (5), a simi- lar magnetic phase diagram is obtained as the exact one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The main difference is that the phase 5 boundaries at the critical angles θc are not as sharp as in the exact case but quite diffuse, as schematically de- picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' This is mainly due to the well-known tendency of SDFT to prefer symmetry breaking, unless highly accurate correlation functionals are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The broadened phase boundary region has a tendency to exhibit “charge sloshing” [51] in the Kohn-Sham self- consistency iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Charge sloshing spoils the con- vergence behavior and must be overcome with special measures, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' charge preconditioning or imaginary time propagation [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' A sufficiently strong external potential Vj can also be applied to one side of the model in or- der to prevent charge sloshing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' A fairly strong external potential in the exchange-only SDFT modeling is also necessary in the vicinity of θc in order to match to the exact initial state because correlation effects tend to be stronger close to the phase boundaries (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' For the simulations of section IV D, where the SDFT calculations are not tethered to an exact initial solu- tion, charge sloshing can arise in the stronger interaction regime, even far from the critical angle θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' We found that replacing the Kohn-Sham self-consistency loop with an imaginary time propagation algorithm [52] for com- puting the SDFT ground state was useful in mitigating charge sloshing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' TIME PROPAGATION AND CHOICE OF INITIAL STATE In order to compare the dynamics of the exact and TD-SDFT solutions, we excite the system with a small, localized magnetic field burst along the y direction dur- ing a brief number of time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' To propagate the full time-dependent many-body Schr¨odinger equation for our Hubbard bowtie we use a standard Crank-Nicolson algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The time-dependent Kohn-Sham equations are also propagated using Crank-Nicolson, including a pre- dictor-corrector scheme (one corrector step suffices) [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Since our interest is predominantly in the dynamical effects comparing KLI and KLIp, we start in both cases from the same ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' This means that the ex- change torques must be included in the calculation of the KLIp initial state, as this is required in order to have KLIp start with the same initial conditions as the full KLI simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' however, these torques are frozen in, effectively in the form of an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' By contrast, in full KLI the exchange torques are time- dependent as the system evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Compared to the differences between exchange-only SDFT and exact many-body benchmarks, the differences between KLI and KLIp are small and can easily be over- shadowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Since we are here interested in relatively sub- tle dynamical exchange torque effects, it is desirable to start from a KLI initial state with external scalar poten- tial Vj and magnetic field Bj chosen to reproduce the exact density and magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' With some effort, Vj and Bj can be numerically constructed by minimizing TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' SOC angle θ and interaction strength U0 for the three ground states considered in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' IV, the total magnitude of the exact xc torque and the exchange-only torque, and the correlation and exchange energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' θ U0 Σj|τxc| Σj|τ KLI x | Ec Ex 30◦ 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='2 × 10−2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='6 × 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='214 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='84 30◦ 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='8 × 10−2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='2 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='236 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='82 60◦ 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='3 × 10−4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='448 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='92 the functional F(Vj, Bj) = � j � (nj − n(0) j )2 + |mj − m(0) j |2� , (10) where n(0) j and m(0) j are the target density and mag- netization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' For each simulation matched to an exact initial state, we minimize F to an accu- racy of at least F = 10−25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The minimization is done via a conjugate gradient method with randomized resets when a local minimum of insufficient accuracy is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Searching over Vj and Bj of only the SDFT simulations to find the minimum of F is extremely computationally expensive due to the high dimensionality of the parame- ter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' In order to overcome this issue, we switch to minimizing F with respect to the external fields of the exact solution once F <∼ 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Minimizing with respect to exact solution parameters is less computationally ex- pensive due to the much smoother response of the exact solution to small changes in the external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' RESULTS AND DISCUSSION The model system shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 1 is simple yet ex- hibits quite a rich range of structural and dynamical be- havior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The parameter space to be explored comprises the hopping strength th, the SOC angle θ, and the inter- action strength U0 (fixing U1 = U0/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' In the following we set th = 1 and limit ourselves to three representative choices of (θ, U0) in the magnetic phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' This will already be sufficient to gain insight into the signifi- cance of the xc torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Table I gives an overview of the three parameter sets, the ground-state exchange and correlation energies Ex and Ec, and the magnitude of the exact xc torque τxc and of the exchange-only torque τx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' These will be further discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' θ = 30◦, U0 = 1 We first consider the case θ = 30◦, which is in the middle of the spin-frustrated region shown in yellow in the phase diagrams of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 2, and for weak interaction strength U0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The magnetization dynamics compari- son of exact, KLI, and KLIp is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 3a, which 6 exact KLI KLIproj b a Frequency Y-axis Magentization (x10−5) Spectral Amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=') FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Comparison of exact, KLI, and KLIp modeling for the case of θ = 30◦ and U0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' (a) Dynamics of the y- component of the magnetization of a corner site exited by a small, short, local burst of magnetic field in the y direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' (b) Associated spectral amplitude (in arbitrary units), calcu- lated via Fourier transform of the data shown in part (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' depicts the magnetization along the y-direction of a cor- ner site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' By construction (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' III), all three methods start from the same initial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' KLI and KLIp stay fairly close to one another for much of the run time due to the relative smallness of the Hubbard interaction, which indicates that the exchange torques are not very important in the chosen regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' For the first few cycles of the precessional motion triggered by the short pulse, exchange-only SDFT is quite close to the exact result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' In spite of that, both KLI and KLIp start to diverge significantly from the exact solution around t = 15, which shows that the correlation effects, although relatively small, eventually start playing a nonnegligible role in the time evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' To gain further insight, we perform a spectral analysis of the time-dependent data via Fourier transformation of the amplitude of the magnetization oscillations, which reveals the spectrum of magnetic excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 3b, KLI and KLIp agree well with the exact spec- trum at low frequencies (up to about a frequency ω = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' At higher frequencies, the SDFT spectra differ from the b a Frequency Y-axis Magentization (x10−5) Spectral Amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=') FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 3 but for U0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' exact spectra, which may be due to the fact that we are using here an adiabatic approximation which does not produce double or higher excitations [53] and hence does not capture all peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' However, KLI and KLIp remain very close to each other throughout, illustrating again that exchange torques are insignificant here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' θ = 30◦, U0 = 3 For the second case, we remain at θ = 30◦, away from the phase boundaries, but increase the interaction strength into the moderately strongly interacting regime, at U0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The real time magnetization dynamics and amplitude spectrum are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Clearly, KLI and KLIp start to differ from each other almost right away, which points to the more important role of the exchange torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' At first glance, it is surprising to see that the projected KLI, which has no torques, agrees better with the exact magnetization oscillations, at least for the first few cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' To explain this, it is helpful to consider the magnitudes of the initial τxc and τx given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' For U0 = 1, the sum of the exchange torques is comparable to the sum of the xc torques (within a factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' at U0 = 3, on the other hand, the exchange torques are much larger than my[0] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='12826 KS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='12828 KSproj 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='12830 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='12832 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 Timemy[O] FFT 10-1 exact KLI 10-2 KLIproj FFT amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=') 10-4 10-5 10-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 10-7, 0 2 4 5 6 Excitation energy (in units of tg)my[o] le-5-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='567e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' exact 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='2 KLI KLIproj 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 Timemy[O] FFT 10-2 amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=') 10 4 10-5 FT 10-6 exact KLI KLIproj 10-7 0 2 3 1 4 5 6 Excitation energy (in units of tg)my[0] 1e-5-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='28e-1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='2 exact 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='4 KLI KLIproj 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 Time7 b a Frequency Y-axis Magentization (x10−5) Spectral Amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=') FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 3 but for θ = 60◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' the xc torques, which suggests that the correlation con- tribution to the torques becomes relatively much more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' In other words, exchange-only overestimates the torques, and correlation compensates for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' KLIp avoids this overestimation (better no exchange torque at all, than too much of it), and brings the dynamics closer to the exact case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Notice that this could have not been anticipated just from looking at the exchange and cor- relation energies Ex and Ec of the initial state, which would have suggested that the exchange is dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The Fourier spectrum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 4b is less clear: while both KLI and KLIp seem to reproduce the rough trends of the exact spectrum, it is difficult to say which one of them agrees better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Neither of them captures the details of the exact spectrum particularly well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' θ = 60◦, U0 = 1 Lastly, we consider the case of θ = 60◦ and U0 = 1, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' This state is at a critical angle of the magnetic phase diagram where artificial charge density symmetry breaking in exchange-only SDFT is prevalent, indicat- ing that strong correlations are needed to reproduce the exact results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' As shown in Table I, Ec is significantly en- hanced relative to Ex, compared to the case of θ = 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' ∑j |τKLI x | 0 1 2 3 4 Fave U0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Red (right axis): Comparison between KLI and KLIp solutions as a function of interaction strength for the case of θ = 30◦ and U0, quantified by the time-averaged dis- tance measure Fave, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Blue (left axis): Σj|τ KLI x | of the Hubbard bowtie ground state versus U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Correspondingly, the exchange torques are lower, due to the localization of the magnetization to one side of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The strong correlation effects at the transition angle result in both KLI and KLIp diverging from the exact solution fairly quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The magnetization oscilla- tions calculated with KLI and KLIp match each other fairly well, at least for the first few cycles, but then dif- ferences start to accumulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The Fourier spectrum, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 5b, has well defined ex- citations, which are fairly well captured by both KLI and KLIp, but some inaccuracies are noticeable at both high and low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Notably, KLIp performs slightly better at estimating the gaps in the spectrum for mid- range frequency excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The better performance of KLIp occurs, similarly to Section IV B, due to the KLI exchange-only approximation substantially overestimat- ing the xc torques, with no correlation to compensate (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Distance between KLI and KLIp versus U0 The effect of the exchange torques can be further quan- tified by introducing the time-averaged distance measure Fave = 1 t � t 0 dt′ � j � � nKLI j − nKLIp j �2 + ���mKLI j − mKLIp j ��� 2 � , (11) where we calculate the time average over a short time (t = 2) after initial excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' This provides an esti- mate of the degree of divergence between the solutions which can be compared with interaction strength and the magnitude of ground state KLI exchange torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Figure 6 shows the time-averaged distance measure (11) between KLI and KLIp as a function of U0 at θ = 30◦, and, for the sake of comparison, the sum of my[O] FFT 10-1 exact KLI 10-2 KLIproj FFT amplitude (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=') 10-4 10-5 10-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 10-7_ 0 2 3 4 5 7 6 Excitation energy (in units of tg)my[o] 1e-5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='521e-1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='2 Y-axis Magnetization Magnitude 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='2 exact KLI 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='4 KLIproj 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='5 5.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content='08 the magnitudes of the KLI exchange torques of the corre- sponding initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Both Fave and Σj|τ KLI x | start out linearly for small interaction strengths U0 and keep in- creasing well into the moderate interaction regime, where Fave appears to start leveling off around U0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' A comparison with exact time-dependent xc torques is, unfortunately, not possible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' even the construction of the exact Σj|τxc| over the whole range of U0 is numeri- cally too demanding, except for the three cases in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Nevertheless, we can infer from the results presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 6 that both exchange and correlation torques must be accounted for even for relatively low interaction strengths in order to accurately describe the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' CONCLUSION We have performed exact and approximate, exchange- only (TD)-SDFT calculations on a half-filled 5-site Hub- bard cluster with varying interaction and SOC strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The purpose of this study was to assess the significance of many-body magnetic torques for the description of spin dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' We considered three scenarios with weak and moderate interactions and close to and away from a tran- sition between different magnetic phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' While this is clearly not an exhaustive exploration of the parameter space, the examples studied here are good representa- tives and allow us to draw meaningful conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' We find that exchange torques become increasingly im- portant as non-local interactions become stronger, with an approximately linear dependence at low interactions (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' 6), but the relationship becomes nonlinear for more general interaction strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Strong correlations in the vicinity of phase boundaries reduce the importance of exchange torques due to localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' When correlations are particularly strong, they appear to counteract the ex- change torques, leading to a net reduction of the total xc torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' This suggests that when lacking a sufficiently ac- curate correlation functional, completely projecting out the xc torques may improve the overall accuracy of TD- SDFT magnetic dynamics, at least for short times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The challenge for future work is clearly to construct correlation functionals that produce accurate torques, and test these against benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' A good starting point will be to do this for similar finite Hubbard models, fol- lowed by tests for the magnetization dynamics in real magnetic materials in the linear and nonlinear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by DOE Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' DE- SC0019109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' The authors wish to thank Aurora Pribram- Jones for helpful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' [1] I.' 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Sahoo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Sinha, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Yang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' M¨unzenberg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Moriyama, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Mizukami, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9AzT4oBgHgl3EQfi_2P/content/2301.01509v1.pdf'} +page_content=' Landeros, R.' metadata={'source': 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b/GtE1T4oBgHgl3EQf_QYB/content/tmp_files/2301.03577v1.pdf.txt @@ -0,0 +1,951 @@ +arXiv:2301.03577v1 [nlin.CD] 9 Jan 2023 +Records and occupation time statistics for area-preserving maps⋆ +Roberto Artuso1,2,∗ Tulio M. de Oliveira3, and Cesar Manchein3† +1Dipartimento di Scienza e Alta Tecnologia and Center for Nonlinear +and Complex Systems, Via Valleggio 11, 22100 Como, Italy; +2I.N.F.N, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy; and +3Departamento de Física, Universidade do Estado de Santa Catarina, 89219-710 Joinville, SC, Brazil +⋆To Giulio Casati, celebrating his birthday and his achievements. +(Dated: January 10, 2023) +A relevant problem in dynamics is to characterize how deterministic systems may exhibit fea- +tures typically associated to stochastic processes. A widely studied example is the study of (normal +or anomalous) transport properties for deterministic systems on a non-compact phase space. We +consider here two examples of area-preserving maps: the Chirikov-Taylor standard map and the +Casati-Prosen triangle map, and we investigate transport properties, records’ statistics and occu- +pation time statistics. While the standard map, when a chaotic sea is present, always reproduces +results expected for simple random walks, the triangle map -whose analysis still displays many elu- +sive points- behaves in a wildly different way, some of the features being compatible with a transient +(non conservative) nature of the dynamics. +Keywords: Area-preserving maps, record statistics, infinite ergodicity. +I. +INTRODUCTION +One of the most remarkable advances in modern dy- +namics lies in the recognition that deterministic sys- +tems may exhibit statistical properties typical of purely +stochastic processes: for instance such systems may dis- +play diffusion properties similar to random walks [1–4]. +Area-preserving maps (see for instance [1]) represent a +prominent example of Hamiltonian systems where subtle +features of dynamics, as integrability vs chaotic proper- +ties, may be studied. In this context one of the most out- +standing example is represented by the Chirikov-Taylor +standard map (SM) (see [1, 5] and references therein): +we also mention the fundamental role of such a map in +the development of quantum chaos, unveiling features +like quantum dynamical localization [6]. Though the SM +has been extensively explored by numerical simulations, +very few rigorous results have been proven (see, for in- +stance, the introduction in [7]): however it is generally +believed that for large nonlinearity parameter this map +typically exhibits good stochastic properties, and sensi- +tive dependence upon initial conditions. Here a remark +is due: such a map can be studied either on a 2-torus +or on an (unbounded) cylinder: the latter representa- +tion is naturally adopted when transport properties are +concerned, and analogies with random walks are taken +into account [1, 3, 8, 9]. While particular nonlinear pa- +rameters in the standard map can be tuned to generate +strong anomalous diffusion [10], here we will only deal +with the case in which diffusion is normal. +Our find- +ings will be confronted with those obtained for another +area-preserving map, characterized by the lack of expo- +nential instability: the so called Casati-Prosen Triangle +∗ roberto.artuso@uninsubria.it +† cesar.manchein@udesc.br +Map (TM) [11], introduced by considering, in an appro- +priate limit, the Birkhoff dynamics of a triangular bil- +liard: apart from its intrinsic interest, such a map is an +ideal benchmark to test whether stochasticity properties, +exhibited by strongly chaotic systems, are showcased also +by systems lacking any exponential instability. It also +turns out that many features about the TM are still de- +bated, starting from basic properties like ergodicity and +mixing (see for instance [12, 13]). +More precisely we will compare different indicators +for both map on the cylinder: though in principle fur- +ther complications are added when one considers a non- +compact phase space [14, 15], this is the appropriate sce- +nario to discuss transport properties and record statis- +tics, and to check whether tools from infinite ergodic +theory may enrich our understanding of such systems. +Our main findings are that the SM, in its typical +chaotic regime, displays all stochastic properties of a +purely stochastic system, while -as expected- results are +far more complicated for the TM, even if we believe that +some new insight is provided by our analysis, in partic- +ular as regards persistence behaviour, occupation time +statistics and the relationship between transport proper- +ties and record statistics. +The paper is organized as follows. +In Sec. II, +the Chirikov-Taylor standard map (1) and the triangle +map (3) -our basic models- are presented and we also +mention the main properties we analyze. Section III is +dedicated to discuss transport properties, records’ statis- +tics and occupation time statistics. We end with a dis- +cussions in Sec. IV. + +2 +II. +THE BASIC SETTING +We recall the definition of the SM +pn+1 = pn + K +2π sin (2πxn), +xn+1 = xn + pn+1 +mod 1; +(1) +K being the nonlinear parameter: when K is sufficiently +big no KAM invariant circles bound the motion and one +can study moments of the diffusing variable p ∈ R: +⟨|pn − p0|q⟩ ∼ nqν(q). +(2) +The typical behaviour observed for the second moment in +simulations is normal diffusion ν(2) = 1/2 [16, 17], while, +for certain parameter values, the existence of stable run- +ning orbits (accelerator modes) induces superdiffusion, +ν(2) > 1/2) [18–20]. We point out that a finer descrip- +tion of anomalous transport is obtained by considering +the full spectrum ν(q): if ν(q) = α · q, for some α ̸= 1/2 +one speaks about weak anomalous diffusion whereas the +case of a nontrivial ν(q) is dubbed strong anomalous dif- +fusion [10]. As far as the SM is concerned we will consider +the case where transport in the stochastic sea is normal +(even if the phase space exhibits a mixture of chaotic and +elliptic components (see Figure 1). +Figure 1. Phase-space portrait for the standard map (1) on +the 2-torus, for K = 2.6. Here 40 uniformly distributed initial +conditions were used for x, while maintaining p0 = 0 fixed: +each initial condition is iterated 104 times. +On the other side the TM is defined (on the cylinder) +as: +pn+1 = pn + 2(xn− ⇂ xn ↿ −µ(−1)⇂xn↿), +xn+1 = xn − 2pn+1 +mod 2, +(3) +where ⇂ · · · ↿ denotes the nearest integer. It was intro- +duced in [11] (see also [21]) as a limit case for the Birkhoff +map of irrational triangular billiards: systems lacking ex- +ponential instability, whose ergodic properties are subtly +related to irrationality properties of the angles [22–25]: +we remark that polygonal billiards represent both a hard +mathematical challenge [26–29], and a natural bench- +mark when trying to assess which microscopic dynam- +ical features lead to macroscopic transport laws [30–32] +( see also [33, 34]): in this respect it is worth mentioning +that anomalous transport has been associated to scaling +exponents of the spectral measure [35], and that general- +ized triangle maps have been investigated recently, both +as connected to dynamical localization [36], and with +respect to slow diffusion [37]. A typical phase portrait +(on the torus) of the TM is shown in Figure 2. Before +Figure 2. Phase-space dynamics for the triangle map (3), for +µ = +1+ +√ +5 +2 +(golden mean). +Here 100 randomly distributed +initial conditions were used for x and p: each initial condition +is iterated 5×104 times. Notice the typical filament structure +in the phase space [23, 24]. +mentioning the numerical experiments we performed, a +crucial observation is in order. When looking at trans- +port properties (and records statistics), considering maps +on the cylinder is quite natural, while from the ergodic +point of view this perspective is somehow delicate, since +no renormalizable invariant density exists [14, 15], and +the appropriate setting is infinite ergodic theory. When +polygonal channels are considered, even establishing re- +current properties of the dynamics is a demanding task +[38]. +The first set of properties we investigated is more con- +ventional, and a few results -as we will mention in the +next section- have already been considered, especially as +far as the SM is considered. We will look at transport +properties, in particular through the first and the sec- +ond moment of the diffusing variable We will also con- +sider records’ statistics, which recently has turned very +popular (see [39, 40] and references therein). Then we +will study statistical properties like persistence probabil- +ity and (generalized) arcsine law [41, 42]: while motion +in the stochastic sea for the SM will exhibit typical prop- +erties of a simple stochastic process like a random walk, + +1.0 +p +0.0 +0.0 +2.00.5 +p +-0.5 +0.03 +our findings are quite different in the case of the TM. +III. +RESULTS +We start by considering properties associated to the +spreading of trajectories over the phase space, then we +will consider occupation time statistics. +A. +Diffusion +This is a warm-up exercise, since transport properties +have been studied both for the SM [1, 16, 17] and for the +TM [25]. We observe normal transport for the case of +the SM (see panels (c) and (d) in Figure 3), while for the +TM are results indicate a superdiffusion, with +⟨(pn − p0)2⟩ ∼ n1.83, +(4) +in agreement with [25]. We remark that by looking at the +power-law exponents of the first two moments, we have +that possibly anomalous diffusion is weak [10], namely if +we consider the full spectrum of moments’ asymptotics: +⟨|pn − p0|q⟩ ∼ nφ(q), +(5) +we have a single scaling, in the sense that +φ(q) = α · q; +(6) +where normal diffusion is recovered when α = 1/2. This +is reasonable since weak anomalous diffusion has been +observed in polygonal billiards [43]. +B. +Average number of records +The statistics of records is very popular in the anal- +ysis of correlated and uncorrelated stochastic time se- +quences [39, 40]: +since this subject has not been ex- +plores thoroughly in the deterministic setting (with the +remarkable exception of [44, 45]), we briefly review the +basic concepts. First of all let us recall the (straightfor- +ward) definition of a record: given a sequence of real data +x0, x1, . . . , xk, . . . the element xm is a record if +xm > xj +j = 0, 1, . . . m − 1, +(7) +(we consider x0 to be the first record). +To the se- +quence of data points we associate the binary string +σ0, σ1, . . . , σk, . . . , where +σl = +� +1 +if xl is a record +0 +otherwise +(8) +The number of records up to time N is then +MN = +N +� +j=0 +σj. +(9) +10−1 +100 +101 +102 +103 +104 +105 +106 +107 +108 +100 +101 +102 +103 +104 +105 +106 +107 +108 +(b) +10−2 +10−1 +100 +101 +102 +103 +104 +105 +100 +101 +102 +103 +104 +105 +106 +(d) +10−1 +100 +101 +102 +103 +104 +100 +101 +102 +103 +104 +105 +106 +107 +108 +(a) +10−2 +10−1 +100 +101 +102 +100 +101 +102 +103 +104 +105 +106 +(c) +σ2(n) +⟨(pn − p0)2⟩ +n +⟨M(n)⟩ +⟨pn − p0⟩ +n +Figure 3. +(a) Average number of records, (b) variance, (c) +first, and (b) second moments of variable p for K = 2.6 in the +standard map (1), as a function of time. These quantities were +computed for 106 initial conditions for x0, arbitrarily chosen +in the chaotic sea along the line p0 = 0. Black-continuous +lines correspond to power-law asymptotics fit F(n) = anγ: +the fitting parameters are, for (a) a = 0.86(0), γ = 0.50(9), for +(b) a = 0.50(7), γ = 1.00(3), for (c) a = 0.77(7), γ = 0.50(1) +and for (d) a = 0.02(1), γ = 0.99(1). +The properties of the average number of records, +⟨MN⟩, and the corresponding variance +V ar(MN) = ⟨M 2 +N⟩ − ⟨MN⟩2 +(10) +are important tools to access the nature of the data se- +quence: as a matter of fact if the different xj are inde- +pendent, identically distributed random variables, then, +for large N we have [46, 47]: +⟨MN⟩ = ln N + γE + O(N −1), +(11) +where γE = 0.5772 . . . is the Euler-Mascheroni constant, +and +V ar(MN) = σ2(N) = ln N + γE − π2 +6 + O(N −1). (12) +We remark that both quantities are independent of the +common distribution of the random variables: this uni- +versality is an important feature of record statistics in +different contexts. +Results are quite different for a correlated sequence, as +when xj denotes the position of a random walker at time +j: +xj+1 = xj + ξj+1, +(13) +where the jumps are taken from a common distribution +℘(ξ). In this case the behaviour is [39, 40]: +⟨MN⟩ ≈ +2 +√π +√ +N, +(14) + +4 +100 +102 +104 +106 +108 +1010 +1012 +100 +101 +102 +103 +104 +105 +106 +107 +(b) +102 +104 +106 +108 +1010 +1012 +100 +101 +102 +103 +104 +105 +106 +107 +(d) +10−1 +100 +101 +102 +103 +104 +105 +106 +100 +101 +102 +103 +104 +105 +106 +107 +(a) +100 +101 +102 +103 +104 +105 +106 +107 +100 +101 +102 +103 +104 +105 +106 +107 +(c) +σ2(n) +⟨(pn − p0)2⟩ +n +⟨M(n)⟩ +⟨pn − p0⟩ +n +Figure 4. (a) Average number of records, (b) variance, (c) +first, and (b) second moments of variable p for the golden +mean µ = +1+ +√ +5 +2 +in the triangular map (3) as a function of +time. +These quantities were computed for 106 initial con- +ditions for x0, arbitrarily chosen in phase space along the +line p0 = 0. Black-continuous lines correspond to power-law +asymptotics F(n) = anγ: the fitting parameters are, for (a) +a = 0.13(9), γ = 0.92(4), for (b) a = 0.04(0), γ = 1.84(9), +for (c) a = 0.65(4), γ = 0.92(4) and for (c) a = 0.67(6), +γ = 1.86(0) in (d). +and +V ar(MN) ≈ 2 +� +1 − 2 +π +� +N, +(15) +so that the standard deviation is of the same order of +magnitude as the average. +Again this is a universal +result, independent of the particular jump distribution +℘(ξ), as long as the distribution is continuous and sym- +metric. The crucial ingredient of the proof is that the +process renews as soon as a new record is achieved and +the appearance of the new record is related to the survival +probability for the process, which is universal in view of +Sparre-Andersen theorem [42, 48, 49] (see also [50]). +Numerical results on records statistics are reported in +Figures 3, 4, panels (a) and (b): for the SM our results +are consistent with early investigations [44, 45], and with +the asymptotic behaviour of a random walk, while for +the TM we observe anomalous scaling w.r.t. (14,15): the +behaviour is related to transport properties, in the sense +that data are consistent with the growths: +⟨MN⟩ ∼ N φ(1), +V ar(MN) ∼ N φ(2). +(16) +A similar behaviour was observed in [44, 45], for the SM +in the presence of accelerator modes. We remark that, +though in the following we will fix our attention of a +particular parameter value for the TM, we checked that +reported experiments do not depend on the particular +parameter choice, as exemplified in Figure 5, where the +growth of the averaged number of records is reported for +different parameters of the TM. +While a general, quantitative relationship (if any) be- +10−1 +100 +101 +102 +103 +104 +105 +106 +100 +101 +102 +103 +104 +105 +106 +107 +⟨Mn⟩ +n +√ +7 +√ +2/2 +( +√ +5 + e)/12 +Figure 5. (a) Average number of records for three additional +parameters µ in the TM. Black-continuous line correspond to +the power-law asymptotic fitting function F(n) = anγ, with +γ = 0.92(4). +tween transport exponents and statistical properties of +records has not been fully developed, to the best of our +knowledge, it is possible, in some cases, to connect φ(1) +to the expected maximum of the walk [51, 52], that, for +random walk with unit jumps, coincides with the num- +ber of records. On the other side we mention that non- +homogeneous random walks offer examples where such +relationship does not hold [53–57]. +C. +Occupation time statistics +When we consider the evolution on the cylinder, both +for the SM and the TM, we are in the presence of in- +finitely ergodic systems [14, 15], since, while Lebesgue +measure is preserved, due to area conservation, the (con- +stant) phase space is unbounded, so the invariant density +cannot be normalized. This has a series of remarkable +consequences, which originally have been considered in +the context of stochastic processes, and then explored in +the deterministic evolution framework. +One of the most striking property that has been in- +vestigated is the generalized arcsine law (see [41] for the +standard formulation for stochastic processes): we briefly +recall the main result that lies at the basis of our analysis, +namely Lamperti’s theorem [58]. The original formula- +tion involves discrete stochastic processes, for which the +infinite set of possible states can be separated into two +sets A and B separated by a single site x0, such that a + +5 +transition from one set to the other can only be achieved +by passing through x0, which can be taken as the start- +ing site, and is supposed to be recurrent (namely the +probability of returning to it is 1). For instance we can +think of one dimensional random walk on an integer lat- +tice, with x0 = 0 and A (B) consists of strictly positive +(negative) lattice sites. We are interested in the limit- +ing distribution of N(n)/n, the fraction of time spent in +the positive semi-axis up to time n. The theorem states +that such a distribution exists in the n → ∞ limit, and +it is characterized by two parameters α and η. η is re- +lated to symmetry properties of the process, being the +expectation value of the fraction of time spent in R+: +η = lim +n→∞ E +�N(n) +n +� +: +(17) +for a symmetric process η = 1/2, and from now on we +will only consider such a case. +−1.0 +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +(b) +−0.4 +−0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +(a) +arccos(2(N(n)/n)−1) +log10P(N(n)/n)) +arccos(2(N(n)/n)−1) +Figure 6. Distribution of the fraction of time spent in the positive axis for the momentum p in the standard (1) (a) and triangle +(3) (b) maps, in semi-logarithmic scale. To enhance readability of the border values, the transformation x → arccos(2x − 1) on +the horizontal axis. The (light blue) points represent the simulation results, the (red) line the Lamperti distribution (20). Data +are obtained by computing 106 initial conditions iterated 106 times for the standard map and 106 initial conditions iterated +108 times for the triangle map. The fitting parameters are α = 0.49(9) for (a) and α = 0.42(0) for (b). In the case of the TM, +data suggest a superposition of a (rescaled) Lamperti distribution and two Dirac’s δ centered of x = 0 and x = 1 (see text). +The other parameter α is instead connected to the be- +haviour of the generating function of first return prob- +abilities to the starting site: it can be shown [59] that +it can be related to the probability Pn of being at the +starting site after n steps in the following way: +Pn ∼ H(n) +n1−α , +(18) +where H(n) is a slowly varying function, namely +lim +n→∞ +H(yn) +n += 1. +(19) +Under such conditions the density of ϕ = N(n)/n in the +infinite time limit is given by Lamperti distribution: +Gα(ϕ) = sin(πα) +π +ϕ1−α(1 − ϕ)1−α +ϕ2α + 2ϕα(1 − ϕ)α cos(πα) + (1 − ϕ)2α , +(20) +that reproduces the usual arcsine law +P ((Nn/n) ≤ ξ) = 2 +π arcsin +�� +ξ +� +(21) +when α = 1/2, in the universality class of Sparre- +Andersen theorem. Deviations from standard arcsine law +have been reported for a number of cases, in the frame- +work of deterministic dynamics [60–67], mainly in the +context of intermittent maps. Numerical experiments for +the SM confirm the validity of the arcsine law, α = 1/2, +see panel (a) in Figure 6: to our knowledge this is the +first time such an indicator has been considered in the +analysis of area preserving maps. +The results, as expected, are quite different for the TM, +and they suggest novel features exhibited by this map. In +particular (see panel (b) in Figure 6) numerical results are +well fitted by a Lamperti distribution (with α ≈ 0.42), +thus different from an ordinary random walk), except for +the endpoints, that present enhanced peaks. Intuitively + +6 +such an additional contribution might be due to a frac- +tion of orbits never returning to the origin: this would +correspond, in stochastic language, to a transient random +walk (we recall that, according to Pólya’s theorem [68] a +simple symmetric random walk is recurrent -so the return +to the starting site is sure- in one and two dimensions, +and transient in higher dimensions). Such a possibility is +indeed not excluded for infinite polygonal channels [38]. +Our last set of simulations concerns the survival proba- +bility [61]: +Pcum(n) = prob (pn ≥ 0 . . . p1 ≥ 0|p0 = 0) . +(22) +10−3 +10−2 +10−1 +100 +100 +101 +102 +103 +104 +105 +106 +(a) +10−2 +10−1 +100 +100 101 102 103 104 105 106 107 108 109 +(b) +Pcum(n) +n +n +Figure 7. Cumulative distribution function for the survival times obtained for the variable p for (a) the standard map (1), and +(b) the triangle map (3), in logarithmic scale. Data are obtained by simulating 106 and 105 initial conditions, respectively. +Continuous-black lines correspond to power-law asymptotic functions F(n) = a + bn−α: the fitting parameters are a = 0, b = +2.80(0), and α = 0.51(5) in (a) and a = 0.021(0), b = 1.62(6), and α = 0.42(0) in (b). +When considering recurrent random walks, the asymp- +totic behaviour of the survival probability is again ruled +by Lamperti exponent [58, 59] (see also [69]): +Pcum(n) ∼ n−α. +(23) +Once again SM simulations (see panel (a) in Figure 7) +agree with expected behaviour for simple random walks +(α = 1/2), while the situation is completely different for +the TM, where the survival probability seems to tend to +a finite limit for large n, see panel (b) in Figure 7. This +is coherent with the transient nature of the TM, which +we conjectured in the analysis of generalized arcsine law. +IV. +DISCUSSION +We have performed a set of extensive numerical experi- +ments on two paradigmatic area-preserving maps, the SM +and the TM, focusing in the case where such maps are +considered on a cylinder, namely a non compact phase +space. Firstly we reproduced known results about nor- +mal diffusion for typical (chaotic) parameters of the SM, +and superdiffusion for the TM. Then we explored records’ +statistics: numerical simulations again confirm that the +SM behave like a simple random walk, while anomalous +growth is exhibited by the TM. The most interesting re- +sults arise in the analysis of occupation times, like gen- +eralized arcsine law and survival probability. While once +again normal stochastic properties are displayed by the +SM, the TM presents more surprising results, which we +conjecture are possibly connected to lack of conserva- +tivity [38] (or transient behaviour, in the language of +random walks). This feature, which we think is worth +of further investigations, might suggest new stochastic +modeling of the TM (see [37]). +AUTHORS’ CONTRIBUTIONS +All authors have contributed substantially to the work. +All authors have read and agreed to the published version +of the manuscript. + +7 +ACKNOWLEDGEMENTS +R.A. acknowledges partial support from PRIN Re- +search Project No. 2017S35EHN “Regular and stochastic +behavior in dynamical systems” of the Italian Ministry of +Education, University and Research (MIUR). R.A. ac- +knowledges an association to the GNFM group of IN- +DAM. R.A thanks Gaia Pozzoli for discussions. C.M. ac- +knowledges the National Council for Scientific and Tech- +nological Development - CNPq (Brazilian agency) for +partial financial support (Grant Number 310228/2020-4). +T.M.O. acknowledges the Coordenação de Aperfeiçoa- +mento de Pessoal de Nível Superior - CAPES (Brazilian +agency ) - Finance Code 001, for partial financial sup- +port. Additionally, T.M.O. and C.M. also acknowledges +the Fundação de Amparo à Pesquisa e Inovação do Es- +tado de Santa Catarina - FAPESC (Brazilian agency) for +partial financial support. +[1] Lichtenberg, A.J.; Lieberman, M.A. Regular and chaotic +dynamics; Springer: Berlin, Germany, 1992. +[2] Ott, E., Chaos in dynamical systems; CUP: Cambridge, +U.K. 2002. +[3] Cvitanović, P.; Artuso, R.; Mainieri, R.; Tanner, G.; Vat- +tay, G., Chaos: Classical and Quantum, ChaosBook.org +; Niels Bohr Institute, Copenhagen 2020. +[4] Artuso, R.; Burioni, R. Anomalous diffusion: determin- +istic and stochastic perspectives. 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J.Stat.Phys. 2006, 123, +883–907. + diff --git a/GtE1T4oBgHgl3EQf_QYB/content/tmp_files/load_file.txt b/GtE1T4oBgHgl3EQf_QYB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..16705c7795568012ff4c03f8e9cb98fb8f80b772 --- /dev/null +++ b/GtE1T4oBgHgl3EQf_QYB/content/tmp_files/load_file.txt @@ -0,0 +1,790 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf,len=789 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='03577v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='CD] 9 Jan 2023 Records and occupation time statistics for area-preserving maps⋆ Roberto Artuso1,2,∗ Tulio M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' de Oliveira3, and Cesar Manchein3† 1Dipartimento di Scienza e Alta Tecnologia and Center for Nonlinear and Complex Systems, Via Valleggio 11, 22100 Como, Italy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' 2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='N, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' and 3Departamento de Física, Universidade do Estado de Santa Catarina, 89219-710 Joinville, SC, Brazil ⋆To Giulio Casati, celebrating his birthday and his achievements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' (Dated: January 10, 2023) A relevant problem in dynamics is to characterize how deterministic systems may exhibit fea- tures typically associated to stochastic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' A widely studied example is the study of (normal or anomalous) transport properties for deterministic systems on a non-compact phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' We consider here two examples of area-preserving maps: the Chirikov-Taylor standard map and the Casati-Prosen triangle map, and we investigate transport properties, records’ statistics and occu- pation time statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' While the standard map, when a chaotic sea is present, always reproduces results expected for simple random walks, the triangle map -whose analysis still displays many elu- sive points- behaves in a wildly different way, some of the features being compatible with a transient (non conservative) nature of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Keywords: Area-preserving maps, record statistics, infinite ergodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' INTRODUCTION One of the most remarkable advances in modern dy- namics lies in the recognition that deterministic sys- tems may exhibit statistical properties typical of purely stochastic processes: for instance such systems may dis- play diffusion properties similar to random walks [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Area-preserving maps (see for instance [1]) represent a prominent example of Hamiltonian systems where subtle features of dynamics, as integrability vs chaotic proper- ties, may be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' In this context one of the most out- standing example is represented by the Chirikov-Taylor standard map (SM) (see [1, 5] and references therein): we also mention the fundamental role of such a map in the development of quantum chaos, unveiling features like quantum dynamical localization [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Though the SM has been extensively explored by numerical simulations, very few rigorous results have been proven (see, for in- stance, the introduction in [7]): however it is generally believed that for large nonlinearity parameter this map typically exhibits good stochastic properties, and sensi- tive dependence upon initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Here a remark is due: such a map can be studied either on a 2-torus or on an (unbounded) cylinder: the latter representa- tion is naturally adopted when transport properties are concerned, and analogies with random walks are taken into account [1, 3, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' While particular nonlinear pa- rameters in the standard map can be tuned to generate strong anomalous diffusion [10], here we will only deal with the case in which diffusion is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Our find- ings will be confronted with those obtained for another area-preserving map, characterized by the lack of expo- nential instability: the so called Casati-Prosen Triangle ∗ roberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='artuso@uninsubria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='it † cesar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='manchein@udesc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='br Map (TM) [11], introduced by considering, in an appro- priate limit, the Birkhoff dynamics of a triangular bil- liard: apart from its intrinsic interest, such a map is an ideal benchmark to test whether stochasticity properties, exhibited by strongly chaotic systems, are showcased also by systems lacking any exponential instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' It also turns out that many features about the TM are still de- bated, starting from basic properties like ergodicity and mixing (see for instance [12, 13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' More precisely we will compare different indicators for both map on the cylinder: though in principle fur- ther complications are added when one considers a non- compact phase space [14, 15], this is the appropriate sce- nario to discuss transport properties and record statis- tics, and to check whether tools from infinite ergodic theory may enrich our understanding of such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Our main findings are that the SM, in its typical chaotic regime, displays all stochastic properties of a purely stochastic system, while -as expected- results are far more complicated for the TM, even if we believe that some new insight is provided by our analysis, in partic- ular as regards persistence behaviour, occupation time statistics and the relationship between transport proper- ties and record statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' II, the Chirikov-Taylor standard map (1) and the triangle map (3) -our basic models- are presented and we also mention the main properties we analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Section III is dedicated to discuss transport properties, records’ statis- tics and occupation time statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' We end with a dis- cussions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' THE BASIC SETTING We recall the definition of the SM pn+1 = pn + K 2π sin (2πxn), xn+1 = xn + pn+1 mod 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' (1) K being the nonlinear parameter: when K is sufficiently big no KAM invariant circles bound the motion and one can study moments of the diffusing variable p ∈ R: ⟨|pn − p0|q⟩ ∼ nqν(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' (2) The typical behaviour observed for the second moment in simulations is normal diffusion ν(2) = 1/2 [16, 17], while, for certain parameter values, the existence of stable run- ning orbits (accelerator modes) induces superdiffusion, ν(2) > 1/2) [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' We point out that a finer descrip- tion of anomalous transport is obtained by considering the full spectrum ν(q): if ν(q) = α · q, for some α ̸= 1/2 one speaks about weak anomalous diffusion whereas the case of a nontrivial ν(q) is dubbed strong anomalous dif- fusion [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' As far as the SM is concerned we will consider the case where transport in the stochastic sea is normal (even if the phase space exhibits a mixture of chaotic and elliptic components (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Phase-space portrait for the standard map (1) on the 2-torus, for K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Here 40 uniformly distributed initial conditions were used for x, while maintaining p0 = 0 fixed: each initial condition is iterated 104 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' On the other side the TM is defined (on the cylinder) as: pn+1 = pn + 2(xn− ⇂ xn ↿ −µ(−1)⇂xn↿), xn+1 = xn − 2pn+1 mod 2, (3) where ⇂ · · · ↿ denotes the nearest integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' It was intro- duced in [11] (see also [21]) as a limit case for the Birkhoff map of irrational triangular billiards: systems lacking ex- ponential instability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' whose ergodic properties are subtly related to irrationality properties of the angles [22–25]: we remark that polygonal billiards represent both a hard mathematical challenge [26–29],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' and a natural bench- mark when trying to assess which microscopic dynam- ical features lead to macroscopic transport laws [30–32] ( see also [33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' 34]): in this respect it is worth mentioning that anomalous transport has been associated to scaling exponents of the spectral measure [35],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' and that general- ized triangle maps have been investigated recently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' both as connected to dynamical localization [36],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' and with respect to slow diffusion [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' A typical phase portrait (on the torus) of the TM is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Before Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Phase-space dynamics for the triangle map (3), for µ = 1+ √ 5 2 (golden mean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Here 100 randomly distributed initial conditions were used for x and p: each initial condition is iterated 5×104 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Notice the typical filament structure in the phase space [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' mentioning the numerical experiments we performed, a crucial observation is in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' When looking at trans- port properties (and records statistics), considering maps on the cylinder is quite natural, while from the ergodic point of view this perspective is somehow delicate, since no renormalizable invariant density exists [14, 15], and the appropriate setting is infinite ergodic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' When polygonal channels are considered, even establishing re- current properties of the dynamics is a demanding task [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' The first set of properties we investigated is more con- ventional, and a few results -as we will mention in the next section- have already been considered, especially as far as the SM is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' We will look at transport properties, in particular through the first and the sec- ond moment of the diffusing variable We will also con- sider records’ statistics, which recently has turned very popular (see [39, 40] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Then we will study statistical properties like persistence probabil- ity and (generalized) arcsine law [41, 42]: while motion in the stochastic sea for the SM will exhibit typical prop- erties of a simple stochastic process like a random walk, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='0 p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='5 p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='03 our findings are quite different in the case of the TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' RESULTS We start by considering properties associated to the spreading of trajectories over the phase space, then we will consider occupation time statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Diffusion This is a warm-up exercise, since transport properties have been studied both for the SM [1, 16, 17] and for the TM [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' We observe normal transport for the case of the SM (see panels (c) and (d) in Figure 3), while for the TM are results indicate a superdiffusion, with ⟨(pn − p0)2⟩ ∼ n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='83, (4) in agreement with [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' We remark that by looking at the power-law exponents of the first two moments, we have that possibly anomalous diffusion is weak [10], namely if we consider the full spectrum of moments’ asymptotics: ⟨|pn − p0|q⟩ ∼ nφ(q), (5) we have a single scaling, in the sense that φ(q) = α · q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' (6) where normal diffusion is recovered when α = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' This is reasonable since weak anomalous diffusion has been observed in polygonal billiards [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Average number of records The statistics of records is very popular in the anal- ysis of correlated and uncorrelated stochastic time se- quences [39, 40]: since this subject has not been ex- plores thoroughly in the deterministic setting (with the remarkable exception of [44, 45]), we briefly review the basic concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' First of all let us recall the (straightfor- ward) definition of a record: given a sequence of real data x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' , xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' the element xm is a record if xm > xj j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' m − 1, (7) (we consider x0 to be the first record).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' To the se- quence of data points we associate the binary string σ0, σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' , σk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' , where σl = � 1 if xl is a record 0 otherwise (8) The number of records up to time N is then MN = N � j=0 σj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' (9) 10−1 100 101 102 103 104 105 106 107 108 100 101 102 103 104 105 106 107 108 (b) 10−2 10−1 100 101 102 103 104 105 100 101 102 103 104 105 106 (d) 10−1 100 101 102 103 104 100 101 102 103 104 105 106 107 108 (a) 10−2 10−1 100 101 102 100 101 102 103 104 105 106 (c) σ2(n) ⟨(pn − p0)2⟩ n ⟨M(n)⟩ ⟨pn − p0⟩ n Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' (a) Average number of records, (b) variance, (c) first, and (b) second moments of variable p for K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='6 in the standard map (1), as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' These quantities were computed for 106 initial conditions for x0, arbitrarily chosen in the chaotic sea along the line p0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Black-continuous lines correspond to power-law asymptotics fit F(n) = anγ: the fitting parameters are, for (a) a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='86(0), γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='50(9), for (b) a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='50(7), γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='00(3), for (c) a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='77(7), γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='50(1) and for (d) a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='02(1), γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='99(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' The properties of the average number of records, ⟨MN⟩, and the corresponding variance V ar(MN) = ⟨M 2 N⟩ − ⟨MN⟩2 (10) are important tools to access the nature of the data se- quence: as a matter of fact if the different xj are inde- pendent, identically distributed random variables, then, for large N we have [46, 47]: ⟨MN⟩ = ln N + γE + O(N −1), (11) where γE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='5772 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' is the Euler-Mascheroni constant, and V ar(MN) = σ2(N) = ln N + γE − π2 6 + O(N −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' (12) We remark that both quantities are independent of the common distribution of the random variables: this uni- versality is an important feature of record statistics in different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Results are quite different for a correlated sequence, as when xj denotes the position of a random walker at time j: xj+1 = xj + ξj+1, (13) where the jumps are taken from a common distribution ℘(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' In this case the behaviour is [39, 40]: ⟨MN⟩ ≈ 2 √π √ N, (14) 4 100 102 104 106 108 1010 1012 100 101 102 103 104 105 106 107 (b) 102 104 106 108 1010 1012 100 101 102 103 104 105 106 107 (d) 10−1 100 101 102 103 104 105 106 100 101 102 103 104 105 106 107 (a) 100 101 102 103 104 105 106 107 100 101 102 103 104 105 106 107 (c) σ2(n) ⟨(pn − p0)2⟩ n ⟨M(n)⟩ ⟨pn − p0⟩ n Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' (a) Average number of records, (b) variance, (c) first, and (b) second moments of variable p for the golden mean µ = 1+ √ 5 2 in the triangular map (3) as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' These quantities were computed for 106 initial con- ditions for x0, arbitrarily chosen in phase space along the line p0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Black-continuous lines correspond to power-law asymptotics F(n) = anγ: the fitting parameters are, for (a) a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='13(9), γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='92(4), for (b) a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='04(0), γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='84(9), for (c) a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='65(4), γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='92(4) and for (c) a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='67(6), γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='86(0) in (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' and V ar(MN) ≈ 2 � 1 − 2 π � N, (15) so that the standard deviation is of the same order of magnitude as the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Again this is a universal result, independent of the particular jump distribution ℘(ξ), as long as the distribution is continuous and sym- metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' The crucial ingredient of the proof is that the process renews as soon as a new record is achieved and the appearance of the new record is related to the survival probability for the process, which is universal in view of Sparre-Andersen theorem [42, 48, 49] (see also [50]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Numerical results on records statistics are reported in Figures 3, 4, panels (a) and (b): for the SM our results are consistent with early investigations [44, 45], and with the asymptotic behaviour of a random walk, while for the TM we observe anomalous scaling w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' (14,15): the behaviour is related to transport properties, in the sense that data are consistent with the growths: ⟨MN⟩ ∼ N φ(1), V ar(MN) ∼ N φ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' (16) A similar behaviour was observed in [44, 45], for the SM in the presence of accelerator modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' We remark that, though in the following we will fix our attention of a particular parameter value for the TM, we checked that reported experiments do not depend on the particular parameter choice, as exemplified in Figure 5, where the growth of the averaged number of records is reported for different parameters of the TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' While a general, quantitative relationship (if any) be- 10−1 100 101 102 103 104 105 106 100 101 102 103 104 105 106 107 ⟨Mn⟩ n √ 7 √ 2/2 ( √ 5 + e)/12 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' (a) Average number of records for three additional parameters µ in the TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Black-continuous line correspond to the power-law asymptotic fitting function F(n) = anγ, with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='92(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' tween transport exponents and statistical properties of records has not been fully developed, to the best of our knowledge, it is possible, in some cases, to connect φ(1) to the expected maximum of the walk [51, 52], that, for random walk with unit jumps, coincides with the num- ber of records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' On the other side we mention that non- homogeneous random walks offer examples where such relationship does not hold [53–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Occupation time statistics When we consider the evolution on the cylinder, both for the SM and the TM, we are in the presence of in- finitely ergodic systems [14, 15], since, while Lebesgue measure is preserved, due to area conservation, the (con- stant) phase space is unbounded, so the invariant density cannot be normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' This has a series of remarkable consequences, which originally have been considered in the context of stochastic processes, and then explored in the deterministic evolution framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' One of the most striking property that has been in- vestigated is the generalized arcsine law (see [41] for the standard formulation for stochastic processes): we briefly recall the main result that lies at the basis of our analysis, namely Lamperti’s theorem [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' The original formula- tion involves discrete stochastic processes, for which the infinite set of possible states can be separated into two sets A and B separated by a single site x0, such that a 5 transition from one set to the other can only be achieved by passing through x0, which can be taken as the start- ing site, and is supposed to be recurrent (namely the probability of returning to it is 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' For instance we can think of one dimensional random walk on an integer lat- tice, with x0 = 0 and A (B) consists of strictly positive (negative) lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' We are interested in the limit- ing distribution of N(n)/n, the fraction of time spent in the positive semi-axis up to time n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' The theorem states that such a distribution exists in the n → ∞ limit, and it is characterized by two parameters α and η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' η is re- lated to symmetry properties of the process, being the expectation value of the fraction of time spent in R+: η = lim n→∞ E �N(n) n � : (17) for a symmetric process η = 1/2, and from now on we will only consider such a case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='5 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='0 (a) arccos(2(N(n)/n)−1) log10P(N(n)/n)) arccos(2(N(n)/n)−1) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Distribution of the fraction of time spent in the positive axis for the momentum p in the standard (1) (a) and triangle (3) (b) maps, in semi-logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' To enhance readability of the border values, the transformation x → arccos(2x − 1) on the horizontal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' The (light blue) points represent the simulation results, the (red) line the Lamperti distribution (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Data are obtained by computing 106 initial conditions iterated 106 times for the standard map and 106 initial conditions iterated 108 times for the triangle map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' The fitting parameters are α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='49(9) for (a) and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='42(0) for (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' In the case of the TM, data suggest a superposition of a (rescaled) Lamperti distribution and two Dirac’s δ centered of x = 0 and x = 1 (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' The other parameter α is instead connected to the be- haviour of the generating function of first return prob- abilities to the starting site: it can be shown [59] that it can be related to the probability Pn of being at the starting site after n steps in the following way: Pn ∼ H(n) n1−α , (18) where H(n) is a slowly varying function, namely lim n→∞ H(yn) n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' (19) Under such conditions the density of ϕ = N(n)/n in the infinite time limit is given by Lamperti distribution: Gα(ϕ) = sin(πα) π ϕ1−α(1 − ϕ)1−α ϕ2α + 2ϕα(1 − ϕ)α cos(πα) + (1 − ϕ)2α , (20) that reproduces the usual arcsine law P ((Nn/n) ≤ ξ) = 2 π arcsin �� ξ � (21) when α = 1/2, in the universality class of Sparre- Andersen theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Deviations from standard arcsine law have been reported for a number of cases, in the frame- work of deterministic dynamics [60–67], mainly in the context of intermittent maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Numerical experiments for the SM confirm the validity of the arcsine law, α = 1/2, see panel (a) in Figure 6: to our knowledge this is the first time such an indicator has been considered in the analysis of area preserving maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' The results, as expected, are quite different for the TM, and they suggest novel features exhibited by this map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' In particular (see panel (b) in Figure 6) numerical results are well fitted by a Lamperti distribution (with α ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='42), thus different from an ordinary random walk), except for the endpoints, that present enhanced peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Intuitively 6 such an additional contribution might be due to a frac- tion of orbits never returning to the origin: this would correspond, in stochastic language, to a transient random walk (we recall that, according to Pólya’s theorem [68] a simple symmetric random walk is recurrent -so the return to the starting site is sure- in one and two dimensions, and transient in higher dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Such a possibility is indeed not excluded for infinite polygonal channels [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Our last set of simulations concerns the survival proba- bility [61]: Pcum(n) = prob (pn ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' p1 ≥ 0|p0 = 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' (22) 10−3 10−2 10−1 100 100 101 102 103 104 105 106 (a) 10−2 10−1 100 100 101 102 103 104 105 106 107 108 109 (b) Pcum(n) n n Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Cumulative distribution function for the survival times obtained for the variable p for (a) the standard map (1), and (b) the triangle map (3), in logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Data are obtained by simulating 106 and 105 initial conditions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Continuous-black lines correspond to power-law asymptotic functions F(n) = a + bn−α: the fitting parameters are a = 0, b = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='80(0), and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='51(5) in (a) and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='021(0), b = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='62(6), and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='42(0) in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' When considering recurrent random walks, the asymp- totic behaviour of the survival probability is again ruled by Lamperti exponent [58, 59] (see also [69]): Pcum(n) ∼ n−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' (23) Once again SM simulations (see panel (a) in Figure 7) agree with expected behaviour for simple random walks (α = 1/2), while the situation is completely different for the TM, where the survival probability seems to tend to a finite limit for large n, see panel (b) in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' This is coherent with the transient nature of the TM, which we conjectured in the analysis of generalized arcsine law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' DISCUSSION We have performed a set of extensive numerical experi- ments on two paradigmatic area-preserving maps, the SM and the TM, focusing in the case where such maps are considered on a cylinder, namely a non compact phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Firstly we reproduced known results about nor- mal diffusion for typical (chaotic) parameters of the SM, and superdiffusion for the TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Then we explored records’ statistics: numerical simulations again confirm that the SM behave like a simple random walk, while anomalous growth is exhibited by the TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' The most interesting re- sults arise in the analysis of occupation times, like gen- eralized arcsine law and survival probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' While once again normal stochastic properties are displayed by the SM, the TM presents more surprising results, which we conjecture are possibly connected to lack of conserva- tivity [38] (or transient behaviour, in the language of random walks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' This feature, which we think is worth of further investigations, might suggest new stochastic modeling of the TM (see [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' AUTHORS’ CONTRIBUTIONS All authors have contributed substantially to the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' All authors have read and agreed to the published version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' 7 ACKNOWLEDGEMENTS R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' acknowledges partial support from PRIN Re- search Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' 2017S35EHN “Regular and stochastic behavior in dynamical systems” of the Italian Ministry of Education, University and Research (MIUR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' ac- knowledges an association to the GNFM group of IN- DAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='A thanks Gaia Pozzoli for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' ac- knowledges the National Council for Scientific and Tech- nological Development - CNPq (Brazilian agency) for partial financial support (Grant Number 310228/2020-4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' acknowledges the Coordenação de Aperfeiçoa- mento de Pessoal de Nível Superior - CAPES (Brazilian agency ) - Finance Code 001, for partial financial sup- port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' Additionally, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' also acknowledges the Fundação de Amparo à Pesquisa e Inovação do Es- tado de Santa Catarina - FAPESC (Brazilian agency) for partial financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' [1] Lichtenberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtE1T4oBgHgl3EQf_QYB/content/2301.03577v1.pdf'} +page_content=' ;' 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b/HtFIT4oBgHgl3EQfXys9/content/tmp_files/2301.11245v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2ecbcff119af0b65bfa5cf9f427059cc3d62a850 --- /dev/null +++ b/HtFIT4oBgHgl3EQfXys9/content/tmp_files/2301.11245v1.pdf.txt @@ -0,0 +1,1378 @@ +arXiv:2301.11245v1 [math.AP] 26 Jan 2023 +Exponential decay of the solutions to nonlinear Schr¨odinger systems +Felipe Angeles∗, M´onica Clapp†, and Alberto Salda˜na (�)‡ +Abstract +We show that the components of finite energy solutions to general nonlinear Schr¨odinger +systems have exponential decay at infinity. +Our results apply to positive or sign-changing +components, and to cooperative, competitive, or mixed-interaction systems. As an application, +we use the exponential decay to derive an upper bound for the least possible energy of a solution +with a prescribed number of positive and nonradial sign-changing components. +Keywords: Exponential decay; Schr¨odinger system; energy bounds; nodal solutions. +MSC2010: 35B40; 35B45; 35J47; 35B06; 35J10; +1 +Introduction +Consider the nonlinear Schr¨odinger system + + + + + + + +−∆ui + Vi(x)ui = +ℓ +� +j=1 +βij|uj|p|ui|p−2ui, +ui ∈ H1(RN), +i = 1, . . . , ℓ, +(1.1) +where N ≥ 1, Vi ∈ L∞(RN), βij ∈ R and 1 < p < 2∗ +2 . Here 2∗ is the usual critical Sobolev +exponent, namely, 2∗ := +2N +N−2 if N ≥ 3 and 2∗ := ∞ for N = 1, 2. +Systems of this type occur as models for various natural phenomena. In physics, for example, +they describe the behavior of standing waves for a mixture of Bose-Einstein condensates of different +hyperfine states which overlap in space [13]. The coefficients βij determine the type of interaction +between the states; if βij > 0, then there is an attractive force between ui and uj, similarly, if +βij < 0, then the force is repulsive, and if βij = 0, then there is no direct interaction between +these components. Whenever all the interaction coefficients are positive, we say that the system is +cooperative. If βii > 0 and βij < 0 for all i ̸= j, then the system is called competitive. And if some +βij are positive and others are negative for i ̸= j, then we say that the system has mixed couplings. +All these regimes exhibit very different qualitative behaviors and have been studied extensively in +recent years, see for instance [5,6,8–12,17,19–24,26] and the references therein. +∗Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Circuito Exterior, Ciudad Universitaria, +04510 Coyoac´an, Ciudad de M´exico, Mexico, felidaujal@im.unam.mx +†Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Campus Juriquilla, Boulevard Juriquilla +3001, 76230 Quer´etaro, Qro., Mexico, monica.clapp@im.unam.mx +‡(Corresponding author �) Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Circuito Ex- +terior, Ciudad Universitaria, 04510 Coyoac´an, Ciudad de M´exico, Mexico, alberto.saldana@im.unam.mx +1 + +System (1.1) has a variational structure, and therefore a natural strategy is to find weak solutions +by minimizing an associated energy functional on a suitable set, under additional assumptions on +the matrix (βij) and on the potentials Vi. Using this approach, several kinds of solutions have been +found in terms of their signs and their symmetries. However, there seems to be no information +available about the decay of these solutions at infinity. In this paper, we show that finite energy +solutions must decay exponentially at infinity, and a rate can be found in terms of the potentials +Vi. Our main result is the following one. +Theorem 1.1. Assume that, for every i = 1, . . . , ℓ, +(V1) Vi : RN → R is H¨older continuous and bounded, +(V2) there exists ρ ≥ 0 such that +σi := +inf +RN∖Bρ(0) Vi > 0. +Let (u1, . . . , uℓ) ∈ +� +H1(RN) +�ℓ be a solution of (1.1) and let µi ∈ (0, √σi). Then, there is C > 0 +such that +|ui(x)| ≤ Ce−µi|x| +for all x ∈ RN and i = 1, . . . , ℓ. +(1.2) +Furthermore, if Vi ≡ 1 for every i = 1, . . . , ℓ, then (1.2) holds true with µi = 1. +We emphasize that each component may have a different decay depending on each potential Vi. +The main obstacle to showing (1.2) is to handle the possibly sublinear term |ui|p−2ui for p ∈ (1, 2) +(which is always the case for N ≥ 4). To explain this point in more detail, assume that (u1, . . . , uℓ) +is a solution of (1.1) and write the i-th equation of the system as +−∆ui + +� +ai(x) − ci(x)|ui(x)|p−2� +ui = 0, +ai := Vi − βii|ui|2p−2, +ci := +ℓ +� +j̸=i +βij|uj|p. +(1.3) +Since every uj ∈ H1(RN)∩C0(RN), we know that ai and ci are bounded in RN, but |ui|p−2 → ∞ as +|x| → ∞ and it is also singular at the nodal set of a sign-changing solution. As a consequence, one +cannot use directly previously known results about exponential decay for scalar equations, such as +those in [1, 3, 18]. In fact, one can easily construct a one dimensional solution of a similar scalar +equation that has a power-type decay. For instance, let w ∈ C2(R) be a positive function such that +w(x) = |x|−2/3 for |x| > 1 and let +c(x) := −w′′(x) + w(x) +w(x) +1 +2 +, +x ∈ R. +Then, w ∈ H1(R) is a solution of −w′′ + w = c w +1 +2 in R, c(x) → 0 as |x| → ∞, and w decays as a +power at infinity. +This shows that the proof of the exponential estimate in Theorem 1.1 must rely on a careful +study of the system structure. In other words, although the sublinear nonlinearity |ui|p−2ui appears +in (1.1), the system is not sublinear. As a whole, it is always superlinear. +With this in mind, we adapt some of the arguments in [1,18] preserving at each step the system +structure of the problem. These arguments rely basically on elliptic regularity and comparison +principles. +2 + +The exponential decay of solutions is a powerful tool in their qualitative study. As an application +of Theorem 1.1, we derive energy bounds of solutions having prescribed positive and nonradial sign- +changing components. For this, power type decay would not be enough. +To be more precise, we consider the autonomous system + + + + + + + +−∆ui + ui = +ℓ +� +j=1 +βij|uj|p|ui|p−2ui, +ui ∈ H1(RN), +i = 1, . . . , ℓ. +(1.4) +where the βij’s satisfy the following condition: +(B1) The matrix (βij) is symmetric and admits a block decomposition as follows: For some 1 ≤ +q ≤ ℓ there exist 0 = ℓ0 < ℓ1 < · · · < ℓq−1 < ℓq = ℓ such that, if we set +Ih := {i ∈ {1, . . . , ℓ} : ℓh−1 < i ≤ ℓh}, +h ∈ {1, . . . , q}, +then βii > 0, βij ≥ 0 if i, j ∈ Ih, and βij < 0 if i ∈ Ih, j ∈ Ik and h ̸= k. +According to this decomposition, a solution u = (u1, . . . , uℓ) to (1.1) may be written in block- +form as +u = (u1, . . . , uq) +with uh = (uℓh−1+1, . . . , uℓh), +h = 1, . . . , q. +We say that u is fully nontrivial if every component ui is different from zero. +Set Q := {1, . . . , q}. Given a partition Q = Q+ ∪ Q− with Q+ ∩ Q− = ∅ we look for solutions +such that every component of uh is positive if h ∈ Q+ and every component of uh is nonradial +and changes sign if h ∈ Q−. To this end, we use variational methods in a space having suitable +symmetries. As shown in [11, Section 3], to guarantee that the solutions obtained are fully nontrivial +we need to assume the following two conditions: +(B2) For each h ∈ Q, the graph whose set of vertices is Ih and whose set of edges is Eh := {{i, j} : +i, j ∈ Ih, i ̸= j, βij > 0} is connected. +(B3) If q ≥ 2 then, for every h ∈ {1, . . . , q} such that ℓh − ℓh−1 ≥ 2, the inequality +� +min +{i,j}∈Eh +βij +� + + +min +h=1,...,q max +i∈Ih βii +� +i,j∈Ih +βij + + +p +p−1 +> C∗ +q +� +k=1 +k̸=h +� +i∈Ih +j∈Ik +|βij| +holds true, where C∗ = C∗(N, p, q, Q+) > 0 is the explicit constant given in (3.7) below. +In [11] it is shown that, for any q, the system (1.1) has a fully nontrivial solution satisfying the +sign requirements described above. Furthermore, an upper bound for its energy is exhibited, but +only for systems with at most 2 blocks, i.e., for q = 1, 2. Here we use Theorem 1.1 to obtain an +energy bound for any number of blocks. +For each h = 1, . . . , q, let RIh := {s = (sℓh−1+1, . . . , sℓh) : si ∈ R for all i ∈ Ih} and define +µh := inf +s∈RIh +s̸=0 + + + +� +i∈Ih s2 +i +� � +i,j∈Ih βij|si|p|sj|p +� 2 +2p + + + +p +p−1 +. +(1.5) +3 + +For any ℓ ∈ N, we write ∥u∥ for the usual norm of u = (u1, . . . , uℓ) in (H1(RN))ℓ, i.e., +∥u∥2 := +ℓ +� +i=1 +� +RN (|∇ui|2 + |ui|2). +We prove the following result. +Theorem 1.2. Let N = 4 or N ≥ 6, and let Q = Q+ ∪ Q− with Q+ ∩ Q− = ∅. Assume (B1), +(B2), and (B3). Then, there exists a fully nontrivial solution u = (u1, . . . , uq) to the system (1.4) +with the following properties: +(a) Every component of uh is positive if h ∈ Q+ and every component of uh is nonradial and +changes sign if h ∈ Q−. +(b) If q = 1, then +∥u∥2 = µ1∥ω∥2 if Q = Q+ +and +∥u∥2 < 10 µ1∥ω∥2 if Q = Q−. +(c) If q ≥ 2 the following estimate holds true +∥u∥2 < + +min +k∈Q +� +akµk + +� +h∈Q∖{k} +bhµh +� + + ∥ω∥2, +(1.6) +where ak := 1 if k ∈ Q+, ak := 12 if k ∈ Q−, bh := 6 if h ∈ Q+, bh := 12 if h ∈ Q−, and ω is the +unique positive radial solution to the equation +− ∆w + w = |w|2p−2w, +w ∈ H1(RN). +(1.7) +To prove Theorem 1.2, we follow the approach in [11] and impose on the variational setting +some carefully constructed symmetries which admit finite orbits. This approach immediately gives +energy estimates but it requires showing a quantitative compactness condition which needs precise +knowledge about the asymptotic decay of the components of the system. Here is where we use +Theorem 1.1. +The paper is organized as follows. Section 2 is devoted to the proof of the exponential decay +stated in Theorem 1.1. The application of this result to derive energy bounds is contained in Section +3, where we also give some concrete examples. +Acknowledgments +We thank Nils Ackermann for helpful comments and suggestions. +F. Angeles and A. Salda˜na +thank the Instituto de Matem´aticas - Campus Juriquilla for the kind hospitality. F. Angeles is +supported by CONACYT (Mexico) through a postdoctoral fellowship under grant A1-S-10457. M. +Clapp is supported by CONACYT (Mexico) through the research grant A1-S-10457. A. Salda˜na +is supported by UNAM-DGAPA-PAPIIT (Mexico) grant IA100923 and by CONACYT (Mexico) +grant A1-S-10457. +4 + +2 +Exponential decay +This section is devoted to the proof of Theorem 1.1. As a first step, we extend the argument +in [2, Lemma 5.3] to systems. Let Br denote the ball of radius r in RN centered at zero. Let σi +and βij as in (V2) and (1.1), then we let σ := (σ1, . . . , σℓ) and β := (βij)ℓ +i,j=1. +Lemma 2.1. Let Vi ∈ L∞(RN) satisfy (V2) and let u = (u1, . . . , uℓ) be a solution of (1.1). Set +ξi(r) := +� +RN∖Br +� +|∇ui|2 + |ui|2� +and +ξ(r) := (ξ1(r), . . . , ξℓ(r)). +Then, there are positive constants C = C(u, σ, β, N, ρ, p) and ϑ = ϑ(σ), with ρ and σi as in (V2), +such that +|ξ(r)|1 := +ℓ +� +i=1 +ξi(r) ≤ Ce−ϑr +for every r ≥ 0. +Proof. Let χ : RN → R be given by χ(r) := 0 if r ≤ 0, χ(r) := r if r ∈ (0, 1) and χ(r) := 1 if +r ≥ 1. Let ur +i (x) := χ(|x| − r)ui(x) for r ≥ 0, x ∈ RN, and i = 1, . . . , ℓ. Then ur +i ∈ H1(RN) and +ur +i (x) = (|x| − r)ui(x), +∇ur +i (x) = (|x| − r)∇ui(x) + x +|x|ui(x), +if x ∈ Br+1 ∖ Br. +Set δ := min{σ1, . . . , σℓ, 1}. Using that |ui x +|x| · ∇ui| ≤ 1 +2(|∇ui|2 + |ui|2) we obtain +� +RN +� +∇ui · ∇ur +i + Vi uiur +i +� +≥ δξi(r + 1) + +� +Br+1∖Br +� +(|x| − r) +� +|∇ui|2 + Vi u2 +i +� ++ ui +x +|x| · ∇ui +� +≥ δξi(r + 1) − 1 +2 +� +Br+1∖Br +� +|∇ui|2 + |ui|2� +≥ (δ + 1 +2)ξi(r + 1) − 1 +2ξi(r) +if r + 1 ≥ ρ. +(2.1) +As u solves (1.1) we have that +���� +� +RN ∇ui · ∇ur +i + Vi uiur +i +���� = +������ +� +RN +ℓ +� +j=1 +βij|uj|p|ui|p−2uiur +i +������ +≤ +ℓ +� +j=1 +� +RN\Br +|βij||uj|p|ui|p−2|ui|2 = +ℓ +� +j=1 +|βij| +� +RN∖Br +|uj|p|ui|p +and since |um|p ≤ +��ℓ +k=1 |uk|2p�1/2 +for every m = 1, . . . , ℓ, we obtain +���� +� +RN ∇ui · ∇ur +i + Vi uiur +i +���� ≤ + + +ℓ +� +j=1 +|βij| + + +ℓ +� +k=1 +� +RN∖Br +|uk|2p. +Given that uk ∈ H1(RN) for all k = 1, . . . , ℓ, Lemma A.1 implies the existence of a constant +C1 = C1(N, p) > 0 such that +���� +� +RN ∇ui · ∇ur +i + Vi uiur +i +���� ≤ C1 + + +ℓ +� +j=1 +|βij| + + +ℓ +� +k=1 +�� +RN∖Br +� +|∇uk|2 + |uk|2��p +(2.2) +5 + +for every r ≥ 1 and i = 1, . . . , ℓ. Set C2 := C1 +�ℓ +i,j=1 |βij|. From (2.1) and (2.2), assuming without +loss of generality that ρ ≥ 2 and adding over i, we get +2δ + 1 +2 +|ξ(r + 1)|1 − 1 +2|ξ(r)|1 ≤ C2 +ℓ +� +k=1 +|ξk(r)|p =: C2 |ξ(r)|p +p +if r + 1 ≥ ρ. +Therefore, +|ξ(r + 1)|1 +|ξ(r)|1 +≤ +1 +2δ + 1 +� +1 + 2C2 +|ξ(r)|p +p +|ξ(r)|1 +� +≤ +1 +2δ + 1 +� +1 + 2C2|ξ(r)|p−1 +1 +� +=: γ(r) +if r + 1 ≥ ρ. (2.3) +Since |ξ(r)|1 → 0 as r → ∞, there is r0 = r0(u, p, β, ρ) ∈ N such that r0 ≥ ρ and γ(r) ≤ γ−1 +0 +for all +r ≥ r0 with γ0 := 2δ+1 +δ+1 > 1. Then, for r > r0 + 1, +|ξ(r)|1 ≤ |ξ(⌊r⌋)|1 = |ξ(r0)|1 +⌊r⌋−1 +� +k=r0 +|ξ(k + 1)|1 +|ξ(k)|1 +≤ |ξ(r0)|1γr0−⌊r⌋ +0 +≤ ∥u∥2γr0−r+1 +0 +, +where ⌊r⌋ denotes the floor of r. Since |ξ(r)|1 ≤ ∥u∥2 ≤ ∥u∥2γr0−r+1 +0 +for r ≤ r0 + 1 we have that +|ξ(r)|1 ≤ ∥u∥2γr0−r+1 +0 += ∥u∥2γr0+1 +0 +e− ln(γ0)r +for every r ≥ 0, +as claimed. +Lemma 2.2. Assume (V1) and let u = (u1, . . . , uℓ) be a solution of (1.1). Then ui ∈ W 2,s(RN) ∩ +C2(RN) for every s ≥ 2 and i = 1, . . . , ℓ. +Proof. Let N ≥ 3. The argument for N = 1, 2 is similar and easier. For each i = 1, . . . , ℓ set +fi := +l +� +j=1 +βij|uj|p|ui|p−2ui. +(2.4) +Since |uk| ≤ |u| := +� +u2 +1 + · · · + u2 +ℓ for every k = 1, . . . ℓ, we have that +|fi| ≤ +ℓ +� +i,j=1 +|βij||uj|p|ui|p−1 ≤ + + +ℓ +� +j=1 +|βij| + + |u|p|u|p−1 ≤ + + +ℓ +� +i,j=1 +|βij| + + |u|2p−1. +(2.5) +Therefore, fi ∈ Ls1(RN) for s1 := +2∗ +2p−1 > 1 and, by the standard Lp-elliptic regularity theory, +ui ∈ W 2,s1(RN) for all i = 1, . . . , ℓ (see, e.g., [14, Chapter 9] or [25, Section 3.2]). +Using a +bootstrapping argument, we conclude the existence of s > max{N +2 , 2} such that ui ∈ W 2,s(RN) for +all i = 1, . . . , ℓ and thus, by the Sobolev embedding theorem, ui ∈ C1,α(RN). Since Vi is H¨older +continuous and bounded, applying the Schauder estimates repeatedly, we deduce that ui is of class +C2 (see [15, Section 1.3]). +In the rest of the paper, we write | · |t for the norm in Lt(RN), 1 ≤ t ≤ ∞. If u = (u1, . . . , uℓ) ∈ +[L∞(RN)]ℓ, then |u|∞ := �ℓ +i=1 supRN |ui|. Moreover, for a proper open subset Ω of RN we denote +the usual Sobolev norm in H1(Ω) by ∥ · ∥H1(Ω), i.e., +∥u∥2 +H1(Ω) := +� +Ω +(|∇u|2 + |u|2). +6 + +Lemma 2.3. Assume (V1). Let u = (u1, . . . , uℓ) be a solution of (1.1), s > max{2, N +2 } and Λ > 0 +be such that |Vi|∞ ≤ Λ for i = 1, . . . , ℓ. Then there is a constant C = C(β, N, p, Λ, s) > 0 such +that, for any x ∈ RN, +∥ui∥W 2,s(B 1 +2 (x)) ≤ C + +|ui| +s−2 +s +∞ ∥ui∥ +2 +s +H1(B1(x)) + |u| +2ps−(s+2) +s +∞ +� +ℓ +� +j=1 +∥uj∥2 +H1(B1(x)) +� p +s + + , +where |u| := +� +u2 +1 + · · · + u2 +ℓ and BR(x) is the ball of radius R centered at x. +Proof. Since ui ∈ W 2,s(RN) ⊂ L∞(RN), we have that +|ui|s = |ui|s−2|ui|2 ≤ |ui|s−2 +∞ |ui|2. +Set fi as in (2.4). By (2.5), there is a constant C2 = C2(β) such that +|fi|s ≤ Cs +2|u|(p−1)s|u|ps = Cs +2|u|(p−1)s+p(s−2)(u2 +1 + · · · + u2 +ℓ)p +≤ Cs +2|u|2ps−(s+2) +∞ +ℓp(u2p +1 + · · · + u2p +ℓ ), +where (p − 1)s + p(s − 2) > 0. Then, by [14, Theorem 9.11], there is a positive constant C1 = +C1(s, N, Λ) such that +∥ui∥W 2,s(B 1 +2 (x)) ≤ C1 +� +|ui|Ls(B1(x)) + |fi|Ls(B1(x)) +� +for any x ∈ RN. +From the previous inequalities we derive +∥ui∥W 2,s(B 1 +2 (x)) ≤ C1 + +|ui| +s−2 +s +∞ ∥ui∥ +2 +s +H1(B1(x))) + C2ℓ +p +s C3|u| +2ps−(s+2) +s +∞ +� +ℓ +� +j=1 +∥uj∥2 +H1(B1(x)) +� p +s + + , +where C3 = C3(N, p) is the constant given by the Sobolev embedding H1(B1) ⊂ L2p(B1). +Lemma 2.4. Assume (V1) − (V2), let u = (u1, . . . , uℓ) be a solution of (1.1) and let fi be as in +(2.4). Then, there are constants η > 0, C1 > 0, and C2 > 0 such that +|ui(x)| ≤ C1e−η|x|, +|fi(x)| ≤ C2e−(2p−1)η|x|, +for all x ∈ RN and i = 1, . . . , ℓ. +Proof. For x ∈ RN with |x| ≥ 2, set r := 1 +2|x|. Then, B1(x) ⊂ RN ∖ Br and, by Lemma 2.1, there +are positive constants K1 = K1(u, σ, β, N, ρ, p) and ϑ = ϑ(σ), with ρ and σi as in (V2), such that +∥uj∥2 +H1(B1(x)) ≤ ∥uj∥2 +H1(RN∖Br) = ξj(r) ≤ +ℓ +� +i=1 +ξi(r) ≤ K1e−ϑr +for every j = 1, . . . , ℓ. +Fix s > max{N +2 , 2}. By Lemma 2.3 there are positive constants K2 = K2(u, β, N, p, Λ, s) and +K3 = K3(u, σ, β, ρ, N, p, s) such that +∥ui∥W 2,s(B 1 +2 (x)) ≤ K2 + +∥ui∥ +2 +s +H1(B1(x))) + +� +ℓ +� +j=1 +∥uj∥2 +H1(B1(x)) +� p +s + + ≤ K2K3e− ϑ +s r. +7 + +Therefore, +|ui(x)| ≤ |ui|L∞(B 1 +2 (x)) ≤ K4∥ui∥W 2,s(B 1 +2 (x)) ≤ K2K3K4e− ϑ +2s|x| +for every x ∈ RN ∖ B2, +where K4 is the positive constant given by the embedding W 2,s(B 1 +2) ⊂ L∞(B 1 +2). +Since ui is +continuous, we may choose C1 ≥ K2K3K4 such that |ui(x)| ≤ C1e− ϑ +s for every x ∈ B2. So, setting +η := ϑ +2s, we obtain +|ui(x)| ≤ C1e−η|x| +for every x ∈ RN. +The estimate for fi follows immediately from (2.5). +The following result is a particular case of [18, Theorem 2.1]. We include a simplified proof for +completeness. +Lemma 2.5. Assume that V : RN → R satisfies σ := infRN∖Bρ(0) V > 0 for some ρ ≥ 0. Let w be +a classical solution of −∆w + V w = f in RN such that +|w(x)| ≤ Ce−η|x| +and +|f(x)| ≤ Ce−δ|x| +for all x ∈ RN +and for some constants C > 0, η ∈ (0, √σ) and δ ∈ (η, √σ]. Then, for any µ ∈ (η, δ), there is +M = M(µ, δ, ρ, σ, C) > 0 such that +|w(x)| ≤ Me−µ|x| +for all x ∈ RN. +Proof. Let ρ, σ, η, δ, µ, and C be as in the statement. Set v(x) := e−µ|x| for x ∈ RN. Then, +∆v(x) = v(x)h(|x|) +for x ∈ RN ∖ {0}, +where h(r) := µ2 − (N − 1)µ +r . +In particular, V (x) − h(|x|) ≥ σ − µ2 =: ε > 0 for |x| > ρ. Fix t ∈ R satisfying +t > C +ε e(µ−δ)ρ +and +w(x) < tv(x) for |x| = ρ. +(2.6) +We claim that w(x) ≤ tv(x) for all |x| > ρ. Indeed, let z := w − tv and assume, by contradiction, +that m := sup|x|≥ρ z(x) > 0. Since lim|x|→∞ z(x) = 0, there is R > ρ such that z(x) ≤ m +2 for +|x| ≥ R. Let Ω := {x ∈ RN : ρ < |x| < R and z(x) > 0}. Then z ≤ m +2 on ∂Ω and, by (2.6), +−∆z(x) = −∆w(x) + t∆v(x) = f(x) − V (x)w(x) + tv(x)h(|x|) += f(x) − V (x)z(x) + tv(x)(h(|x|) − V (x)) +< Ce−δ|x| − εtv(x) = Ce−δ|x| − εte−µ|x| < 0 +for every x ∈ Ω. +Then, by the maximum principle, m = maxΩ z = max∂Ω z ≤ m +2 . This is a contradiction. Therefore +m ≤ 0, namely, w(x) ≤ te−µ|x| for all |x| ≥ ρ. Arguing similarly for −w and using that w ∈ L∞(RN) +we obtain that |w(x)| ≤ Me−µ|x| for all x ∈ RN, as claimed. +We are ready to prove Theorem 1.1. +Proof of Theorem 1.1. Iterating Lemmas 2.4 and 2.5, using that 2p − 1 > 1, one shows that, for +any µi ∈ (0, √σi), there is C > 0 such that |ui(x)| ≤ Ce−µi|x| for all x ∈ RN and for all i = 1, . . . , ℓ. +Now, assume that Vi ≡ 1 for every i = 1, . . . , ℓ and let µ ∈ (0, 1) be such that (2p − 1)µ > 1. +By Lemma 2.4, we have that |fi(x)| ≤ C2e−(2p−1)µ|x| for all x ∈ RN. +The claim now follows +from [1, Theorem 2.3(c)]. +8 + +3 +Energy estimates for seminodal solutions +In this section we prove Theorem 1.2. +Consider the autonomous system (1.4) where N ≥ 4, +1 < p < +N +N−2 and βij satisfy the assumption (B1) stated in the Introduction. According to the +decomposition given by (B1), a solution u = (u1, . . . , uℓ) to (1.4) may be written in block-form as +u = (u1, . . . , uq) +with uh = (uℓh−1+1, . . . , uℓh), +h = 1, . . . , q. +u is called fully nontrivial if every component ui is different from zero. We say that u is block-wise +nontrivial if at least one component in each block uh is nontrivial. +Following [11], we introduce suitable symmetries to produce a change of sign in some compo- +nents. Let G be a finite subgroup of the group O(N) of linear isometries of RN and denote by +Gx := {gx : g ∈ G} the G-orbit of x ∈ RN. Let φ : G → Z2 := {−1, 1} be a homomorphism of +groups. A function u : RN → R is called G-invariant if it is constant on Gx for every x ∈ RN and +it is called φ-equivariant if +u(gx) = φ(g)u(x) for all g ∈ G, x ∈ RN. +(3.1) +Note that, if φ ≡ 1 is the trivial homomorphism and u satisfies (3.1), then u is G-invariant. On +the other hand, if φ is surjective every nontrivial function satisfying (3.1) is nonradial and changes +sign. Define +H1(RN)φ := {u ∈ H1(RN) : u is φ-equivariant}. +For each h = 1, . . . , q, fix a homomorphism φh : G → Z2. Take φi := φh for all i ∈ Ih and set +φ = (φ1, . . . , φℓ). Denote by +Hφ := H1(RN)φ1 × · · · × H1(RN)φℓ, +and let J φ : Hφ → R be the functional given by +J φ(u) := 1 +2 +ℓ +� +i=1 +∥ui∥2 − 1 +2p +ℓ +� +i,j=1 +βij +� +RN |ui|p|uj|p. +This functional is of class C1 and its critical points are the solutions to the system (1.4) satisfying +(3.1). The block-wise nontrivial solutions belong to the Nehari set +N φ := {u ∈ Hφ : ∥uh∥ ̸= 0 and ∂uhJ φ(u)uh = 0 for every h = 1, . . . , ℓ}. +Note that +∂uhJ φ|K(u)uh = ∥uh∥2 − +ℓ +� +k=1 +� +(i,j)∈Ih×Ik +βij +� +RN |ui|p|uj|p, +and that J φ(u) = p−1 +2p ∥u∥2 if u ∈ N φ. Let +cφ := +inf +u∈N φ J φ(u). +If s = (s1, . . . , sq) ∈ Rq and u = (u1, . . . , uq) ∈ Hφ we write su := (s1u1, . . . , squq). The following +facts were proved in [8]. +9 + +Lemma 3.1. +(i) cφ > 0. +(ii) If the coordinates of u ∈ Hφ satisfy +q +� +k=1 +� +(i,j)∈Ih×Ik +� +RN βij|ui|p|uj|p > 0 +for every h = 1, . . . , q, +(3.2) +then there exists a unique su ∈ (0, ∞)q such that suu ∈ N φ. Furthermore, +J φ(suu) = +max +s∈(0,∞)q J φ(su). +Proof. See [8, Lemma 2.2] or [11, Lemma 2.2]. +Lemma 3.2. If cφ is attained, then the system (1.4) has a block-wise nontrivial solution u = +(u1, . . . , uℓ) ∈ Hφ. Furthermore, if ui is nontrivial, then ui is positive if φi ≡ 1 and ui is nonradial +and changes sign if φi is surjective. +Proof. It is shown in [8, Lemma 2.4] that any minimizer of J φ on N φ is a block-wise nontrivial +solution to (1.4). If ui ̸= 0 and φi is surjective, then ui is nonradial and changes sign. If φi ≡ 1 then +|ui| is G-invariant and replacing ui with |ui| we obtain a solution with the required properties. +Set Q := {1, . . . , q} and fix a decomposition Q = Q+ ∪ Q− with Q+ ∩ Q− = ∅. From now +on, we consider the following symmetries. We write RN ≡ C × C × RN−4 and a point in RN as +(z1, z2, y) ∈ C × C × RN−4. +Definitions 3.3. Let i denote the imaginary unit. For each m ∈ N, let +Km := {e2πij/m : j = 0, . . . , m − 1}, +Gm be the group generated by Km ∪{τ}∪O(N −4), acting on each point (z1, z2, y) ∈ C×C×RN−4 +as +e2πij/m(z1, z2, y) := (e2πij/mz1, e2πij/mz2, y), +τ(z1, z2, y) := (z2, z1, y), +α(z1, z2, y) := (z1, z2, αy) +if α ∈ O(N − 4), +and θ : Gm → Z2 be the homomorphism satisfying +θ(e2πij/m) = 1, +θ(τ) = −1, +and +θ(α) = 1 +for every α ∈ O(N − 4). +Define φh : Gm → Z2 by +φh := +� +1 +if h ∈ Q+, +θ +if h ∈ Q−. +(3.3) +Due to the lack of compactness, cφ is not always attained; see e.g. [11, Corollary 2.8(i)]. A +sufficient condition for this to happen is given by the next lemma. We use the following notation. +If Q′ ⊂ Q := {1, . . . , q} we consider the subsystem of (1.4) obtained by deleting all components of +uh for every h /∈ Q′, and we denote by J φ +Q′ and N φ +Q′ the functional and the Nehari set associated +to this subsystem. We write +cφ +Q′ := +inf +u∈N φ +Q′ +J φ +Q′(u). +If Q′ = {h} we omit the curly brackets and write, for instance, cφ +h or J φ +h . +10 + +Lemma 3.4 (Compactness). Let N ̸= 5, m ≥ 5 and φh : Gm → Z2 be as in (3.3). If, for each +h ∈ Q := {1, . . . , q}, the strict inequality +cφ < + + + +cφ +Q∖{h} + mµh +p−1 +2p ∥ω∥2, +if h ∈ Q+, +cφ +Q∖{h} + 2mµh +p−1 +2p ∥ω∥2, +if h ∈ Q−, +(3.4) +holds true, then cφ is attained, where ω is the positive radial solution to (1.7) and µh is given by +(1.5). +Proof. This statement follows by combining [11, Corollary 2.8(ii)] with [11, Equation (5.1)]. +To verify condition (3.4) we introduce a suitable test function. Fix m ≥ 5 and let Km be as in +Definitions 3.3. If h ∈ Q+, we take ζh := ( 1 +√ +2, +1 +√ +2, 0) and, for each R > 1, we define +�σhR(x) := +� +g∈Km +ω(x − Rgζh), +x ∈ RN. +If h ∈ Q− we take ζh := (1, 0, 0) and we define +�σhR(x) := +� +g∈G′m +φh(g) ω(x − Rgζh), +x ∈ RN, +where ω is the positive radial solution to (1.7) and G′ +m is the subgroup of Gm generated by Km∪{τ}. +Note that �σhR(gx) = φh(g)�σhR(x) for every g ∈ Gm, x ∈ RN. Let +σhR := thR�σhR, +(3.5) +where thR > 0 is chosen so that ∥σhR∥2 = +� +RN |σhR|2p. +Lemma 3.5. If m ≥ 5, then, for each h ∈ {1, . . . , q}, there exist th = (tℓh−1+1, . . . , tℓh) ∈ +(0, ∞)ℓh−ℓh−1 and C0, R0 > 0 such that thσhR := (tℓh−1+1σhR, . . . , tℓhσhR) ∈ N φ +h and +J φ +h (thσhR) ≤ |Gmζh| µh +p−1 +2p ∥ω∥2 − C0e−Rdm +for every R ≥ R0, +where |Gmζh| is the cardinality of the Gm-orbit of ζh, i.e., |Gmζh| = m if h ∈ Q+ and |Gmζh| = 2m +if h ∈ Q−, and +dm := |1 − e2πi/m|. +(3.6) +Proof. Take th = (tℓh−1+1, . . . , tℓh) ∈ (0, ∞)ℓh−ℓh−1 such that +� +i∈Ih +t2 +i = +� +i,j∈Ih +βijtp +jtp +i = µh +and apply [11, Proposition 4.1(i) and Lemma 4.4]. +11 + +Proof of Theorem 1.2. Assume (B1) and let φh : Gm → Z2 be given by (3.3). For q = 1 and m ≥ 5 +it is proved in [11, Corollary 4.2 and Proposition 4.5] that cφ is attained at u ∈ N φ satisfying +∥u∥2 = µ1∥ω∥2 if Q+ = {1} +and +∥u∥2 < 2m µ1∥ω∥2 if Q− = {1}. +Taking m = 5 gives statement (b). +Fix m = 6. We claim that cφ is attained and that the estimate (c) holds true for every q ≥ 2. +To prove this claim, we proceed by induction. Assume it is true for q − 1 with q ≥ 2. +We will show that the compactness condition (3.4) holds true. Using a change of coordinates, it +suffices to argue for h = q. By induction hypothesis there exists w = (w1, . . . , wq−1) ∈ N φ +Q∖{q} such +that J φ +Q∖{q}(w) = cφ +Q∖{q}. For each R > 1 let σqR be as in (3.5) and take tq ∈ (0, ∞)ℓ−ℓq−1 as in +Lemma 3.5. Set whR = wh for h = 1, . . . , q−1 and wqR = tqσqR, and define wR = (w1R, . . . , wℓR) := +(w1R, . . . , wqR). Then, as w ∈ N φ +Q∖{q} and the interaction between the components of w and σqR +tends to 0 as R → ∞, we have that wR satisfies (3.2) for large enough R and, as a consequence, +there exist R1 > 0 and (s1R, . . . , sqR) ∈ [1/2, 2]q such that (s1Rw1R, . . . , sqRwqR) ∈ N φ if R ≥ R1. +Set uR = (u1R, . . . , uℓR) := (s1Rw1R, . . . , sqRwqR). Using that w ∈ N φ +Q∖{q} and tqσqR ∈ N φ +q , from +the last statement in Lemma 3.1(ii) and Lemma 3.5 we derive +J φ(uR) = 1 +2 +ℓ +� +i=1 +∥uiR∥2 − 1 +2p +ℓ +� +i,j=1 +βij +� +RN |uiR|p|ujR|p +≤ J φ +Q∖{q}(w) + J φ +q (tqσqR) − 1 +p +q−1 +� +h=1 +� +(i,j)∈Ih×Iq +βij +� +RN |shRwiR|p|sqRwjR|p +≤ cφ +Q∖{q} + |Gmζh| µq +p−1 +2p ∥ω∥2 − C0e−Rdm + C1 +q−1 +� +h=1 +� +i∈Ih +� +RN |wiR|p|σqR|p, +if R ≥ max{R0, R1}, where C0 and C1 are positive constants and dm is given in (3.6). +It is well known that |ω(x)| ≤ Ce−|x| and, as w solves a subsystem of (1.4), Theorem 1.1 asserts +that +|wiR(x)| ≤ Ce−|x| +for every i ∈ Ih with h = 1, . . . , q − 1. +Therefore, for every g ∈ Gm, +� +RN |wiR|p|ω( · − Rgζh)|p ≤ C +� +RN e−p|x| e−p|x−Rgζh| dx ≤ Ce−Rp. +So, if p > dm, we conclude that +cφ < cφ +Q∖{q} + |Gmζh| µq +p−1 +2p ∥ω∥2 +and, by Lemmas 3.4 and 3.2, cφ is attained at a block-wise nontrivial solution u of (1.4) such +that every component of uh is positive if h ∈ Q+ and every component of uh is nonradial and +changes sign if h ∈ Q−. Furthermore, since we are assuming (B2) and (B3) with C∗ as in (3.7) +below, [11, Theorem 3.3] asserts that u is fully nontrivial. +Finally, note that p > 1 = dm because m = 6. As |Gmζh| = 6 if h ∈ Q+ and |Gmζh| = 12 if +h ∈ Q−, the estimate in statement (c) follows by induction. +12 + +Remark 3.6. If m = 5 and p > dm we arrive to a similar conclusion, where, in this case, the +constant bh in statement (b) is 5 if h ∈ Q+ and it is 10 if h ∈ Q−. Note, however, that numbers p +satisfying d5 = 2 sin π +5 < p < +N +N−2 exist only for N ≤ 13. +Remark 3.7. For φh as in (3.3), the constant C∗ > 0 appearing in (B3) depends on N, p, q, and +Q+. It is explicitly defined in [11, Equation (3.1)] as +C∗ := + + +pdφ +(p − 1)S +p +p−1 +φ + + +p +, +(3.7) +where +dφ := p − 1 +2p +inf +(v1,...,vq)∈Uφ +q +� +h=1 +∥vh∥2 +with Uφ := {(v1, . . . , vq) : vh ∈ H1(RN)φh ∖ {0}, ∥vh∥2 = |vh|2p +2p, vhvk = 0 if h ̸= k}, and +Sφ := +min +h=1,...,q +inf +v∈H1(RN)φh∖{0} +∥v∥2 +|v|2 +2p +. +Remark 3.8. In the proof of Theorem 1.2 we use [1, Theorem 2.3], which also characterizes the +sharp decay rate for positive components by providing a bound from below. This kind of information +can be useful to show uniqueness of positive solutions for some problems, see [4, Section 8.2]. +To conclude, we discuss some special cases. +Examples 3.9. Assume (B1) and let p ∈ (1, 2∗ +2 ). +(a) If q = 1 the system (1.4) is cooperative and more can be said. Indeed, it is shown in [11, +Corollary 4.2 and Proposition 4.5] that, if (B2) is satisfied, then (1.4) has a synchronized +solution u = (t1u, . . . , tℓu), where (t1, . . . , tℓ) ∈ (0, ∞)ℓ is a minimizer for (1.5) and u is a +nontrivial φ-equivariant least energy solution of the equation +−∆u + u = |u|2p−2u, +u ∈ H1(RN)φ. +(3.8) +Here, if Q+ = {1}, then φ ≡ 1 (and therefore u = ω) and ∥u∥2 ≤ µ1∥ω∥2. On the other +hand, if Q− = {1}, then φ : Gm → Z2 is the homomorphism θ given in Definitions 3.3 and +∥u∥2 ≤ 10µ1∥ω∥2. +(b) If q = ℓ ≥ 2 the system (1.4) is competitive, i.e., βii > 0 and βij < 0 if i ̸= j. Assumptions +(B2) and (B3) are automatically satisfied and, as µi = β +− +1 +p−1 +ii +, the estimate in Theorem 1.2(c) +becomes +∥u∥2 < + +min +j∈Q +� +ajβ +− +1 +p−1 +jj ++ +� +i∈Q∖{i} +biβ +− +1 +p−1 +ii +� + + ∥ω∥2 +≤ + + + +(6 |Q+| + 12 |Q−| − 5) β +− +1 +p−1 +0 +∥ω∥2 +if Q+ ̸= ∅, +12 |Q−|β +− +1 +p−1 +0 +∥ω∥2 +if Q+ = ∅, +where |Q±| denotes the cardinality of Q± and β0 := min{β11, . . . , βℓℓ}. +13 + +(c) Similarly, for any q ≥ 2, the estimate in Theorem 1.2(c) yields +∥u∥2 ≤ +� +(6 |Q+| + 12 |Q−| − 5) µ∗∥ω∥2 +if Q+ ̸= ∅, +12 |Q−| µ∗∥ω∥2 +if Q+ = ∅. +where µ∗ = max{µ1, . . . , µq}. +Assumptions (B2) and (B3) guarantee that u is fully nontrivial. Note that the left-hand side of +the inequality in (B3) depends only on the entries of the submatrices (βij)i,j∈Ih, h = 1, . . . , q, +whereas the right-hand side only depends on the other entries. So, if the former are large +enough with respect to the absolute values of the latter, (B3) is satisfied. For example, if we +take ℓ = 2q and the matrix is + + + + + + + + + + + +λ +λ +β13 +β14 +β15 +. . . +β1ℓ +λ +λ +β23 +β24 +β25 +. . . +β2ℓ +β31 +β32 +λ +λ +β35 +. . . +β3ℓ +β41 +β42 +λ +λ +β45 +. . . +β4ℓ +... +... +... +... +βℓ−1 1 +. . . +βℓ−1 ℓ−2 +λ +λ +βℓ1 +. . . +βℓ ℓ−2 +λ +λ + + + + + + + + + + + +. +with λ > 0 and βji = βij < 0, then (B1) and (B2) are satisfied. If, additionally, +λ > 4 +2p−1 +p−1 (q − 1)C∗ +and +|βij| ≤ 1, +then, for any h = 1, . . . , q, +� +min +{i,j}∈Eh +βij +� + + +min +h=1,...,q max +i∈Ih +βii +� +i,j∈Ih +βij + + +p +p−1 += λ +� λ +4λ +� +p +p−1 +> C∗4(q − 1) ≥ C∗ +q +� +k=1 +k̸=h +� +i∈Ih +j∈Ik +|βij| +so (B3) is satisfied. +A +An auxiliary result +Lemma A.1. For every r ≥ 1 there is a linear operator Er : H1(RN ∖ Br) → H1(RN) such that, +for every u ∈ H1(RN ∖ Br), +(i) Eru = u a.e. in RN ∖ Br, +(ii) |Eru|2 +2 ≤ C1|u|2 +L2(RN∖Br) +(iii) ∥Eru∥2 ≤ C1∥u∥2 +H1(RN∖Br) +for some positive constant C1 depending only on N and not on r. As a consequence, given p ∈ (1, 2∗ +2 ) +there is a positive constant C depending only on N and p such that +|u|L2p(RN∖Br) ≤ C∥u∥H1(RN∖Br) +for every u ∈ H1(RN ∖ Br) and every r ≥ 1. +14 + +Proof. Fix a linear (extension) operator E1 : H1(RN ∖ B1) → H1(RN) and a positive constant C1 +satisfying (i), (ii) and (iii) for r = 1; see e.g. [16, Theorem 2.3.2]. For r > 1, set �u(x) := u(rx) +and, for u ∈ H1(RN ∖ Br), define +(Eru)(y) := (E1�u) +�y +r +� +. +Then, � +Eru = E1�u. Clearly, Er satisfies (i). Note that |�u|2 +L2(RN ∖B1) = r−N|u|2 +L2(RN∖Br) and that +∥�u∥2 +H1(RN∖B1) = r−N +�� +RN∖Br +� +r2|∇u|2 + |u|2�� +. +Similar identities hold true when we replace RN ∖ B1 and RN ∖ Br with RN. Therefore, +r−N|Eru|2 +2 = |� +Eru|2 +2 = |E1�u|2 +2 ≤ C1∥�u∥2 +L2(RN∖B1) = r−NC1|u|2 +L2(RN ∖Br), +which yields (ii). Furthermore, +r−N +�� +RN +� +r2|∇(Eru)|2 + |Eru|2�� += ∥� +Eru∥2 = ∥E1�u∥2 +≤ C1∥�u∥2 +H1(RN∖B1) = r−NC1 +�� +RN∖Br +� +r2|∇u|2 + |u|2�� +. +This inequality, combined with (ii), yields +r2∥Eru∥2 = +� +RN +� +r2|∇(Eru)|2 + |Eru|2� ++ (r2 − 1) +� +RN |Eru|2 +≤ C1 +� +RN∖Br +� +r2|∇u|2 + |u|2� ++ C1(r2 − 1) +� +RN∖Br +|u|2 = r2C1∥u∥2 +H1(RN∖Br), +which gives (iii). +For p ∈ (1, +N +N−2) let C2 = C2(N, p) be the constant for the Sobolev embedding H1(RN) ⊂ +L2p(RN). 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(9) 141 (2020), 50–88. +17 + diff --git a/HtFIT4oBgHgl3EQfXys9/content/tmp_files/load_file.txt b/HtFIT4oBgHgl3EQfXys9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ca272eb1b86af5d1378599cb519e069b52f5a7e2 --- /dev/null +++ b/HtFIT4oBgHgl3EQfXys9/content/tmp_files/load_file.txt @@ -0,0 +1,899 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf,len=898 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='11245v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='AP] 26 Jan 2023 Exponential decay of the solutions to nonlinear Schr¨odinger systems Felipe Angeles∗, M´onica Clapp†, and Alberto Salda˜na (�)‡ Abstract We show that the components of finite energy solutions to general nonlinear Schr¨odinger systems have exponential decay at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Our results apply to positive or sign-changing components, and to cooperative, competitive, or mixed-interaction systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' As an application, we use the exponential decay to derive an upper bound for the least possible energy of a solution with a prescribed number of positive and nonradial sign-changing components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Keywords: Exponential decay;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Schr¨odinger system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' energy bounds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' nodal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' MSC2010: 35B40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 35B45;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 35J47;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 35B06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 35J10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 1 Introduction Consider the nonlinear Schr¨odinger system \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 −∆ui + Vi(x)ui = ℓ � j=1 βij|uj|p|ui|p−2ui, ui ∈ H1(RN), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1) where N ≥ 1, Vi ∈ L∞(RN), βij ∈ R and 1 < p < 2∗ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Here 2∗ is the usual critical Sobolev exponent, namely, 2∗ := 2N N−2 if N ≥ 3 and 2∗ := ∞ for N = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Systems of this type occur as models for various natural phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' In physics, for example, they describe the behavior of standing waves for a mixture of Bose-Einstein condensates of different hyperfine states which overlap in space [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' The coefficients βij determine the type of interaction between the states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' if βij > 0, then there is an attractive force between ui and uj, similarly, if βij < 0, then the force is repulsive, and if βij = 0, then there is no direct interaction between these components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Whenever all the interaction coefficients are positive, we say that the system is cooperative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' If βii > 0 and βij < 0 for all i ̸= j, then the system is called competitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' And if some βij are positive and others are negative for i ̸= j, then we say that the system has mixed couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' All these regimes exhibit very different qualitative behaviors and have been studied extensively in recent years, see for instance [5,6,8–12,17,19–24,26] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' ∗Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Circuito Exterior, Ciudad Universitaria, 04510 Coyoac´an, Ciudad de M´exico, Mexico, felidaujal@im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='mx †Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Campus Juriquilla, Boulevard Juriquilla 3001, 76230 Quer´etaro, Qro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=', Mexico, monica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='clapp@im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='mx ‡(Corresponding author �) Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Circuito Ex- terior, Ciudad Universitaria, 04510 Coyoac´an, Ciudad de M´exico, Mexico, alberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='saldana@im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='mx 1 System (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1) has a variational structure, and therefore a natural strategy is to find weak solutions by minimizing an associated energy functional on a suitable set, under additional assumptions on the matrix (βij) and on the potentials Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Using this approach, several kinds of solutions have been found in terms of their signs and their symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' However, there seems to be no information available about the decay of these solutions at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' In this paper, we show that finite energy solutions must decay exponentially at infinity, and a rate can be found in terms of the potentials Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Our main result is the following one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Assume that, for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ, (V1) Vi : RN → R is H¨older continuous and bounded, (V2) there exists ρ ≥ 0 such that σi := inf RN∖Bρ(0) Vi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uℓ) ∈ � H1(RN) �ℓ be a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1) and let µi ∈ (0, √σi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then, there is C > 0 such that |ui(x)| ≤ Ce−µi|x| for all x ∈ RN and i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2) Furthermore, if Vi ≡ 1 for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2) holds true with µi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' We emphasize that each component may have a different decay depending on each potential Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' The main obstacle to showing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2) is to handle the possibly sublinear term |ui|p−2ui for p ∈ (1, 2) (which is always the case for N ≥ 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' To explain this point in more detail, assume that (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uℓ) is a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1) and write the i-th equation of the system as −∆ui + � ai(x) − ci(x)|ui(x)|p−2� ui = 0, ai := Vi − βii|ui|2p−2, ci := ℓ � j̸=i βij|uj|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3) Since every uj ∈ H1(RN)∩C0(RN), we know that ai and ci are bounded in RN, but |ui|p−2 → ∞ as |x| → ∞ and it is also singular at the nodal set of a sign-changing solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' As a consequence, one cannot use directly previously known results about exponential decay for scalar equations, such as those in [1, 3, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' In fact, one can easily construct a one dimensional solution of a similar scalar equation that has a power-type decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' For instance, let w ∈ C2(R) be a positive function such that w(x) = |x|−2/3 for |x| > 1 and let c(x) := −w′′(x) + w(x) w(x) 1 2 , x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then, w ∈ H1(R) is a solution of −w′′ + w = c w 1 2 in R, c(x) → 0 as |x| → ∞, and w decays as a power at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' This shows that the proof of the exponential estimate in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1 must rely on a careful study of the system structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' In other words, although the sublinear nonlinearity |ui|p−2ui appears in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1), the system is not sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' As a whole, it is always superlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' With this in mind, we adapt some of the arguments in [1,18] preserving at each step the system structure of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' These arguments rely basically on elliptic regularity and comparison principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 2 The exponential decay of solutions is a powerful tool in their qualitative study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' As an application of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1, we derive energy bounds of solutions having prescribed positive and nonradial sign- changing components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' For this, power type decay would not be enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' To be more precise, we consider the autonomous system \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 −∆ui + ui = ℓ � j=1 βij|uj|p|ui|p−2ui, ui ∈ H1(RN), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4) where the βij’s satisfy the following condition: (B1) The matrix (βij) is symmetric and admits a block decomposition as follows: For some 1 ≤ q ≤ ℓ there exist 0 = ℓ0 < ℓ1 < · · · < ℓq−1 < ℓq = ℓ such that, if we set Ih := {i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ} : ℓh−1 < i ≤ ℓh}, h ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q}, then βii > 0, βij ≥ 0 if i, j ∈ Ih, and βij < 0 if i ∈ Ih, j ∈ Ik and h ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' According to this decomposition, a solution u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uℓ) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1) may be written in block- form as u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uq) with uh = (uℓh−1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uℓh), h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' We say that u is fully nontrivial if every component ui is different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Set Q := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Given a partition Q = Q+ ∪ Q− with Q+ ∩ Q− = ∅ we look for solutions such that every component of uh is positive if h ∈ Q+ and every component of uh is nonradial and changes sign if h ∈ Q−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' To this end, we use variational methods in a space having suitable symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' As shown in [11, Section 3], to guarantee that the solutions obtained are fully nontrivial we need to assume the following two conditions: (B2) For each h ∈ Q, the graph whose set of vertices is Ih and whose set of edges is Eh := {{i, j} : i, j ∈ Ih, i ̸= j, βij > 0} is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (B3) If q ≥ 2 then, for every h ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q} such that ℓh − ℓh−1 ≥ 2, the inequality � min {i,j}∈Eh βij � \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 min h=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=',q max i∈Ih βii � i,j∈Ih βij \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb p p−1 > C∗ q � k=1 k̸=h � i∈Ih j∈Ik |βij| holds true, where C∗ = C∗(N, p, q, Q+) > 0 is the explicit constant given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='7) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' In [11] it is shown that, for any q, the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1) has a fully nontrivial solution satisfying the sign requirements described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Furthermore, an upper bound for its energy is exhibited, but only for systems with at most 2 blocks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=', for q = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Here we use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1 to obtain an energy bound for any number of blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' For each h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q, let RIh := {s = (sℓh−1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , sℓh) : si ∈ R for all i ∈ Ih} and define µh := inf s∈RIh s̸=0 \uf8eb \uf8ec \uf8ed � i∈Ih s2 i � � i,j∈Ih βij|si|p|sj|p � 2 2p \uf8f6 \uf8f7 \uf8f8 p p−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='5) 3 For any ℓ ∈ N, we write ∥u∥ for the usual norm of u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uℓ) in (H1(RN))ℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=', ∥u∥2 := ℓ � i=1 � RN (|∇ui|2 + |ui|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' We prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let N = 4 or N ≥ 6, and let Q = Q+ ∪ Q− with Q+ ∩ Q− = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Assume (B1), (B2), and (B3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then, there exists a fully nontrivial solution u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uq) to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4) with the following properties: (a) Every component of uh is positive if h ∈ Q+ and every component of uh is nonradial and changes sign if h ∈ Q−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (b) If q = 1, then ∥u∥2 = µ1∥ω∥2 if Q = Q+ and ∥u∥2 < 10 µ1∥ω∥2 if Q = Q−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (c) If q ≥ 2 the following estimate holds true ∥u∥2 < \uf8eb \uf8edmin k∈Q � akµk + � h∈Q∖{k} bhµh � \uf8f6 \uf8f8 ∥ω∥2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='6) where ak := 1 if k ∈ Q+, ak := 12 if k ∈ Q−, bh := 6 if h ∈ Q+, bh := 12 if h ∈ Q−, and ω is the unique positive radial solution to the equation − ∆w + w = |w|2p−2w, w ∈ H1(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='7) To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2, we follow the approach in [11] and impose on the variational setting some carefully constructed symmetries which admit finite orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' This approach immediately gives energy estimates but it requires showing a quantitative compactness condition which needs precise knowledge about the asymptotic decay of the components of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Here is where we use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Section 2 is devoted to the proof of the exponential decay stated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' The application of this result to derive energy bounds is contained in Section 3, where we also give some concrete examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Acknowledgments We thank Nils Ackermann for helpful comments and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Angeles and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Salda˜na thank the Instituto de Matem´aticas - Campus Juriquilla for the kind hospitality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Angeles is supported by CONACYT (Mexico) through a postdoctoral fellowship under grant A1-S-10457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Clapp is supported by CONACYT (Mexico) through the research grant A1-S-10457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Salda˜na is supported by UNAM-DGAPA-PAPIIT (Mexico) grant IA100923 and by CONACYT (Mexico) grant A1-S-10457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 4 2 Exponential decay This section is devoted to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' As a first step, we extend the argument in [2, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3] to systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let Br denote the ball of radius r in RN centered at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let σi and βij as in (V2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1), then we let σ := (σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , σℓ) and β := (βij)ℓ i,j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let Vi ∈ L∞(RN) satisfy (V2) and let u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uℓ) be a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Set ξi(r) := � RN∖Br � |∇ui|2 + |ui|2� and ξ(r) := (ξ1(r), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ξℓ(r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then, there are positive constants C = C(u, σ, β, N, ρ, p) and ϑ = ϑ(σ), with ρ and σi as in (V2), such that |ξ(r)|1 := ℓ � i=1 ξi(r) ≤ Ce−ϑr for every r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let χ : RN → R be given by χ(r) := 0 if r ≤ 0, χ(r) := r if r ∈ (0, 1) and χ(r) := 1 if r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let ur i (x) := χ(|x| − r)ui(x) for r ≥ 0, x ∈ RN, and i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then ur i ∈ H1(RN) and ur i (x) = (|x| − r)ui(x), ∇ur i (x) = (|x| − r)∇ui(x) + x |x|ui(x), if x ∈ Br+1 ∖ Br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Set δ := min{σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , σℓ, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Using that |ui x |x| · ∇ui| ≤ 1 2(|∇ui|2 + |ui|2) we obtain � RN � ∇ui · ∇ur i + Vi uiur i � ≥ δξi(r + 1) + � Br+1∖Br � (|x| − r) � |∇ui|2 + Vi u2 i � + ui x |x| · ∇ui � ≥ δξi(r + 1) − 1 2 � Br+1∖Br � |∇ui|2 + |ui|2� ≥ (δ + 1 2)ξi(r + 1) − 1 2ξi(r) if r + 1 ≥ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1) As u solves (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1) we have that ���� � RN ∇ui · ∇ur i + Vi uiur i ���� = ������ � RN ℓ � j=1 βij|uj|p|ui|p−2uiur i ������ ≤ ℓ � j=1 � RN\\Br |βij||uj|p|ui|p−2|ui|2 = ℓ � j=1 |βij| � RN∖Br |uj|p|ui|p and since |um|p ≤ ��ℓ k=1 |uk|2p�1/2 for every m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ, we obtain ���� � RN ∇ui · ∇ur i + Vi uiur i ���� ≤ \uf8eb \uf8ed ℓ � j=1 |βij| \uf8f6 \uf8f8 ℓ � k=1 � RN∖Br |uk|2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Given that uk ∈ H1(RN) for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1 implies the existence of a constant C1 = C1(N, p) > 0 such that ���� � RN ∇ui · ∇ur i + Vi uiur i ���� ≤ C1 \uf8eb \uf8ed ℓ � j=1 |βij| \uf8f6 \uf8f8 ℓ � k=1 �� RN∖Br � |∇uk|2 + |uk|2��p (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2) 5 for every r ≥ 1 and i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Set C2 := C1 �ℓ i,j=1 |βij|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2), assuming without loss of generality that ρ ≥ 2 and adding over i, we get 2δ + 1 2 |ξ(r + 1)|1 − 1 2|ξ(r)|1 ≤ C2 ℓ � k=1 |ξk(r)|p =: C2 |ξ(r)|p p if r + 1 ≥ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Therefore, |ξ(r + 1)|1 |ξ(r)|1 ≤ 1 2δ + 1 � 1 + 2C2 |ξ(r)|p p |ξ(r)|1 � ≤ 1 2δ + 1 � 1 + 2C2|ξ(r)|p−1 1 � =: γ(r) if r + 1 ≥ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3) Since |ξ(r)|1 → 0 as r → ∞, there is r0 = r0(u, p, β, ρ) ∈ N such that r0 ≥ ρ and γ(r) ≤ γ−1 0 for all r ≥ r0 with γ0 := 2δ+1 δ+1 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then, for r > r0 + 1, |ξ(r)|1 ≤ |ξ(⌊r⌋)|1 = |ξ(r0)|1 ⌊r⌋−1 � k=r0 |ξ(k + 1)|1 |ξ(k)|1 ≤ |ξ(r0)|1γr0−⌊r⌋ 0 ≤ ∥u∥2γr0−r+1 0 , where ⌊r⌋ denotes the floor of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Since |ξ(r)|1 ≤ ∥u∥2 ≤ ∥u∥2γr0−r+1 0 for r ≤ r0 + 1 we have that |ξ(r)|1 ≤ ∥u∥2γr0−r+1 0 = ∥u∥2γr0+1 0 e− ln(γ0)r for every r ≥ 0, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Assume (V1) and let u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uℓ) be a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then ui ∈ W 2,s(RN) ∩ C2(RN) for every s ≥ 2 and i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let N ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' The argument for N = 1, 2 is similar and easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ set fi := l � j=1 βij|uj|p|ui|p−2ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4) Since |uk| ≤ |u| := � u2 1 + · · · + u2 ℓ for every k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' ℓ, we have that |fi| ≤ ℓ � i,j=1 |βij||uj|p|ui|p−1 ≤ \uf8eb \uf8ed ℓ � j=1 |βij| \uf8f6 \uf8f8 |u|p|u|p−1 ≤ \uf8eb \uf8ed ℓ � i,j=1 |βij| \uf8f6 \uf8f8 |u|2p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='5) Therefore, fi ∈ Ls1(RN) for s1 := 2∗ 2p−1 > 1 and, by the standard Lp-elliptic regularity theory, ui ∈ W 2,s1(RN) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=', [14, Chapter 9] or [25, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Using a bootstrapping argument, we conclude the existence of s > max{N 2 , 2} such that ui ∈ W 2,s(RN) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ and thus, by the Sobolev embedding theorem, ui ∈ C1,α(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Since Vi is H¨older continuous and bounded, applying the Schauder estimates repeatedly, we deduce that ui is of class C2 (see [15, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' In the rest of the paper, we write | · |t for the norm in Lt(RN), 1 ≤ t ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' If u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uℓ) ∈ [L∞(RN)]ℓ, then |u|∞ := �ℓ i=1 supRN |ui|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Moreover, for a proper open subset Ω of RN we denote the usual Sobolev norm in H1(Ω) by ∥ · ∥H1(Ω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=', ∥u∥2 H1(Ω) := � Ω (|∇u|2 + |u|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 6 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Assume (V1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uℓ) be a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1), s > max{2, N 2 } and Λ > 0 be such that |Vi|∞ ≤ Λ for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then there is a constant C = C(β, N, p, Λ, s) > 0 such that, for any x ∈ RN, ∥ui∥W 2,s(B 1 2 (x)) ≤ C \uf8eb \uf8ed|ui| s−2 s ∞ ∥ui∥ 2 s H1(B1(x)) + |u| 2ps−(s+2) s ∞ � ℓ � j=1 ∥uj∥2 H1(B1(x)) � p s \uf8f6 \uf8f8 , where |u| := � u2 1 + · · · + u2 ℓ and BR(x) is the ball of radius R centered at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Since ui ∈ W 2,s(RN) ⊂ L∞(RN), we have that |ui|s = |ui|s−2|ui|2 ≤ |ui|s−2 ∞ |ui|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Set fi as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='5), there is a constant C2 = C2(β) such that |fi|s ≤ Cs 2|u|(p−1)s|u|ps = Cs 2|u|(p−1)s+p(s−2)(u2 1 + · · · + u2 ℓ)p ≤ Cs 2|u|2ps−(s+2) ∞ ℓp(u2p 1 + · · · + u2p ℓ ), where (p − 1)s + p(s − 2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then, by [14, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='11], there is a positive constant C1 = C1(s, N, Λ) such that ∥ui∥W 2,s(B 1 2 (x)) ≤ C1 � |ui|Ls(B1(x)) + |fi|Ls(B1(x)) � for any x ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' From the previous inequalities we derive ∥ui∥W 2,s(B 1 2 (x)) ≤ C1 \uf8eb \uf8ed|ui| s−2 s ∞ ∥ui∥ 2 s H1(B1(x))) + C2ℓ p s C3|u| 2ps−(s+2) s ∞ � ℓ � j=1 ∥uj∥2 H1(B1(x)) � p s \uf8f6 \uf8f8 , where C3 = C3(N, p) is the constant given by the Sobolev embedding H1(B1) ⊂ L2p(B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Assume (V1) − (V2), let u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uℓ) be a solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1) and let fi be as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then, there are constants η > 0, C1 > 0, and C2 > 0 such that |ui(x)| ≤ C1e−η|x|, |fi(x)| ≤ C2e−(2p−1)η|x|, for all x ∈ RN and i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' For x ∈ RN with |x| ≥ 2, set r := 1 2|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then, B1(x) ⊂ RN ∖ Br and, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1, there are positive constants K1 = K1(u, σ, β, N, ρ, p) and ϑ = ϑ(σ), with ρ and σi as in (V2), such that ∥uj∥2 H1(B1(x)) ≤ ∥uj∥2 H1(RN∖Br) = ξj(r) ≤ ℓ � i=1 ξi(r) ≤ K1e−ϑr for every j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Fix s > max{N 2 , 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3 there are positive constants K2 = K2(u, β, N, p, Λ, s) and K3 = K3(u, σ, β, ρ, N, p, s) such that ∥ui∥W 2,s(B 1 2 (x)) ≤ K2 \uf8eb \uf8ed∥ui∥ 2 s H1(B1(x))) + � ℓ � j=1 ∥uj∥2 H1(B1(x)) � p s \uf8f6 \uf8f8 ≤ K2K3e− ϑ s r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 7 Therefore, |ui(x)| ≤ |ui|L∞(B 1 2 (x)) ≤ K4∥ui∥W 2,s(B 1 2 (x)) ≤ K2K3K4e− ϑ 2s|x| for every x ∈ RN ∖ B2, where K4 is the positive constant given by the embedding W 2,s(B 1 2) ⊂ L∞(B 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Since ui is continuous, we may choose C1 ≥ K2K3K4 such that |ui(x)| ≤ C1e− ϑ s for every x ∈ B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' So, setting η := ϑ 2s, we obtain |ui(x)| ≤ C1e−η|x| for every x ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' The estimate for fi follows immediately from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' The following result is a particular case of [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' We include a simplified proof for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Assume that V : RN → R satisfies σ := infRN∖Bρ(0) V > 0 for some ρ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let w be a classical solution of −∆w + V w = f in RN such that |w(x)| ≤ Ce−η|x| and |f(x)| ≤ Ce−δ|x| for all x ∈ RN and for some constants C > 0, η ∈ (0, √σ) and δ ∈ (η, √σ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then, for any µ ∈ (η, δ), there is M = M(µ, δ, ρ, σ, C) > 0 such that |w(x)| ≤ Me−µ|x| for all x ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let ρ, σ, η, δ, µ, and C be as in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Set v(x) := e−µ|x| for x ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then, ∆v(x) = v(x)h(|x|) for x ∈ RN ∖ {0}, where h(r) := µ2 − (N − 1)µ r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' In particular, V (x) − h(|x|) ≥ σ − µ2 =: ε > 0 for |x| > ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Fix t ∈ R satisfying t > C ε e(µ−δ)ρ and w(x) < tv(x) for |x| = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='6) We claim that w(x) ≤ tv(x) for all |x| > ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Indeed, let z := w − tv and assume, by contradiction, that m := sup|x|≥ρ z(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Since lim|x|→∞ z(x) = 0, there is R > ρ such that z(x) ≤ m 2 for |x| ≥ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let Ω := {x ∈ RN : ρ < |x| < R and z(x) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then z ≤ m 2 on ∂Ω and, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='6), −∆z(x) = −∆w(x) + t∆v(x) = f(x) − V (x)w(x) + tv(x)h(|x|) = f(x) − V (x)z(x) + tv(x)(h(|x|) − V (x)) < Ce−δ|x| − εtv(x) = Ce−δ|x| − εte−µ|x| < 0 for every x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then, by the maximum principle, m = maxΩ z = max∂Ω z ≤ m 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Therefore m ≤ 0, namely, w(x) ≤ te−µ|x| for all |x| ≥ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Arguing similarly for −w and using that w ∈ L∞(RN) we obtain that |w(x)| ≤ Me−µ|x| for all x ∈ RN, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' We are ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Iterating Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='5, using that 2p − 1 > 1, one shows that, for any µi ∈ (0, √σi), there is C > 0 such that |ui(x)| ≤ Ce−µi|x| for all x ∈ RN and for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Now, assume that Vi ≡ 1 for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ and let µ ∈ (0, 1) be such that (2p − 1)µ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4, we have that |fi(x)| ≤ C2e−(2p−1)µ|x| for all x ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' The claim now follows from [1, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 8 3 Energy estimates for seminodal solutions In this section we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Consider the autonomous system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4) where N ≥ 4, 1 < p < N N−2 and βij satisfy the assumption (B1) stated in the Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' According to the decomposition given by (B1), a solution u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uℓ) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4) may be written in block-form as u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uq) with uh = (uℓh−1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uℓh), h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' u is called fully nontrivial if every component ui is different from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' We say that u is block-wise nontrivial if at least one component in each block uh is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Following [11], we introduce suitable symmetries to produce a change of sign in some compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let G be a finite subgroup of the group O(N) of linear isometries of RN and denote by Gx := {gx : g ∈ G} the G-orbit of x ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let φ : G → Z2 := {−1, 1} be a homomorphism of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' A function u : RN → R is called G-invariant if it is constant on Gx for every x ∈ RN and it is called φ-equivariant if u(gx) = φ(g)u(x) for all g ∈ G, x ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1) Note that, if φ ≡ 1 is the trivial homomorphism and u satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1), then u is G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' On the other hand, if φ is surjective every nontrivial function satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1) is nonradial and changes sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Define H1(RN)φ := {u ∈ H1(RN) : u is φ-equivariant}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' For each h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q, fix a homomorphism φh : G → Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Take φi := φh for all i ∈ Ih and set φ = (φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , φℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Denote by Hφ := H1(RN)φ1 × · · · × H1(RN)φℓ, and let J φ : Hφ → R be the functional given by J φ(u) := 1 2 ℓ � i=1 ∥ui∥2 − 1 2p ℓ � i,j=1 βij � RN |ui|p|uj|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' This functional is of class C1 and its critical points are the solutions to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4) satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' The block-wise nontrivial solutions belong to the Nehari set N φ := {u ∈ Hφ : ∥uh∥ ̸= 0 and ∂uhJ φ(u)uh = 0 for every h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Note that ∂uhJ φ|K(u)uh = ∥uh∥2 − ℓ � k=1 � (i,j)∈Ih×Ik βij � RN |ui|p|uj|p, and that J φ(u) = p−1 2p ∥u∥2 if u ∈ N φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let cφ := inf u∈N φ J φ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' If s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , sq) ∈ Rq and u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uq) ∈ Hφ we write su := (s1u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , squq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' The following facts were proved in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 9 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (i) cφ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (ii) If the coordinates of u ∈ Hφ satisfy q � k=1 � (i,j)∈Ih×Ik � RN βij|ui|p|uj|p > 0 for every h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2) then there exists a unique su ∈ (0, ∞)q such that suu ∈ N φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Furthermore, J φ(suu) = max s∈(0,∞)q J φ(su).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' See [8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2] or [11, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' If cφ is attained, then the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4) has a block-wise nontrivial solution u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uℓ) ∈ Hφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Furthermore, if ui is nontrivial, then ui is positive if φi ≡ 1 and ui is nonradial and changes sign if φi is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' It is shown in [8, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4] that any minimizer of J φ on N φ is a block-wise nontrivial solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' If ui ̸= 0 and φi is surjective, then ui is nonradial and changes sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' If φi ≡ 1 then |ui| is G-invariant and replacing ui with |ui| we obtain a solution with the required properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Set Q := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q} and fix a decomposition Q = Q+ ∪ Q− with Q+ ∩ Q− = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' From now on, we consider the following symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' We write RN ≡ C × C × RN−4 and a point in RN as (z1, z2, y) ∈ C × C × RN−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Definitions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let i denote the imaginary unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' For each m ∈ N, let Km := {e2πij/m : j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , m − 1}, Gm be the group generated by Km ∪{τ}∪O(N −4), acting on each point (z1, z2, y) ∈ C×C×RN−4 as e2πij/m(z1, z2, y) := (e2πij/mz1, e2πij/mz2, y), τ(z1, z2, y) := (z2, z1, y), α(z1, z2, y) := (z1, z2, αy) if α ∈ O(N − 4), and θ : Gm → Z2 be the homomorphism satisfying θ(e2πij/m) = 1, θ(τ) = −1, and θ(α) = 1 for every α ∈ O(N − 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Define φh : Gm → Z2 by φh := � 1 if h ∈ Q+, θ if h ∈ Q−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3) Due to the lack of compactness, cφ is not always attained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' [11, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='8(i)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' A sufficient condition for this to happen is given by the next lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' We use the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' If Q′ ⊂ Q := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q} we consider the subsystem of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4) obtained by deleting all components of uh for every h /∈ Q′, and we denote by J φ Q′ and N φ Q′ the functional and the Nehari set associated to this subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' We write cφ Q′ := inf u∈N φ Q′ J φ Q′(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' If Q′ = {h} we omit the curly brackets and write, for instance, cφ h or J φ h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 10 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4 (Compactness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let N ̸= 5, m ≥ 5 and φh : Gm → Z2 be as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' If, for each h ∈ Q := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q}, the strict inequality cφ < \uf8f1 \uf8f2 \uf8f3 cφ Q∖{h} + mµh p−1 2p ∥ω∥2, if h ∈ Q+, cφ Q∖{h} + 2mµh p−1 2p ∥ω∥2, if h ∈ Q−, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4) holds true, then cφ is attained, where ω is the positive radial solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='7) and µh is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' This statement follows by combining [11, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='8(ii)] with [11, Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' To verify condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4) we introduce a suitable test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Fix m ≥ 5 and let Km be as in Definitions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' If h ∈ Q+, we take ζh := ( 1 √ 2, 1 √ 2, 0) and, for each R > 1, we define �σhR(x) := � g∈Km ω(x − Rgζh), x ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' If h ∈ Q− we take ζh := (1, 0, 0) and we define �σhR(x) := � g∈G′m φh(g) ω(x − Rgζh), x ∈ RN, where ω is the positive radial solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='7) and G′ m is the subgroup of Gm generated by Km∪{τ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Note that �σhR(gx) = φh(g)�σhR(x) for every g ∈ Gm, x ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Let σhR := thR�σhR, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='5) where thR > 0 is chosen so that ∥σhR∥2 = � RN |σhR|2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' If m ≥ 5, then, for each h ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q}, there exist th = (tℓh−1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , tℓh) ∈ (0, ∞)ℓh−ℓh−1 and C0, R0 > 0 such that thσhR := (tℓh−1+1σhR, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , tℓhσhR) ∈ N φ h and J φ h (thσhR) ≤ |Gmζh| µh p−1 2p ∥ω∥2 − C0e−Rdm for every R ≥ R0, where |Gmζh| is the cardinality of the Gm-orbit of ζh, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=', |Gmζh| = m if h ∈ Q+ and |Gmζh| = 2m if h ∈ Q−, and dm := |1 − e2πi/m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Take th = (tℓh−1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , tℓh) ∈ (0, ∞)ℓh−ℓh−1 such that � i∈Ih t2 i = � i,j∈Ih βijtp jtp i = µh and apply [11, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1(i) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 11 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Assume (B1) and let φh : Gm → Z2 be given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' For q = 1 and m ≥ 5 it is proved in [11, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='5] that cφ is attained at u ∈ N φ satisfying ∥u∥2 = µ1∥ω∥2 if Q+ = {1} and ∥u∥2 < 2m µ1∥ω∥2 if Q− = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Taking m = 5 gives statement (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Fix m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' We claim that cφ is attained and that the estimate (c) holds true for every q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' To prove this claim, we proceed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Assume it is true for q − 1 with q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' We will show that the compactness condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Using a change of coordinates, it suffices to argue for h = q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' By induction hypothesis there exists w = (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , wq−1) ∈ N φ Q∖{q} such that J φ Q∖{q}(w) = cφ Q∖{q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' For each R > 1 let σqR be as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='5) and take tq ∈ (0, ∞)ℓ−ℓq−1 as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Set whR = wh for h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q−1 and wqR = tqσqR, and define wR = (w1R, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , wℓR) := (w1R, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , wqR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then, as w ∈ N φ Q∖{q} and the interaction between the components of w and σqR tends to 0 as R → ∞, we have that wR satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2) for large enough R and, as a consequence, there exist R1 > 0 and (s1R, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , sqR) ∈ [1/2, 2]q such that (s1Rw1R, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , sqRwqR) ∈ N φ if R ≥ R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Set uR = (u1R, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , uℓR) := (s1Rw1R, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , sqRwqR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Using that w ∈ N φ Q∖{q} and tqσqR ∈ N φ q , from the last statement in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1(ii) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='5 we derive J φ(uR) = 1 2 ℓ � i=1 ∥uiR∥2 − 1 2p ℓ � i,j=1 βij � RN |uiR|p|ujR|p ≤ J φ Q∖{q}(w) + J φ q (tqσqR) − 1 p q−1 � h=1 � (i,j)∈Ih×Iq βij � RN |shRwiR|p|sqRwjR|p ≤ cφ Q∖{q} + |Gmζh| µq p−1 2p ∥ω∥2 − C0e−Rdm + C1 q−1 � h=1 � i∈Ih � RN |wiR|p|σqR|p, if R ≥ max{R0, R1}, where C0 and C1 are positive constants and dm is given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' It is well known that |ω(x)| ≤ Ce−|x| and, as w solves a subsystem of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4), Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1 asserts that |wiR(x)| ≤ Ce−|x| for every i ∈ Ih with h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Therefore, for every g ∈ Gm, � RN |wiR|p|ω( · − Rgζh)|p ≤ C � RN e−p|x| e−p|x−Rgζh| dx ≤ Ce−Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' So, if p > dm, we conclude that cφ < cφ Q∖{q} + |Gmζh| µq p−1 2p ∥ω∥2 and, by Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2, cφ is attained at a block-wise nontrivial solution u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4) such that every component of uh is positive if h ∈ Q+ and every component of uh is nonradial and changes sign if h ∈ Q−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Furthermore, since we are assuming (B2) and (B3) with C∗ as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='7) below, [11, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3] asserts that u is fully nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Finally, note that p > 1 = dm because m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' As |Gmζh| = 6 if h ∈ Q+ and |Gmζh| = 12 if h ∈ Q−, the estimate in statement (c) follows by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 12 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' If m = 5 and p > dm we arrive to a similar conclusion, where, in this case, the constant bh in statement (b) is 5 if h ∈ Q+ and it is 10 if h ∈ Q−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Note, however, that numbers p satisfying d5 = 2 sin π 5 < p < N N−2 exist only for N ≤ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' For φh as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3), the constant C∗ > 0 appearing in (B3) depends on N, p, q, and Q+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' It is explicitly defined in [11, Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1)] as C∗ := \uf8eb \uf8ed pdφ (p − 1)S p p−1 φ \uf8f6 \uf8f8 p , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='7) where dφ := p − 1 2p inf (v1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=',vq)∈Uφ q � h=1 ∥vh∥2 with Uφ := {(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , vq) : vh ∈ H1(RN)φh ∖ {0}, ∥vh∥2 = |vh|2p 2p, vhvk = 0 if h ̸= k}, and Sφ := min h=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=',q inf v∈H1(RN)φh∖{0} ∥v∥2 |v|2 2p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' In the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2 we use [1, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3], which also characterizes the sharp decay rate for positive components by providing a bound from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' This kind of information can be useful to show uniqueness of positive solutions for some problems, see [4, Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' To conclude, we discuss some special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Examples 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Assume (B1) and let p ∈ (1, 2∗ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (a) If q = 1 the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4) is cooperative and more can be said.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Indeed, it is shown in [11, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='5] that, if (B2) is satisfied, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4) has a synchronized solution u = (t1u, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , tℓu), where (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , tℓ) ∈ (0, ∞)ℓ is a minimizer for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='5) and u is a nontrivial φ-equivariant least energy solution of the equation −∆u + u = |u|2p−2u, u ∈ H1(RN)φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='8) Here, if Q+ = {1}, then φ ≡ 1 (and therefore u = ω) and ∥u∥2 ≤ µ1∥ω∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' On the other hand, if Q− = {1}, then φ : Gm → Z2 is the homomorphism θ given in Definitions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3 and ∥u∥2 ≤ 10µ1∥ω∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' (b) If q = ℓ ≥ 2 the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='4) is competitive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=', βii > 0 and βij < 0 if i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Assumptions (B2) and (B3) are automatically satisfied and, as µi = β − 1 p−1 ii , the estimate in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2(c) becomes ∥u∥2 < \uf8eb \uf8edmin j∈Q � ajβ − 1 p−1 jj + � i∈Q∖{i} biβ − 1 p−1 ii � \uf8f6 \uf8f8 ∥ω∥2 ≤ \uf8f1 \uf8f2 \uf8f3 (6 |Q+| + 12 |Q−| − 5) β − 1 p−1 0 ∥ω∥2 if Q+ ̸= ∅, 12 |Q−|β − 1 p−1 0 ∥ω∥2 if Q+ = ∅, where |Q±| denotes the cardinality of Q± and β0 := min{β11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , βℓℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 13 (c) Similarly, for any q ≥ 2, the estimate in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2(c) yields ∥u∥2 ≤ � (6 |Q+| + 12 |Q−| − 5) µ∗∥ω∥2 if Q+ ̸= ∅, 12 |Q−| µ∗∥ω∥2 if Q+ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' where µ∗ = max{µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , µq}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Assumptions (B2) and (B3) guarantee that u is fully nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Note that the left-hand side of the inequality in (B3) depends only on the entries of the submatrices (βij)i,j∈Ih, h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q, whereas the right-hand side only depends on the other entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' So, if the former are large enough with respect to the absolute values of the latter, (B3) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' For example, if we take ℓ = 2q and the matrix is \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed λ λ β13 β14 β15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' β1ℓ λ λ β23 β24 β25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' β2ℓ β31 β32 λ λ β35 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' β3ℓ β41 β42 λ λ β45 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' β4ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' βℓ−1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' βℓ−1 ℓ−2 λ λ βℓ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' βℓ ℓ−2 λ λ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' with λ > 0 and βji = βij < 0, then (B1) and (B2) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' If, additionally, λ > 4 2p−1 p−1 (q − 1)C∗ and |βij| ≤ 1, then, for any h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' , q, � min {i,j}∈Eh βij � \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 min h=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=',q max i∈Ih βii � i,j∈Ih βij \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb p p−1 = λ � λ 4λ � p p−1 > C∗4(q − 1) ≥ C∗ q � k=1 k̸=h � i∈Ih j∈Ik |βij| so (B3) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' A An auxiliary result Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' For every r ≥ 1 there is a linear operator Er : H1(RN ∖ Br) → H1(RN) such that, for every u ∈ H1(RN ∖ Br), (i) Eru = u a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' in RN ∖ Br, (ii) |Eru|2 2 ≤ C1|u|2 L2(RN∖Br) (iii) ∥Eru∥2 ≤ C1∥u∥2 H1(RN∖Br) for some positive constant C1 depending only on N and not on r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' As a consequence, given p ∈ (1, 2∗ 2 ) there is a positive constant C depending only on N and p such that |u|L2p(RN∖Br) ≤ C∥u∥H1(RN∖Br) for every u ∈ H1(RN ∖ Br) and every r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Fix a linear (extension) operator E1 : H1(RN ∖ B1) → H1(RN) and a positive constant C1 satisfying (i), (ii) and (iii) for r = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' [16, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' For r > 1, set �u(x) := u(rx) and, for u ∈ H1(RN ∖ Br), define (Eru)(y) := (E1�u) �y r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then, � Eru = E1�u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Clearly, Er satisfies (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Note that |�u|2 L2(RN ∖B1) = r−N|u|2 L2(RN∖Br) and that ∥�u∥2 H1(RN∖B1) = r−N �� RN∖Br � r2|∇u|2 + |u|2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Similar identities hold true when we replace RN ∖ B1 and RN ∖ Br with RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Therefore, r−N|Eru|2 2 = |� Eru|2 2 = |E1�u|2 2 ≤ C1∥�u∥2 L2(RN∖B1) = r−NC1|u|2 L2(RN ∖Br), which yields (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Furthermore, r−N �� RN � r2|∇(Eru)|2 + |Eru|2�� = ∥� Eru∥2 = ∥E1�u∥2 ≤ C1∥�u∥2 H1(RN∖B1) = r−NC1 �� RN∖Br � r2|∇u|2 + |u|2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' This inequality, combined with (ii), yields r2∥Eru∥2 = � RN � r2|∇(Eru)|2 + |Eru|2� + (r2 − 1) � RN |Eru|2 ≤ C1 � RN∖Br � r2|∇u|2 + |u|2� + C1(r2 − 1) � RN∖Br |u|2 = r2C1∥u∥2 H1(RN∖Br), which gives (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' For p ∈ (1, N N−2) let C2 = C2(N, p) be the constant for the Sobolev embedding H1(RN) ⊂ L2p(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Then, for any u ∈ H1(RN ∖ Br), using statements (i) and (iii) we obtain |u|2 L2p(RN∖Br) ≤ |Eru|2 2p ≤ C2∥Eru∥2 ≤ C2C1∥u∥2 H1(RN∖Br), as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' References [1] Ackermann, Nils;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Dancer, Norman: Precise exponential decay for solutions of semilinear elliptic equations and its effect on the structure of the solution set for a real analytic nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Differential Integral Equations 29 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 7-8, 757–774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' [2] Ackermann, Nils;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Weth, Tobias: Multibump solutions of nonlinear periodic Schr¨odinger equa- tions in a degenerate setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 7 (2005), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 3, 269–298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' [3] Berezin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' Shubin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=': The Schr¨odinger equation.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} +page_content=' 17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtFIT4oBgHgl3EQfXys9/content/2301.11245v1.pdf'} diff --git a/IdAzT4oBgHgl3EQfU_z6/content/tmp_files/2301.01278v1.pdf.txt b/IdAzT4oBgHgl3EQfU_z6/content/tmp_files/2301.01278v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8fab4deb0dc1589a2f2e22a957fb9e43a1f5cd75 --- /dev/null +++ b/IdAzT4oBgHgl3EQfU_z6/content/tmp_files/2301.01278v1.pdf.txt @@ -0,0 +1,1460 @@ +AE +Students’ Perceptions of Sustainable Universities in Hungary: An Importance- +Performance Analysis + +496 +Amfiteatru Economic +STUDENTS’ PERCEPTIONS OF SUSTAINABLE UNIVERSITIES +IN HUNGARY: AN IMPORTANCE-PERFORMANCE ANALYSIS + +Szabolcs Nagy1* and Mariann Veresné Somosi2 + 1)2) University of Miskolc, Miskolc, Hungary + + + +Please cite this article as: +Nagy, S. and Somosi, M.V., 2020. Students’ +Perceptions of Sustainable Universities in Hungary: +An Importance-Performance Analysis. Amfiteatru +Economic, 22(54), pp. 496-515. + +DOI: 10.24818/EA/2020/54/496 + +Article History +Received: 29 December 2019 +Revised: 3 February 2020 +Accepted: 30 March 2020 + +Abstract +In order to succeed, universities are forced to respond to the new challenges in the rapidly +changing world. The recently emerging fourth-generation universities should meet +sustainability objectives to better serve their students and their communities. It is essential +for universities to measure their sustainability performance to capitalise on their core +strengths and to overcome their weaknesses. In line with the stakeholder theory, the +objective of this study was to investigate students’ perceptions of university sustainability +including their expectations about and satisfaction with the efforts that universities make +towards sustainability. This paper proposes a new approach that combines the sustainable +university scale, developed by the authors, with the importance-performance analysis to +identify key areas of university sustainability. To collect data, an online survey was +conducted in Hungary in 2019. The sustainable university scale was found to be a reliable +construct to measure different aspects of university sustainability. Results of the +importance-performance analysis suggest that students consider Hungarian universities +unsustainable. Research findings indicate that Hungarian universities perform poorly in +sustainable purchasing and renewable energy use, but their location and their efforts +towards separate waste collection are their major competitive advantages. The main +domains of university sustainability were also discussed. This study provides university +decision-makers and researchers with insightful results supporting the transformation of +traditional universities into sustainable, fourth-generation higher education institutions. + +Keywords: sustainable university, students’ perception, importance-performance analysis, +Hungary, student satisfaction, student expectation + +JEL Classification: I23, Q56 + + + +* Corresponding author, Szabolcs Nagy – nagy.szabolcs@uni-miskolc.hu + + +Sustainable University +AE + +Vol. 22 • No. 54 • May 2020 +497 +Introduction +We live in the age of rapid changes to which higher education institutions should adopt. A +university that wants to succeed needs to respond to the challenges of the new era. One of +them is the urgency to meet sustainability objectives (Filho, Manolas and Pace, 2015; Soini, +et al., 2018; Olalla and Merino, 2019). Universities are undergoing a rapid transformation +as they are not only traditionally engaged in education but are also playing an increasingly +important role in the society (Papp-Váry and Lukács, 2019). Nowadays, the emergence of +the so-called Fourth Generation universities, which actively shape their socio-economic +environment, can be seen (Pawłowski, 2009; Lukovics and Zuti, 2017). +The topic of sustainable development is increasingly present among the major concerns of +the international academic community (Grecu and Ipiña, 2014). Universities must take +steps to achieve the United Nations Sustainable Development Goals (Paletta, et al., 2019). +Target 4.7 declares that students have the right to acquire the knowledge and skills needed +to promote sustainable development (UN, 2019). Globally, the proliferation of the efforts to +assess universities’ responses to the challenges of sustainability can be seen (Li, Gu and +Liu, 2018). Adams, Martin, and Boom (2018) draw the attention to the importance of the +university sustainability culture. +Adaptation of the stakeholder theory is essential for higher education institutions +(Mainardes, et al., 2010) as stakeholders can create opportunities for or pose threats to an +organisation (Chapleo and Sims, 2017). Students as stakeholders have a serious impact on +the future development of universities (Degtjarjova, Lapina and Freidenfelds, 2018). +Commitment to sustainability of leaders and important stakeholders play a key role in the +effectiveness of sustainable development initiatives in higher education institutions +(Wright, 2010.) +The position of Hungarian higher education institutes in the world rankings is not very +favourable. The best Hungarian university can be found around the 500th place in global +rankings. There are only seven or eight Hungarian institutions that are ranked at all +(Polónyi and Kozma, 2019). The weak performance of the Hungarian higher education +institutions in sustainability rankings explains the need for a comprehensive analysis of +university sustainability in Hungary from the students as stakeholders’ perspective, which +is one of the main objectives of this study. +Students as stakeholders form expectations regarding university sustainability not only +generally, but also very specifically, and how those expectations are met determines the +level of their satisfaction. This study aims to investigate student expectations about and +satisfaction with the attributes of the sustainable university by using the sustainable +university scale (SUS) combined with the importance-performance analysis (IPA). SUS, +the items of which are the determinants of university sustainability, was developed by the +authors. IPA has been widely used to examine the relationship between importance, +performance, and satisfaction in many areas (Yuvinatileng, et al., 2013; Wyród-Wróbel and +Biesok, 2017, Kim, et al., 2018) However, no previous study has investigated it in the +context of university sustainability in spite of the fact that universities should use +managerial tools to develop their sustainability strategy. This study seeks to address this +research gap. + + +AE +Students’ Perceptions of Sustainable Universities in Hungary: An Importance- +Performance Analysis + +498 +Amfiteatru Economic +1. Literature review +1.1. Perceptions of the sustainable university +In the UI GreenMetric World University Ranking 2019, which provides information about +the current conditions and policies related to Green Campus and Sustainability, only seven +Hungarian universities can be found. The University of Szeged is in the best position, +ranked first in Hungary, and 74th in the world. It is followed by the University of Pecs, +ranked 100th globally and the University of Debrecen, in the 202nd position in the world +ranking. The University of Miskolc, for which the authors work, can be found only in the +605th place in this ranking of 780 universities globally (Greenmetric, 2019). Students’ +perceptions of university sustainability were assumed to be in line with this poor ranking +performance. It is therefore hypothesized that students are not satisfied with the +sustainability performance of the Hungarian higher education institutions (H1). Mention +must be made of some of the shortcomings of the GreenMetric Ranking, i.e. non- +compliance with the Berlin Principles (Ragazzi and Ghidini, 2017), however, it is still one +of the best tools to quantify university sustainability. +The perceptions of university students towards factors of a sustainable university was first +discussed by Nejati and Nejati (2013). The authors developed a reliable scale to assess the +university practices towards sustainability. They identified four main dimensions of the +sustainable university scale, which are respectively: 1) community outreach, 2) +sustainability commitment and monitoring, 3) waste and energy, and 4) land use and +planning. Their initial scale contained 28 items, which they reduced to a 12-item scale, +which could be a key instrument for university decision-makers and stakeholders to +measure the university’s performance regarding the implementation of the transition +strategy towards sustainability. Their construct measuring sustainability practices of +universities contains 1) community outreach programs; 2) green community centres; 3) +partnerships with government, non-governmental organizations, and industry working +toward sustainability; 4-5) written commitment to sustainability (university and department +level); 6-7) sustainability audits on the surrounding community and on campus; 8) reuse of +campus waste; 9) use of renewable and safe energy sources; 10) sustainable support +services (e.g. recycling bins on campus, efficient public transport throughout the +university); 11) sustainable campus building planning and 12) sustainable campus land-use. +Dagiliute, Liobikiene and Minelgaite (2018) were the first to investigate the differences in +the perceived sustainability performance between the ‘green’ and the ‘non-green’ +universities. They compared the students' attitudes towards sustainability in two Lithuanian +universities. They did not find any significant differences in sustainability aspects in +general, however, students of the green university sought more information about +sustainability and were more often involved in sustainability activities. They also found that +campus sustainability and environmental information have a significant impact on students’ +sustainable behaviour. In their study, they used a scale to measure perceptions made up of +16 items, grouped into four main constructs: 1) ‘campus sustainability’, 2) ‘environmental +information’, 3) ‘students’ sustainability involvement’, and 4) ‘university's role in +sustainable development. The item ‘university's self-representation as a green university’ +was also involved in their construct. Their 17-item scale involves 1) environmental student +organization(s); 2) use of public transport, bikes; 3) possibility to recycle waste at the +university; 4) use one's own non-disposable cup; 5) availability of strategic documents and +their implementation reports; 6) sustainability-related information during lectures; + +Sustainable University +AE + +Vol. 22 • No. 54 • May 2020 +499 +7) university website on environmental objectives; 8) participation in environmental, social +activities; 9) involvement in activities at the university; 10) energy and resource saving; +11) contribution to social well-being, tolerance; 12) environmental education; 13) +cooperation with other national and foreign universities and businesses; 14) inclusion of +sustainability aspects in study programmes; 15) sustainability research; 16) university's +self-representation as a green university; and 17) declared environmental objectives. They +found that students considered social aspects, i.e. social well-being, tolerance the most +important attribute of the sustainable university. However, students considered +environmental aspects, such as energy saving, environmental education, and actions less +important. +Li, Gu and Liu (2018) established a new scoring system for campus sustainability in +Australia. They suggest that sustainable campus performance indicators should be +identified from the different perspectives of the economy, environment and society. In +order to identify and prioritise the key sustainability indicators for university campuses, +they proposed a new approach combining the qualitative scoring method and an analytical +hierarchical process. After thorough literature review, they identified 54 indicators and +quantified the weight coefficients for the criteria, sub-criteria and elements, and proposed a +model that can be a flexible tool for university decision-makers. +It is hypothesized that combining the most relevant items of the constructs developed by +Nejati and Nejati (2013), Dagiliute, Liobikiene and Minelgaite (2018) and Li, Gu and Liu +(2018), a new, reliable scale to measure perceived university sustainability, i.e. the +sustainable university scale, can be developed (H2). +Shuqin, et al. (2019) aimed to assess and compare the sustainability performance of +different Chinese universities. The authors developed a campus sustainability evaluation +system that is made up of the five main domains of campus sustainability, which are +respectively: organization and management, energy and resource saving, friendly +environment, campus culture, and social outreach. Their evaluation system included 14 +mandatory indicators and 69 optional indicators. They found that the most problematic +fields are organization management, resource saving and campus culture. For example, +there are issues with green education, green research and green humanities as they are not +so developed there. The assessment tool proposed by the authors can be used to guide the +green campus revolution in China and could be adopted by the rest of the world. +Wakkee, et al. (2019) demonstrated how (entrepreneurial) universities can drive regional +sustainable development in developing countries. They found that local campus leadership, +a holistic teaching and research programme, and student involvement can have significant +local effects. +1.2. Importance-Performance Analysis (IPA) +The importance-performance analysis (IPA) was developed by Martilla and James (1977). +The original version of IPA defines consumer satisfaction as the function of two +components that are respectively: the importance of an attribute of the product/service, and +the perceived performance of the company on this attribute. The mean of importance and +performance ratings of each attribute determines its position on the importance- +performance matrix or grid, which is also often called the Cartesian diagram (Figure no. 1). +The overall mean of the performance/importance ratings is used as a delimiter of high and +low performance/importance (Yuvinatileng, Utomo and Latuperissa, 2013). + +AE +Students’ Perceptions of Sustainable Universities in Hungary: An Importance- +Performance Analysis + +500 +Amfiteatru Economic +The 2x2 IPA matrix can be divided into four quadrants. Each quadrant requires a different +approach and strategy (Wyród-Wróbel and Biesok, 2017): + Quadrant 1: Keep up the good work. This is the best possible position for an attribute. +This quadrant contains the competitive advantages and major strengths of a company. The +organization must defend all of them to succeed. These are high importance/high +performance items. + + +Figure no. 1: The modified Importance Performance Matrix +Source: Kim, Jeon, Cho and Kim, 2018. + + Quadrant 2: The territory of Possible overkill. Here low importance/high performance +attributes, i.e. items of overperformance, can be found. Organizations should deploy +business resources used here somewhere else (e.g. in Quadrant 1) or should increase the +importance of those attributes that can be found here to turn them into competitive +advantages. + Quadrant 3: The area of Low priority. Low importance/low performance attributes can +be seen here. Those are minor weaknesses that require no additional resources. +Organizations are suggested to avoid investing in this quadrant. + Quadrant 4: Concentrate here. High importance/low performance attributes can be +found here. Those are the major weaknesses of an organization that require immediate +corrective actions to increase consumer satisfaction and to avoid customer churn. +1.3. Stakeholder theory +The stakeholder theory originates from the 1980s. Freeman (1984) was the first to coin the +phrase as an opposite to the shareholder theory or Friedman’s doctrine, which suggests that a +company’s sole responsibility is to make money for its shareholders (Friedman, 1965). + +High +Quadrant 2 +Quadrant1 +Possible overkill +Keep up the good work +Performance +Quadrant3 +Quadrant 4 +Low priority +Concentrate here +Low +Low +Importance +HighSustainable University +AE + +Vol. 22 • No. 54 • May 2020 +501 +According to the stakeholder theory, shareholders are only one of many stakeholders in a +company, and an organization’s key to market success is how it satisfies all the stakeholders, +not only its shareholders (Freeman, 2010). The stakeholder theory says that the stakeholder +ecosystem is made up of all parties that invested and involved in, or affected by, the company. +Therefore, companies must pay special attention to their employees, vendors, suppliers, +owners, community/neighbours, community groups, competitors, governmental bodies, +oversight organizations and the local ecology (Freeman, 2010). +The stakeholder theory is intertwined with the domains of ethics and sustainability. Carroll and +Buchholtz (2014) suggest that successful businesses in society adopt a stakeholder +management approach. The stakeholder theory is solid ground for corporate social +responsibility and business ethics inside the company (Kakabadse, Rozules and Davies, 2005). +The stakeholder ecosystem of a university comprises current, former (alumni) and potential +students, parents, municipalities, academics, faculties, management (Rector, the Senate, +Chancellor), administrative staff, governmental organisations, Academy of Sciences, +research partners and companies. In higher education institutions, students and employees +are always the major stakeholders in terms of their number. According to the stakeholder +theory, universities are service providers to students and students are one of the most +important stakeholders (Degtjarjova, Lapina, and Freidenfelds, 2018). The more satisfied +students are, the more likely it is that the university could succeed, also in the field of +sustainability. It is therefore assumed that IPA as a strategic tool should be used to +maximize student satisfaction with the efforts that universities make towards sustainability. + +2. Methodology +2.1. Methodology and research questions +Based on the literature review presented above, and in line with the main objectives of the +research, this study aims to address the following research questions respectively: + R1: What are the student expectations about university sustainability in Hungary? +(student expectations) + R2: To what extent are students satisfied with the sustainability performance of +universities? (student satisfaction). H1 refers to this question. + R3: Is combining sustainable university scale (SUS) with importance-performance +analysis (IPA) a powerful strategic tool for university decision-makers to identify key areas +of university sustainability? + R4: What are the main components of the perceived university sustainability? + R5: Is sustainable university scale (SUS) a reliable construct to measure students’ +perceptions of university sustainability? H2 refers to this question. +In line with the research questions, the following hypotheses were developed: + H1: Students are not satisfied with the sustainability performance of the Hungarian +higher education institutions. + +AE +Students’ Perceptions of Sustainable Universities in Hungary: An Importance- +Performance Analysis + +502 +Amfiteatru Economic + H2: Combining the most relevant items of the constructs developed by Nejati and +Nejati (2013), Dagiliute, Liobikiene and Minelgaite (2018) and Li, Gu and Liu (2018), a +new, reliable scale for measuring perceived university sustainability, i.e. the Sustainable +University Scale (SUS), can be developed. +To answer the research questions, and to thoroughly investigate students’ perceptions of the +sustainable university, a questionnaire made up of 47 questions grouped into three sections +were designed: + Section 1: Importance of the sustainable university scale (SUS) items. It contains 21 +statements measured on a five-point importance scale (1. not at all important … 5. very +important). Respondents were asked to answer the following question: “How important are +the followings to you regarding a sustainable university?”. SUS items can be seen in Table +no. 1. + Section 2: Perceived performance of the sustainable university scale (SUS) items: The +very same 21 statements as in Section 1, measured on a five-point rating scale (1 – very +poor ... 5 – excellent), answering the question “How do you rate the sustainability +performance of your university?”. + Section 3: Demographic variables. It contains 5 questions including gender, age, +study level, branch of sciences and the university where they study (Table no. 2). +The sustainable university scale (SUS), which contains 21 items, is a construct developed +by the authors. It is based on the domains of university sustainability discussed in the +literature review. More specifically, in our construct we combined 9 items (item 4, 5, 7, 9, +10, 15, 17, 18 and 20) from Dagiliute, Liobikiene and Minelgaite (2018) with 9 items (item +1, 3, 6, 7, 8, 11, 13, 14 and 16) used by Nejati and Nejati (2013), with 3 items (item 9, 11 +and 16) from Li, Gu and Liu (2018). It must be noted that four items are overlapping. They +were found in not only one but two of the three reference studies (item 7, 9, 11 and 16). +Moreover, we added four new items to SUS (item 2, 12, 19 and 21). The newly added items +are 1) the awareness of the sustainability strategy of the university; 2) green location; 3) the +inclusion of sustainability information into normal courses and 4) the integration of +sustainability research results into the curricula. The sustainable university scale makes it +possible that university decision-makers could gain deep insight into how students perceive +their efforts towards sustainability. +Eight of 21 items were used without any modifications in its original form (referred as +‘original’), nine items were modified to be unambiguous (referred to as ‘revised’), and the +four new items that we added are labelled as ‘New’ (Table no 1.). +Table no. 1: The items of the sustainable university scale (SUS) + +Sustainable university scale items +S* +Type +1 +The university has a sustainability strategy +2 +R +2 +All the students, researchers, academic and non-academic staff are +aware of the sustainability strategy of the university +4 +N +3 +Regular sustainability audits are performed on campus +2 +O +4 +Sustainability information is readily available on the university's +website, newsletter, Neptun messages, etc. +1 +R + +Sustainable University +AE + +Vol. 22 • No. 54 • May 2020 +503 + +Sustainable university scale items +S* +Type +5 +The university distinguishes itself as sustainable/green from other +higher education institutions. +1 +R +6 +The university established environmentally and socially responsible +purchasing practices +2 +O +7 +Separate waste collection is possible on campus, and the university +encourages everyone to do so. +1, 2 +R +8 +The university uses renewable energy sources (e.g. solar panels). +2 +O +9 +The university saves water and energy (e.g. LED lighting) +1, 3 +R +10 +The university encourages use of public transport, bikes. +1 +O +11 +The university buildings are designed / converted in an energy +efficient and sustainable way (e.g. windows, doors, insulation) +2, 3 +R +12 +The university buildings are located in a natural setting (quiet, +green area with many trees where the air quality is excellent) +4 +N +13 +The university engages in community outreach programs that +benefit the local environment. +2 +O +14 +The university has created partnerships with government, non- +governmental organizations, and industry working toward +sustainability. +2 +O +15 +The university has active environmental student organization(s) +1 +O +16 +There are many green actions, projects running / available at the +university to support the achievement of sustainability goals +2, 3 +R +17 +The university offers a lot of study programmes related to +sustainability. +1 +R +18 +The university offers a lot of subjects/courses related to +sustainability. +1 +R +19 +There is also a lot of information about sustainability in normal +courses +4 +N +20 +The university promotes sustainability research +1 +O +21 +Sustainability research results are integrated into the curricula +4 +N +Notes: S* (Source) = 1: Dagiliute, Liobikiene and Minelgaite (2018); 2: Nejati and Nejati (2013); 3: +Li, Gu and Liu (2018); 4: New variables added by the authors. +Type = N: new, O: original, R: revised. +An online survey, designed in Google Form, was conducted to collect data in October and +November 2019. Current student status (ongoing studies) was the one and only eligibility +criterion for students to participate in the study. Convenience sampling method was used. +Students of nine Hungarian universities located in different regions of Hungary were asked +to fill in the online questionnaire. The internal messaging systems of the universities were +used to reach their students. Due to the low response rate, the sample size is 297. +SPSS 24 was used for data analysis (Babbie, Wagner and Zaino, 2019), and MS Excel for +data visualisation (Walkenbach, 2016). Means were calculated to quantify the importance +(R1) and performance (R2) of each item of the sustainable university scale. Importance- +performance matrix was drawn to illustrate the position of SUS items to answer R3 (Kim et +al., 2018.). To answer R4, Principle Component Analysis was run to understand patterns in + +AE +Students’ Perceptions of Sustainable Universities in Hungary: An Importance- +Performance Analysis + +504 +Amfiteatru Economic +SUS items (Jolliffe, 2011). The reliability of the sustainable university scale (SUS) was +measured by Cronbach's alpha to answer R5 (DeVellis, 2017). Frequency tables of +demographic variables were also calculated (Babbie, Wagner and Zaino, 2019). +2.2. The sample +Of the sample of 297 respondents, 61.3% was female, 38.7% male (Table no. 2). Mostly +undergraduate students (77.1%) participated in this study, however some graduate students +(16.8%) and doctoral students (6.1%) contributed to the survey. The majority of the +respondents (54.2%) fell into the category ‘aged 18-24’. Most of the students in the sample +study social sciences (51.1%), engineering (23.9%) or humanities (13.87%). A significant +part of them study in Miskolc (75.4%), the rest (24.6%) in other Hungarian universities. +Therefore, this convenience sample is not representative, which is a limitation of this study. +Table no. 2: Distribution of demographic variables (N=297) +Demographic variables +Values +Frequency +Percent +Gender +male +115 +38.7 + +female +182 +61.3 +Study level +bachelor +229 +77.1 + +master +50 +16.8 + +PhD +18 +6.1 +Age +18-24 +161 +54.2 + +25-31 +60 +20.2 + +32-38 +29 +9.8 + +39-45 +27 +9.1 + +46- +20 +6.7 +Branch of science +agricultural sciences +2 +0.7 + +arts +4 +1.3 + +engineering +71 +23.9 + +humanities +41 +13.8 + +medicine +19 +6.4 + +natural sciences +7 +2.4 + +social sciences +153 +51.5 +University +Corvinus University +2 +0.7 + +Eszterhazy Uni. Eger +14 +4.7 + +METU Budapest +3 +1.0 + +National Uni. of Public Service +2 +0.7 + +Szechenyi Uni. Gyor +7 +2.4 + +University of Miskolc +224 +75.4 + +University of Pannonia +27 +9.1 + +University of Pecs +15 +5.1 + +University of Szeged +3 +1.0 + + +Sustainable University +AE + +Vol. 22 • No. 54 • May 2020 +505 + +3. Results and discussion +This chapter is divided into five main sections, and each of them discusses the results +related to one of the research questions. +3.1. Importance of the sustainable university scale items (student expectations) +In order to answer the first research question (R1), and to investigate student expectations +about university sustainability in Hungary, the items of SUS were measured on a five-point +importance scale. The lowest value (1) means the item is not at all important, whereas the +highest value (5) indicates the item is very important. The importance of SUS items refers +to the students’ expectations regarding university sustainability. It expresses their opinion +on what a university should do in order to be sustainable. +It was found that the opportunity for separate waste collection on campus and +encouragement of this activity by the university is the most important attribute of university +sustainability (4.54), whereas regular sustainability audits performed on campus is the least +important for university students (3.51). They consider water and energy savings (e.g. the +use of LEDs) as well as sustainable university buildings that are designed or converted in +an energy efficient and sustainable way (e.g. windows, doors, insulation) extremely +important (4.43). +If a university intends to be more sustainable, it must make efforts to provide the necessary +infrastructure for sperate waste collection and promote this activity. The sustainable +university should save water and energy and invest in sustainable, energy efficient +buildings on campus. These findings are not fully consistent with those of Dagiliute, +Liobikiene and Minelgaite (2018), who found recycling is less important for students. +It is also crucial for the students that university buildings must be located in a natural +setting, e.g. in a quiet, green area with many trees where the air quality is excellent (4.39). +Students, therefore, expect sustainable universities not only to be green, but to be located in +a green environment. For the most important stakeholders, it is also essential that the +sustainable university should use renewable energy sources, e.g. solar panels (4.35), it has a +sustainability strategy (4.1) and promotes sustainability research (4.07). +It was also found that students think it important that the sustainable university carries out +environmentally and socially responsible purchasing practices (4.0) and encourages the use +of public transport, bikes (4.0). In a sustainable university, it is important that all the +students, researchers, academic and non-academic staff should be aware of the +sustainability strategy of the university (3.95) and sustainability information should be +readily available on the university’s website, newsletters, etc. (3.94), as well as the +university should create partnerships with government, non-governmental organizations, +and industry working toward sustainability (3.94). Green actions and projects (3.9) and +community outreach programs (3.89) were found to be even less important. +The existence of environmental student organization(s) (3.76), the integration of +sustainability research results into the curricula (3.72) as well as sustainability-focused +positioning, when the university distinguishes itself as sustainable/green from other higher +education institutions (3.71) are even less central for the students. There is only moderate +demand for subjects/courses about sustainability (3.71) Students do not require that a lot of + +AE +Students’ Perceptions of Sustainable Universities in Hungary: An Importance- +Performance Analysis + +506 +Amfiteatru Economic +information about sustainability should be integrated into normal courses (3.61) or the +university should offer a lot of study programmes related to sustainability issues (3.6). +These results match those observed in earlier studies (Dagiliute, Liobikiene and Minelgaite, +2018). The overall mean of the importance items is 3.98. (Table no. 3.) +3.2. Perceived performance of the sustainable university items (student satisfaction) +In order to answer the second research question (R2), and to find out to what extent +students are satisfied with the performance of the Hungarian universities towards +sustainability, students were asked to rate the performance of the universities on a five- +point rating scale. The lowest score (1) indicates very poor rating (dissatisfaction), whereas +the highest score (5) means excellent rating (very high satisfaction). The rating scores of the +sustainable university scale items refer to how students are satisfied with the sustainability +performance of the university where they study. It expresses their opinion on how +sustainable the university is perceived regarding each attribute (item) of the sustainable +university scale. It allows decision-makers to get more insight into how their efforts +towards sustainability are seen by their students, their most important stakeholders. +As far as the perceived sustainability performance of the Hungarian universities is +concerned, their overall performance rating is only 3.23, which means that that students are +not satisfied with it and consider Hungarian universities unsustainable (Table no. 3). These +results provide support for the first hypothesis (H1), therefore it has been accepted. +Students are most satisfied with the location of the university buildings, the rating of which +is very good (4.17). It suggests that Hungarian universities have preferred locations that are +mostly situated in quiet, green areas with many trees where the air quality is excellent. This +could be a strength they capitalise on. Student are also satisfied with the separate waste +collection opportunities on campus (3.7), community outreach programs benefiting the +local environment (3.5) and the promotion of sustainability research (3.47). Students are +somewhat satisfied with the efforts made towards sustainability strategy (3.35), +partnerships with government, non-governmental organizations, and industry working +towards sustainability (3.34), as well as sustainable university buildings (3.33), water and +energy savings in the university (3.29) and the use of public transport and bikes (3.27). +However, students are not really satisfied with how much information about sustainability +is integrated into normal courses (3.14) and the mostly unsustainable purchasing practices +of universities (3.13). They are not convinced by the green actions/projects (3.12) and the +integration of sustainability research results into the curricula (3.11). +Moreover, students think only limited information on sustainability is available for them on +the website or in the newsletters of the universities (3.08). This is a serious problem as the +lack of information is usually one of the greatest barriers towards sustainability (Avila et al, +2017). Also, students think that they and other important stakeholders (researchers, +academic and non-academic staff) are not aware of the sustainability strategy of the +university (3.06), however their participation would be essential in the implementation. +Students do not think that universities position themselves as sustainable/green (3.03) or +use solar panels or other renewable energy sources (3.02). They are not content with the +number of subjects/courses about sustainability (3.0), green/environmental student +organizations (2.99) and the number of study programmes related to sustainability (2.93). +Students were found to be the least satisfied with the sustainability audits on campus (2.82). + +Sustainable University +AE + +Vol. 22 • No. 54 • May 2020 +507 +3.3. Importance-performance analysis (IPA) +In this section, in line with research question 3 (R3), it is discussed whether combining the +sustainable university scale (SUS) with importance-performance analysis (IPA) could be a +useful strategic tool for university decision-makers to identify key areas of university +sustainability. In order to determine the position of each item of the sustainable university +scale in the quadrants of the importance-performance matrix, deviations of the means from +the overall mean of importance (Δ IMP) and performance (Δ PER) were calculated. Table +no. 3 shows the results and the position of each item in the quadrants of IPA. +Seven attributes of the sustainable university scale including location, separate waste +collection, strategy, energy and water savings, public transport, research and sustainable +buildings fall into the ‘Keep up the good work” quadrant (Q1), which contains the +competitive advantages (strengths) of the Hungarian universities with regard to +sustainability. It is suggested that universities should use all of them in communication +campaigns targeted at students who are concerned about sustainability. +Table no. 3: Importance and performance of the sustainable university scale items + + +IMP +means +PER +means +Δ IMP +Δ PER +Quad- +rant +1 +Sustainability strategy +4.10 +3.35 +0.12 +0.12 +Q1 +2 +Awareness of the sust. strategy +3.95 +3.06 +-0.03 +-0.17 +Q3 +3 +Sustainability audits +3.51 +2.82 +-0.47 +-0.41 +Q3 +4 +Sustainability information +3.94 +3.08 +-0.04 +-0.15 +Q3 +5 +Green positioning +3.71 +3.03 +-0.27 +-0.20 +Q3 +6 +Green purchasing +4.00 +3.13 +0.02 +-0.10 +Q4 +7 +Separate waste collection +4.54 +3.70 +0.56 +0.47 +Q1 +8 +Renewable energy sources +4.35 +3.02 +0.37 +-0.21 +Q4 +9 +Water and energy savings +4.43 +3.29 +0.45 +0.06 +Q1 +10 +Public transport, bikes +4.00 +3.27 +0.02 +0.04 +Q1 +11 +Sustainable buildings +4.43 +3.33 +0.45 +0.10 +Q1 +12 +Green location +4.39 +4.17 +0.41 +0.94 +Q1 +13 +Community outreach programs +3.89 +3.50 +-0.09 +0.27 +Q2 +14 +Sustainability partnerships +3.94 +3.34 +-0.04 +0.11 +Q2 +15 +Green student organization(s) +3.76 +2.99 +-0.22 +-0.24 +Q3 +16 +Green actions, projects +3.90 +3.12 +-0.08 +-0.11 +Q3 +17 +Green study programmes +3.60 +2.93 +-0.38 +-0.30 +Q3 +18 +Green subjects/courses +3.71 +3.00 +-0.27 +-0.23 +Q3 +19 +Greening normal courses +3.61 +3.14 +-0.37 +-0.09 +Q3 +20 +Sustainability research +4.07 +3.47 +0.09 +0.24 +Q1 +21 +Sustainability research integration +3.72 +3.11 +-0.26 +-0.12 +Q3 + +Total +3.98 +3.23 + + + + Notes: IMP: importance; PER: performance; Quadrants: (1) Keep up the good work (2) Possible +overkill (3) Low priority (4) Concentrate here + + +AE +Students’ Perceptions of Sustainable Universities in Hungary: An Importance- +Performance Analysis + +508 +Amfiteatru Economic +Campus location is found to be the biggest strength. The favourable location is very +important for the students. They require that university buildings should be situated in a +quiet, green environment, and for most of them, this expectation is fully met. Separate +waste collection, which is the most important aspect of the sustainable university from the +students’ perspective, is also a major strength as students are quite satisfied with it. +Hungarian universities must communicate that they provide the infrastructure for separate +waste collection and promote this activity. +Based on our findings, it is advisable for universities to emphasize that their students are +satisfied with their efforts towards energy and water savings and appreciate their +endeavours to increase energy efficiency on campus. Also, students are content with how +sustainable the design of the university buildings is. It can also be suggested that Hungarian +higher education institutions should communicate that they promote sustainability research, +encourages the use of public transport, bikes and they have a written sustainability strategy. +Two items can be found in Q2, which is the possible overkill quadrant. It contains items +that are not important for the students, however they, the most important stakeholders of the +universities are satisfied with it (performance ratings are better than the overall average). +The performance of universities concerning community outreach programs, partnership +with governmental, non-governmental organizations, and industry is better than required. In +this case, it is suggested that universities should make a communication campaign to +increase the importance of their community outreach programs and sustainability +partnership to turn those activities into competitive advantages. +Ten items – nearly the half of the sustainable university scale items – can be found in Q3, +which represents “Low priority” attributes having low importance and low perceived +performance. The items that fall into this quadrant are respectively: 1) awareness of the +sustainability strategy; 2) regular sustainability audits; 3) information regarding +sustainability (website, newsletters, etc.); 4) sustainability-focused positioning of the +universities; 5) active green student organizations; 6) sustainability-related projects/actions; +7) study programs related to sustainability; 8) subjects/courses related to sustainability; 9) +integration of sustainability into normal/traditional courses; and finally, 10) integration of +sustainability research results into the curricula. Hungarian universities are strongly advised +to avoid any investments in those activities. +Last but not at least, two sustainable university items can be found in Q4. This is the +“Concentrate here” quadrant representing attributes that universities should immediately +improve to achieve higher student satisfaction with regard to their attempts to be more +sustainable. The items listed here have high importance and low perceived performance +suggesting that students are really dissatisfied with them in spite of the fact that those items +are really important for them. On the one hand, they do not believe that universities have +environmentally and socially responsible purchasing practices, on the other hand they are +disappointed with the use renewable energy sources (e.g. solar panels) on campus. It is +therefore suggested that universities should concentrate more on green/socially responsible +procurement and should increase the use of renewable energy sources to make students +who are concerned about sustainability more satisfied. Universities should consider more +the sustainability performance of their suppliers. They should be greening their tenders, +prefer local suppliers, and install more solar panels, etc. (Figure no. 2). + + +Sustainable University +AE + +Vol. 22 • No. 54 • May 2020 +509 + +Figure no. 2: Importance-performance of the sustainable university scale items +As no research has been found that surveyed the perceived importance and performance of +the attributes of university sustainability, it is therefore not possible to compare the results +discussed here to the findings of previous works. However, this study fills this gap in the +literature and propose a new methodology to investigate the attributes of university +sustainability. As an answer to R3, it can be concluded that importance performance +analysis (IPA) is a strong strategic tool for university decision-makers to identify key areas +of university sustainability when combined with the sustainable university scale (SUS). +Using the results of IPA, universities could implement corrective actions to make students +as stakeholders more satisfied with their efforts to be more sustainable. +3.4. Factor analysis of the sustainable university scale items +In order to investigate patterns in perceived university sustainability, and answer R4, factor +analysis was used. The dataset of the importance of SUS items were analysed as it refers to +the students’ expectation. The very high Kaiser-Meyer-Olkin Measure of Sampling +Adequacy value (KMO=0.938) indicates that a factor analysis is a useful method with our +data. The Bartlett's Test of Sphericity (Approx. Chi-Square = 4400.484; df = 210; Sig.= +0.000) also reconfirms it (Jolliffe, 2011). +The extraction communalities are acceptable, although the lower values of Green Location +and Green Positioning show that they don't fit as well as the others. Only three factors in +the initial solution have eigenvalues greater than 1. Together, they account for almost 65% +of the variability in the original variables (Table no. 4). This suggests that three latent +influences are associated with sustainable university perceptions, but there remains room +for a lot of unexplained variation (Babbie, Wagner and Zaino, 2019). The scree plot also +confirmed the choice of three components. +Table no. 4: Total variance explained + +greenlocation +4.1 +separatewaste +collection +3.6 +Performance +communityoutreach +programs +sustainabilityresearch +sustainabilitypartnerships +sustainablebuildings +sustainabilitystrategy +publictransport,bikes +water andenergy savings +greeningnormal courses +greenactions,projects +sustainability +green purchasing +3.1 +researchintegration +sustainabilityinformation +green subjects/courses +green positioning +awarenessofthesust.strategy +renewableenergysources +green studentorganization(s) +greenstudyprogranmimes +sustainabilityaudits +2.6 +3.4 +3.6 +3.8 +4 +4.2 +4.4 +4.6 +ImportanceAE +Students’ Perceptions of Sustainable Universities in Hungary: An Importance- +Performance Analysis + +510 +Amfiteatru Economic +Compo- +nent +Initial Eigenvalues +Extraction Sums of +Squared Loadings +Rotation Sums of Squared +Loadings +Total +% of +Variance +Cumulative +% +Total +% of +Variance +Cumulative +% +Total % of +Variance +Cumulative +% +1 +10.648 50.706 +50.706 +10.648 50.706 +50.706 +5.782 27.535 +27.535 +2 +1.650 7.856 +58.563 +1.650 7.856 +58.563 +3.937 18.746 +46.281 +3 +1.333 6.346 +64.909 +1.333 6.346 +64.909 +3.912 18.628 +64.909 +4 +.916 4.360 +69.269 + + + + + + +21 +.139 .663 +100.000 + + + + + + +Notes: Extraction Method: Principal Component Analysis. +To rotate the factor components, Varimax rotation with Kaiser normalization was used. The +first rotated factor component is most highly correlated with community outreach +programs, sustainability partnerships, green study programmes, green subjects/courses, +greening normal courses, sustainability research and sustainability research integration +items (Table no. 5). These variables are not particularly correlated with the other two factor +components, and each of them refers to actions towards meeting sustainability objectives, +or related to education or research, it is therefore the first component called as Sustainable +Actions, Education & Research (SAER). +Table no. 5: Rotated component matrix +Items +1. Sust. Actions, +Education & +Research +2. Sust. +Operation/ +Infrastructure +3. Sust. +Strategy +Type +Sustainability strategy +0.20 +0.23 +0.76 +ST1 +Awareness of the sust. strategy +0.24 +0.24 +0.80 +ST2 +Sustainability audits +0.39 +0.06 +0.73 +ST3 +Sustainability information +0.25 +0.32 +0.74 +ST4 +Green positioning +0.31 +0.29 +0.48 +ST5 +Green purchasing +0.51 +0.37 +0.44 +PU1 +Separate waste collection +0.15 +0.78 +0.24 +WE1 +Renewable energy sources +0.26 +0.75 +0.30 +WE2 +Water and energy savings +0.13 +0.82 +0.31 +WE3 +Public transport, bikes +0.50 +0.56 +-0.01 +WE4 +Sustainable buildings +0.28 +0.73 +0.26 +WE5 +Green location +0.39 +0.49 +0.08 +LO1 +Community outreach programs +0.60 +0.31 +0.33 +SA1 +Sustainability partnerships +0.68 +0.25 +0.33 +SA2 +Green student organization(s) +0.57 +0.33 +0.41 +SA3 +Green actions, projects +0.63 +0.28 +0.45 +SA4 +Green study programmes +0.83 +0.16 +0.19 +SER1 +Green subjects/courses +0.82 +0.17 +0.21 +SER2 +Greening normal courses +0.71 +0.16 +0.27 +SER3 +Sustainability research +0.71 +0.34 +0.30 +SER4 +Sustainability research +0.78 +0.22 +0.26 +SER5 + +Sustainable University +AE + +Vol. 22 • No. 54 • May 2020 +511 +Items +1. Sust. Actions, +Education & +Research +2. Sust. +Operation/ +Infrastructure +3. Sust. +Strategy +Type +integration +Notes: (1) Extraction Method: Principal Component Analysis. Rotation Method: Varimax with Kaiser +Normalization. Rotation converged in 6 iterations. (2) ST: Strategy, commitment & monitoring; PU: +Purchasing; WE: Waste & energy; LO: Location; SA: Sustainability actions; SER: Sustainable +education & research +The second factor component, which is called Sustainable Operation/Infrastructure, are +made up of separate waste collection, renewable energy sources, water and energy savings +and sustainable buildings. All those items are related to the domain of waste and energy. +The third component, Sustainable Strategy, has been named after the items that correlated +with it the most. All of them are related to the sustainability strategy including the written +sustainability strategy, and its awareness, regular sustainability audits and sustainability +information. Because of their moderately large correlations with the first and the third +components, green student organizations and green action/projects bridges Sustainable +Actions, Education & Research and Sustainable Strategy. Public transport, bikes and green +location variables bridge the first and the second components, whereas green positioning +and green purchasing are highly correlated with all the three factor components. +These results suggest that students form expectations about the three main domains of +university sustainability: 1) sustainable strategy, 2) sustainable operations/infrastructure, +and 3) sustainable actions/education/research. These are the main topics of the university +sustainability in the mind of the most important stakeholder. These findings are not in line +with those of previous studies (Nejati and Nejati, 2013; Dagiliute, Liobikiene and +Minelgaite, 2018). In both of the earlier studies, the number of factor components was +higher, and the structure of the component was different from our results. +It is proposed that universities should deal with all the three components separately, and it +would be beneficial for them to assign managers in charge to each domain to fully meet +student expectations. +3.5. Reliability of the sustainable university scale +In order to test H2 and to investigate whether all the 21 items of the sustainable university +scale reliably measure the same latent variable, a Cronbach's alpha was run on both SUS +importance and SUS performance datasets. +In the reliability statistics table of SUS importance, Cronbach's alpha was 0.95, which +indicates a very high level of internal consistency for our scale with this specific sample +(DeVellis, 2017.). The "Cronbach's Alpha “If Item Deleted" column showed that removal +of any item would result in a lower Cronbach's alpha, so no items were removed from the +21 item-scale. In the reliability statistics table of SUS performance dataset, Cronbach's +alpha was even higher (0.985), which indicates an even higher level of internal consistency. +Here also no items were removed as the “If Item Deleted" column showed that removal of +any item would result in a lower Cronbach's alpha. +Also, a reliability analysis was run in order to ensure internal consistency of the identified +constructs after the principle component analysis. The high Cronbach’s alpha values +confirmed the reliability of the constructs (α Sustainable Strategy = 0.850, no. of items = 5; + +AE +Students’ Perceptions of Sustainable Universities in Hungary: An Importance- +Performance Analysis + +512 +Amfiteatru Economic +α Sustainable Operation & Infrastructure = 0.861, no. of items = 6; and α Sustainable +Actions, Education & Research= 0.938, no. of items =10). +All these findings support H2. It is therefore accepted that the sustainable university scale +(SUS) is a reliable construct to measure perceived university sustainability. + +Conclusions +The stakeholder theory suggests that organizations should fully meet stakeholders’ +expectations to be successful (Freeman, 2010). Students are one of the biggest and most +important stakeholders of universities (Degtjarjova, Lapina and Freidenfelds, 2018), and +could have a significant impact on the environment (Emanuel and Adams, 2011). +Nowadays, the public demand for more sustainable universities is growing (Md Shahbudin, +et al., 2011.). More and more students want to study about sustainability, expect the +integration of sustainability research into curricula and prefer universities that make efforts +to operate in a more sustainable manner (Dagiliute, Liobikiene and Minelgaite, 2018). +University decision-makers (Rector, Chancellor, Deans and the Senate) should consider +sustainability issues to a greater extent when developing organizational strategy. This study +extends the knowledge of the above decision-makers regarding students’ perception of +university sustainability in many aspects. +The current study found that separate waste collection on campus is the most important +student expectation about sustainability. However, it is not in line with the result of +previous studies. Dagiliute, Liobikiene and Minelgaite (2018) found recycling less +important for students. Nonetheless, our findings are consistent with those of other studies +suggesting that students expect water and energy savings and energy efficient, sustainable +university buildings in a sustainable university. Also, it is important for the students that the +buildings should be located in a green environment. Universities are therefore advised to +promote separate waste collection, save water and energy, and maintain sustainable, energy +efficient buildings that are situated in green parks (R1). +In the current study, the low value of general satisfaction with the performance of +universities towards sustainability (3.23) confirmed H1 and suggests that students are not +satisfied with it and consider Hungarian universities rather unsustainable. Students’ +perceptions of university sustainability are in line with the weak positions of the Hungarian +higher education institution in green rankings (Greenmetric, 2019). Our findings show that +students are most satisfied with the location of the university buildings, which suggests that +Hungarian universities have preferred locations. Also, students are content with the +opportunity to collect waste separately on campus, the community outreach programs that +universities offer, and the promotion of research on sustainability (R2). +The findings of this research confirmed H3. By combining the importance-performance +analysis (IPA) with the sustainable university scale (SUS), a simple but powerful strategic +managerial tool can be developed. It could be widely used by university decision-makers to +investigate the key areas of university sustainability. IPA helps to identify competitive +advantages and major weaknesses in the domains of sustainability and make it possible for +decision-makers to implement corrective actions to make students as stakeholders more +satisfied with the university's efforts to address sustainability. Two major weaknesses were +found in our study. Hungarian universities perform poorly in sustainable purchasing and + +Sustainable University +AE + +Vol. 22 • No. 54 • May 2020 +513 +use less renewable energy (e.g. solar panels) on campus than it is expected by their +students. It is therefore suggested that universities should immediately make both their +energy use and purchasing process more sustainable. On the other hand, it was also found +that campus location and separate waste collection are the major competitive advantages. It +is suggested that the major strengths are used in the marketing campaigns of universities to +make their green positioning more effective and to build the sustainable university brand +image. Strategy, energy and water savings, public transport, sustainable buildings and +research are also strengths of the Hungarian universities that should be communicated (R3). +The three main domains of university sustainability were also identified. These are the +strategy towards sustainability, actions to promote sustainability including education and +research, and the sustainable infrastructure/operations. This is a unique structure and +different from those presented in earlier studies (Nejati and Nejati, 2013; Dagiliute, +Liobikiene and Minelgaite, 2018), which suggests that Hungarian universities should use a +nation-specific approach to university sustainability. Future studies on this topic are +therefore recommended to investigate it in different cultural and national contexts (R4). +The sustainable university scale (SUS) was found to be a reliable construct to measure +perceived university sustainability (H2 accepted). The adaptation of this construct is +therefore proposed to both researchers and university decision-makers to investigate how +students do perceive the efforts that universities make towards sustainability. Combined +with IPA, it could be a powerful benchmarking tool, which is an important practical +implication (R5). +Further research should be done to compare the perceived university sustainability of green +and non-green universities in different cultural settings. + +References +Adams, R., Martin, S. and Boom, K., 2018. University culture and sustainability: Designing +and implementing an enabling framework. Journal of Cleaner Production, 171, pp.434- +445. +Avila, L.V., Filho, W.L., Brandli, L., Macgregor, C.J., Molthan-Hill, P., Özuyar, P.G. and +Moreira, R.M., 2017. Barriers to innovation and sustainability at universities around the +world. Journal of Cleaner Production, 164, pp.1268-1278. +Babbie, E.R., Wagner, W.E. and Zaino, J., 2019. Adventures in social research: data +analysis using IBM® SPSS® statistics. Los Angeles, CA: Sage. +Carroll, A.B., and Buchholtz, A.K., 2014. Business and society: ethics, sustainability and +stakeholder management. Cengage Learning. +Chapleo, C. and Sims, C., 2017. Stakeholder analysis in higher education: a case study of +the University of Portsmouth. Perspectives: Policy and Practice in Higher Education +14(1), pp. 12-20. +Dagiliūtė, R., Liobikienė, G. and Minelgaitė, A., 2018. Sustainability at universities: +Students’ perceptions from Green and Non-Green universities. Journal of Cleaner +Production, 181, pp.473-482. + +AE +Students’ Perceptions of Sustainable Universities in Hungary: An Importance- +Performance Analysis + +514 +Amfiteatru Economic +Degtjarjova, I., Lapina, I., and Freidenfelds, D., 2018. Student as stakeholder: "voice of +customer" in higher education quality development. Marketing and Management of +Innovations, 2, pp. 388-398. +DeVellis, R.F., 2017. Scale Development: theory and applications. Los Angeles: Sage. +Emanuel, R. and Adams, J., 2011. 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London: Transnational Press, pp. 179-183. +Pawłowski, K., 2009. The ‘Fourth Generation University’ as a Creator of the Local and +Regional Development. Higher Education in Europe, 34(1), pp.51-64. +Polónyi, I. and Kozma, T., 2019. Az egyetemfejlesztés alternatívái. (Alternatives to +university development). Magyar Tudomány, 180(9), pp. 1326-1336. +Ragazzi, M. and Ghidini, F., 2017. Environmental sustainability of universities: critical +analysis of a green ranking. Energy Procedia, 119, pp.111-120. +Shuqin, C., Minyan, L., Hongwei, T., Xiaoyu, L. and Jian, G., 2019. Assessing +sustainability on Chinese university campuses: Development of a campus sustainability +evaluation system and its application with a case study. Journal of Building +Engineering, 24, p.100747. +Soini, K., Jurgilevich, A., Pietikäinen, J. and Korhonen-Kurki, K., 2018. Universities +responding to the call for sustainability: A typology of sustainability centres. Journal of +Cleaner Production, 170, pp.1423-1432. +Sterling, S., 2014. The sustainable university: challenge and response. The sustainable +university: progress and prospects. London: Routledge. +United Nations, 2019. Sustainable Development Goals. [online] Available at: + [Accessed 28 October 2019]. +Yuvinatileng, M., Utomo, W.H. and Latuperissa, R., 2013. Analysis of Service Quality +using Servqual Method and Importance Performance Analysis (IPA) in Population +Department, Tomohon City. International Journal of Computer Applications, 70(19), +pp.23-30. +Wakkee, I., Sijde, P.V.D., Vaupell, C. and Ghuman, K., 2019. The universities role in +sustainable +development: +Activating +entrepreneurial +scholars +as +agents +of +change. Technological Forecasting and Social Change, 141, pp.195-205. +Walkenbach, J., 2016. Microsoft® Excel® 2016 bible. Indianapolis: Wiley. +Wright, T., 2010. University presidents’ conceptualizations of sustainability in higher +education. International Journal of Sustainability in Higher Education, 11(1), pp.61-73. +Wyród-Wróbel, J. and Biesok, G., 2017. Decision making on various approaches to +Importance-Performance Analysis (IPA). European Journal of Business Science and +Technology, 3(2), pp. 123-131. + diff --git a/IdAzT4oBgHgl3EQfU_z6/content/tmp_files/load_file.txt b/IdAzT4oBgHgl3EQfU_z6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f99d0c2b1b872cb563729780cf0d0bd2c4501ef9 --- /dev/null +++ b/IdAzT4oBgHgl3EQfU_z6/content/tmp_files/load_file.txt @@ -0,0 +1,1091 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf,len=1090 +page_content='AE Students’ Perceptions of Sustainable Universities in Hungary: An Importance- Performance Analysis 496 Amfiteatru Economic STUDENTS’ PERCEPTIONS OF SUSTAINABLE UNIVERSITIES IN HUNGARY: AN IMPORTANCE-PERFORMANCE ANALYSIS Szabolcs Nagy1* and Mariann Veresné Somosi2 1)2) University of Miskolc, Miskolc, Hungary Please cite this article as: Nagy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' and Somosi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Students’ Perceptions of Sustainable Universities in Hungary: An Importance-Performance Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Amfiteatru Economic, 22(54), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 496-515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='24818/EA/2020/54/496 Article History Received: 29 December 2019 Revised: 3 February 2020 Accepted: 30 March 2020 Abstract In order to succeed, universities are forced to respond to the new challenges in the rapidly changing world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The recently emerging fourth-generation universities should meet sustainability objectives to better serve their students and their communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It is essential for universities to measure their sustainability performance to capitalise on their core strengths and to overcome their weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' In line with the stakeholder theory, the objective of this study was to investigate students’ perceptions of university sustainability including their expectations about and satisfaction with the efforts that universities make towards sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' This paper proposes a new approach that combines the sustainable university scale, developed by the authors, with the importance-performance analysis to identify key areas of university sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' To collect data, an online survey was conducted in Hungary in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The sustainable university scale was found to be a reliable construct to measure different aspects of university sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Results of the importance-performance analysis suggest that students consider Hungarian universities unsustainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Research findings indicate that Hungarian universities perform poorly in sustainable purchasing and renewable energy use, but their location and their efforts towards separate waste collection are their major competitive advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The main domains of university sustainability were also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' This study provides university decision-makers and researchers with insightful results supporting the transformation of traditional universities into sustainable, fourth-generation higher education institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Keywords: sustainable university, students’ perception, importance-performance analysis, Hungary, student satisfaction, student expectation JEL Classification: I23, Q56 Corresponding author, Szabolcs Nagy – nagy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='szabolcs@uni-miskolc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='hu Sustainable University AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 22 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 54 • May 2020 497 Introduction We live in the age of rapid changes to which higher education institutions should adopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' A university that wants to succeed needs to respond to the challenges of the new era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' One of them is the urgency to meet sustainability objectives (Filho, Manolas and Pace, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Soini, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Olalla and Merino, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Universities are undergoing a rapid transformation as they are not only traditionally engaged in education but are also playing an increasingly important role in the society (Papp-Váry and Lukács, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Nowadays, the emergence of the so-called Fourth Generation universities, which actively shape their socio-economic environment, can be seen (Pawłowski, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Lukovics and Zuti, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The topic of sustainable development is increasingly present among the major concerns of the international academic community (Grecu and Ipiña, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Universities must take steps to achieve the United Nations Sustainable Development Goals (Paletta, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Target 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='7 declares that students have the right to acquire the knowledge and skills needed to promote sustainable development (UN, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Globally, the proliferation of the efforts to assess universities’ responses to the challenges of sustainability can be seen (Li, Gu and Liu, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Adams, Martin, and Boom (2018) draw the attention to the importance of the university sustainability culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Adaptation of the stakeholder theory is essential for higher education institutions (Mainardes, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=', 2010) as stakeholders can create opportunities for or pose threats to an organisation (Chapleo and Sims, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Students as stakeholders have a serious impact on the future development of universities (Degtjarjova, Lapina and Freidenfelds, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Commitment to sustainability of leaders and important stakeholders play a key role in the effectiveness of sustainable development initiatives in higher education institutions (Wright, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=') The position of Hungarian higher education institutes in the world rankings is not very favourable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The best Hungarian university can be found around the 500th place in global rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' There are only seven or eight Hungarian institutions that are ranked at all (Polónyi and Kozma, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The weak performance of the Hungarian higher education institutions in sustainability rankings explains the need for a comprehensive analysis of university sustainability in Hungary from the students as stakeholders’ perspective, which is one of the main objectives of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Students as stakeholders form expectations regarding university sustainability not only generally, but also very specifically, and how those expectations are met determines the level of their satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' This study aims to investigate student expectations about and satisfaction with the attributes of the sustainable university by using the sustainable university scale (SUS) combined with the importance-performance analysis (IPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' SUS, the items of which are the determinants of university sustainability, was developed by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' IPA has been widely used to examine the relationship between importance, performance, and satisfaction in many areas (Yuvinatileng, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Wyród-Wróbel and Biesok, 2017, Kim, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=', 2018) However, no previous study has investigated it in the context of university sustainability in spite of the fact that universities should use managerial tools to develop their sustainability strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' This study seeks to address this research gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' AE Students’ Perceptions of Sustainable Universities in Hungary: An Importance- Performance Analysis 498 Amfiteatru Economic 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Literature review 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Perceptions of the sustainable university In the UI GreenMetric World University Ranking 2019, which provides information about the current conditions and policies related to Green Campus and Sustainability, only seven Hungarian universities can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The University of Szeged is in the best position, ranked first in Hungary, and 74th in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It is followed by the University of Pecs, ranked 100th globally and the University of Debrecen, in the 202nd position in the world ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The University of Miskolc, for which the authors work, can be found only in the 605th place in this ranking of 780 universities globally (Greenmetric, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Students’ perceptions of university sustainability were assumed to be in line with this poor ranking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It is therefore hypothesized that students are not satisfied with the sustainability performance of the Hungarian higher education institutions (H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Mention must be made of some of the shortcomings of the GreenMetric Ranking, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' non- compliance with the Berlin Principles (Ragazzi and Ghidini, 2017), however, it is still one of the best tools to quantify university sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The perceptions of university students towards factors of a sustainable university was first discussed by Nejati and Nejati (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The authors developed a reliable scale to assess the university practices towards sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' They identified four main dimensions of the sustainable university scale, which are respectively: 1) community outreach, 2) sustainability commitment and monitoring, 3) waste and energy, and 4) land use and planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Their initial scale contained 28 items, which they reduced to a 12-item scale, which could be a key instrument for university decision-makers and stakeholders to measure the university’s performance regarding the implementation of the transition strategy towards sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Their construct measuring sustainability practices of universities contains 1) community outreach programs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 2) green community centres;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 3) partnerships with government, non-governmental organizations, and industry working toward sustainability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 4-5) written commitment to sustainability (university and department level);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 6-7) sustainability audits on the surrounding community and on campus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 8) reuse of campus waste;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 9) use of renewable and safe energy sources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 10) sustainable support services (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' recycling bins on campus, efficient public transport throughout the university);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 11) sustainable campus building planning and 12) sustainable campus land-use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Dagiliute, Liobikiene and Minelgaite (2018) were the first to investigate the differences in the perceived sustainability performance between the ‘green’ and the ‘non-green’ universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=" They compared the students' attitudes towards sustainability in two Lithuanian universities." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' They did not find any significant differences in sustainability aspects in general, however, students of the green university sought more information about sustainability and were more often involved in sustainability activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' They also found that campus sustainability and environmental information have a significant impact on students’ sustainable behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=" In their study, they used a scale to measure perceptions made up of 16 items, grouped into four main constructs: 1) ‘campus sustainability’, 2) ‘environmental information’, 3) ‘students’ sustainability involvement’, and 4) ‘university's role in sustainable development." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=" The item ‘university's self-representation as a green university’ was also involved in their construct." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Their 17-item scale involves 1) environmental student organization(s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 2) use of public transport, bikes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 3) possibility to recycle waste at the university;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=" 4) use one's own non-disposable cup;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 5) availability of strategic documents and their implementation reports;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 6) sustainability-related information during lectures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Sustainable University AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 22 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 54 • May 2020 499 7) university website on environmental objectives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 8) participation in environmental, social activities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 9) involvement in activities at the university;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 10) energy and resource saving;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 11) contribution to social well-being, tolerance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 12) environmental education;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 13) cooperation with other national and foreign universities and businesses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 14) inclusion of sustainability aspects in study programmes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 15) sustainability research;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=" 16) university's self-representation as a green university;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' and 17) declared environmental objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' They found that students considered social aspects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' social well-being, tolerance the most important attribute of the sustainable university.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' However, students considered environmental aspects, such as energy saving, environmental education, and actions less important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Li, Gu and Liu (2018) established a new scoring system for campus sustainability in Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' They suggest that sustainable campus performance indicators should be identified from the different perspectives of the economy, environment and society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' In order to identify and prioritise the key sustainability indicators for university campuses, they proposed a new approach combining the qualitative scoring method and an analytical hierarchical process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' After thorough literature review, they identified 54 indicators and quantified the weight coefficients for the criteria, sub-criteria and elements, and proposed a model that can be a flexible tool for university decision-makers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It is hypothesized that combining the most relevant items of the constructs developed by Nejati and Nejati (2013), Dagiliute, Liobikiene and Minelgaite (2018) and Li, Gu and Liu (2018), a new, reliable scale to measure perceived university sustainability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' the sustainable university scale, can be developed (H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Shuqin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' (2019) aimed to assess and compare the sustainability performance of different Chinese universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The authors developed a campus sustainability evaluation system that is made up of the five main domains of campus sustainability, which are respectively: organization and management, energy and resource saving, friendly environment, campus culture, and social outreach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Their evaluation system included 14 mandatory indicators and 69 optional indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' They found that the most problematic fields are organization management, resource saving and campus culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' For example, there are issues with green education, green research and green humanities as they are not so developed there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The assessment tool proposed by the authors can be used to guide the green campus revolution in China and could be adopted by the rest of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Wakkee, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' (2019) demonstrated how (entrepreneurial) universities can drive regional sustainable development in developing countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' They found that local campus leadership, a holistic teaching and research programme, and student involvement can have significant local effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Importance-Performance Analysis (IPA) The importance-performance analysis (IPA) was developed by Martilla and James (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The original version of IPA defines consumer satisfaction as the function of two components that are respectively: the importance of an attribute of the product/service, and the perceived performance of the company on this attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The mean of importance and performance ratings of each attribute determines its position on the importance- performance matrix or grid, which is also often called the Cartesian diagram (Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The overall mean of the performance/importance ratings is used as a delimiter of high and low performance/importance (Yuvinatileng, Utomo and Latuperissa, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' AE Students’ Perceptions of Sustainable Universities in Hungary: An Importance- Performance Analysis 500 Amfiteatru Economic The 2x2 IPA matrix can be divided into four quadrants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Each quadrant requires a different approach and strategy (Wyród-Wróbel and Biesok, 2017): Quadrant 1: Keep up the good work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' This is the best possible position for an attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' This quadrant contains the competitive advantages and major strengths of a company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The organization must defend all of them to succeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' These are high importance/high performance items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 1: The modified Importance Performance Matrix Source: Kim, Jeon, Cho and Kim, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Quadrant 2: The territory of Possible overkill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Here low importance/high performance attributes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' items of overperformance, can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Organizations should deploy business resources used here somewhere else (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' in Quadrant 1) or should increase the importance of those attributes that can be found here to turn them into competitive advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Quadrant 3: The area of Low priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Low importance/low performance attributes can be seen here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Those are minor weaknesses that require no additional resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Organizations are suggested to avoid investing in this quadrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Quadrant 4: Concentrate here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' High importance/low performance attributes can be found here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Those are the major weaknesses of an organization that require immediate corrective actions to increase consumer satisfaction and to avoid customer churn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Stakeholder theory The stakeholder theory originates from the 1980s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Freeman (1984) was the first to coin the phrase as an opposite to the shareholder theory or Friedman’s doctrine, which suggests that a company’s sole responsibility is to make money for its shareholders (Friedman, 1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' High Quadrant 2 Quadrant1 Possible overkill Keep up the good work Performance Quadrant3 Quadrant 4 Low priority Concentrate here Low Low Importance HighSustainable University AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 22 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 54 • May 2020 501 According to the stakeholder theory, shareholders are only one of many stakeholders in a company, and an organization’s key to market success is how it satisfies all the stakeholders, not only its shareholders (Freeman, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The stakeholder theory says that the stakeholder ecosystem is made up of all parties that invested and involved in, or affected by, the company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Therefore, companies must pay special attention to their employees, vendors, suppliers, owners, community/neighbours, community groups, competitors, governmental bodies, oversight organizations and the local ecology (Freeman, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The stakeholder theory is intertwined with the domains of ethics and sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Carroll and Buchholtz (2014) suggest that successful businesses in society adopt a stakeholder management approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The stakeholder theory is solid ground for corporate social responsibility and business ethics inside the company (Kakabadse, Rozules and Davies, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The stakeholder ecosystem of a university comprises current, former (alumni) and potential students, parents, municipalities, academics, faculties, management (Rector, the Senate, Chancellor), administrative staff, governmental organisations, Academy of Sciences, research partners and companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' In higher education institutions, students and employees are always the major stakeholders in terms of their number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' According to the stakeholder theory, universities are service providers to students and students are one of the most important stakeholders (Degtjarjova, Lapina, and Freidenfelds, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The more satisfied students are, the more likely it is that the university could succeed, also in the field of sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It is therefore assumed that IPA as a strategic tool should be used to maximize student satisfaction with the efforts that universities make towards sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Methodology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Methodology and research questions Based on the literature review presented above, and in line with the main objectives of the research, this study aims to address the following research questions respectively: R1: What are the student expectations about university sustainability in Hungary?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' (student expectations) R2: To what extent are students satisfied with the sustainability performance of universities?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' (student satisfaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' H1 refers to this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' R3: Is combining sustainable university scale (SUS) with importance-performance analysis (IPA) a powerful strategic tool for university decision-makers to identify key areas of university sustainability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' R4: What are the main components of the perceived university sustainability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' R5: Is sustainable university scale (SUS) a reliable construct to measure students’ perceptions of university sustainability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' H2 refers to this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' In line with the research questions, the following hypotheses were developed: H1: Students are not satisfied with the sustainability performance of the Hungarian higher education institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' AE Students’ Perceptions of Sustainable Universities in Hungary: An Importance- Performance Analysis 502 Amfiteatru Economic H2: Combining the most relevant items of the constructs developed by Nejati and Nejati (2013), Dagiliute, Liobikiene and Minelgaite (2018) and Li, Gu and Liu (2018), a new, reliable scale for measuring perceived university sustainability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' the Sustainable University Scale (SUS), can be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' To answer the research questions, and to thoroughly investigate students’ perceptions of the sustainable university, a questionnaire made up of 47 questions grouped into three sections were designed: Section 1: Importance of the sustainable university scale (SUS) items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It contains 21 statements measured on a five-point importance scale (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' not at all important … 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' very important).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Respondents were asked to answer the following question: “How important are the followings to you regarding a sustainable university?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' SUS items can be seen in Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Section 2: Perceived performance of the sustainable university scale (SUS) items: The very same 21 statements as in Section 1, measured on a five-point rating scale (1 – very poor .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 5 – excellent), answering the question “How do you rate the sustainability performance of your university?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Section 3: Demographic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It contains 5 questions including gender, age, study level, branch of sciences and the university where they study (Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The sustainable university scale (SUS), which contains 21 items, is a construct developed by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It is based on the domains of university sustainability discussed in the literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' More specifically, in our construct we combined 9 items (item 4, 5, 7, 9, 10, 15, 17, 18 and 20) from Dagiliute, Liobikiene and Minelgaite (2018) with 9 items (item 1, 3, 6, 7, 8, 11, 13, 14 and 16) used by Nejati and Nejati (2013), with 3 items (item 9, 11 and 16) from Li, Gu and Liu (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It must be noted that four items are overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' They were found in not only one but two of the three reference studies (item 7, 9, 11 and 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Moreover, we added four new items to SUS (item 2, 12, 19 and 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The newly added items are 1) the awareness of the sustainability strategy of the university;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 2) green location;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 3) the inclusion of sustainability information into normal courses and 4) the integration of sustainability research results into the curricula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The sustainable university scale makes it possible that university decision-makers could gain deep insight into how students perceive their efforts towards sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Eight of 21 items were used without any modifications in its original form (referred as ‘original’), nine items were modified to be unambiguous (referred to as ‘revised’), and the four new items that we added are labelled as ‘New’ (Table no 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=" 1: The items of the sustainable university scale (SUS) Sustainable university scale items S* Type 1 The university has a sustainability strategy 2 R 2 All the students, researchers, academic and non-academic staff are aware of the sustainability strategy of the university 4 N 3 Regular sustainability audits are performed on campus 2 O 4 Sustainability information is readily available on the university's website, newsletter, Neptun messages, etc." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 1 R Sustainable University AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 22 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 54 • May 2020 503 Sustainable university scale items S* Type 5 The university distinguishes itself as sustainable/green from other higher education institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 1 R 6 The university established environmentally and socially responsible purchasing practices 2 O 7 Separate waste collection is possible on campus, and the university encourages everyone to do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 1, 2 R 8 The university uses renewable energy sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' solar panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 2 O 9 The university saves water and energy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' LED lighting) 1, 3 R 10 The university encourages use of public transport, bikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 1 O 11 The university buildings are designed / converted in an energy efficient and sustainable way (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' windows, doors, insulation) 2, 3 R 12 The university buildings are located in a natural setting (quiet, green area with many trees where the air quality is excellent) 4 N 13 The university engages in community outreach programs that benefit the local environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 2 O 14 The university has created partnerships with government, non- governmental organizations, and industry working toward sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 2 O 15 The university has active environmental student organization(s) 1 O 16 There are many green actions, projects running / available at the university to support the achievement of sustainability goals 2, 3 R 17 The university offers a lot of study programmes related to sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 1 R 18 The university offers a lot of subjects/courses related to sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 1 R 19 There is also a lot of information about sustainability in normal courses 4 N 20 The university promotes sustainability research 1 O 21 Sustainability research results are integrated into the curricula 4 N Notes: S* (Source) = 1: Dagiliute, Liobikiene and Minelgaite (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 2: Nejati and Nejati (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 3: Li, Gu and Liu (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 4: New variables added by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Type = N: new, O: original, R: revised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' An online survey, designed in Google Form, was conducted to collect data in October and November 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Current student status (ongoing studies) was the one and only eligibility criterion for students to participate in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Convenience sampling method was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Students of nine Hungarian universities located in different regions of Hungary were asked to fill in the online questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The internal messaging systems of the universities were used to reach their students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Due to the low response rate, the sample size is 297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' SPSS 24 was used for data analysis (Babbie, Wagner and Zaino, 2019), and MS Excel for data visualisation (Walkenbach, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Means were calculated to quantify the importance (R1) and performance (R2) of each item of the sustainable university scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Importance- performance matrix was drawn to illustrate the position of SUS items to answer R3 (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' To answer R4, Principle Component Analysis was run to understand patterns in AE Students’ Perceptions of Sustainable Universities in Hungary: An Importance- Performance Analysis 504 Amfiteatru Economic SUS items (Jolliffe, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=" The reliability of the sustainable university scale (SUS) was measured by Cronbach's alpha to answer R5 (DeVellis, 2017)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Frequency tables of demographic variables were also calculated (Babbie, Wagner and Zaino, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The sample Of the sample of 297 respondents, 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='3% was female, 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='7% male (Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Mostly undergraduate students (77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='1%) participated in this study, however some graduate students (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='8%) and doctoral students (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='1%) contributed to the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The majority of the respondents (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='2%) fell into the category ‘aged 18-24’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Most of the students in the sample study social sciences (51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='1%), engineering (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='9%) or humanities (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='87%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' A significant part of them study in Miskolc (75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='4%), the rest (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='6%) in other Hungarian universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Therefore, this convenience sample is not representative, which is a limitation of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 2: Distribution of demographic variables (N=297) Demographic variables Values Frequency Percent Gender male 115 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='7 female 182 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='3 Study level bachelor 229 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='1 master 50 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='8 PhD 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='1 Age 18-24 161 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='2 25-31 60 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='2 32-38 29 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='8 39-45 27 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='1 46- 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='7 Branch of science agricultural sciences 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='7 arts 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='3 engineering 71 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='9 humanities 41 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='8 medicine 19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='4 natural sciences 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='4 social sciences 153 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='5 University Corvinus University 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='7 Eszterhazy Uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Eger 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='7 METU Budapest 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='0 National Uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' of Public Service 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='7 Szechenyi Uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Gyor 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='4 University of Miskolc 224 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='4 University of Pannonia 27 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='1 University of Pecs 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='1 University of Szeged 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='0 Sustainable University AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 22 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 54 • May 2020 505 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Results and discussion This chapter is divided into five main sections, and each of them discusses the results related to one of the research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Importance of the sustainable university scale items (student expectations) In order to answer the first research question (R1), and to investigate student expectations about university sustainability in Hungary, the items of SUS were measured on a five-point importance scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The lowest value (1) means the item is not at all important, whereas the highest value (5) indicates the item is very important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The importance of SUS items refers to the students’ expectations regarding university sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It expresses their opinion on what a university should do in order to be sustainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It was found that the opportunity for separate waste collection on campus and encouragement of this activity by the university is the most important attribute of university sustainability (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='54), whereas regular sustainability audits performed on campus is the least important for university students (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' They consider water and energy savings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' the use of LEDs) as well as sustainable university buildings that are designed or converted in an energy efficient and sustainable way (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' windows, doors, insulation) extremely important (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' If a university intends to be more sustainable, it must make efforts to provide the necessary infrastructure for sperate waste collection and promote this activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The sustainable university should save water and energy and invest in sustainable, energy efficient buildings on campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' These findings are not fully consistent with those of Dagiliute, Liobikiene and Minelgaite (2018), who found recycling is less important for students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It is also crucial for the students that university buildings must be located in a natural setting, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' in a quiet, green area with many trees where the air quality is excellent (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Students, therefore, expect sustainable universities not only to be green, but to be located in a green environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' For the most important stakeholders, it is also essential that the sustainable university should use renewable energy sources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' solar panels (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='35), it has a sustainability strategy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='1) and promotes sustainability research (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='07).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It was also found that students think it important that the sustainable university carries out environmentally and socially responsible purchasing practices (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='0) and encourages the use of public transport, bikes (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' In a sustainable university, it is important that all the students, researchers, academic and non-academic staff should be aware of the sustainability strategy of the university (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='95) and sustainability information should be readily available on the university’s website, newsletters, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='94), as well as the university should create partnerships with government, non-governmental organizations, and industry working toward sustainability (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Green actions and projects (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='9) and community outreach programs (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='89) were found to be even less important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The existence of environmental student organization(s) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='76), the integration of sustainability research results into the curricula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='72) as well as sustainability-focused positioning, when the university distinguishes itself as sustainable/green from other higher education institutions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='71) are even less central for the students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' There is only moderate demand for subjects/courses about sustainability (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='71) Students do not require that a lot of AE Students’ Perceptions of Sustainable Universities in Hungary: An Importance- Performance Analysis 506 Amfiteatru Economic information about sustainability should be integrated into normal courses (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='61) or the university should offer a lot of study programmes related to sustainability issues (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' These results match those observed in earlier studies (Dagiliute, Liobikiene and Minelgaite, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The overall mean of the importance items is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' (Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=') 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Perceived performance of the sustainable university items (student satisfaction) In order to answer the second research question (R2), and to find out to what extent students are satisfied with the performance of the Hungarian universities towards sustainability, students were asked to rate the performance of the universities on a five- point rating scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The lowest score (1) indicates very poor rating (dissatisfaction), whereas the highest score (5) means excellent rating (very high satisfaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The rating scores of the sustainable university scale items refer to how students are satisfied with the sustainability performance of the university where they study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It expresses their opinion on how sustainable the university is perceived regarding each attribute (item) of the sustainable university scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It allows decision-makers to get more insight into how their efforts towards sustainability are seen by their students, their most important stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' As far as the perceived sustainability performance of the Hungarian universities is concerned, their overall performance rating is only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='23, which means that that students are not satisfied with it and consider Hungarian universities unsustainable (Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' These results provide support for the first hypothesis (H1), therefore it has been accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Students are most satisfied with the location of the university buildings, the rating of which is very good (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It suggests that Hungarian universities have preferred locations that are mostly situated in quiet, green areas with many trees where the air quality is excellent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' This could be a strength they capitalise on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Student are also satisfied with the separate waste collection opportunities on campus (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='7), community outreach programs benefiting the local environment (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='5) and the promotion of sustainability research (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Students are somewhat satisfied with the efforts made towards sustainability strategy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='35), partnerships with government, non-governmental organizations, and industry working towards sustainability (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='34), as well as sustainable university buildings (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='33), water and energy savings in the university (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='29) and the use of public transport and bikes (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' However, students are not really satisfied with how much information about sustainability is integrated into normal courses (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='14) and the mostly unsustainable purchasing practices of universities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' They are not convinced by the green actions/projects (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='12) and the integration of sustainability research results into the curricula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Moreover, students think only limited information on sustainability is available for them on the website or in the newsletters of the universities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='08).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' This is a serious problem as the lack of information is usually one of the greatest barriers towards sustainability (Avila et al, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Also, students think that they and other important stakeholders (researchers, academic and non-academic staff) are not aware of the sustainability strategy of the university (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='06), however their participation would be essential in the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Students do not think that universities position themselves as sustainable/green (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='03) or use solar panels or other renewable energy sources (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' They are not content with the number of subjects/courses about sustainability (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='0), green/environmental student organizations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='99) and the number of study programmes related to sustainability (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='93).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Students were found to be the least satisfied with the sustainability audits on campus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Sustainable University AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 22 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 54 • May 2020 507 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Importance-performance analysis (IPA) In this section, in line with research question 3 (R3), it is discussed whether combining the sustainable university scale (SUS) with importance-performance analysis (IPA) could be a useful strategic tool for university decision-makers to identify key areas of university sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' In order to determine the position of each item of the sustainable university scale in the quadrants of the importance-performance matrix, deviations of the means from the overall mean of importance (Δ IMP) and performance (Δ PER) were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 3 shows the results and the position of each item in the quadrants of IPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Seven attributes of the sustainable university scale including location, separate waste collection, strategy, energy and water savings, public transport, research and sustainable buildings fall into the ‘Keep up the good work” quadrant (Q1), which contains the competitive advantages (strengths) of the Hungarian universities with regard to sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It is suggested that universities should use all of them in communication campaigns targeted at students who are concerned about sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 3: Importance and performance of the sustainable university scale items IMP means PER means Δ IMP Δ PER Quad- rant 1 Sustainability strategy 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='12 Q1 2 Awareness of the sust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' strategy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='17 Q3 3 Sustainability audits 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='51 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='41 Q3 4 Sustainability information 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='94 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='15 Q3 5 Green positioning 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='71 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='20 Q3 6 Green purchasing 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='10 Q4 7 Separate waste collection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='54 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='47 Q1 8 Renewable energy sources 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='21 Q4 9 Water and energy savings 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='06 Q1 10 Public transport, bikes 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='04 Q1 11 Sustainable buildings 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='10 Q1 12 Green location 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='94 Q1 13 Community outreach programs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='89 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='27 Q2 14 Sustainability partnerships 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='94 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='11 Q2 15 Green student organization(s) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='76 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='24 Q3 16 Green actions, projects 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='90 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='11 Q3 17 Green study programmes 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='30 Q3 18 Green subjects/courses 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='71 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='23 Q3 19 Greening normal courses 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='61 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='09 Q3 20 Sustainability research 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='07 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='24 Q1 21 Sustainability research integration 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='72 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='12 Q3 Total 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='98 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='23 Notes: IMP: importance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' PER: performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Quadrants: (1) Keep up the good work (2) Possible overkill (3) Low priority (4) Concentrate here AE Students’ Perceptions of Sustainable Universities in Hungary: An Importance- Performance Analysis 508 Amfiteatru Economic Campus location is found to be the biggest strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The favourable location is very important for the students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' They require that university buildings should be situated in a quiet, green environment, and for most of them, this expectation is fully met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Separate waste collection, which is the most important aspect of the sustainable university from the students’ perspective, is also a major strength as students are quite satisfied with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Hungarian universities must communicate that they provide the infrastructure for separate waste collection and promote this activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Based on our findings, it is advisable for universities to emphasize that their students are satisfied with their efforts towards energy and water savings and appreciate their endeavours to increase energy efficiency on campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Also, students are content with how sustainable the design of the university buildings is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It can also be suggested that Hungarian higher education institutions should communicate that they promote sustainability research, encourages the use of public transport, bikes and they have a written sustainability strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Two items can be found in Q2, which is the possible overkill quadrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It contains items that are not important for the students, however they, the most important stakeholders of the universities are satisfied with it (performance ratings are better than the overall average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The performance of universities concerning community outreach programs, partnership with governmental, non-governmental organizations, and industry is better than required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' In this case, it is suggested that universities should make a communication campaign to increase the importance of their community outreach programs and sustainability partnership to turn those activities into competitive advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Ten items – nearly the half of the sustainable university scale items – can be found in Q3, which represents “Low priority” attributes having low importance and low perceived performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The items that fall into this quadrant are respectively: 1) awareness of the sustainability strategy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 2) regular sustainability audits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 3) information regarding sustainability (website, newsletters, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 4) sustainability-focused positioning of the universities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 5) active green student organizations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 6) sustainability-related projects/actions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 7) study programs related to sustainability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 8) subjects/courses related to sustainability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 9) integration of sustainability into normal/traditional courses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' and finally, 10) integration of sustainability research results into the curricula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Hungarian universities are strongly advised to avoid any investments in those activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Last but not at least, two sustainable university items can be found in Q4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' This is the “Concentrate here” quadrant representing attributes that universities should immediately improve to achieve higher student satisfaction with regard to their attempts to be more sustainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The items listed here have high importance and low perceived performance suggesting that students are really dissatisfied with them in spite of the fact that those items are really important for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' On the one hand, they do not believe that universities have environmentally and socially responsible purchasing practices, on the other hand they are disappointed with the use renewable energy sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' solar panels) on campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It is therefore suggested that universities should concentrate more on green/socially responsible procurement and should increase the use of renewable energy sources to make students who are concerned about sustainability more satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Universities should consider more the sustainability performance of their suppliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' They should be greening their tenders, prefer local suppliers, and install more solar panels, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' (Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Sustainable University AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 22 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 54 • May 2020 509 Figure no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 2: Importance-performance of the sustainable university scale items As no research has been found that surveyed the perceived importance and performance of the attributes of university sustainability, it is therefore not possible to compare the results discussed here to the findings of previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' However, this study fills this gap in the literature and propose a new methodology to investigate the attributes of university sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' As an answer to R3, it can be concluded that importance performance analysis (IPA) is a strong strategic tool for university decision-makers to identify key areas of university sustainability when combined with the sustainable university scale (SUS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Using the results of IPA, universities could implement corrective actions to make students as stakeholders more satisfied with their efforts to be more sustainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Factor analysis of the sustainable university scale items In order to investigate patterns in perceived university sustainability, and answer R4, factor analysis was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The dataset of the importance of SUS items were analysed as it refers to the students’ expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The very high Kaiser-Meyer-Olkin Measure of Sampling Adequacy value (KMO=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='938) indicates that a factor analysis is a useful method with our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=" The Bartlett's Test of Sphericity (Approx." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Chi-Square = 4400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='484;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' df = 210;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Sig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='000) also reconfirms it (Jolliffe, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=" The extraction communalities are acceptable, although the lower values of Green Location and Green Positioning show that they don't fit as well as the others." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Only three factors in the initial solution have eigenvalues greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Together, they account for almost 65% of the variability in the original variables (Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' This suggests that three latent influences are associated with sustainable university perceptions, but there remains room for a lot of unexplained variation (Babbie, Wagner and Zaino, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The scree plot also confirmed the choice of three components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 4: Total variance explained greenlocation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='1 separatewaste collection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='6 Performance communityoutreach programs sustainabilityresearch sustainabilitypartnerships sustainablebuildings sustainabilitystrategy publictransport,bikes water andenergy savings greeningnormal courses greenactions,projects sustainability green purchasing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='1 researchintegration sustainabilityinformation green subjects/courses green positioning awarenessofthesust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='strategy renewableenergysources green studentorganization(s) greenstudyprogranmimes sustainabilityaudits 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='8 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='6 ImportanceAE Students’ Perceptions of Sustainable Universities in Hungary: An Importance- Performance Analysis 510 Amfiteatru Economic Compo- nent Initial Eigenvalues Extraction Sums of Squared Loadings Rotation Sums of Squared Loadings Total % of Variance Cumulative % Total % of Variance Cumulative % Total % of Variance Cumulative % 1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='648 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='706 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='706 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='648 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='706 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='706 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='782 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='535 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='535 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='650 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='856 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='563 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='650 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='856 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='563 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='937 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='746 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='281 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='333 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='346 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='909 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='333 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='346 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='909 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='912 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='628 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='909 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='916 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='360 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='269 21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='139 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='663 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='000 Notes: Extraction Method: Principal Component Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' To rotate the factor components, Varimax rotation with Kaiser normalization was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The first rotated factor component is most highly correlated with community outreach programs, sustainability partnerships, green study programmes, green subjects/courses, greening normal courses, sustainability research and sustainability research integration items (Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' These variables are not particularly correlated with the other two factor components, and each of them refers to actions towards meeting sustainability objectives, or related to education or research, it is therefore the first component called as Sustainable Actions, Education & Research (SAER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Table no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 5: Rotated component matrix Items 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Sust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Actions, Education & Research 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Sust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Operation/ Infrastructure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Sust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Strategy Type Sustainability strategy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='76 ST1 Awareness of the sust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' strategy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='80 ST2 Sustainability audits 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='73 ST3 Sustainability information 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='74 ST4 Green positioning 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='48 ST5 Green purchasing 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='44 PU1 Separate waste collection 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='24 WE1 Renewable energy sources 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='30 WE2 Water and energy savings 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='31 WE3 Public transport, bikes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='01 WE4 Sustainable buildings 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='26 WE5 Green location 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='08 LO1 Community outreach programs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='33 SA1 Sustainability partnerships 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='33 SA2 Green student organization(s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='41 SA3 Green actions, projects 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='45 SA4 Green study programmes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='19 SER1 Green subjects/courses 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='21 SER2 Greening normal courses 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='27 SER3 Sustainability research 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='30 SER4 Sustainability research 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='26 SER5 Sustainable University AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 22 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 54 • May 2020 511 Items 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Sust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Actions, Education & Research 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Sust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Operation/ Infrastructure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Sust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Strategy Type integration Notes: (1) Extraction Method: Principal Component Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Rotation Method: Varimax with Kaiser Normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Rotation converged in 6 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' (2) ST: Strategy, commitment & monitoring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' PU: Purchasing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' WE: Waste & energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' LO: Location;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' SA: Sustainability actions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' SER: Sustainable education & research The second factor component, which is called Sustainable Operation/Infrastructure, are made up of separate waste collection, renewable energy sources, water and energy savings and sustainable buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' All those items are related to the domain of waste and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The third component, Sustainable Strategy, has been named after the items that correlated with it the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' All of them are related to the sustainability strategy including the written sustainability strategy, and its awareness, regular sustainability audits and sustainability information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Because of their moderately large correlations with the first and the third components, green student organizations and green action/projects bridges Sustainable Actions, Education & Research and Sustainable Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Public transport, bikes and green location variables bridge the first and the second components, whereas green positioning and green purchasing are highly correlated with all the three factor components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' These results suggest that students form expectations about the three main domains of university sustainability: 1) sustainable strategy, 2) sustainable operations/infrastructure, and 3) sustainable actions/education/research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' These are the main topics of the university sustainability in the mind of the most important stakeholder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' These findings are not in line with those of previous studies (Nejati and Nejati, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Dagiliute, Liobikiene and Minelgaite, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' In both of the earlier studies, the number of factor components was higher, and the structure of the component was different from our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It is proposed that universities should deal with all the three components separately, and it would be beneficial for them to assign managers in charge to each domain to fully meet student expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=" Reliability of the sustainable university scale In order to test H2 and to investigate whether all the 21 items of the sustainable university scale reliably measure the same latent variable, a Cronbach's alpha was run on both SUS importance and SUS performance datasets." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=" In the reliability statistics table of SUS importance, Cronbach's alpha was 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='95, which indicates a very high level of internal consistency for our scale with this specific sample (DeVellis, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The "Cronbach\'s Alpha “If Item Deleted" column showed that removal of any item would result in a lower Cronbach\'s alpha, so no items were removed from the 21 item-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=" In the reliability statistics table of SUS performance dataset, Cronbach's alpha was even higher (0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='985), which indicates an even higher level of internal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Here also no items were removed as the “If Item Deleted" column showed that removal of any item would result in a lower Cronbach\'s alpha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Also, a reliability analysis was run in order to ensure internal consistency of the identified constructs after the principle component analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The high Cronbach’s alpha values confirmed the reliability of the constructs (α Sustainable Strategy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='850, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' of items = 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' AE Students’ Perceptions of Sustainable Universities in Hungary: An Importance- Performance Analysis 512 Amfiteatru Economic α Sustainable Operation & Infrastructure = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='861, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' of items = 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' and α Sustainable Actions, Education & Research= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='938, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' of items =10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' All these findings support H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It is therefore accepted that the sustainable university scale (SUS) is a reliable construct to measure perceived university sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Conclusions The stakeholder theory suggests that organizations should fully meet stakeholders’ expectations to be successful (Freeman, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Students are one of the biggest and most important stakeholders of universities (Degtjarjova, Lapina and Freidenfelds, 2018), and could have a significant impact on the environment (Emanuel and Adams, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Nowadays, the public demand for more sustainable universities is growing (Md Shahbudin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' More and more students want to study about sustainability, expect the integration of sustainability research into curricula and prefer universities that make efforts to operate in a more sustainable manner (Dagiliute, Liobikiene and Minelgaite, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' University decision-makers (Rector, Chancellor, Deans and the Senate) should consider sustainability issues to a greater extent when developing organizational strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' This study extends the knowledge of the above decision-makers regarding students’ perception of university sustainability in many aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The current study found that separate waste collection on campus is the most important student expectation about sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' However, it is not in line with the result of previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Dagiliute, Liobikiene and Minelgaite (2018) found recycling less important for students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Nonetheless, our findings are consistent with those of other studies suggesting that students expect water and energy savings and energy efficient, sustainable university buildings in a sustainable university.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Also, it is important for the students that the buildings should be located in a green environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Universities are therefore advised to promote separate waste collection, save water and energy, and maintain sustainable, energy efficient buildings that are situated in green parks (R1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' In the current study, the low value of general satisfaction with the performance of universities towards sustainability (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='23) confirmed H1 and suggests that students are not satisfied with it and consider Hungarian universities rather unsustainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Students’ perceptions of university sustainability are in line with the weak positions of the Hungarian higher education institution in green rankings (Greenmetric, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Our findings show that students are most satisfied with the location of the university buildings, which suggests that Hungarian universities have preferred locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Also, students are content with the opportunity to collect waste separately on campus, the community outreach programs that universities offer, and the promotion of research on sustainability (R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The findings of this research confirmed H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' By combining the importance-performance analysis (IPA) with the sustainable university scale (SUS), a simple but powerful strategic managerial tool can be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It could be widely used by university decision-makers to investigate the key areas of university sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=" IPA helps to identify competitive advantages and major weaknesses in the domains of sustainability and make it possible for decision-makers to implement corrective actions to make students as stakeholders more satisfied with the university's efforts to address sustainability." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Two major weaknesses were found in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Hungarian universities perform poorly in sustainable purchasing and Sustainable University AE Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 22 • No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' 54 • May 2020 513 use less renewable energy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' solar panels) on campus than it is expected by their students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It is therefore suggested that universities should immediately make both their energy use and purchasing process more sustainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' On the other hand, it was also found that campus location and separate waste collection are the major competitive advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' It is suggested that the major strengths are used in the marketing campaigns of universities to make their green positioning more effective and to build the sustainable university brand image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Strategy, energy and water savings, public transport, sustainable buildings and research are also strengths of the Hungarian universities that should be communicated (R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The three main domains of university sustainability were also identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' These are the strategy towards sustainability, actions to promote sustainability including education and research, and the sustainable infrastructure/operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' This is a unique structure and different from those presented in earlier studies (Nejati and Nejati, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Dagiliute, Liobikiene and Minelgaite, 2018), which suggests that Hungarian universities should use a nation-specific approach to university sustainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' Future studies on this topic are therefore recommended to investigate it in different cultural and national contexts (R4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/IdAzT4oBgHgl3EQfU_z6/content/2301.01278v1.pdf'} +page_content=' The sustainable university scale (SUS) was found to be a reliable construct to measure perceived 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[online] Available at: 1013 cm-2) [12]. Notable +exceptions to the cleanness-implies-high-RT-mobility scenario are suspended graphene samples, where flexural +phonons dramatically contribute to carrier scattering leading to a T2 behaviour of the resistivity [13], and +rotationally faulted graphene bilayers close to magic-angle, showing strong phonon-driven T-linear resistivity +[14]. The difference between freely suspended graphene and graphene encapsulated in hBN is due to the fact +that in the latter case van der Waals interaction between graphene and substrate makes flexural phonons +harder, suppressing an intrinsic rippling instability [15]. +In this work, we address the fundamental question whether the e-ph mechanism in clean graphene could also +govern the electrical transport in the QH regime [16] at temperatures close to RT. In this sense, we note that + +3 + +previous literature on the RT-QH effect in graphene [17–20] exclusively includes experiments on SiO2- +supported devices, precluding such investigation. + +Results +The QH effect in 2DESs manifests when the Fermi level (EF) lies on the localised states between two LLs, formed +in a perpendicular magnetic field and separated by an energy gap ΔLL. The interplay between this energy scale +and the thermal energy kT governs the basic phenomenology of the electrical transport in the QH regime. +When 𝑘𝑇 ≪ 𝛥𝐿𝐿, no conduction takes place in the 2D bulk, while 1D chiral edge states carry the electrical +current ballistically, leading to zero longitudinal resistivity (ρxx) when measured in four-probe configuration +(Figure 1a, upper panel). As the temperature increases and 𝑘𝑇~𝛥𝐿𝐿, thermal excitation of extended bulk states +(close to the LLs centre) exponentially restores bulk conduction and carrier scattering (Figure 1a, lower panel), +resulting in a finite value of the longitudinal resistivity minimum according to 𝜌𝑥𝑥 = 𝜌0 exp (− 𝛥𝐿𝐿 2𝑘𝑇 +⁄ +). This +relation is vastly employed to estimate the inter-LL separation via T-dependent measurements of the local +resistivity minimum (under the precaution that the activation energy underestimates ΔLL due to disorder- +broadening of the LLs [21]). The pre-factor to the exponential term, ρ0, which is often not considered explicitly, +determines the magnitude of the T-activated resistivity (shaded yellow area Figure 1a, lower) and contains +information regarding the disorder potential [22, 23]. In perpendicular magnetic fields, e-ph scattering requires +lattice vibrations with a wave-vector in the order of the inverse of the magnetic length (𝑙𝐵~ 25 nm √𝐵[T] +⁄ +) +[24], which defines a third energy scale relevant to our problem 𝐸𝑝ℎ = ℏv𝑠 𝑙𝐵 +⁄ + (where vs is the sound velocity in +the material). In conventional 2DESs, the small ΔLL leads to a complete suppression of the QH effect within a +few K [25], where the Eph-controlled phonon population can be considered negligible. Although the low +electronic mass in 2DESs such as InSb [26] and HgTe [27–29] enables the observation of the QHE up to liquid- +nitrogen temperature, this is insufficient to ensure 𝑘𝑇 ≫ 𝐸𝑝ℎ and therefore insufficient to realize a +predominance of e-ph interaction. This condition, as sketched in Figure 1b, is instead fulfilled by graphene in + +4 + +the RT-QH regime (the field dependence of Eph and the corresponding T-dependent excitation probability for +acoustic phonons in graphene at B = 30 T are shown in SI Figure S1). Under this circumstance, the T-activated +resistivity (shaded dark cyan area in Figure 1b) should directly relate to e-ph scattering [24]. +Figure 1c shows a representative measurement of the RT-QH effect, acquired at B = 30 T in a +hBN/graphene/hBN back-gated Hall bar (sample D2). The Hall conductivity (σxy) presents weak slope changes +around filling factors ν = ±2 (Vg ~ ±20 V), while the shelves-like features at low carrier concentration originate +from the onset of electron-hole coexistence in the highly-degenerate N = 0 LL [30]. ρxx, in addition to the +pronounced maximum around the charge neutrality point (CNP), shows two sizable minima (Figure 1c, inset), +indicative of T-activated QH states. Notably, the overall robustness of the RT-QH signatures dramatically differs +in high-mobility graphene with respect to SiO2-supported samples [17]; we thoroughly address this striking +observation in a separate work, where we study the suppression of the σxy plateaus in ultra-high-quality +devices at temperatures significantly lower than RT. In the following, we will focus on the magnitude of ρxx in +the RT-QH regime and identify the underlying mechanism employing a collection of dry-assembled hBN/ +graphene/hBN heterostructures. + +In Figure 2 we present the main transport characteristics of our devices (details on the fabrication are given in +Methods), measured at zero magnetic field and at elevated temperatures. Figure 2a shows the RT mobility of +three hBN-encapsulated devices, calculated according to the Drude model ( 𝜇 = 1 (𝑛𝑒𝜌𝑥𝑥) +⁄ + ), as a function of +the carrier density n. All the mobility curves are well above the typical values for SiO2-supported graphene +(grey shaded area) over the whole n range. Importantly, sample D3 shows a μ(n) dependence comparable to +the data of Ref. [5] (dash-dotted line), demonstrating the standard fingerprint of phonon-limited RT mobility in +zero magnetic field [11, 12] (as confirmed by temperature-dependent resistivity data shown in SI Figure S2). +We note that, although Wang et al. employed a 15 μm-wide van der Pauw device, e-ph scattering imposes a ~1 +μm upper bound to the electronic mean free path at B = 0 and RT [5]. Therefore, the zero-field e-ph limit can +also be realized using narrow Hall bars, provided that their channel width exceeds 1 μm (1.5 μm to 2.3 μm in + +5 + +our devices). The overall high quality of the samples is also supported by the observation of fractional QH +states at liquid-helium temperature (see data for sample D2 in SI Figure S3, and Ref. [31] for sample D4, +fabricated using CVD-grown graphene). In Figure 2b we explore the correlation between the carrier mobility +(calculated using the field-effect formula [32]) and the charge inhomogeneity in the CNP region, estimated as +the usual n* parameter [33] (see Figure 2b inset for an example of the extraction). We consider data at T = 220 +K, where clear thermal activation is observed in the RT-QH regime. n* values above the intrinsic CNP thermal +broadening (~2.6 × 1010 cm-2 at 220 K, beginning of the x-axis in Figure 2b) quantify the residual disorder, +which, in our devices, remains well below the typical observations for graphene on SiO2 (n* in the few-1011 cm-2 +range). In addition, as for Refs. [33, 34], the linear μ-1(n*) dependence (see shaded area in Figure 2b) indicates +scattering from long-range potentials, attributed to random strain variations generic to graphene on substrates +[35]. We can therefore conclude that the devices at disposal (i) span a low-disorder range unexplored in +previous RT-QH experiments, and (ii) present a well-defined disorder type, with increasing impact along the D4- +to-D1 sequence. + +We then employ the sample temperature as an experimental knob to control the excitation of both phonons +(see SI Figure S1) and bulk-extended electronic states in strong magnetic fields. In Figure 3a we sketch the +effect of increasing T on the Landau-quantized electrons in graphene at B = 30 T. Toward RT, the broadening of +the Fermi-Dirac distribution around EF (experimentally set by Vg) ensures excited charge carriers from both the +N = 0 and N = 1 LLs, across the giant gap ΔLL. Accordingly, the local resistivity minimum at filling factor ν = 2 +leaves zero and displays increasing finite values, as shown in the experimental curves of Figure 3b. In Figure 3c, +we present a complete picture of the T-dependence of ρxx (ν = 2) for samples D1-4, at selected magnetic fields +(30 T and 25 T in the main panel and inset, respectively; data at ν = -2 are shown in SI Figure S4). In addition to +our data, we show reference points from Ref. [20] (black diamonds, ρxx (ν = 2) in graphene on SiO2), and two +theoretical calculations defining different dissipation limits (continuous lines). In both cases we take an +activation energy equal to ΔLL/2: this was shown to be accurate for high B-fields in Ref. [20] and should hold + +6 + +true for clean graphene with reduced LL broadening. The upper line (yellow) assumes the universal +conductivity pre-factor due to long-range disorder (2e2/h) [23], multiplied by a factor 4 to take into account the +LL degeneracy of graphene. The lower line (dark cyan) is based on the work by Alexeev et al. [24], who +calculated the conductivity mediated by two-phonon scattering for graphene in the RT-QH regime. The +relevant e-ph process conserves the LL number, but modifies the in-plane electronic momentum. We note that +this phenomenology is fundamentally different from that of magneto-phonons oscillations, recently discovered +in extra-wide graphene devices [36], which rely on resonant inter-LL scattering at T < 200 K. Here, two-phonon +scattering within each LL contributes with a conductivity pre-factor σ0 = σN(T/300 K)(B/10 T)1/2 , which depends +both on temperature and magnetic field (in contrast to the constant pre-factor commonly assumed in QH +studies). In the ν = 2 state, the predominant contribution to the σN terms comes from the N = 0 LL (0.65 e2/h, +one order of magnitude larger with respect to N = 1, 0.06 e2/h) [24]. Strikingly, the resulting activated +behaviour, not including any free parameter, is well approximated by our devices, while the reference data +from graphene on SiO2 follow the long-range disorder limit. The qualitative agreement between theoretical +calculations and experimental data, together with the contrasting behaviour with respect to previous reports +[20], indicate that graphene/hBN heterostructures support an e-ph-dominated transport in the RT-QH regime. +Arrhenius-type fits to the conductivity [37], shown in SI Figure S5, confirm the contrasting magnitude of the +pre-factor for the two generations of graphene devices (as well as the correctness of the assumed gap size). + +Despite the presence of long-range potentials (Figure 2b), our data clearly indicate that the e-ph pre-factor +does not simply add up to the standard long-range disorder term. To elucidate this point, we quantitatively +analyse the deviation from the phonon-mediated limit in the different devices. We proceed by fitting the data +from samples D1-3 (only two high T curves are acquired for D4 due to experimental limitations) with a +generalized relation (Figure 4, inset), which adds to the theoretical e-ph dependence from Ref. [24] an +activation part with a constant pre-factor (ρD). This term is intended to account for the effect of residual +disorder, and it is the only free parameter in the fits. In Figure 4 we plot the extracted ρD for the three samples + +7 + +at different magnetic fields, as a function of the n* parameter (averaged between the electron and hole-side). +The linear ρD(n*) behaviour observed here (shaded area in Figure 4) indicates that the random strain variations +inducing the CNP broadening are also responsible for ρxx exceeding the e-ph limit in the RT-QH regime. Notably, +the only device to display an exact e-ph-type dependence (D3, ρD ~ 0), is also the one to show a Drude mobility +comparable to the zero-field e-ph limit [5]. Taking into account the sample-dependent correction due to +residual disorder, in SI (Figure S6) we proceed to a quantitative investigation of the field and temperature +dependence of the conductivity pre-factor in our samples, revealing the expected B1/2 behaviour of the e-ph +term. However, we note that the simplified pre-factor proposed in Ref. [24] is the result of several +approximations and, more importantly, it neglects the effect of disorder. To better understand the interplay +between the different scattering mechanisms underlying the activated resistivity, in SI (Figures S7 and S8) we +discuss additional data at lower temperature (down to 50 K) and magnetic field (down to 1 T). We find that ρD +drastically increases toward low T, with the activated resistivity exceeding the e-ph limit by more than one +order of magnitude in a clean sample. However, as the temperature and magnetic field are increased, ρD +progressively drops (i. e., the activated resistivity tends toward the e-ph limit), suggesting a temperature-driven +crossover between regimes dominated by either disorder or e-ph interaction (the latter being realized only +close to RT). While it is not surprising that the e-ph limit works as a lower bound to the activated resistivity of +real samples, the non-universality (i.e., the sample and temperature dependence) of the disorder contribution +deserves particular attention in future theoretical treatments of the RT-QH in graphene. + +Discussion +The physics of graphene is essentially determined by its deviations from flatness (that is, ripples), due to either +thermal fluctuations associated to flexural phonons for freely suspended samples or to roughness of substrate +like for graphene on SiO2 [15]. In both cases, ripples induce inhomogeneity of electron density with electron +and hole puddles in the vicinity of the CNP [38, 39]. In particular, for the case of graphene on SiO2 the + +8 + +amplitude of induced inhomogeneity of charge-carrier density is estimated as 3×1011 cm-2 [39], in agreement +with the above cited experimental values of n*. This makes the system strongly disordered, and any intrinsic +scattering mechanisms become irrelevant. Oppositely, the hBN substrate is atomically flat [1] and at the same +time suppresses intrinsic ripples which increases the RT carrier mobility by an order of magnitude and makes +intrinsic scattering mechanisms dominant [15]. Indeed, experimentally measured n* for our samples is an +order-of-magnitude smaller than what is supposed to be induced by ripples at RT. This results in an essentially +different picture of QH physics at high enough temperatures. + +In conclusion, we showed experimental evidence of predominant e-ph scattering in the QH regime. This is +realized by uniquely combining strong magnetic fields, high temperatures and hBN-encapsulation of graphene. +Although the RT-QH in graphene has long been known, we showed that mitigation of disorder via van der +Waals engineering provides novel insights on the transport mechanisms in this phenomenon. + + + + + + + + + + + + + + + + +9 + +Methods +Graphene-hBN van der Waals assembly and device fabrication +hBN/graphene/hBN samples D1-3 are assembled using the standard van der Waals dry pick-up [5], starting +from micromechanically exfoliated graphene flakes previously identified by optical and Raman microscopy. +Sample D4 is obtained by CVD growth on Cu foil and direct hBN-mediated pick-up after controlled decoupling +via Cu surface oxidation [31]. All the devices are fabricated making use of electron beam lithography, reactive +ion etching and e-beam evaporation of Cr/Au 1D edge contacts [5]. +Magnetotransport measurements +We use standard lock-in acquisition at low frequency (13 Hz), with simultaneous ρxx and ρxy measurements in +four-probe configuration, either under a constant current excitation (12.5 nA, sample D1-D3) or a constant +voltage bias (300 µV, sample D4). The devices are mounted in a VTI system with low-pressure 4He serving as +exchange gas, coupling the samples to a liquid-N2 reservoir. The cryogenic system is accommodated in the +access bore of a resistive Bitter magnet at HFML-EMFL, with a maximum field of 33 T. + +Data Availability +The data presented in this study are available at https://doi.org/10.5281/zenodo.7352031 . + +References +[1] Yankowitz, M., Ma, Q., Jarillo-Herrero, P. & LeRoy B. J. van der Waals heterostructures combining graphene +and hexagonal boron nitride. Nat. Rev. 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This work was supported by HFML-RU/NWO-I, member +of the European Magnetic Field Laboratory (EMFL). This work was also supported by CENTERA Laboratories in +the frame of the International Research Agendas Program for the Foundation for Polish Sciences co-financed by + +13 + +the European Union under the European Regional Development Fund (no. MAB/2018/9). D.V. acknowledges +financial support from the Ministry of Universities (Spain) (Ph.D. contract FPU19/04224). J.A.D-N thanks the +support from the Universidad de Salamanca for the María Zambrano postdoctoral grant funded by the Next +Generation EU Funding for the Requalification of the Spanish University System 2021–23, Spanish Ministry of +Universities. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, +Japan (Grant Number JPMXP0112101001) and JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and +21H05233). + +This version of the article has been accepted for publication, after peer review, but is not the Version of Record +and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available +online at: https://doi.org/10.1038/s41467-023-35986-3 . + +Author Contributions Statement +U.Z., S.W. and S.P. conceived the experiment and coordinated the collaboration. D.V., V.C. and M.S. fabricated +the graphene devices and performed the transport measurements. J.A.D.-N., A.M.-R. and J.S.-S. provided +technical assistance in the cleanroom processing. C.S.A.M. and K.R. provided technical assistance during the +high-field experiments. K.W. and T.T. provided single crystals of hBN. B.B., C.S. and E.D. supervised the +experimental work. D.V., V.C., M.S., and S.P. performed the data analysis. M.I.K. provided theoretical input for +the interpretation of the results. S.P. wrote the manuscript with input from all the co-authors. + +Competing Interests Statement +The authors declare no competing interests. + + + +14 + +Figures and Captions + +Figure 1 | Dissipation regimes in the quantum Hall phase: high-quality graphene at RT. a, Schematics of +temperature-dependent transport in conventional quantum Hall systems, such as 2DESs in semiconductors. At +low T (relative to the LL separation, upper part), the electrical current is carried by chiral edge states, leading to +zero longitudinal resistance. At higher T (lower part), thermally-excited bulk states give a finite resistivity due to +disorder scattering (yellow shading), with negligible contribution from lattice vibrations. b, At RT, graphene +supports both the QH effect (due to large inter-LL spacing) and predominant e-ph scattering in high-mobility +samples, enabling the realization of a different transport regime, with phonon-mediated dissipation at high +magnetic fields (dark cyan shading). c, ρxx (black) and σxy (red) as a function of the back-gate voltage (corrected +by a 5.2 V offset from the CNP), measured in hBN-encapsulated sample D2 at B = 30 T and T = 295 K. Inset: +zoom-in of ρxx in the vicinity of filling factor ν = 2 (the dark cyan shading indicates the finite value of the +resistivity minimum). + + +a +kT << △LL +c +kT 0, then the causal effects of X on +Y are identifiable and given by the adjustment formula: +P(yx) = +� +z +P(z|x) +� +x′ +P(y|x′, z)P(x′). +If causal effects are not identifiable, Tian and Pearl +[Tian and Pearl, 2000] provided the following bounds for the +causal effects. +P(x, y) ≤ P(yx) ≤ 1 − P(x, y′). +(2) +Finally, we review the identification conditions for PNS, +PS, and PN [Tian and Pearl, 2000]. +Definition 7. (Monotonicity) A Variable Y is said to be mono- +tonic relative to variable X in a causal model M iff +y′ +x ∧ yx′ = false. +Theorem 8. If Y is monotonic relative to X, then PNS, PN, +and PS are all identifiable, and +PNS = P(yx) − P(yx′), +PN = P(y) − P(yx′) +P(x, y) +, +PS = P(yx) − P(y) +P(x′, y′) +. +If PNS, PN, and PS are not identifiable, informative bounds +are given by Tian and Pearl [Tian and Pearl, 2000]. +max + + + + + +0, +P(yx) − P(yx′), +P(y) − P(yx′), +P(yx) − P(y) + + + + + +≤ PNS +(3) +min + + + + + + + + + +P(yx), +P(y′ +x′), +P(x, y) + P(x′, y′), +P(yx) − P(yx′)+ +P(x, y′) + P(x′, y) + + + + + + + + + +≥ PNS +(4) +max +� +0, +P (y)−P (yx′) +P (x,y) +� +≤ PN +(5) +min +� +1, +P (y′ +x′)−P (x′,y′) +P (x,y) +� +≥ PN +(6) +max +� +0, +P (y′)−P (y′ +x) +P (x′,y′) +� +≤ PS +(7) +min +� +1, +P (yx)−P (x,y) +P (x′,y′) +� +≥ PS +(8) +The identification conditions mentioned above (i.e., back- +door and front-door criteria and monotonicity) are robust. +However, it may still be hard to achieve in real-world appli- +cations. In this work, we extend the definition of identifia- +bility, in which a sufficiently small interval is allowed. By +the new definition, the estimates of causal quantities are still +near point estimations, and more conditions for identifiability +could be discovered. If nothing is specified, the discussion +in this paper will be restricted to binary treatment and effect +(i.e., X and Y are binary). + +3 +Main Results +First, we extend the definition of identifiability, which we call +ǫ-identifiability. +Definition 9 (ǫ-Identifiability). Let Q(M) be any computable +quantity of a class of SCM M that is compatible with graph +G. We say that Q is ǫ-identifiable in M (and ǫ-identified to +q) if, there exists q s.t. for any model m from M, Q(m) ∈ +[q − ǫ, q + ǫ] with statistical data PM(v), where P(v) is the +statistical data over the set V of observed variables. If our +observations are limited and permit only a partial set FM +of features (of PM(v)) to be estimated, we define Q to be ǫ- +identifiable from FM if Q(m) ∈ [q − ǫ, q + ǫ] with statistical +data FM. +With the above definition, the causal quantity is at a max- +imum distance of ǫ from its true value. We will use the in- +fix operator symbol ≈ǫ to represent its left-hand side being +within ǫ of its right-hand side: +r ≈ǫ q ⇐⇒ r ∈ [q − ǫ, q + ǫ]. +(9) +The +following +sections +explicate +conditions +for +ǫ- +identifiability of causal effects, PNS, PS, and PN. +3.1 +ǫ-Identifiability of Causal Effects +The causal effect P(YX) can be ǫ-identified with information +about the observational joint distribution P(X, Y ). This can +be seen by rewriting Equation (2) as: +P(x, y) ⩽ P(yx) ⩽ P(x, y) + P(x′). +(10) +Here, P(yx) is ǫ-identified to P(x, y) + ǫ when P(x′) ⩽ 2ǫ. +This ǫ-identification indicates a lower bound of P(x, y) and +an upper bound of P(x, y) + 2ǫ. Since P(x′) ⩽ 2ǫ, these +bounds are equivalent to (10). Notably, only P(x, y) and an +upper bound on P(x′) are necessary to ǫ-identify P(yx). This +is generalized in Theorem 10, without any assumptions of the +causal structure. +Theorem 10. The causal effect P(YX) is ǫ-identified as fol- +lows: +P(yx) ≈ǫ P(x, y) + ǫ +if P(x′) ⩽ 2ǫ, +(11) +P(y′ +x) ≈ǫ P(x, y′) + ǫ +if P(x′) ⩽ 2ǫ, +(12) +P(yx′) ≈ǫ P(x′, y) + ǫ +if P(x) ⩽ 2ǫ, +(13) +P(y′ +x′) ≈ǫ P(x′, y′) + ǫ +if P(x) ⩽ 2ǫ. +(14) +Proof. See Appendix 8.1. +When the complete distribution P(X, Y ) is known, The- +orem 10 provides no extra precision over Equation (10). Its +power comes from when only part of the distribution is known +and only an upper bound on P(X) is available or able to be +assumed. +Knowledge of a causal structure can aid ǫ-identification. In +Figure 1, there is a binary confounder U. If the full joint dis- +tribution P(X, Y, U) was available, the causal effect P(YX) +could be computed simply through the backdoor adjustment +formula. In the absence of the full joint distribution, Theo- +rem 11 allows ǫ-identification of P(yx) with only knowledge +of P(x) and the conditional probability P(y|x) as well as an +upper bound on P(u). +U +X +Y +Figure 1: The causal graph, where X is a binary treatment, Y is a +binary effect, and U is a binary confounder. +Theorem 11. Given the causal graph in Figure 1 and +P(u) ≤ P(x) − c for some constant c, where 0 < c ⩽ P(x), +P(yx) ≈ǫ P(y|x) + +P(x) − c +2cP(x) + P(x) + c · ǫ +if P(u) ≤ +2cP(x) +2cP(x) + P(x) + c · ǫ. +(15) +Specifically, if P(x) ≥ 0.5, then the causal effect P(yx) is +ǫ-identified to P(y|x) + +ǫ +13 if P(u) < +4 +13ǫ. +Proof. See Appendix 8.2. +Note that x ∈ {x, x′}, y ∈ {y, y′}, and u ∈ {u, u′} in +Theorem 11. The constant c should be maximized satisfying +both c ⩽ P(x) − P(u) and the condition in Equation (15) for +a given ǫ. The larger c is, the closer P(yx) is ǫ-identified to +P(y|x). This needs to be balanced with minimizing ǫ. +As an example, if P(x) ≥ 0.5 and P(u) ⩽ 0.1, then the +causal effect P(yx) is ǫ-identified to P(y|x) + +ǫ +13 if P(u) ⩽ +4 +13ǫ. +Essentially, P(yx) is ǫ-identified to P(y|x) plus some frac- +tion of ǫ when P(u) is sufficiently small. +Therefore, the +causal effect P(yx) is near P(y|x) if P(U) is specific (i.e., +P(u) or P(u′) is minimal). In this case, Theorem 11 can be +advantageous over the backdoor adjustment formula to com- +pute P(yx), even when data on X, Y , and U are available, +because P(Y |X, U), required for the adjustment formula, is +impractical to estimate with P(U) close to 0. +3.2 +ǫ-Identifiability of PNS +Even though Tian and Pearl derived tight bounds on PNS +[Tian and Pearl, 2000], the PNS can be potentially further +narrowed when taking into account particular upper bound +assumptions on causal effects or observational probabilities. +This can be seen by analyzing the bounds of PNS in Equa- +tions (3) and (4). Picking any of the arguments to the max +function of the lower bound and any of the arguments to the +min function of the upper bound, we can make a condition +that the range of those two values is less than 2ǫ. For ex- +ample, let us pick the second argument of the max function, +P(yx) − P(yx′), and the first argument of the min function, +P(yx): +P(yx) − [P(yx) − P(yx′)] ⩽ 2ǫ, +P(yx′) ⩽ 2ǫ. +(16) +Equation (16) is the assumption and the PNS is the +ǫ-identified to ǫ above the lower bound or ǫ below the upper +bound: +PNS ≈ǫ P(yx) − P(yx′) + ǫ, or +(17) +PNS ≈ǫ P(yx) − ǫ. +(18) + +Since it is assumed that P(yx′) ⩽ 2ǫ, Equation (17) is equiv- +alent to Equation (18). The complete set of ǫ-identifications +and associated conditions are stated in Theorem 12. +Theorem 12. The PNS is ǫ-identified as follows: +PNS ≈ǫ ǫ +if P(yx) ⩽ 2ǫ, +(19) +PNS ≈ǫ ǫ +if P(y′ +x′) ⩽ 2ǫ, +(20) +PNS ≈ǫ ǫ +if P(x, y) + P(x′, y′) ⩽ 2ǫ, +(21) +PNS ≈ǫ ǫ +if P(yx) − P(yx′)+ +P(x, y′) + P(x′, y) ⩽ 2ǫ, +(22) +PNS ≈ǫ P(yx) − ǫ +if P(yx′) ⩽ 2ǫ, +(23) +PNS ≈ǫ P(y′ +x′) − ǫ +if P(y′ +x) ⩽ 2ǫ, +(24) +PNS ≈ǫ P(yx)− +P(yx′) + ǫ +if P(x, y′) + P(x′, y) ⩽ 2ǫ, +(25) +PNS ≈ǫ P(yx)− +P(yx′) + ǫ +if P(yx′) − P(yx)+ +P(x, y) + P(x′, y′) ⩽ 2ǫ, +(26) +PNS ≈ǫ P(x, y)− +P(x′, y′) − ǫ +if P(yx′) − P(yx)+ +P(x, y) + P(x′, y′) ⩽ 2ǫ, +(27) +PNS ≈ǫ P(y′ +x′) − ǫ +if P(y′) ⩽ 2ǫ, +(28) +PNS ≈ǫ P(yx) − ǫ +if P(yx) + P(yx′)− +P(y) ⩽ 2ǫ, +(29) +PNS ≈ǫ P(y) − P(yx′) + ǫ +if P(yx) + P(yx′)− +P(y) ⩽ 2ǫ, +(30) +PNS ≈ǫ P(x, y)+ +P(x′, y′) − ǫ +if P(x′, y′) + P(yx′)− +P(x′, y) ⩽ 2ǫ, +(31) +PNS ≈ǫ P(y) − P(yx′) + ǫ +if P(x′, y′) + P(yx′)− +P(x′, y) ⩽ 2ǫ, +(32) +PNS ≈ǫ P(y) − P(yx′) + ǫ +if P(x′, y) + P(y′ +x′)− +P(x′, y′) ⩽ 2ǫ, +(33) +PNS ≈ǫ P(yx) − ǫ +if P(y) ⩽ 2ǫ, +(34) +PNS ≈ǫ P(y′ +x′) − ǫ +if P(y′ +x′) − P(yx)+ +P(y) ⩽ 2ǫ, +(35) +PNS ≈ǫ P(y) − P(yx′) + ǫ +if P(y′ +x′) − P(yx)+ +P(y) ⩽ 2ǫ, +(36) +PNS ≈ǫ P(x, y)+ +P(x′, y′) − ǫ +if P(x, y) + P(y′ +x)− +P(x, y′) ⩽ 2ǫ, +(37) +PNS ≈ǫ P(yx) − P(y) + ǫ +if P(x, y) + P(y′ +x)− +P(x, y′) ⩽ 2ǫ, +(38) +PNS ≈ǫ P(yx) − P(y) + ǫ +if P(x′, y) + P(y′ +x′)− +P(x′, y′) ⩽ 2ǫ. +(39) +Proof. See Appendix 8.3. +Note that in the above theorem, eight conditions consist +solely of experimental probabilities or solely of observational +probabilities. This potentially eliminates the need for some +types of studies, at least partially, even when estimating +a counterfactual quantity such as PNS. For example, if a +decision-maker knows that P(y) is large (P(y) ⩾ 0.95), they +can immediately conclude PNS ≈0.05 P(y′ +x′) − 0.05 with- +out knowing the specific value of P(y). Thus, only a control +group study would be sufficient. +3.3 +ǫ-Identifiability of PN and PS +Tian and Pearl derived tight bounds on PN and PS in addi- +tion to PNS. Similar to the derivation of Theorem 12, we can +potentially narrow those bounds by taking into account upper +bound assumptions on causal effects or observational proba- +bilities. The set of ǫ-identifications and associated conditions +are stated in Theorems 13 and 14. +Theorem 13. The PN is ǫ-identified as follows: +PN ≈ǫ ǫ +if P(y′ +x′) − P(x′, y′) +⩽ 2ǫP(x, y), +(40) +PN ≈ǫ 1 − ǫ +if P(yx′) − P(x′, y) +⩽ 2ǫP(x, y), +(41) +PN ≈ǫ +P(y) − P(yx′) +P(x, y) ++ ǫ +if P(yx′) − P(x′, y) +⩽ 2ǫP(x, y), +(42) +PN ≈ǫ +P(y′ +x′) − P(x′, y′) +P(x, y) +− ǫ +if P(x, y′) +⩽ 2ǫP(x, y), +(43) +PN ≈ǫ +P(y) − P(yx′) +P(x, y) ++ ǫ +if P(x, y′) +⩽ 2ǫP(x, y). +(44) +Proof. See Appendix 8.4. + +Table 1: Results of an observational study with 1500 individuals +who have access to the medicine, where 1260 individuals chose to +receive the medicine and 240 individuals chose not to. +Take the medicine +Take no medicine +Recovered +780 +210 +Not recovered +480 +30 +Theorem 14. The PS is ǫ-identified as follows: +PS ≈ǫ ǫ +if P(yx) − P(x, y) +⩽ 2ǫP(x′, y′), +(45) +PS ≈ǫ 1 − ǫ +if P(y′ +x) − P(x, y′) +⩽ 2ǫP(x′, y′), +(46) +PS ≈ǫ +P(y′) − P(y′ +x) +P(x′, y′) ++ ǫ +if P(y′ +x) − P(x, y′) +⩽ 2ǫP(x′, y′), +(47) +PS ≈ǫ +P(yx) − P(x, y) +P(x′, y′) +− ǫ +if P(x′, y) +⩽ 2ǫP(x′, y′), +(48) +PS ≈ǫ +P(y′) − P(y′ +x) +P(x′, y′) ++ ǫ +if P(x′, y) +⩽ 2ǫP(x′, y′). +(49) +Proof. See Appendix 8.5. +4 +Examples +Here, we illustrate how to apply ǫ-Identifiability in real appli- +cations by two simulated examples. +4.1 +Causal Effects of Medicine +Consider a medicine manufacturer who wants to know the +causal effect of a new medicine on a disease. They conducted +an observational study where 1500 patients were given access +to the medicine; the results of the study are summarized in Ta- +ble 1. In addition, the expert from the medicine manufacturer +acknowledged that family history is the only confounder of +taking medicine and recovery, and the family history of the +disease is extremely rare; only 1% of the people have the fam- +ily history. +Let X = x denote that a patient chose to take the medicine, +and X = x′ denote that a patient chose not to take the +medicine. Let Y = y denote that a patient recovered, and +Y = y′ denote that a patient did not recover. Let U = u de- +note that a patient has the family history, and U = u′ denote +that a patient has no family history. +To obtain the causal effect of the medicine (i.e., using ad- +justment formula (1)), we have to know the observational data +associated with family history, which is difficult to obtain. +Fortunately, from Table 1, we obtain that P(x) = 0.84 and +P(y|x) = 0.62. We also have the prior that P(u) = 0.01. +Since 0.01 = P(u) ≤ P(x) − 0.8 (let c = 0.8) and +0.01 = P(u) < +2c∗0.025P (x) +2cP (x)+P (x)+c = 0.0113, we can ap- +ply Theorem 11 to obtain that P(yx) is 0.025-identified to +P(y|x)+ +P (x)−c +2cP (x)+P (x)+c0.025 = 0.62. This means the causal +effect of the medicine is very close to 0.62 (i.e., 0.025 close), +which can not be 0.025 far from 0.62. Then the medicine man- +ufacturer can conclude that the causal effect of the medicine +is roughly 0.62 without knowing the observational data asso- +ciated with the family history. +Or even simpler, note that P(x) = 0.84 > 0.5 and P(u) = +0.01 < 0.1, P(u) = 0.01 < +4 +13 ∗ 0.035 = 0.0108. We obtain +that P(yx) is 0.035-identified to P(y|x) + 0.035 +13 += 0.62. The +decision-maker can make the same conclusion as above. +4.2 +PNS of Flu Shot +Consider a newly invented flu shot. After a vaccination com- +pany introduced a new flu shot, the number of people infected +by flu reached the lowest point in 20 years (i.e., less than 5% +of people infected by flu). The government concluded that +the new flu shot is the key to success. However, some anti- +vaccination associations believe it is because people’s physi- +cal quality increases yearly. Therefore, they all want to know +how many percentages of people are uninfected because of +the flu shot. The PNS of the flu shot (i.e., the percentage of +individuals who would not infect by the flu if they had taken +the flu shot and would infect otherwise) is indeed what they +want. +Let X = x denote that an individual has taken the flu shot +and X = x′ denote that an individual has not taken the flu +shot. Let Y = y denote an individual infected by the flu and +Y = y′ denote an individual not infected by the flu. +If they want to apply the bounds of PNS in Equations (3) +and (4), they must conduct both experimental and observa- +tional studies. However, note that P(y) < 0.05, one could +apply Equation (34) in Theorem 12, which PNS is 0.025- +identified to P(yx)− 0.025 (i.e., PNS is very close to P(yx)). +Thus, according to [Li et al., 2022], only an experimental +study for the treated group with a sample size of 385 is ad- +equate for estimating PNS. +5 +ǫ-Identifiability in Unit Selection Problem +One utility of the causal quantities is the unit selection prob- +lem [Li and Pearl, 2022b; Li and Pearl, 2019], in which Li +and Pearl defined an objective causal function to select a set +of individuals that have the desired mode of behavior. +Let X denote the binary treatment and Y denote the bi- +nary effect. According to Li and Pearl, individuals were di- +vided into four response types: Complier (i.e., P(yx, y′ +x′)), +always-taker (i.e., P(yx, yx′)), never-taker (i.e., P(y′ +x, y′ +x′)), +and defier (i.e., P(y′ +x, yx′)). Suppose the payoff of selecting +a complier, always-taker, never-taker, and defier is β, γ, θ, δ, +respectively (i.e., benefit vector). The objective function (i.e., +benefit function) that optimizes the composition of the four +types over the selected set of individuals c is as follows: +f(c) = βP(yx, y′ +x′|c) + γP(yx, yx′|c) + +θP(y′ +x, y′ +x′|c) + δP(y′ +x, yx′|c). +Li and Pearl provided two types of identifiability condi- +tions for the benefit function. One is about the response type +such that there is no defier in the population (i.e., monotonic- +ity). Another is about the benefits vector’s relations, such that +β + δ = γ + θ (i.e., gain equality). These two conditions + +Table 2: Results of an experimental study with 1500 randomly se- +lected customers were forced to apply the discount, and 1500 ran- +domly selected customers were forced not to. +Discount +No discount +Bought the purchase +900 +750 +No purchase +600 +750 +are helpful but still too specific and challenging to satisfy in +real-world applications. If the benefit function is not identifi- +able, it can be bounded using experimental and observational +data. Here in this paper, we extend the gain equality to the +ǫ-identifiability as stated in the following theorem. +Theorem 15. Given a causal diagram G and distribution +compatible with G, let C be a set of variables that does not +contain any descendant of X in G, then the benefit function +f(c) = βP(yx, y′ +x′|c) + γP(yx, yx′|c) + θP(y′ +x, y′ +x′|c) + +δP(yx′, y′ +x|c) is |β−γ−θ+δ| +2 +-identified to (γ − δ)P(yx|c) + +δP(yx′|c) + θP(y′ +x′|c) + β−γ−θ+δ +2 +. +One critical use case of the above theorem is that decision- +makers usually only care about the sign (gain or lose) of the +benefit function. Decision-makers can apply the above theo- +rem before conducting any observational study to see if the +sign of the benefit function can be determined, as we will il- +lustrate in the next section. +5.1 +Example: Non-immediate Profit +Consider the most common example in [Li and Pearl, 2019]. +A sale company proposed a discount on a purchase in +order to increase the total non-immediate profit. +The +company assessed that the profit of offering the dis- +count to complier, always-taker, never-taker, and defier is +$100, −$60, $0, −$140, respectively. Let X = x denote that +a customer applied the discount, and X = x denote that a +customer did not apply the discount. Let Y = y denote that a +customer bought the purchase and Y = y′ denote that a cus- +tomer did not. The benefit function is then (here c denote all +customers) +f(c) = 100P(yx, y′ +x′|c) − 60P(yx, yx′|c) + +0P(y′ +x, y′ +x′|c) − 140P(y′ +x, yx′|c). +The company conducted an experimental study where 1500 +randomly selected customers were forced to apply the dis- +count, and 1500 randomly selected customers were forced not +to. The results are summarized in Table 2. The experimental +data reads P(yx|c) = 0.6 and P(yx′|c) = 0.5. +Before conducting any observational study, one can con- +clude that the benefit function is 10-identified to −12 using +Theorem 15. This result indicates that the benefit function is +at most 10 away from −12; thus, the benefit function is nega- +tive regardless of the observational data. The decision-maker +then can easily conclude that the discount should not offer to +the customers. +6 +Discussion +We have defined the ǫ-identifiability of causal quantities and +provided a list of ǫ-identifiable conditions for causal effects, +PNS, PN, and PS. We still have some further discussions +about the topic. +First, all conditions except Theorem 11 are conditions from +observational or experimental data. In other words, if some of +the observational or experimental distributions satisfied a par- +ticular condition, then the causal quantities are ǫ-identifiable. +These conditions are advantageous in real-world applications +as no specific causal graph is needed. +However, we still +love to discover more graphical conditions of ǫ-identifiability, +such as back-door or front-door criterion. +Second, the bounds of PNS, PS, PN, and the benefit func- +tion can be narrowed by covariates information with their +causal structure [Dawid et al., 2017; Li and Pearl, 2022d; +Mueller et al., 2021]. +The ǫ-identifiability can also be ex- +tended if covariates information and their causal structure are +available, which should be an exciting direction in the future. +Third, monotonicity is defined using a causal quantity, and +in the meantime, monotonicity is also an identifiable condi- +tion for other causal quantities (e.g., PNS). Thus, another +charming direction is how the ǫ-identifiability of monotonic- +ity affects the ǫ-identifiability of other causal quantities. +7 +Conclusion +In this paper, we defined the ǫ-identifiability of causal quan- +tities, which is easier to satisfy in real-world applications. +We provided the ǫ-identifiability conditions for causal effects, +PNS, PS, and PN. We further illustrated the use cases of the +proposed conditions by simulated examples. +References +[Balke and Pearl, 1997] Alexander A Balke and Judea Pearl. +Probabilistic counterfactuals: +Semantics, computation, +and applications. Technical report, UCLA Dept. of Com- +puter Science, 1997. +[Bareinboim and Pearl, 2012] E. Bareinboim and J. Pearl. +Causal +inference +by +surrogate +experiments: +z- +identifiability. +In Nando de Freitas and Kevin Murphy, +editors, Proceedings of the Twenty-Eighth Conference +on Uncertainty in Artificial Intelligence, pages 113–120, +Corvallis, OR, 2012. AUAI Press. +[Dawid et al., 2017] Philip Dawid, +Monica Musio, +and +Rossella Murtas. The probability of causation. Law, Prob- +ability and Risk, (16):163–179, 2017. +[Galles and Pearl, 1998] David Galles and Judea Pearl. An +axiomatic characterization of causal counterfactuals. Foun- +dations of Science, 3(1):151–182, 1998. +[Halpern, 2000] Joseph Y Halpern. +Axiomatizing causal +reasoning. +Journal of Artificial Intelligence Research, +12:317–337, 2000. +[Li and Pearl, 2019] Ang Li and Judea Pearl. +Unit selec- +tion based on counterfactual logic. +In Proceedings of +the Twenty-Eighth International Joint Conference on Ar- +tificial Intelligence, IJCAI-19, pages 1793–1799. Interna- +tional Joint Conferences on Artificial Intelligence Organi- +zation, 7 2019. + +[Li and Pearl, 2022a] A. +Li +and +J. +Pearl. +Prob- +abilities +of +causation +with +non-binary +treat- +ment +and +effect. +Technical +Report +R-516, +, +De- +partment of Computer Science, University of California, +Los Angeles, CA, 2022. +[Li and Pearl, 2022b] A. Li and J. Pearl. +Unit selection +with nonbinary treatment and effect. +Technical Report +R-517, , De- +partment of Computer Science, University of California, +Los Angeles, CA, 2022. +[Li and Pearl, 2022c] Ang Li and Judea Pearl. Bounds on +causal effects and application to high dimensional data. In +Proceedings of the AAAI Conference on Artificial Intelli- +gence, volume 36, pages 5773–5780, 2022. +[Li and Pearl, 2022d] Ang Li and Judea Pearl. Unit selec- +tion with causal diagram. In Proceedings of the AAAI Con- +ference on Artificial Intelligence, volume 36, pages 5765– +5772, 2022. +[Li et al., 2020] Ang Li, Suming J. Chen, Jingzheng Qin, +and Zhen Qin. +Training machine learning models with +causal logic. In Companion Proceedings of the Web Con- +ference 2020, pages 557–561, 2020. +[Li et al., 2022] A. Li, R. Mao, and J. Pearl. +Prob- +abilities of causation: +Adequate size of experimen- +tal and observational samples. +Technical Report R- +518, , De- +partment of Computer Science, University of California, +Los Angeles, CA, 2022. +[Mueller and Pearl, 2022] Mueller and Pearl. Personalized +decision making – a conceptual introduction. Technical +Report R-513, Department of Computer Science, Univer- +sity of California, Los Angeles, CA, 2022. +[Mueller et al., 2021] S. Mueller, +A. Li, +and J. Pearl. +Causes +of +effects: +Learning +individual +responses +from +population +data. +Technical +Report +R-505, +, +De- +partment of Computer Science, University of California, +Los Angeles, CA, 2021. +Forthcoming, Proceedings of +IJCAI-2022. +[Pearl, 1993] J Pearl. Aspects of graphical models connected +with causality. Proceedings of the 49th Session of the inter- +national Statistical Institute, Italy, pages 399–401, 1993. +[Pearl, 1995] Judea Pearl. Causal diagrams for empirical re- +search. Biometrika, 82(4):669–688, 1995. +[Pearl, 1999] Judea Pearl. Probabilities of causation: Three +counterfactual interpretations and their identification. Syn- +these, pages 93–149, 1999. +[Pearl, 2009] Judea Pearl. Causality. Cambridge university +press, 2nd edition, 2009. +[Shpitser and Pearl, 2009] I. Shpitser and J Pearl. +Effects +of treatment on the treated: Identification and generaliza- +tion. In Proceedings of the Twenty-Fifth Conference on Un- +certainty in Artificial Intelligence, pages 514–521. AUAI +Press, Montreal, Quebec, 2009. +[Tian and Pearl, 2000] Jin Tian and Judea Pearl. Probabili- +ties of causation: Bounds and identification. Annals of +Mathematics and Artificial Intelligence, 28(1-4):287–313, +2000. + +8 +Appendix +8.1 +Proof of Theorem 10 +Proof. From Equation (2) we have, +P(x, y) ≤ P(yx) ≤ 1 − P(x, y′). +Let 1 − P(x, y′) − P(x, y) ≤ 2ǫ, we obtain P(x′) ≤ 2ǫ. +Therefore, P(yx) is ǫ-identified to P(x, y) + ǫ if P(x′) ≤ 2ǫ, +Equation (11) holds. Similarily, we can substitute x, y with +x′, y′, respectively. Equations (12) to (14) hold. +8.2 +Proof of Theorem 11 +Proof. First, by adjustment formula in Equation (1), we have, +P(yx) = P(y|x, u)P(u) + P(y|x, u′)P(u′). +Thus, +P(yx) +≥ +P(y|x, u′)P(u′) += +P(y|x, u′)(1 − P(u)) += +P(x, y, u′) +P(x, u′) (1 − P(u)) +≥ +P(x, y) − P(u) +P(x) +(1 − P(u)) += +P(y|x) − P(y|x)P(u) − P(u) +P(x) + P 2(u) +P(x) +≥ +P(y|x) − P(u) − P(u) +P(x) += +P(y|x) − (1 + +1 +P(x))P(u). +Also if P(x) ≥ P(u) + c for some constant c > 0, we have, +P(yx) +≤ +P(u) + P(y|x, u′)(1 − P(u)) +≤ +P(u) + P(x, y, u′) +P(x, u′) (1 − P(u)) +≤ +P(u) + +P(x, y) +P(x) − P(u)(1 − P(u)) +≤ +P(u) + +P(x, y) +P(x) − P(u) += +P(u) + +P(x, y) +P(x)(1 − P (u) +P (x)) += +P(u) + +P(x, y)(1 − P (u) +P (x)) + P(y|x)P(u) +P(x)(1 − P (u) +P (x)) += +P(u) + P(y|x) + P(y|x)P(u) +P(x) − P(u) +≤ +P(y|x) + P(u) + +P(u) +P(x) − P(u) +≤ +P(y|x) + P(u) + P(u) +c += +P(y|x) + P(u)(1 + 1 +c ) +Therefore, we have, +P(y|x) − (1 + +1 +P(x))P(u) ≤ P(yx) ≤ P(y|x) + (1 + 1 +c)P(u). +Let +(1 + 1 +c)P(u) + (1 + +1 +P(x))P(u) ≤ 2ǫ. +We have, +P(u) +≤ +2 +2 + 1 +c + +1 +P (x) +ǫ += +2cP(x) +2cP(x) + P(x) + cǫ. +Then we know that if P(u) ≤ +2cP (x) +2cP (x)+P (x)+cǫ, +P(y|x) − (1 + +1 +P(x)) +2cP(x) +2cP(x) + P(x) + cǫ ≤ +P(yx), +P(y|x) + (1 + 1 +c ) +2cP(x) +2cP(x) + P(x) + cǫ ≥ +P(yx), +P(y|x) − +2cP(x) + 2c +2cP(x) + P(x) + cǫ ≤ +P(yx), +P(y|x) + +2cP(x) + 2P(x) +2cP(x) + P(x) + cǫ ≥ +P(yx). +Therefore, P(yx) is ǫ-identified to P(y|x)− +2cP (x)+2c +2cP (x)+P (x)+cǫ+ +ǫ = P(y|x) + +P (x)−c +2cP (x)+P (x)+cǫ. +Besides, if P(x) ≥ 0.5 and P(u) ≤ 0.1, let c = 0.4, we have +P(y|x) − (1 + +1 +P(x))P(u) ≤ P(yx), +P(y|x) + (1 + 1 +c )P(u) ≥ P(yx). +P(y|x) − (1 + 1 +0.5)P(u) ≤ P(yx), +P(y|x) + (1 + 1 +0.4)P(u) ≥ P(yx). +P(y|x) − 3P(u) ≤ P(yx) ≤ P(y|x) + 3.5P(u). +Let 3.5P(u) + 3P(u) ≤ 2ǫ, we have P(u) ≤ +4 +13ǫ, and +P(y|x) − 12 +13ǫ ≤ +P(yx) +≤ P(y|x) + 14 +13ǫ. +Therefore, P(yx) is ǫ-identified to P(y|x) − 12 +13ǫ + ǫ = +P(y|x) + +ǫ +13. +8.3 +Proof of Theorem 12 +Proof. From the bounds of PNS in Equations (3) and (4) is +as follows: +max + + + + + +0, +P(yx) − P(yx′), +P(y) − P(yx′), +P(yx) − P(y) + + + + + +≤ PNS +min + + + + + + + + + +P(yx), +P(y′ +x′), +P(x, y) + P(x′, y′), +P(yx) − P(yx′)+ ++P(x, y′) + P(x′, y) + + + + + + + + + +≥ PNS. + +Let P(yx) − 0 ≤ 2ǫ, we obtain that PNS is ǫ-identified to ǫ if +P(yx) ≤ 2ǫ, Equation (19) holds. +Similarly, the rest of 20 equations can be obtained by letting +P(y′ +x′) − 0 +≤ +2ǫ, +P(x, y) + P(x′, y′) − 0 +≤ +2ǫ, +P(yx) − P(yx′) + P(x, y′) + P(x′, y) − 0 +≤ +2ǫ, +P(yx) − (P(yx) − P(yx′)) +≤ +2ǫ, +P(y′ +x′) − (P(yx) − P(yx′)) +≤ +2ǫ, +P(x, y) + P(x′, y′) − (P(yx) − P(yx′)) +≤ +2ǫ, +P(yx) − P(yx′) + P(x, y′) + P(x′, y)− +(P(yx) − P(yx′)) +≤ +2ǫ, +P(yx) − (P(y) − P(yx′)) +≤ +2ǫ, +P(y′ +x′) − (P(y) − P(yx′)) +≤ +2ǫ, +P(x, y) + P(x′, y′) − (P(y) − P(yx′)) +≤ +2ǫ, +P(yx) − P(yx′) + P(x, y′) + P(x′, y)− +(P(y) − P(yx′)) +≤ +2ǫ, +P(yx) − (P(yx) − P(y)) +≤ +2ǫ, +P(y′ +x′) − (P(yx) − P(y)) +≤ +2ǫ, +P(x, y) + P(x′, y′) − (P(yx) − P(y)) +≤ +2ǫ, +P(yx) − P(yx′) + P(x, y′) + P(x′, y)− +(P(yx) − P(y)) +≤ +2ǫ. +8.4 +Proof of Theorem 13 +Proof. From the bounds of PN in Equations (5) and (6) is as +follows: +max +� +0, +P (y)−P (yx′) +P (x,y) +� +≤ PN ≤ min +� +1, +P (y′ +x′)−P (x′,y′) +P (x,y) +� +Let +P (y′ +x′)−P (x′,y′) +P (x,y) +−0 ≤ 2ǫ, we obtain that PN is ǫ-identified +to ǫ if P(y′ +x′) − P(x′, y′) ≤ 2P(x, y)ǫ, Equation (40) holds. +Similarly, the rest of 4 equations can be obtained by letting +1 − P(y) − P(yx′) +P(x, y) +≤ +2ǫ, +P(y′ +x′) − P(x′, y′) +P(x, y) +− P(y) − P(yx′) +P(x, y) +≤ +2ǫ. +8.5 +Proof of Theorem 14 +Proof. From the bounds of PS in Equations (7) and (8) is as +follows: +max +� +0, +P (y′)−P (y′ +x) +P (x′,y′) +� +≤ PS ≤ min +� +1, +P (yx)−P (x,y) +P (x′,y′) +� +Let P (yx)−P (x,y) +P (x′,y′) +− 0 ≤ 2ǫ, we obtain that PS is ǫ-identified +to ǫ if P(yx) − P(x, y) ≤ 2P(x′, y′)ǫ, Equation (45). +Similarly, the rest of 4 conditions can be obtained by letting +1 − P(y′) − P(y′ +x) +P(x′, y′) +≤ +2ǫ, +P(yx) − P(x, y) +P(x′, y′) +− P(y′) − P(y′ +x) +P(x′, y′) +≤ +2ǫ. +8.6 +Proof of Theorem 15 +Proof. +f(c) += +βP(yx, y′ +x′|c) + γP(yx, yx′|c) + +θP(y′ +x, y′ +x′|c) + δP(y′ +x, yx′|c) += +βP(yx, y′ +x′|c) + γ[P(yx|c) − P(yx, y′ +x′|c)] + +θ[P(y′ +x′) − P(yx, y′ +x′|c)] + δP(y′ +x, yx′|c) += +γP(yx|c) + θP(y′ +x′|c) + (β − γ − θ)P(yx, y′ +x′|c) + +δP(y′ +x, yx′|c). +(50) +Note that, we have, +P(y′ +x, yx′|c) = P(yx, y′ +x′|c) − P(yx|c) + P(yx′|c). +(51) +Substituting Equation (51) into Equation (50), we have, +f(c) += +γP(yx|c) + θP(y′ +x′|c) + (β − γ − θ)P(yx, y′ +x′|c) + +δP(y′ +x, yx′|c) += +γP(yx|c) + θP(y′ +x′|c) + (β − γ − θ)P(yx, y′ +x′|c) + +δ[P(yx, y′ +x′|c) − P(yx|c) + P(yx′|c)] += +(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′ +x′|c) + +(β − γ − θ + δ)P(yx, y′ +x′|c). +Case 1: If β − γ − θ + δ ≥ 0, +f(c) +≤ +(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′ +x′|c) + +β − γ − θ + δ +2 ++ |β − γ − θ + δ| +2 += +(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′ +x′|c) + +β − γ − θ + δ. +and, +f(c) +≥ +(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′ +x′|c) + +β − γ − θ + δ +2 +− |β − γ − θ + δ| +2 += +(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′ +x′|c). +Therefore, f(c) is |β−γ−θ+δ| +2 +-identified to (γ − δ)P(yx|c) + +δP(yx′|c) + θP(y′ +x′|c) + β−γ−θ+δ +2 +. +Case 2: If β − γ − θ + δ < 0, +f(c) +≤ +(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′ +x′|c) + +β − γ − θ + δ +2 ++ |β − γ − θ + δ| +2 += +(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′ +x′|c). +and, +f(c) +≥ +(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′ +x′|c) + +β − γ − θ + δ +2 +− |β − γ − θ + δ| +2 += +(γ − δ)P(yx|c) + δP(yx′|c) + θP(y′ +x′|c) + +β − γ − θ + δ. + +Therefore, f(c) is |β−γ−θ+δ| +2 +-identified to (γ − δ)P(yx|c) + +δP(yx′|c) + θP(y′ +x′|c) + β−γ−θ+δ +2 +. + diff --git a/NdFLT4oBgHgl3EQfOC8n/content/tmp_files/load_file.txt b/NdFLT4oBgHgl3EQfOC8n/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..df83c9d1f4344551053dd8ebd7ff87ef41002788 --- /dev/null +++ b/NdFLT4oBgHgl3EQfOC8n/content/tmp_files/load_file.txt @@ -0,0 +1,579 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf,len=578 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='12022v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='AI] 27 Jan 2023 ǫ-Identifiability of Causal Quantities Ang Li , Scott Mueller and Judea Pearl Cognitive Systems Laboratory, Department of Computer Science, University of California, Los Angeles, Los Angeles, California, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' {angli, scott, judea}@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='edu Abstract Identifying the effects of causes and causes of ef- fects is vital in virtually every scientific field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Of- ten, however, the needed probabilities may not be fully identifiable from the data sources available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' This paper shows how partial identifiability is still possible for several probabilities of causation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' We term this ǫ-identifiability and demonstrate its use- fulness in cases where the behavior of certain sub- populations can be restricted to within some nar- row bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' In particular, we show how unidentifi- able causal effects and counterfactual probabilities can be narrowly bounded when such allowances are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Often those allowances are easily measured and reasonably assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Finally, ǫ-identifiability is applied to the unit selection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' 1 Introduction Both Effects of Causes (EoC) and Causes of Effects (CoE) play an important role in several fields, such as health science, social science, and business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' For example, the causal effects identified by the adjustment [Pearl, 1993] formula helps decision-maker avoid randomized controlled trial using purely observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' For another exam- ple, probabilities of causation have been proven critical in personalized decision-making [Mueller and Pearl, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Be- sides, a linear combination of probabilities of causation has been used to solve the unit selection problem defined by Li and Pearl [Li and Pearl, 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Li and Pearl, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Li and Pearl, 2022d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Causal quantities can also increase the accuracy of machine learning models by combining causal quantities with the model’s label [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The causal quantities have been studied for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Pearl first defined the causal quantities such as causal effects [Pearl, 1993], probability of necessity and suffi- ciency (PNS), probability of sufficiency (PS), and prob- ability of necessity (PN) [Pearl, 1999] and their identi- fiability [Pearl, 2009] using the structural causal model (SCM) [Galles and Pearl, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Halpern, 2000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Pearl also proposed the identification conditions of the causal ef- fects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', back-door and front-door criteria) [Pearl, 1993].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Pearl, Bareinboim, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' have studied more conditions for identifying the causal effects [Bareinboim and Pearl, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Shpitser and Pearl, 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' If the causal effects are not iden- tifiable, the informative bounds are given by Li and Pearl using non-linear programming [Li and Pearl, 2022c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Then, Tian and Pearl proposed the identification conditions of the binary probabilities of causation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', monotonicity) [Tian and Pearl, 2000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' If the probabilities of causation are not identifiable, Tian and Pearl [Tian and Pearl, 2000] also have informative tight bounds for them using Balke’s Linear programming [Balke and Pearl, 1997].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Mueller, Li, and Pearl [Mueller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', 2021], as well as Dawid [Dawid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', 2017], increased those bounds using additional covariate informa- tion and the corresponding causal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Recently, Li and Pearl also proposed the theoretical work for non-binary prob- abilities of causation [Li and Pearl, 2022a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' In real-world applications, decision-makers are more likely to have identifiable cases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', the causal quantities have point estimations) because the bounds under unidentifiable cases may be less informative (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='1 ≤ PNS ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Besides, estimating the bounds often requires a combination of exper- imental and observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' So we wonder if something is sitting between the identifiable and the bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Inspired by the idea of the confidence interval, in this paper, we proposed the definition of ǫ-identifiability, in which more conditions of ǫ-identifiability can be found while the estimations of the causal quantities are still near point estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' 2 Preliminaries Here, we review the definition of PNS, PS, and PN de- fined by Pearl [Pearl, 1999], as well as the definition of identifiable and the conditions for identifying PNS, PS, and PN [Tian and Pearl, 2000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Besides, we review the tight bounds of PNS, PS, and PN when they are unidentifiable [Tian and Pearl, 2000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Readers who are familiar with the above knowledge may skip this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Similarly to any works mentioned above, we used the causal language of the SCM [Galles and Pearl, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Halpern, 2000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The introductory counterfactual sentence “Variable Y would have the value y, had X been x” in this language is denoted by Yx = y, and shorted as yx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' We have two types of data: experimental data, which is in the form of causal effects (denoted as P(yx)), and observational data, which is in the form of a joint probability function (denoted as P(x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' First, the definition of identifiable for any causal quantities defined using SCM is as follows: Definition 1 (Identifiability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Let Q(M) be any computable quantity of a class of SCM M that is compatible with graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' We say that Q is identifiable in M if, for any pairs of models M1 and M2 from M, Q(M1) = Q(M2) whenever PM1(v) = PM2(v), where P(v) is the statistical data over the set V of observed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' If our observations are lim- ited and permit only a partial set FM of features (of PM(v)) to be estimated, we define Q to be identifiable from FM if Q(M1) = Q(M2) whenever FM1 = FM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' [Pearl, 2009] Second, the definitions of three binary probabilities of cau- sation defined using SCM are as follow [Pearl, 1999]: Definition 2 (Probability of necessity (PN)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Let X and Y be two binary variables in a causal model M, let x and y stand for the propositions X = true and Y = true, respec- tively, and x′ and y′ for their complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The probability of necessity is defined as the expression PN = ∆ P(Yx′ = false|X = true, Y = true) = ∆ P(y′ x′|x, y) Definition 3 (Probability of sufficiency (PS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Let X and Y be two binary variables in a causal model M, let x and y stand for the propositions X = true and Y = true, respec- tively, and x′ and y′ for their complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The probability of sufficiency is defined as the expression PS = ∆ P(yx|y′, x′) Definition 4 (Probability of necessity and sufficiency (PNS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Let X and Y be two binary variables in a causal model M, let x and y stand for the propositions X = true and Y = true, respectively, and x′ and y′ for their complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The proba- bility of necessity and sufficiency is defined as the expression PNS = ∆ P(yx, y′ x′) Third, we review the identification conditions for causal effects [Pearl, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Pearl, 1995].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Definition 5 (Back-door criterion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Given an ordered pair of variables (X, Y ) in a directed acyclic graph G, a set of vari- ables Z satisfies the back-door criterion relative to (X, Y ), if no node in Z is a descendant of X, and Z blocks every path between X and Y that contains an arrow into X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' If a set of variables Z satisfies the back-door criterion for X and Y , the causal effects of X on Y are identifiable and given by the adjustment formula: P(yx) = � z P(y|x, z)P(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (1) Definition 6 (Front-door criterion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' A set of variables Z is said to satisfy the front-door criterion relative to an ordered pair of variables (X, Y ) if: Z intercepts all directed paths from X to Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' there is no back-door path from X to Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' and all back-door paths from Z to Y are blocked by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' If a set of variables Z satisfies the front-door criterion for X and Y , and P(x, Z) > 0, then the causal effects of X on Y are identifiable and given by the adjustment formula: P(yx) = � z P(z|x) � x′ P(y|x′, z)P(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' If causal effects are not identifiable, Tian and Pearl [Tian and Pearl, 2000] provided the following bounds for the causal effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' P(x, y) ≤ P(yx) ≤ 1 − P(x, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (2) Finally, we review the identification conditions for PNS, PS, and PN [Tian and Pearl, 2000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (Monotonicity) A Variable Y is said to be mono- tonic relative to variable X in a causal model M iff y′ x ∧ yx′ = false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' If Y is monotonic relative to X, then PNS, PN, and PS are all identifiable, and PNS = P(yx) − P(yx′), PN = P(y) − P(yx′) P(x, y) , PS = P(yx) − P(y) P(x′, y′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' If PNS, PN, and PS are not identifiable, informative bounds are given by Tian and Pearl [Tian and Pearl, 2000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' max \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' P(yx) − P(yx′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' P(y) − P(yx′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' P(yx) − P(y) \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe ≤ PNS (3) min \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 P(yx),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' P(y′ x′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y) + P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' P(yx) − P(yx′)+ P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′) + P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y) \uf8fc \uf8f4 \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8f4 \uf8fe ≥ PNS (4) max � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' P (y)−P (yx′) P (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='y) � ≤ PN (5) min � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' P (y′ x′)−P (x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='y′) P (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='y) � ≥ PN (6) max � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' P (y′)−P (y′ x) P (x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='y′) � ≤ PS (7) min � 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' P (yx)−P (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='y) P (x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='y′) � ≥ PS (8) The identification conditions mentioned above (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', back- door and front-door criteria and monotonicity) are robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' However, it may still be hard to achieve in real-world appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' In this work, we extend the definition of identifia- bility, in which a sufficiently small interval is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' By the new definition, the estimates of causal quantities are still near point estimations, and more conditions for identifiability could be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' If nothing is specified, the discussion in this paper will be restricted to binary treatment and effect (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', X and Y are binary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' 3 Main Results First, we extend the definition of identifiability, which we call ǫ-identifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Definition 9 (ǫ-Identifiability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Let Q(M) be any computable quantity of a class of SCM M that is compatible with graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' We say that Q is ǫ-identifiable in M (and ǫ-identified to q) if, there exists q s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' for any model m from M, Q(m) ∈ [q − ǫ, q + ǫ] with statistical data PM(v), where P(v) is the statistical data over the set V of observed variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' If our observations are limited and permit only a partial set FM of features (of PM(v)) to be estimated, we define Q to be ǫ- identifiable from FM if Q(m) ∈ [q − ǫ, q + ǫ] with statistical data FM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' With the above definition, the causal quantity is at a max- imum distance of ǫ from its true value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' We will use the in- fix operator symbol ≈ǫ to represent its left-hand side being within ǫ of its right-hand side: r ≈ǫ q ⇐⇒ r ∈ [q − ǫ, q + ǫ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (9) The following sections explicate conditions for ǫ- identifiability of causal effects, PNS, PS, and PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='1 ǫ-Identifiability of Causal Effects The causal effect P(YX) can be ǫ-identified with information about the observational joint distribution P(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' This can be seen by rewriting Equation (2) as: P(x, y) ⩽ P(yx) ⩽ P(x, y) + P(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (10) Here, P(yx) is ǫ-identified to P(x, y) + ǫ when P(x′) ⩽ 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' This ǫ-identification indicates a lower bound of P(x, y) and an upper bound of P(x, y) + 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Since P(x′) ⩽ 2ǫ, these bounds are equivalent to (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Notably, only P(x, y) and an upper bound on P(x′) are necessary to ǫ-identify P(yx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' This is generalized in Theorem 10, without any assumptions of the causal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The causal effect P(YX) is ǫ-identified as fol- lows: P(yx) ≈ǫ P(x, y) + ǫ if P(x′) ⩽ 2ǫ, (11) P(y′ x) ≈ǫ P(x, y′) + ǫ if P(x′) ⩽ 2ǫ, (12) P(yx′) ≈ǫ P(x′, y) + ǫ if P(x) ⩽ 2ǫ, (13) P(y′ x′) ≈ǫ P(x′, y′) + ǫ if P(x) ⩽ 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (14) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' See Appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' When the complete distribution P(X, Y ) is known, The- orem 10 provides no extra precision over Equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Its power comes from when only part of the distribution is known and only an upper bound on P(X) is available or able to be assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Knowledge of a causal structure can aid ǫ-identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' In Figure 1, there is a binary confounder U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' If the full joint dis- tribution P(X, Y, U) was available, the causal effect P(YX) could be computed simply through the backdoor adjustment formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' In the absence of the full joint distribution, Theo- rem 11 allows ǫ-identification of P(yx) with only knowledge of P(x) and the conditional probability P(y|x) as well as an upper bound on P(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' U X Y Figure 1: The causal graph, where X is a binary treatment, Y is a binary effect, and U is a binary confounder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Given the causal graph in Figure 1 and P(u) ≤ P(x) − c for some constant c, where 0 < c ⩽ P(x), P(yx) ≈ǫ P(y|x) + P(x) − c 2cP(x) + P(x) + c · ǫ if P(u) ≤ 2cP(x) 2cP(x) + P(x) + c · ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (15) Specifically, if P(x) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='5, then the causal effect P(yx) is ǫ-identified to P(y|x) + ǫ 13 if P(u) < 4 13ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' See Appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Note that x ∈ {x, x′}, y ∈ {y, y′}, and u ∈ {u, u′} in Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The constant c should be maximized satisfying both c ⩽ P(x) − P(u) and the condition in Equation (15) for a given ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The larger c is, the closer P(yx) is ǫ-identified to P(y|x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' This needs to be balanced with minimizing ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' As an example, if P(x) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='5 and P(u) ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='1, then the causal effect P(yx) is ǫ-identified to P(y|x) + ǫ 13 if P(u) ⩽ 4 13ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Essentially, P(yx) is ǫ-identified to P(y|x) plus some frac- tion of ǫ when P(u) is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Therefore, the causal effect P(yx) is near P(y|x) if P(U) is specific (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', P(u) or P(u′) is minimal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' In this case, Theorem 11 can be advantageous over the backdoor adjustment formula to com- pute P(yx), even when data on X, Y , and U are available, because P(Y |X, U), required for the adjustment formula, is impractical to estimate with P(U) close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='2 ǫ-Identifiability of PNS Even though Tian and Pearl derived tight bounds on PNS [Tian and Pearl, 2000], the PNS can be potentially further narrowed when taking into account particular upper bound assumptions on causal effects or observational probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' This can be seen by analyzing the bounds of PNS in Equa- tions (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Picking any of the arguments to the max function of the lower bound and any of the arguments to the min function of the upper bound, we can make a condition that the range of those two values is less than 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' For ex- ample, let us pick the second argument of the max function, P(yx) − P(yx′), and the first argument of the min function, P(yx): P(yx) − [P(yx) − P(yx′)] ⩽ 2ǫ, P(yx′) ⩽ 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (16) Equation (16) is the assumption and the PNS is the ǫ-identified to ǫ above the lower bound or ǫ below the upper bound: PNS ≈ǫ P(yx) − P(yx′) + ǫ, or (17) PNS ≈ǫ P(yx) − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (18) Since it is assumed that P(yx′) ⩽ 2ǫ, Equation (17) is equiv- alent to Equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The complete set of ǫ-identifications and associated conditions are stated in Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The PNS is ǫ-identified as follows: PNS ≈ǫ ǫ if P(yx) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (19) PNS ≈ǫ ǫ if P(y′ x′) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (20) PNS ≈ǫ ǫ if P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y) + P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (21) PNS ≈ǫ ǫ if P(yx) − P(yx′)+ P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′) + P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (22) PNS ≈ǫ P(yx) − ǫ if P(yx′) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (23) PNS ≈ǫ P(y′ x′) − ǫ if P(y′ x) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (24) PNS ≈ǫ P(yx)− P(yx′) + ǫ if P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′) + P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (25) PNS ≈ǫ P(yx)− P(yx′) + ǫ if P(yx′) − P(yx)+ P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y) + P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (26) PNS ≈ǫ P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y)− P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′) − ǫ if P(yx′) − P(yx)+ P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y) + P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (27) PNS ≈ǫ P(y′ x′) − ǫ if P(y′) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (28) PNS ≈ǫ P(yx) − ǫ if P(yx) + P(yx′)− P(y) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (29) PNS ≈ǫ P(y) − P(yx′) + ǫ if P(yx) + P(yx′)− P(y) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (30) PNS ≈ǫ P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y)+ P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′) − ǫ if P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′) + P(yx′)− P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (31) PNS ≈ǫ P(y) − P(yx′) + ǫ if P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′) + P(yx′)− P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (32) PNS ≈ǫ P(y) − P(yx′) + ǫ if P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y) + P(y′ x′)− P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (33) PNS ≈ǫ P(yx) − ǫ if P(y) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (34) PNS ≈ǫ P(y′ x′) − ǫ if P(y′ x′) − P(yx)+ P(y) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (35) PNS ≈ǫ P(y) − P(yx′) + ǫ if P(y′ x′) − P(yx)+ P(y) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (36) PNS ≈ǫ P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y)+ P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′) − ǫ if P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y) + P(y′ x)− P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (37) PNS ≈ǫ P(yx) − P(y) + ǫ if P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y) + P(y′ x)− P(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′) ⩽ 2ǫ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (38) PNS ≈ǫ P(yx) − P(y) + ǫ if P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y) + P(y′ x′)− P(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' y′) ⩽ 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (39) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' See Appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Note that in the above theorem, eight conditions consist solely of experimental probabilities or solely of observational probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' This potentially eliminates the need for some types of studies, at least partially, even when estimating a counterfactual quantity such as PNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' For example, if a decision-maker knows that P(y) is large (P(y) ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='95), they can immediately conclude PNS ≈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='05 P(y′ x′) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='05 with- out knowing the specific value of P(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Thus, only a control group study would be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='3 ǫ-Identifiability of PN and PS Tian and Pearl derived tight bounds on PN and PS in addi- tion to PNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Similar to the derivation of Theorem 12, we can potentially narrow those bounds by taking into account upper bound assumptions on causal effects or observational proba- bilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The set of ǫ-identifications and associated conditions are stated in Theorems 13 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The PN is ǫ-identified as follows: PN ≈ǫ ǫ if P(y′ x′) − P(x′, y′) ⩽ 2ǫP(x, y), (40) PN ≈ǫ 1 − ǫ if P(yx′) − P(x′, y) ⩽ 2ǫP(x, y), (41) PN ≈ǫ P(y) − P(yx′) P(x, y) + ǫ if P(yx′) − P(x′, y) ⩽ 2ǫP(x, y), (42) PN ≈ǫ P(y′ x′) − P(x′, y′) P(x, y) − ǫ if P(x, y′) ⩽ 2ǫP(x, y), (43) PN ≈ǫ P(y) − P(yx′) P(x, y) + ǫ if P(x, y′) ⩽ 2ǫP(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (44) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' See Appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Table 1: Results of an observational study with 1500 individuals who have access to the medicine, where 1260 individuals chose to receive the medicine and 240 individuals chose not to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Take the medicine Take no medicine Recovered 780 210 Not recovered 480 30 Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The PS is ǫ-identified as follows: PS ≈ǫ ǫ if P(yx) − P(x, y) ⩽ 2ǫP(x′, y′), (45) PS ≈ǫ 1 − ǫ if P(y′ x) − P(x, y′) ⩽ 2ǫP(x′, y′), (46) PS ≈ǫ P(y′) − P(y′ x) P(x′, y′) + ǫ if P(y′ x) − P(x, y′) ⩽ 2ǫP(x′, y′), (47) PS ≈ǫ P(yx) − P(x, y) P(x′, y′) − ǫ if P(x′, y) ⩽ 2ǫP(x′, y′), (48) PS ≈ǫ P(y′) − P(y′ x) P(x′, y′) + ǫ if P(x′, y) ⩽ 2ǫP(x′, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' (49) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' See Appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' 4 Examples Here, we illustrate how to apply ǫ-Identifiability in real appli- cations by two simulated examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='1 Causal Effects of Medicine Consider a medicine manufacturer who wants to know the causal effect of a new medicine on a disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' They conducted an observational study where 1500 patients were given access to the medicine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' the results of the study are summarized in Ta- ble 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' In addition, the expert from the medicine manufacturer acknowledged that family history is the only confounder of taking medicine and recovery, and the family history of the disease is extremely rare;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' only 1% of the people have the fam- ily history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Let X = x denote that a patient chose to take the medicine, and X = x′ denote that a patient chose not to take the medicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Let Y = y denote that a patient recovered, and Y = y′ denote that a patient did not recover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Let U = u de- note that a patient has the family history, and U = u′ denote that a patient has no family history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' To obtain the causal effect of the medicine (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', using ad- justment formula (1)), we have to know the observational data associated with family history, which is difficult to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Fortunately, from Table 1, we obtain that P(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='84 and P(y|x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' We also have the prior that P(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Since 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='01 = P(u) ≤ P(x) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='8 (let c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='8) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='01 = P(u) < 2c∗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='025P (x) 2cP (x)+P (x)+c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='0113, we can ap- ply Theorem 11 to obtain that P(yx) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='025-identified to P(y|x)+ P (x)−c 2cP (x)+P (x)+c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='025 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' This means the causal effect of the medicine is very close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='62 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='025 close), which can not be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='025 far from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Then the medicine man- ufacturer can conclude that the causal effect of the medicine is roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='62 without knowing the observational data asso- ciated with the family history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Or even simpler, note that P(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='84 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='5 and P(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='01 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='1, P(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='01 < 4 13 ∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='035 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='0108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' We obtain that P(yx) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='035-identified to P(y|x) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='035 13 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The decision-maker can make the same conclusion as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='2 PNS of Flu Shot Consider a newly invented flu shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' After a vaccination com- pany introduced a new flu shot, the number of people infected by flu reached the lowest point in 20 years (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', less than 5% of people infected by flu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The government concluded that the new flu shot is the key to success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' However, some anti- vaccination associations believe it is because people’s physi- cal quality increases yearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Therefore, they all want to know how many percentages of people are uninfected because of the flu shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The PNS of the flu shot (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', the percentage of individuals who would not infect by the flu if they had taken the flu shot and would infect otherwise) is indeed what they want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Let X = x denote that an individual has taken the flu shot and X = x′ denote that an individual has not taken the flu shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Let Y = y denote an individual infected by the flu and Y = y′ denote an individual not infected by the flu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' If they want to apply the bounds of PNS in Equations (3) and (4), they must conduct both experimental and observa- tional studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' However, note that P(y) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='05, one could apply Equation (34) in Theorem 12, which PNS is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='025- identified to P(yx)− 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='025 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', PNS is very close to P(yx)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Thus, according to [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', 2022], only an experimental study for the treated group with a sample size of 385 is ad- equate for estimating PNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' 5 ǫ-Identifiability in Unit Selection Problem One utility of the causal quantities is the unit selection prob- lem [Li and Pearl, 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Li and Pearl, 2019], in which Li and Pearl defined an objective causal function to select a set of individuals that have the desired mode of behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Let X denote the binary treatment and Y denote the bi- nary effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' According to Li and Pearl, individuals were di- vided into four response types: Complier (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', P(yx, y′ x′)), always-taker (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', P(yx, yx′)), never-taker (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', P(y′ x, y′ x′)), and defier (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', P(y′ x, yx′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Suppose the payoff of selecting a complier, always-taker, never-taker, and defier is β, γ, θ, δ, respectively (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', benefit vector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The objective function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', benefit function) that optimizes the composition of the four types over the selected set of individuals c is as follows: f(c) = βP(yx, y′ x′|c) + γP(yx, yx′|c) + θP(y′ x, y′ x′|c) + δP(y′ x, yx′|c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Li and Pearl provided two types of identifiability condi- tions for the benefit function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' One is about the response type such that there is no defier in the population (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', monotonic- ity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Another is about the benefits vector’s relations, such that β + δ = γ + θ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', gain equality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' These two conditions Table 2: Results of an experimental study with 1500 randomly se- lected customers were forced to apply the discount, and 1500 ran- domly selected customers were forced not to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Discount No discount Bought the purchase 900 750 No purchase 600 750 are helpful but still too specific and challenging to satisfy in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' If the benefit function is not identifi- able, it can be bounded using experimental and observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Here in this paper, we extend the gain equality to the ǫ-identifiability as stated in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Given a causal diagram G and distribution compatible with G, let C be a set of variables that does not contain any descendant of X in G, then the benefit function f(c) = βP(yx, y′ x′|c) + γP(yx, yx′|c) + θP(y′ x, y′ x′|c) + δP(yx′, y′ x|c) is |β−γ−θ+δ| 2 identified to (γ − δ)P(yx|c) + δP(yx′|c) + θP(y′ x′|c) + β−γ−θ+δ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' One critical use case of the above theorem is that decision- makers usually only care about the sign (gain or lose) of the benefit function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Decision-makers can apply the above theo- rem before conducting any observational study to see if the sign of the benefit function can be determined, as we will il- lustrate in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='1 Example: Non-immediate Profit Consider the most common example in [Li and Pearl, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' A sale company proposed a discount on a purchase in order to increase the total non-immediate profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The company assessed that the profit of offering the dis- count to complier, always-taker, never-taker, and defier is $100, −$60, $0, −$140, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Let X = x denote that a customer applied the discount, and X = x denote that a customer did not apply the discount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Let Y = y denote that a customer bought the purchase and Y = y′ denote that a cus- tomer did not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The benefit function is then (here c denote all customers) f(c) = 100P(yx, y′ x′|c) − 60P(yx, yx′|c) + 0P(y′ x, y′ x′|c) − 140P(y′ x, yx′|c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The company conducted an experimental study where 1500 randomly selected customers were forced to apply the dis- count, and 1500 randomly selected customers were forced not to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The results are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The experimental data reads P(yx|c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='6 and P(yx′|c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Before conducting any observational study, one can con- clude that the benefit function is 10-identified to −12 using Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' This result indicates that the benefit function is at most 10 away from −12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' thus, the benefit function is nega- tive regardless of the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The decision-maker then can easily conclude that the discount should not offer to the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' 6 Discussion We have defined the ǫ-identifiability of causal quantities and provided a list of ǫ-identifiable conditions for causal effects, PNS, PN, and PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' We still have some further discussions about the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' First, all conditions except Theorem 11 are conditions from observational or experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' In other words, if some of the observational or experimental distributions satisfied a par- ticular condition, then the causal quantities are ǫ-identifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' These conditions are advantageous in real-world applications as no specific causal graph is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' However, we still love to discover more graphical conditions of ǫ-identifiability, such as back-door or front-door criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Second, the bounds of PNS, PS, PN, and the benefit func- tion can be narrowed by covariates information with their causal structure [Dawid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Li and Pearl, 2022d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Mueller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The ǫ-identifiability can also be ex- tended if covariates information and their causal structure are available, which should be an exciting direction in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Third, monotonicity is defined using a causal quantity, and in the meantime, monotonicity is also an identifiable condi- tion for other causal quantities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', PNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Thus, another charming direction is how the ǫ-identifiability of monotonic- ity affects the ǫ-identifiability of other causal quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' 7 Conclusion In this paper, we defined the ǫ-identifiability of causal quan- tities, which is easier to satisfy in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' We provided the ǫ-identifiability conditions for causal effects, PNS, PS, and PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' We further illustrated the use cases of the proposed conditions by simulated examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' References [Balke and Pearl, 1997] Alexander A Balke and Judea Pearl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Probabilistic counterfactuals: Semantics, computation, and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Technical report, UCLA Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' of Com- puter Science, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' [Bareinboim and Pearl, 2012] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Bareinboim and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Pearl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Causal inference by surrogate experiments: z- identifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' In Nando de Freitas and Kevin Murphy, editors, Proceedings of the Twenty-Eighth Conference on Uncertainty in Artificial Intelligence, pages 113–120, Corvallis, OR, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' AUAI Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' [Dawid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=', 2017] Philip Dawid, Monica Musio, and Rossella Murtas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' The probability of causation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Law, Prob- ability and Risk, (16):163–179, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' [Galles and Pearl, 1998] David Galles and Judea Pearl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' An axiomatic characterization of causal counterfactuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Foun- dations of Science, 3(1):151–182, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' [Halpern, 2000] Joseph Y Halpern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Axiomatizing causal reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Journal of Artificial Intelligence Research, 12:317–337, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' [Li and Pearl, 2019] Ang Li and Judea Pearl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Unit selec- tion based on counterfactual logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' In Proceedings of the Twenty-Eighth International Joint Conference on Ar- tificial Intelligence, IJCAI-19, pages 1793–1799.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Interna- tional Joint Conferences on Artificial Intelligence Organi- zation, 7 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' [Li and Pearl, 2022a] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Li and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Pearl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Prob- abilities of causation with non-binary treat- ment and effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFLT4oBgHgl3EQfOC8n/content/2301.12022v1.pdf'} +page_content=' Technical Report R-516, 0, we say a product strat- +egy ˆσ is an ǫ-approximate Nash equilibrium (ǫ-NE) if no one can achieve more than ǫ utility gain by +deviating from her current strategy. Formally, +ui(σi, ˆσ−i) ≤ ui(ˆσi, ˆσ−i) + ǫ, ∀i ∈ [n], σi ∈ ∆Ai. +(ǫ-NE) +The definition of ǫ-NE reflects the idea that players might not be willing to deviate from their strategies +when the amount of utility they could gain by doing so is tiny (not more than ǫ). +Coarse Correlated Equilibrium (CCE) +We say a joint (possibly correlated) strategy π∗ is a CCE +if no player can receive a higher payoff by acting independently, i.e., +ui(σi, π∗ +−i) ≤ ui(π∗), ∀i ∈ [n], σi ∈ ∆Ai, +(CCE) +and we say ˆπ is an ǫ-approximate coarse correlated equilibrium (ǫ-CCE) for ǫ > 0 if +ui(σi, ˆπ−i) ≤ ui(ˆπ) + ǫ, ∀i ∈ [n], σi ∈ ∆Ai, +(ǫ-CCE) +The difference between NE and CCE is that in an NE, players execute their strategy individu- +ally in a decentralized way, while in a CCE, players’ strategies are possibly correlated. +A stan- +dard technique to correlate the strategy is sending each player a signal from a centralized controller +[Shoham and Leyton-Brown, 2008]. +Correlated Equilibrium (CE) +CE is similar to CCE, except that in a CE, each player can observe +her recommended action before she acts. Thus, player i deviates her strategy through strategy mod- +ification φi : Ai → Ai. φi maps actions in Ai to possibly different actions in Ai. Based on strategy +modification, we say a joint (possibly correlated) strategy π∗ is a CE if +� +a∈A +π∗(a)ui(φi(ai), a−i) ≤ ui(π∗), ∀i, ∀φi, +(CE) +and a joint strategy ˆπ is an ǫ-approximate correlated equilibrium (ǫ-CE) for ǫ > 0 if +� +a∈A +ˆπ(a)ui(φi(ai), a−i) ≤ ui(ˆπ) + ǫ, ∀i, ∀φi, +(ǫ-CE) +Note that for a finite n-player normal-form game, at least one NE, CE, and CCE must exist. This +is because NE always exists [Nash et al., 1950] and NE ⊆ CE ⊆ CCE. +Equilibrium Approximation +To evaluate the quality of a joint strategy to approximate an equilib- +rium, we define approximation based on exploitability [Lockhart et al., 2019, Goktas and Greenwald, +2022]. +Definition 2.1 (Exploitability and Approximation). Given a joint strategy π, the exploitability (or +regret) Ei(π, u) of player i is the maximum payoff gain of i by deviating from her current strategy, i.e., +Ei(π, u) := max +σ′ +i +ui(σ′ +i, π−i) − ui(π) = max +a′ +i +ui(a′ +i, π−i) − ui(π) +and the exploitability under strategy modification ECE +i +(π, u) of player i is the maximum payoff gain of +i by deviating through strategy modification, i.e., +ECE +i +(π, u) := max +φi +� +a∈A +π(a)ui(φi(ai), a−i) − ui(π). +3 + +Algorithm 1 Example: learning NE/CCE approximator via minibatch SGD +1: Input: Training set S +2: Parameters: Number of iterations T > 0, batch size B > 0, learning rate η > 0, initial parameters +w0 ∈ Rd of the approximator model. +3: for t = 0 to T do +4: +Receive minibatch St = {u(1), . . . , u(B)} ⊂ S +5: +Compute the empirical average approximation of St: +6: +LSt(f wt) ← 1 +B +�B +i=1 E(f wt(u(i)), u(i)) +7: +Update model parameters: +8: +wt+1 ← wt − η∇wtLSt(f wt) +9: end for +The equilibrium approximation is defined as the maximum exploitability over all players 2, i.e., +E(π, u) := +� +maxi∈[n] Ei(π, u) +, for NE and CCE +maxi∈[n] ECE +i +(π, u) +, for CE +Based on approximation, we can restate the definition of solution concepts. A product strategy σ +is an NE of game Γu if E(σ, u) = 0 and is an ǫ-NE if E(σ, u) ≤ ǫ. A joint strategy π is a (C)CE of Γu +if E(π, u) = 0 and is an ǫ-(C)CE if E(π, u) ≤ ǫ. +2.2 +Equilibrium Approximator +The equilibrium approximators, including NE, CE, and CCE approximators, aim to predict solution +concepts from game representations. In our paper, we fix the number of players n and action space A. +We denote by U the set of all the possible game payoffs. The NE approximator f NE : U → ×i∈[n]∆Ai +maps a game payoff to a product strategy, where f NE(u)i ∈ ∆Ai is player i’s predicted strategy. We +define FNE as the function class of the NE approximator. Similarly, we define (C)CE approximator +as f (C)CE : U → ∆A and (C)CE approximator class as F(C)CE. +An equilibrium approximator can be learned through machine learning paradigms based on empir- +ical data. For instance, Feng et al. [2021] learn the NE approximator using the game payoffs generated +in the learning process of PSRO, and optimize the approximator by gradient descent in exploitability. +Marris et al. [2022] learn the CE and CCE approximators using the games i.i.d. generated from a +manually designed distribution, and optimize the approximators using maximum welfare minimum +relative entropy loss. Such a loss balances the equilibrium approximation, the social welfare, and the +relative entropy of the joint strategy. Additionally, another simple way to learn NE or CCE equilibrium +approximator is to apply minibatch stochastic gradient descent (SGD) on approximation. Specifically, +we denote w ∈ Rd as the d-dimensional parameter of the approximator, such as the weights of the +neural network. We can optimize w by the standard minibatch SGD algorithm on approximation (See +Algorithm 1). +3 +Equivariant Equilibrium Approximator +In this section, we introduce the equivariance of the equilibrium approximators and the way how +we apply orbit averaging [Elesedy and Zaidi, 2021] to construct equivariant approximators. Equiv- +ariant approximator has been developed in many literatures [Hartford et al., 2016, Feng et al., 2021, +Marris et al., 2022, Wu and Lisser, 2022], which we will discuss latter. +We first define the permutation of a game. Let ρi : Ai → Ai be a permutation function of player i, +which is a bijection from Ai to Ai itself. Define Gi ∋ ρi as the class of permutation function for player +i, which forms an Abelian group. +Definition 3.1 (Permutation of a game). For a normal-form game Γu = (n, u, A), we define the +ρi-permutation of payoff tensor u as ρiu = (ρiuj)j∈[n], in which +(ρiuj)(ai, a−i) = uj(ρ−1 +i +(ai), a−i), ∀a ∈ A. +2A similar metric of equilibrium approximation is called Nikaido-Isoda function [Nikaidˆo and Isoda, 1955] or Nash- +Conv [Lockhart et al., 2019], which is the sum of exploitability over all players, i.e., � +i∈[n] Ei(π, u). +4 + +We also define the ρi-permutation of joint strategy π as ρiπ, where +(ρiπ)(ai, a−i) = π(ρ−1 +i +(ai), a−i), ∀a ∈ A, +and the ρi-permutation of product strategy σ as ρiσ = (ρiσj)j∈[n], where +∀aj ∈ Aj, ρiσj(aj) = +� +σj(aj) +, if j ̸= i +σi(ρ−1 +i ai) +, if j = i +Equivariant NE Approximator +Considering the relationship of ρi-permuted game and ρi-permuted +product strategy, we have the following result for NE: +Lemma 3.2. In a normal-form game Γu = (n, u, A), for arbitrary player i ∈ [n] and any (ǫ-)NE +strategy σ = (σi, σ−i), ρiσ = (ρiσi, σ−i) is also an (ǫ-)NE for the ρi-permuted game Γρiu. +Lemma 3.2 tells the unimportance of action order with respect to NE approximation. Based on it, +we can summarize two ideal properties for NE approximators, which are defined as follows: +Definition 3.3 (Player-Permutation-Equivariance, PPE). We say an NE approximator f NE satisfies +player i permutation-equivariant (i-PE) if for arbitrary permutation function ρi ∈ Gi we have +f NE(ρiu)i = ρif NE(u)i, +(i-PE) +Moreover, we say f NE is player-permutation-equivariant (PPE) if f NE satisfies i-PE for all player +i ∈ [n]. +Definition 3.4 (Opponent-Permutation-Invariance, OPI). We say an NE approximator f NE is oppo- +nent i permutation-invariant (i-PI) if for any other player j ∈ [n] − {i} and arbitrary permutation +function ρi ∈ Gi we have +f NE(ρiu)j = f NE(u)j, ∀j ̸= i +(i-PI) +and we say f NE is opponent-permutation-invariant (OPI) if f NE satisfies i-PI for all player i ∈ [n]. +Equivariant (C)CE approximator +Considering the relationship of ρi-permuted game and ρi- +permuted joint strategy, we have a similar result for CE and CCE: +Lemma 3.5. In a normal-form game Γu = (n, u, A), for an arbitrary player i ∈ [n] and any (ε-)CE +or (ǫ-)CCE strategy π, ρiπ is also an (ε-)CE or (ǫ-)CCE for the ρi-permuted game Γρiu. +Inspired by Lemma 3.5, we can also summarize an ideal property for CE and CCE approximators +defined as follows. +Definition 3.6 (Permutation-Equivariance,PE). We say an (C)CE approximator f (C)CE is player i +permutation-equivariant (i-PE) if for permutation function ρi ∈ Gi we have +f (C)CE(ρiu) = ρif (C)CE(u), +and we say f (C)CE is permutation-equivariant (PE) if f (C)CE satisfies i-PE for all player i ∈ [n]. +Equivariant Approximators in Literature +For two-player games, Feng et al. [2021] propose an +MLP-based NE approximator that satisfies both PPE and OPI for zero-sum games. Additionally, they +also design a Conv1d-based NE approximator that satisfies PPE only. Hartford et al. [2016] give a PPE +approximator to predict players’ strategies. The traditional algorithms Tsaknakis and Spirakis [2007] +and Deligkas et al. [2022], which approximate NE by optimization, are also PPE and OPI to payoff +and the initial strategies. For n-player general games, Marris et al. [2022] provide a permutation- +equivariant approximator to approximate CE and CCE. Equivariant architectures are also adopted +in optimal auction design [Rahme et al., 2021, Duan et al., 2022, Ivanov et al., 2022], and Qin et al. +[2022] theoretically characterize the benefits of permutation-equivariant in auction mechanisms. We +follow the rough idea of Qin et al. [2022] when we analyze the benefits of equivariant equilibrium +approximators. +5 + +3.1 +Orbit Averaging +Orbit averaging is a well-known method to enforce equivariance or invariance for a function [Schulz-Mirbach, +1994]. It averages the inputs of a function over the orbit of a group (e.g., the permutation group in +our paper). +Orbit Averaging for NE Approximator +For an NE approximator f NE and any player i ∈ [n], +we can construct a i-PI or i-PE NE approximator by averaging with respect to all the permutations +of player i. Specifically, we construct an i-PI NE approximator by operator Oi with +(Oif NE)(u)j = +� +f NE(u)i +, if j = i +1 +|Ai|! +� +ρi∈Gi f NE(ρiu)j +, otherwise +and we construct an i-PE NE approximator by operator Pi with: +(Pif NE)(u)j = +� +1 +|Ai|! +� +ρi∈Gi ρ−1 +i f NE(ρiu)i +, if j = i +f NE(u)j +, otherwise +Lemma 3.7. Oif NE is i-PI and Pif NE is i-PE. Specially, if f NE is already i-PI or i-PE, then we +have Oif NE = f NE or Pif NE = f NE, respectively. +To construct a PPE or OPI NE approximator, we composite the operators with respect to all +players. Let O = O1 ◦ O2 ◦ · · · ◦ On and P = P1 ◦ P2 ◦ · · · ◦ Pn, we get the following corollary: +Lemma 3.8. Of NE is OPI and Pf NE is PPE. If f NE is already OPI or PPE, we have Of NE = f NE +or Pf NE = f NE, respectively. +Furthermore, we can also compose P ◦O to construct a NE approximator with both PPE and OPI. +Orbit Averaging for (C)CE Approximator +For CE or CCE approximator f, we define Qi- +project for player i ∈ [n] to construct an i-PE approximator, which averages with respect to all the +permutations of player i. +(Qif (C)CE)(u) = +1 +|Ai|! +� +ρi∈Gi +ρ−1 +i f (C)CE(ρiu) +Similarly, we define Q = Q1 ◦ Q2 ◦ · · · ◦ Qn as the composite operator. +Lemma 3.9. Qif (C)CE is i-PE and Qf (C)CE is PE. Specifically, If f (C)CE is already i-PE or PE, +then we have Qif (C)CE = f (C)CE or Qf (C)CE = f (C)CE, respectively. +Combined with Lemma 3.7, Lemma 3.8 and Lemma 3.9, we can have the following corollary directly. +Corollary 3.10. O2 = O, P2 = P, Q2 = Q. +The benefit of using orbit averaging is shown in the following lemma: +Lemma 3.11. Denote X as an idempotent operator, i.e. X 2 = X (e.g. O, P or Q). For function +class F of NE, CE or CCE approximator, let FX be any subset of F that is closed under X, then XFX +is the largest subset of FX that is invariant under X. +According to Lemma 3.8, Lemma 3.9 and Lemma 3.11, OFNE(or PFNE/QF(C)CE) is the largest +subset of FNE(or FNE/F(C)CE) with the corresponding property (OPI, PPE, or PE) if FNE(or +FNE/F(C)CE) is closed operator under O(or P/Q). The result tells that the orbit averaging oper- +ators, while enforcing the operated function to be equivariance or invariance, keep as large capacity +of the function class as possible. Therefore, we believe that orbit averaging is an ideal approach to +constructing equivariant or invariant functions. +6 + +4 +Theoretical Analysis of Benefits +In this section, we theoretically analyze the benefits of equivariant approximators with respect to +generalizability and approximation. +4.1 +Benefits for Generalization +We first derive the generalization bound and sample complexity for general approximator classes, +and then we show the benefits of equivariant approximators by applying orbit averaging to the ap- +proximators. +The representativeness of an approximator class is measured by the covering numbers [Shalev-Shwartz and Ben-David, +2014] under ℓ∞-distance, which are defined as follows: +Definition 4.1 (ℓ∞-distance). The ℓ∞-distance between two equilibrium approximators f, g is: +ℓ∞(f, g) = max +u∈U ∥f(u) − g(u)∥, +where we define the distance of two product strategies σ and σ′ as +∥σ1 − σ2∥ = max +i∈[n] +� +ai∈Ai +|σ1 +i (ai) − σ2 +i (ai)| +and the distance of two joint strategy π and π′ as +∥π1 − π2∥ = +� +a∈A +|π1(a) − π2(a)| +Definition 4.2 (r-covering number). For r > 0, we say function class Fr r-covers another function +class F under ℓ∞-distance if for all function f ∈ F, there exists fr ∈ Fr such that ∥f − fr∥∞ ≤ r. The +r-covering number N∞(F, r) of F is the cardinality of the smallest function class Fr that r-covers F +under ℓ∞-distance. +Based on covering numbers, we provide the generalization bounds of NE, CE and CCE approxima- +tors. The bounds describe the difference between the expected testing approximation and empirical +training approximation. +Theorem 4.3 (Generalization bound). For function class F of NE, CE or CCE approximator, with +probability at least 1 − δ over draw of the training set S (with size m) from payoff distribution D, for +all approximator f ∈ F we have +Eu∼D[E(f(u), u)] − 1 +m +� +u∈S +E(f(u), u) ≤ 2 · inf +r>0{ +� +2 ln N∞(F, r) +m ++ Lr} + 4 +� +2 ln(4/δ) +m +, +where L = 2n for NE approximator, and L = 2 for CE and CCE approximators. +To get the theorem, we first show that all three equilibrium approximations are Lipschitz continuous +with respect to strategies. Afterward, we derive the Rademacher complexity [Bartlett and Mendelson, +2002] of the expected approximation based on the Lipschitz continuity and covering numbers. See +Appendix A.6 for the detailed proof. +We can see from Theorem 4.3 that, with a large enough training set, the generalization gaps of +equilibrium approximators go to zero if the covering number N∞(F, r) is bounded. As a result, we +can estimate the expected testing performance through the empirical training performance. +We can also derive the sample complexities of equilibrium approximators to achieve the desirable +generalizability. +Theorem 4.4 (Sample complexity). For ǫ, δ ∈ (0, 1), function class F of NE, CE or CCE approxi- +mator and distribution D, with probability at least 1 − δ over draw of the training set S with +m ≥ +9 +2ǫ2 +� +ln 2 +δ + ln N∞(F, ǫ +3L) +� +7 + +games sampled from D, ∀f ∈ F we have +Eu∼D[E(f(u), u)] ≤ 1 +m +� +u∈S +E(f(u), u) + ǫ, +where L = 2n for NE approximators, and L = 2 for CE and CCE approximators. +The proof is based on the Lipschitz continuity of approximation, uniform bound, and concentration +inequality. See Appendix A.7 for details. Theorem 4.4 is also called the uniform convergence of function +class F, which is a sufficient condition for agnostic PAC learnable [Shalev-Shwartz and Ben-David, +2014]. +As for the benefits of equivariant approximators for generalizability, the following result indicates +that the projected equilibrium approximators have smaller covering numbers. +Theorem 4.5. The O-projected, P-projected and Q-projected approximator classes have smaller cov- +ering numbers, i.e., ∀r > 0 we have +N∞(OFNE, r) ≤ N∞(FNE, r), +N∞(PFNE, r) ≤ N∞(FNE, r), +N∞(QF(C)CE, r) ≤ N∞(F(C)CE, r) +The proof is done by showing all the operators are contraction mappings. See Appendix A.8 for +details. +Both the generalization bounds in Theorem 4.3 and the sample complexities in Theorem 4.4 decrease +with the decrease of covering numbers N∞(F, r). Thus, we can see from Theorem 4.5 that both PPE +and OPI can improve the generalizability of NE approximators, and PE can improve the generalizability +of CE and CCE approximators. +4.2 +Benefits for Approximation +We then show the benefits of equivariance for approximation when the payoff distribution is invari- +ant under permutation. The permutation-invariant distribution holds when the action is anonymous +or indifferent or when we pre-train the equilibrium approximators using a manually designed distribu- +tion [Marris et al., 2022]. +(C)CE Approximator +The following theorem tells the benefit of permutation-equivariance in de- +creasing the exploitability of (C)CE approximators. +Theorem 4.6. When the payoff distribution D is invariant under the permutation of payoffs, the +Q-projected (C)CE approximator has a smaller expected equilibrium approximation. Formally, for all +f (C)CE ∈ F(C)CE and permutation-invariant distribution D, we have +Eu∼D[E(Qf (C)CE(u), u)] ≤ Eu∼D[E(f (C)CE(u), u)], +The proof is done by using the convexity of approximation. See Appendix A.10 for details. We can +see from Theorem 4.6 that, when payoff distribution is invariant under permutation, it is beneficial to +use equivariant architecture as the CE or CCE approximators. +NE Approximator +As for NE approximator, we have similar results. +Theorem 4.7. For bimatrix constant-sum games, when the payoff distribution D is invariant under the +permutation of payoffs, then the X-projected (X ∈ {O, P}) NE approximator has a smaller expected +exploitability. Formally, for all f NE ∈ FNE and permutation-invariant distribution D for bimatrix +constant-sum games, we have +Eu∼D[ +� +i +Ei((Xf NE)(u), u)] ≤ Eu∼D[ +� +i +Ei(f NE(u), u)] +8 + +Theorem 4.8. When the payoff distribution D is invariant under the permutation of payoffs, and +f NE satisfies OPI, then the P-projected NE approximator has a smaller expected NE approximation. +Formally, for all f NE ∈ FNE that is OPI and permutation-invariant distribution D, we have +Eu∼D[E((Pf NE)(u), u)] ≤ Eu∼D[E(f NE(u), u)]. +Theorem 4.9. For bimatrix games, when the payoff distribution D is invariant under the permutation +of payoffs, and f NE satisfies PPE, then the O-projected NE approximator has a smaller expected NE +approximation. Formally, for all f NE ∈ FNE that is PPE and permutation-invariant distribution D of +bimatrix games, we have +Eu∼D[E((Of NE)(u), u)] ≤ Eu∼D[E(f NE(u), u)]. +Theorem 4.8 and Theorem 4.9 tell that PPE and OPI approximators can achieve better approxi- +mation than ones with only PPE or OPI. Meanwhile, we can see from Theorem 4.7 that for bimatrix +constant-sum games (such as zero-sum games), it can be preferred to introduce PPE or OPI to the +architectures. +5 +Theoretical Analysis of Limitations +As we discussed in Section 4, equivariant approximators enjoy better generalizability and better +approximation sometimes. However, as we will show, they have some limitations regarding equilibrium +selection and social welfare. Such limitations attribute to the limited representativeness caused by +equivariance. +5.1 +Equilibrium Selection +We first show that there may be equilibria points that equivariant approximators will never find. +We illustrate such limitation in permutation-invariant games, which is defined as follows: +Definition 5.1 (Permutation-ρ-Invariant Game). We say a game Γu is permutation-ρ-invariant, where +ρ = ◦i∈[n]ρi, if the payoff u is permutation-invariant with respect to ρ. That is, ρu = u. +Permutation-ρ-invariance indicates that one cannot distinguish joint action a from ρa using only +the payoff u. We’d like to provide an example to show more insight of permutation-ρ-invariant games: +Example 5.2. For a 2-player game Γu = (2, u = (u1, u2), A = ([m1], [m2])) , Let ρi = (mi, mi − +1, . . . , 1) and ρ = ρ1 ◦ ρ2. If one of the following conditions holds, then u is permutation-ρ-invariant: +1. u1 and u2 are symmetric and persymmetric (i.e., symmetric with respect to the northeast-to- +southwest diagonal) squares. +2. Both u1 and u2 are centrosymmetric, i.e., ui(x, y) = ui(m1 +1−x, m2 +1−y) for i ∈ {1, 2}, x ∈ +[m1] and y ∈ [m2]. +For permutation ρ = (◦i∈[n]ρi) and player k ∈ [n], we denote the set of non-fixed actions of player +k under ρk as +V (ρk) := {ak|ak ∈ Ak, ρk(ak) ̸= ak}. +Based on V (ρk), we find some equilibria points of permutation-ρ-invariant games that any equivariant +approximators will never find. +Theorem 5.3. For a permutation-ρ-invariant game Γu. if there is a pure NE a∗ = (a∗ +i )i∈[n] and at +least one player k ∈ [n] such that a∗ +k ∈ V (ρk), then a∗ will never be found by any NE approximator +with both PPE and OPI. Besides, a∗ (as a pure CE or CCE) will also never be found by any CE or +CCE approximator with PE. +We illustrate Theorem 5.3 by the following example: +9 + +Example 5.4. Consider a bimatrix game with identity utility +u = +� +1, 1 +0, 0 +0, 0 +1, 1 +� +There are two pure NE (bolded in the above matrix) and one mixed NE of σ1 = (0.5, 0.5) and σ2 = +(0.5, 0.5). Let ρi be the unique permute function (except for identity function) of player i ∈ [2], and +ρ = ρ1 ◦ ρ2. The game is permutation-ρ-invariant. +Case 1: Let f be a permutation-equivariant CE or CCE approximator, and denote π = f(u). We +have +π = f(u) +(a) += f(ρu) +(b) += ρf(u), +where (a) holds by permutation-ρ-invariance of u, and (b) holds by PE of f. Thus, we have π1,1 = +π2,2 ∈ [0, 1 +2] and π1,2 = π2,1 ∈ [0, 1 +2]. As a result, the two pure (C)CEs cannot be found. +Case 2: Let f be a NE approximator that holds PPE and OPI. Denote f(u) = (σ1, σ2), where +σ1 = (p1, 1 − p1) and σ2 = (p2, 1 − p2). By PPE and OPI of f, we have +f(u)1 = (p1, 1 − p1) +(a) += f(ρ1ρ2u)1 +(b) += ρ1f(ρ2u)1 +(c) += ρ1f(u)1 = (1 − p1, p1), +where (a) holds by permutaion-ρ-invariance of u, (b) holds by PPE of f, and (c) holds by OPI of f. +As a result, the only NE that f could find is the mixed NE. +As we can see from the example and Theorem 5.3, the equivariance, while introducing inductive bias +to the approximator architecture, is also a strong constraint. Such a constraint is why the equivariant +approximators cannot find all the equilibria points. +5.2 +Social Welfare +The social welfare of a joint strategy π is defined as the sum of all players’ utilities, i.e., +SW(π, u) = +� +i∈[n] +ui(π). +The equilibrium with higher social welfare is usually preferred [Marris et al., 2022]. +To analyze the social welfare of equivariant approximators, we define the worst social welfare ratio +as follows: +Definition 5.5. For any N, M ≥ 2 and two NE (or CE/CCE) approximator classes F1, F2 that target +on games with number of players n ≤ N and |Ai| ≤ M, we define the worst social welfare ratio of F1 +over F2 as: +SWRN,M(F1, F2) := inf +D +maxf1∈F1 Eu∼DSW(f1(u), u) +maxf2∈F2 Eu∼DSW(f2(u), u) +SWRN,M(F1, F2) measures the relative representativeness of F1 over F2 in terms of social welfare. +Based on that, we have the following result for equivariant CE and CCE approximator classes: +Theorem 5.6. Given N, M ≥ 2, let F(C)CE +PE +be the function class (target on games with number of +players n ≤ N and |Ai| ≤ M) of all the (C)CE approximators with PE. Denote by F(C)CE +general the function +class of all the (C)CE approximators. Then we have +SWRN,M(F(C)CE +PE +, F(C)CE +general) = 1. +Theorem 5.6 tells that, while the permutation-equivariant (C)CE approximator class may not be +able to find all the (C)CE in a game, it can keep the social welfare of the output solutions. +However, when considering equivariant NE approximators, we have the following negative result: +10 + +Theorem 5.7. Given N, M ≥ 2, let FNE +OPI, FNE +PPE and FNE +both be the function classes (target on games +with number of players n ≤ N and |Ai| ≤ M) of all the NE approximators with OPI, PPE and both. +Denote the function class of all the NE approximators as FNE +general. Then we have +SWRN,M(FNE +OPI, FNE +general) = +1 +M N−1 , +(1) +SWRN,M(FNE +PPE, FNE +general) ≤ 1 +M , +(2) +SWRN,M(FNE +both, FNE +general) = +1 +M N−1 . +(3) +Additionally, when M ≥ 3, denote by �FNE +both the function class of all the NE oracles (functions that +always output exact NE solutions of the input games) with both PPE and OPI, and by � +FNE +general the +function class of all the NE oracles. Then we have +SWRN,M( �FNE +both, �FNE +general) = 0. +(4) +The proof is done by construction (See Appendix A.15 for details). As an illustration of Equa- +tion (4), consider a bimatrix game with the following payoff: +u = + + +1, 1 +0, 0 +0, 1 +2 + ε +0, 0 +1, 1 +0, 1 +2 + ε +1 +2 + ε, 0 +1 +2 + ε, 0 +ε, ε + + +for ǫ ∈ (0, 1 +2). The maximum NE (the upper-left corner of u) social welfare is 2, which can be found +by at least one NE oracle in �FNE +general. However, the only NE (the lower-right corner of u) that the NE +oracles in �FNE +both could find only has a social welfare of 2ǫ. As a result, +SWR2,3( �FNE +both, �FNE +general) ≤ 2ǫ +2 = ǫ, +which goes to zero as ǫ → 0. Recall that we always have SWRN,M ≥ 0, thus Equation (4) holds when +N = 2 and M = 3. +Theorem 5.7 tells that equivariant NE approximators may lose some social welfare while enjoying +better generalizability. Such a result inspires us to balance generalizability and social welfare when +designing the NE approximator architecture. +6 +Conclusion and Future Work +In this paper, we theoretically analyze the benefits and limitations of equivariant equilibrium +approximators, including player-permutation-equivariant (PPE) and opponent-permutation-invariant +(OPI) NE approximator, and permutation-equivariant (PE) CE and CCE approximators. For the +benefits, we first show that these equivariant approximators enjoy better generalizability. To get the +result, we derive the generalization bounds and sample complexities based on covering numbers, and +then we prove that the symmetric approximators have lower covering numbers. We then show that +the equivariant approximators can decrease the exploitability when the payoff distribution is invariant +under permutation. For the limitations, we find the equivariant approximators may fail to find some +equilibria points due to their limited representativeness caused by equivariance. Besides, while equiv- +ariant (C)CE approximators can keep the social welfare, the equivariant NE approximators reach a +small worst social welfare ratio comparing to the general approximators. Such a result indicates that +equivariance may reduce social welfare; therefore, we’d better balance the generalizability and social +welfare when we design the architectures of NE approximators. +As for future work, since in our paper we assume the training and testing payoff distribution are +the same, an interesting topic is to study the benefits of equivariant approximators under the payoff +distribution shift. Moreover, since we consider fixed and discrete action space, another interesting +future direction is to analyze the benefits of equivariant approximators in varying or continuous action +space. +11 + +References +Peter L Bartlett and Shahar Mendelson. Rademacher and gaussian complexities: Risk bounds and +structural results. Journal of Machine Learning Research, 3(Nov):463–482, 2002. +Nicolo Cesa-Bianchi and G´abor Lugosi. Prediction, learning, and games. Cambridge university press, +2006. +Xi Chen, Xiaotie Deng, and Shang-Hua Teng. 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Information Sciences, 2023. +13 + +A +Omitted Proof +A.1 +Useful Lemma +We first introduce a lemma, which will be frequently used in the following proofs. +Lemma A.1. ∀i, j ∈ [n], ρi ∈ Gi we have (ρiu)j(σi, σ−i) = uj(ρ−1 +i σi, σ−i) and (ρiu)j(π) = uj(ρ−1 +i π) +Proof. Define �ai := ρ−1 +i ai. For product strategy σ = (σi)i∈[n], +(ρiu)j(σi, σ−i) = +� +ai∈Ai +� +a−i∈A−i +(ρiu)j(ai, a−i) · σi(ai) · σ−i(a−i) += +� +ai∈Ai +� +a−i∈A−i +uj(ρ−1 +i ai, a−i) · σi(ai) · σ−i(a−i) += +� +ai∈Ai +� +a−i∈A−i +uj(ρ−1 +i ai, a−i) · (ρ−1 +i +σi)(ρ−1 +i ai) · σ−i(a−i) += +� +�ai∈Ai +� +a−i∈A−i +uj(�ai, a−i) · (ρ−1 +i +σi)(�ai) · σ−i(a−i) +=uj(ρ−1 +i σi, σ−i) +For joint strategy π, +(ρiu)j(π) = +� +ai∈Ai +� +a−i∈A−i +(ρiuj)(ai, a−i) · π(ai, a−i) += +� +ai∈Ai +� +a−i∈A−i +uj(ρ−1 +i ai, a−i) · π(ai, a−i) += +� +ai∈Ai +� +a−i∈A−i +uj(ρ−1 +i ai, a−i) · (ρ−1 +i +π)(ρ−1 +i +ai, a−i) += +� +�ai∈Ai +� +a−i∈A−i +uj(�ai, a−i) · (ρ−1 +i +π)(�ai, a−i) +=uj(ρ−1 +i π) +A.2 +Proof of Lemma 3.2 +Proof. For player i, we have +Ei(ρiσ, ρiu) = max +ai∈Ai ρiui(ai, ρiσ−i) − ρiui(ρiσ) = max +ai∈Ai ρiui(ai, σ−i) − ρiui(ρiσi, σ−i) += max +ai∈Ai ui(ρ−1 +i ai, σ−i) − ui(ρ−1 +i ρiσi, σ−i) +(a) += max +ai∈Ai ui(ai, σ−i) − ui(σi, σ−i) = Ei(σ, u), +where (a) holds since ρi is a bijection on Ai. For player j ̸= i, we have +Ej(ρiσ, ρiu) = max +aj∈A ρiuj(aj, ρiσ−j) − ρiuj(ρiσ) = max +aj∈Aj uj(aj, ρ−1 +i ρiσ−j) − uj(ρ−1 +i ρiσ) += max +aj∈Aj uj(aj, σ−j) − uj(σ) = Ej(σ, u) +From above, we have E(ρiσ, ρiu) = E(σ, u), thus if σ is a ε-NE of Γu, then ρiσ must be a ε-NE of +Γρiu. +14 + +A.3 +Proof of Lemma 3.5 +CCE +For player i, we have +Ei(ρiπ, ρiu) = max +ai∈Ai(ρiui)(ai, (ρiπ)−i) − (ρiui)(ρiπi) += max +ai∈Ai(ρiui)(ai, (ρiπ)−i) − ui(ρ−1 +i ρiπi) += max +ai∈Ai(ρiui)(ai, (ρiπ)−i) − ui(πi) += max +ai∈Ai +� +b∈A +(ρiui)(ai, b−i) · (ρiπ)(b) − ui(πi) += max +ai∈Ai +� +bi∈Ai,b−i∈A−i +ui(ρ−1 +i +ai, b−i) · π(ρ−1 +i +bi, b−i) − ui(πi) += max +ai∈Ai +� +bi∈Ai,b−i∈A−i +ui(ai, b−i) · π(bi, b−i) − ui(πi) +, ρi is a bijection on Ai +=Ei(π, u) +For player j ̸= i, we have +Ej(ρiπ, ρiu) = max +aj∈Aj(ρiuj)(aj, (ρiπ)−j) − (ρiuj)(ρiπj) += max +aj∈Aj(ρiuj)(aj, (ρiπ)−j) − uj(ρ−1 +i ρiπj) += max +aj∈Aj(ρiuj)(aj, (ρiπ)−j) − uj(πj) += max +aj∈Aj +� +b∈A +(ρiuj)(aj, b−j) · (ρiπ)(b) − uj(πj) += max +aj∈Aj +� +bi∈Ai,b−i∈A−i +uj(aj, (b−j)−i, ρ−1 +i bi) · π(ρ−1 +i +bi, b−i) − uj(πj) += max +aj∈Aj +� +bi∈Ai,b−i∈A−i +uj(aj, (b−j)−i, bi) · π(bi, b−i) − uj(πj) +, ρi is a bijection on Ai +=Ej(π, u) +Thus, we have E(ρiπ, ρiu) = E(π, u). Thus, if π is a ε-CCE of Γu, then ρiπ must be a ε-CCE of Γρiu. +CE +For player j ̸= i, we have +ECE +j +(ρiπ, ρiu) = +max +φj:Aj→Aj +� +a∈A +(ρiπ)(a) · (ρiuj)(φj(aj), a−j) − (ρiuj)(ρiπ) += +max +φj:Aj→Aj +� +a∈A +π(ρ−1 +i +ai, a−i) · uj(φj(aj), a−i,j, ρ−1 +i ai) − uj(π) += +max +φj:Aj→Aj +� +a∈A +π(ai, a−i) · uj(φj(aj), a−i,j, ai) − uj(π) +, ρi is a bijection on Ai +=ECE +j +(π, u) +For player i, we define operator ¯ρi as (¯ρiφi)(ai) = ρ−1 +i φi(ρiai). We can verify that ¯ρi is a bijection +on {φi : Ai → Ai}, because ¯· is a homomorphism in the sense that ρ1 +i ◦ ρ2 +i = ρ2 +i ρ1 +i and ¯· maps the +identity mapping of Ai to the identity mapping of {Ai → Ai}. Specifically, +ρ1 +i ◦ ρ2 +i φi(ai) = (ρ1 +i )−1(ρ2 +i φi)(ρ1 +i ai) = (ρ1 +i )−1(ρ2 +i )−1φi(ρ2 +i ρ1 +i ai) = ρ2 +i ρ1 +i φi(ai), +and +eiφi(ai) = e−1 +i φi(eiai) = φi(ai). +15 + +Based on ¯ρi, we have +ECE +i +(ρiπ, ρiu) += +max +φi:Ai→Ai +� +a∈A +(ρiπ)(a) · (ρiui)(φi(ai), a−i) − ui(π) += +max +φi:Ai→Ai +� +a∈A +π(ρ−1 +i ai, a−i)ui(ρ−1 +i φi(ai), a−i) − ui(π) += +max +φi:Ai→Ai +� +a∈A +π(ρ−1 +i ai, a−i)ui(ρ−1 +i φi(ρi(ρ−1 +i ai)), a−i) − ui(π) += +max +φi:Ai→Ai +� +a∈A +π(ai, a−i)ui(ρ−1 +i φi(ρiai), a−i) − ui(π) +, ρi is a bijection on Ai += +max +φi:Ai→Ai +� +a∈A +π(ai, a−i)ui((¯ρiφi)(ai), a−i) − ui(π) += +max +φi:Ai→Ai +� +a∈A +π(ai, a−i)ui(φi(ai), a−i) − ui(π) +, ¯ρi is a bijection on {Ai → Ai} +=ECE +i +(π, u) +Thus, we have E(ρiπ, ρiu) = E(π, u), thus if π is a ε-CE of Γu, then ρiπ must be a ε-CE of Γρiu. +A.4 +Proof of Lemma 3.7 to Lemma 3.9 +Proof of Lemma 3.7. ∀j ̸= i, ρ0 ∈ Gi, for operator Oi we have +(Oif NE)(ρ0u)j = +1 +|Ai|! +� +ρi∈Gi +f NE(ρiρ0u)j +(a) += +1 +|Ai|! +� +�ρi∈Gi +f NE(�ρiu)j = (Oif NE)(u)j +where in (a) we define �ρi = ρiρ0, and (a) holds since ρ0 is a bijection on Gi. As a result, Oif NE is i-PI. +For operator Pi we have +(Pif NE)(ρ0u)i = +1 +|Ai|! +� +ρi∈Gi +ρ−1 +i f NE(ρiρ0u)j = ρ0 +1 +|Ai|! +� +ρi∈Gi +ρ−1 +0 ρ−1 +i f NE(ρiρ0u)j +=ρ0 +1 +|Ai|! +� +�ρi∈Gi +�ρ−1 +i f NE(�ρiu)j = ρ0(Pif NE)(u)i, +therefore Pif NE is i-PE. +If f NE is already i-PI, ∀j ̸= i we have +Oif NE(u)j = +1 +|Ai|! +� +ρi∈Gi +f NE(ρiu)j = +1 +|Ai|! +� +ρi∈Gi +f NE(u)j = f NE(u)j, +and Oif NE(u)i = f NE(u)i according to definition of Oi. Therefore, Oif NE = f NE for i-PI f NE. +If f NE is already i-PE, we have +Pif NE(u)i = +1 +|Ai|! +� +ρi∈Gi +ρ−1 +i f NE(ρiu)i = +1 +|Ai|! +� +ρi∈Gi +ρ−1 +i ρif NE(u)i = +1 +|Ai|! +� +ρi∈Gi +f NE(u)i = f NE(u)i, +and ∀j ̸= i, Pif NE(u)j = f NE(u)j according to definition of Pi. Therefore, Pif NE = f NE for i-PE +f NE. +Proof of Lemma 3.8. A direct inference from Lemma 3.7 +Proof of Lemma 3.9. ∀ρ0 ∈ Gi, we have +16 + +(Qif (C)CE)(ρ0u) = +1 +|Ai|! +� +ρi∈Gi +ρ−1 +i f (C)CE(ρiρ0u) = ρ0 +1 +|Ai|! +� +ρi∈Gi +ρ−1 +0 ρ−1 +i +f (C)CE(ρiρ0u) +=ρ0 +1 +|Ai|! +� +�ρi∈Gi +�ρ−1 +i f (C)CE(�ρiu) = ρ0(Qif (C)CE)(u) +If f (C)CE is already i-PE, we have +Qif (C)CE(u) = +1 +|Ai|! +� +ρi∈Gi +ρ−1 +i f (C)CE(ρiu) = +1 +|Ai|! +� +ρi∈Gi +ρ−1 +i ρif (C)CE(u) = +1 +|Ai|! +� +ρi∈Gi +f (C)CE(u) = f (C)CE(u) +A.5 +Proof of Lemma 3.11 +We prove the three claims below. +1. XFX ⊆ FX . +2. X 2FX = XFX . +3. If XY = Y ⊆ FX , then Y ⊆ XFX +The first claim holds because FX is closed under X, and the second claim holds because X is +idempotent. For the third claim, from Y ⊆ FX we know XY ⊆ XFX , then Y = XY ⊆ XFX . +We immediately know XFX is the largest subset of FX that is invariant under X. +A.6 +Proof of Theorem 4.3 +Some of the techniques come from D¨utting et al. [2019] and Duan et al. [2021]. We first introduce +some useful lemmas. Denote ℓ : F × U → R as the loss function (such as ℓ(f, u) := E(f(u), u)). We +measure the capacity of the composite function class ℓ ◦ F using the empirical Rademacher complex- +ity [Bartlett and Mendelson, 2002] on the training set S, which is defined as: +RS(ℓ ◦ F) := 1 +mEx∼{+1,−1}m +� +sup +f∈F +m +� +i=1 +xi · ℓ(f, u(i)) +� +, +where x is distributed i.i.d. according to uniform distribution in {+1, −1}. We have +Lemma A.2 (Shalev-Shwartz and Ben-David [2014]). Let S be a training set of size m drawn i.i.d. +from distribution D over U. Then with probability at least 1 − δ over draw of S from D, for all f ∈ F, +Eu∼D[ℓ(f, u)] − 1 +m +� +u∈S +ℓ(l, u) ≤ 2RS(ℓ ◦ F) + 4 +� +2 ln(4/δ) +m +Lemma A.3. If |ℓ(·)| ≤ c for constant c > 0 and ∀f, f ′ ∈ F, |ℓ(f, u) − ℓ(f ′, u)| ≤ L∥f − f ′∥∞, then +we have +Eu∼D[ℓ(f, u)] − 1 +m +� +u∈S +ℓ(l, u) ≤ 2 inf +r>0 +� +c +� +2 ln N∞(F, r) +m ++ Lr +� ++ 4 +� +2 ln(4/δ) +m +Proof. For function class F, let Fr with |Fr| = N∞(F, r) be the function class that r-covers F for +17 + +some r > 0. Similarly, ∀f ∈ F, denote fr ∈ Fr be the function that r-covers f. We have +RS(ℓ ◦ F) = 1 +mEx +� +sup +f∈F +m +� +i=1 +xi · ℓ(f, u(i)) +� += 1 +mEx +� +sup +f∈F +m +� +i=1 +xi · +� +ℓ(fr, u(i)) + ℓ(f, u(i)) − ℓ(fr, u(i)) +�� +≤ 1 +mEx +� +sup +fr∈Fr +m +� +i=1 +xi · ℓ(fr, u(i)) +� ++ 1 +mEx +� +sup +f∈F +m +� +i=1 +|xi · Lr| +� +, |ℓ(f, u) − ℓ(fr, u)| ≤ L∥f − fr∥∞ = Lr +≤ sup +fr∈Fr +� +� +� +� +m +� +i=1 +ℓ2(fr, u(i)) · +� +2 ln N∞(F, r) +m ++ Lr +m Ex∥x∥ +, the first term holds by Massart’s lemma +≤ +√ +c2m · +� +2 ln N∞(F, r) +m ++ Lr +m Ex∥x∥ +≤c +� +2 ln N∞(F, r) +m ++ Lr, +(5) +Combining Lemma A.2 and Equation (5), we get +Eu∼D[ℓ(f, u)] − 1 +m +� +u∈S +ℓ(l, u) ≤ 2 inf +r>0 +� +c +� +2 ln N∞(F, r) +m ++ Lr +� ++ 4 +� +2 ln(4/δ) +m +NE Approximator +Lemma A.4. For arbitrary product mixed strategy σ and σ′, we have +|E(σ, u) − E(σ′, u)| ≤ 2n∥σ − σ′∥, +Proof. ∀σ, σ′, we define y−j := (σ1, . . . , σj−1, σ′ +j+1, . . . , σ′ +n). Then, ∀i ∈ [n] we have +|ui(σ) − ui(σ′)| =|ui(σ1, σ2, . . . , σn) − ui(σ′, σ′ +2, . . . , σ′ +n)| += +��� +n +� +j=1 +� +ui(σ1, . . . , σj, σ′ +j+1, . . . , σ′ +n) − ui(σ1, . . . , σ′ +j, σ′ +j+1, . . . , σ′ +n) +���� += +��� +n +� +j=1 +� +ui(σj, y−j) − ui(σ′ +j, y−j) +���� += +��� +n +� +j=1 +� +aj +(σj(aj) − σ′ +j(aj)) +� +a−j +ui(aj, a−j)y−j(a−j) +��� +≤ +n +� +j=1 +� +aj +���σj(aj) − σ′ +j(aj) +��� +� +a−j +ui(aj, a−j)y−j(a−j) +≤ +n +� +j=1 +� +aj +���σj(aj) − σ′ +j(aj) +��� +� +a−j +y−j(a−j) +, ui(·) ∈ [0, 1] +≤ +n +� +j=1 +� +aj∈Aj +���σj(aj) − σ′ +j(aj) +��� ≤ n max +j∈[n] +� +aj∈Aj +���σj(aj) − σ′ +j(aj) +��� +=n∥σ − σ′∥, +Therefore, ∀ai ∈ Ai, +ui(ai, σ−i) − ui(σ) =ui(ai, σ−i) − ui(ai, σ′ +−i) + ui(ai, σ′ +−i) − ui(σ′) + ui(σ′) − ui(σ) +≤n∥σ − σ′∥ + E(σ′, u) + n∥σ − σ′∥ +=E(σ′, u) + 2n∥σ − σ′∥. +18 + +Based on that, we get +E(σ, u) = +max +i∈N,ai∈Ai[ui(ai, σ−i) − ui(σ)] ≤ E(σ′, u) + 2n∥σ − σ′∥ +Similarly, we also have +E(σ′, u) ≤ E(σ, u) + 2n∥σ − σ′∥ +Based on Lemma A.4, ∀f, f ′ ∈ FNE, we have +E(f(u), u) − E(f ′(u), u) ≤ 2∥f(u) − f ′(u)∥ ≤ 2∥f − f ′∥∞ +Considering that |E(·)| ≤ 1, according to Lemma A.3, we have: +Eu∼D[E(f NE(u), u)] − 1 +m +� +u∈S +E(f NE(u), u) ≤ 2 · inf +r>0 +�� +2 ln N∞(FNE, r) +m ++ 2nr +� ++ 4 +� +2 ln(4/δ) +m +CCE Approximator +Lemma A.5. For arbitrary joint mixed strategy π and π′, we have +|E(π, u) − E(π′, u)| ≤ 2∥π − π′∥, +Proof. ∀π, π′, ∀i ∈ [n] we have +|ui(π) − ui(π′)| = +� +a∈A +(π(a) − π′(a))ui(a) +(a) +≤ +� +a∈A +|π(a) − π′(a)| = ∥π − π′∥ +(6) +where (a) holds since ui(·) ∈ [0, 1]. Therefore, ∀ai ∈ Ai, +ui(ai, π−i) − ui(π) =ui(ai, π−i) − ui(ai, π′ +−i) + ui(ai, π′ +−i) − ui(π′) + ui(π′) − ui(π) +≤∥π − π′∥ + E(π′, u) + ∥π − π′∥ +=E(π′, u) + 2∥π − π′∥. +Based on that, we get +E(π, u) = +max +i∈N,ai∈Ai[ui(ai, π−i) − ui(π)] ≤ E(π′, u) + 2∥π − π′∥ +Similarly, we also have +E(π′, u) ≤ E(π, u) + 2∥π − π′∥ +Based on Lemma A.5, ∀f, f ′ ∈ FCCE, we have +E(f(u), u) − E(f ′(u), u) ≤ 2∥f(u) − f ′(u)∥ ≤ 2∥f − f ′∥∞ +Considering that |E(·)| ≤ 1, according to Lemma A.3, we have: +Eu∼D[E(f CCE(u), u)] − 1 +m +� +u∈S +E(f CCE(u), u) ≤ 2 · inf +r>0 +�� +2 ln N∞(FCCE, r) +m ++ 2r +� ++ 4 +� +2 ln(4/δ) +m +19 + +CE Approximator +Lemma A.6. For arbitrary joint mixed strategy π and π′, we have +|ECE(π, u) − ECE(π′, u)| ≤ 2∥π − π′∥, +Proof. ∀ai ∈ Ai, ∀φi, we have +� +a∈A +π(a)ui(φ(ai), a−i) − ui(π) = +� +a∈A +π(a)ui(φ(ai), a−i) − +� +a∈A +π′(a)ui(φ(ai), a−i) ++ +� +a∈A +π′(a)ui(φ(ai), a−i) − ui(π′) + ui(π′) − ui(π) +≤∥π − π′∥ + ECE(π′, u) + ∥π − π′∥ +=ECE(π′, u) + 2∥π − π′∥. +Based on that, we get +ECE(π, u) = max +i∈N max +φi +� +a∈A +π(a)ui(φ(ai), a−i) − ui(π) ≤ ECE(π′, u) + 2∥π − π′∥ +Similarly, we also have +ECE(π′, u) ≤ ECE(π, u) + 2∥π − π′∥ +Based on Lemma A.5, ∀f, f ′ ∈ FCE, we have +ECE(f(u), u) − ECE(f ′(u), u) ≤ 2∥f(u) − f ′(u)∥ ≤ 2∥f − f ′∥∞ +Considering that |E(·)| ≤ 1, according to Lemma A.3, we have: +Eu∼D[ECE(f CE(u), u)] − 1 +m +� +u∈S +ECE(f CE(u), u) ≤ 2 · inf +r>0 +�� +2 ln N∞(FCE, r) +m ++ 2r +� ++ 4 +� +2 ln(4/δ) +m +A.7 +Proof of Theorem 4.4 +For function class F of NE, CE or CCE approximators, according to Lemma A.4, Lemma A.5 and +Lemma A.6, ∀f, g ∈ F we have +E(CE)(f(u), u) − E(CE)(g(u), u) ≤ L∥f(u) − g(u)∥ ≤ L∥f − g∥∞, +(7) +where L = 2n for NE approximators, and L = 2 for CE and CCE approximators. +For simplicity, we denote LS(f) = +1 +m +� +u∈S E(CE)(f(u), u) and LD(f) = Eu∼D[E(CE)(f(u), u)]. let +Fr with |Fr| = N∞(F, r) be the function class that r-covers F for some r > 0. ∀ǫ ∈ (0, 1), by setting +r = +ǫ +3L we have +PS∼Dm +� +∃f ∈ F, +��LS(f) − LD(f) +�� > ǫ +� +≤PS∼Dm +� +∃f ∈ F, +��LS(f) − LS(fr) +�� + +��LS(fr) − LD(fr) +�� + +��LD(fr) − LD(f) +�� > ǫ +� +(a) +≤PS∼Dm +� +∃f ∈ F, Lr + +��LS(fr) − LD(fr) +�� + Lr > ǫ +� +≤PS∼Dm +� +∃fr ∈ Fr, +��LS(fr) − LD(fr) +�� > ǫ − 2Lr +� +(b) +≤N∞(F, r)PS∼Dm +���LS(f) − LD(f) +�� > ǫ − 2Lr +� +(c) +≤2N∞(F, r) exp(−2m(ǫ − 2Lr)2), +=2N∞(F, ǫ +3L) exp(−2 +9mǫ2) +where (a) holds by Equation (7), (b) holds by union bound, and (c) holds by Hoeffding inequality. As +a result, when m ≥ +9 +2ǫ2 +� +ln 2 +δ + ln N∞(F, +ǫ +3L) +� +, we have PS∼Dm +� +∃f ∈ F, +���LS(f) − LD(f) +��� > ǫ +� +< δ. +20 + +A.8 +Proof of Theorem 4.5 +We first provide an auxiliary lemma. +Lemma A.7. For function class F and orbit averaging operator X, if ∀f, g ∈ F, ℓ∞(Xf, Xg) ≤ +ℓ∞(f, g), then N∞(XF, r) ≤ N∞(F, r) for any r > 0. +Proof. ∀r > 0, Denote Fr as the smallest r-covering set that covers F with size N∞(F, r). ∀f ∈ F, +let fr ∈ Fr be the function that r-covers f. We have ℓ∞(Xfr, Xf) ≤ ℓ∞(fr, f) ≤ r. Therefore, XFr +is a r-covering set of XF, and we have N∞(XF, r) ≤ |XFr| ≤ |Fr| = N∞. +For player i ∈ [n] and ∀f NE, gNE ∈ FNE, assuming U is closed under any ρi ∈ Gi. For Oi, +l∞(Oif NE, OigNE) = max +u∈U ∥Oif NE(u) − OigNE(u)∥ += max +j∈[n] max +u∈U ∥(Oif NE)(u)j − (OigNE)(u)j∥ += max +� +max +u∈U ∥f NE(u)i − gNE(u)i∥, max +j̸=i max +u∈U ∥ +1 +|Ai|! +� +ρi∈Gi +(f NE(ρiu)j − gNE(ρiu)j)∥ +� +≤ max +� +max +u∈U ∥f NE(u)i − gNE(u)i∥, max +j̸=i +1 +|Ai|! +� +ρi∈Gi +max +u∈U ∥f NE(ρiu)j − gNE(ρiu)j∥ +� += max +� +max +u∈U ∥f NE(u)i − gNE(u)i∥, max +j̸=i +1 +|Ai|! +� +ρi∈Gi +max +u∈U ∥f NE(u)j − gNE(u)j∥ +� += max +� +max +u∈U ∥f NE(u)i − gNE(u)i∥, max +j̸=i max +u +∥f NE(u)j − gNE(u)j∥ +� +=l∞(f NE, gNE) +Since O = O1 ◦ · · · ◦ On, we have +ℓ∞(Of NE, OgNE) ≤ ℓ∞(f NE, gNE). +(8) +For Pi, +l∞(Pif NE, PigNE) = max +u∈U max +j∈[n] ∥(Pif NE)(u)j − (PigNE)(u)j∥ += max +� +max +j̸=i max +u +∥f NE(u)j − gNE(u)j∥, max +u +∥ +1 +|Ai|! +� +ρi∈Gi +ρ−1 +i (f NE(ρiu)i − gNE(ρiu)i)∥ +� += max +� +max +j̸=i max +u +∥f NE(u)j − gNE(u)j∥, max +u +∥ +1 +|Ai|! +� +ρi∈Gi +(f NE(ρiu)i − gNE(ρiu)i)∥ +� +≤ max +� +max +j̸=i max +u +∥f NE(u)j − gNE(u)j∥, +1 +|Ai|! +� +ρi∈Gi +max +u +∥f NE(ρiu)i − gNE(ρiu)i∥ +� += max +� +max +j̸=i max +u +∥f NE(u)j − gNE(u)j∥, +1 +|Ai|! +� +ρi∈Gi +max +u +∥f NE(u)i − gNE(u)i∥ +� += max +� +max +j̸=i max +u +∥f NE(u)j − gNE(u)j∥, max +u +∥f NE(u)i − gNE(u)i∥ +� +=l∞(f NE, gNE) +Since P = P1 ◦ · · · ◦ Pn, we have +ℓ∞(Pf NE, PgNE) ≤ ℓ∞(f NE, gNE). +(9) +21 + +For CE or CCE approximator f (C)CE ∈ F(C)CE and Qi, we have +l∞(Qif (C)CE, Qig(C)CE) = max +u∈U ∥(Qif (C)CE)(u) − (Qig(C)CE)(u)∥ += max +u +∥ +1 +|Ai|! +� +ρi∈Gi +ρ−1 +i (f (C)CE(ρiu) − g(C)CE(ρiu))∥ +≤ max +u +1 +|Ai|! +� +ρi∈Gi +∥ρ−1 +i (f (C)CE(ρiu) − g(C)CE(ρiu))∥ +≤ +1 +|Ai|! +� +ρi∈Gi +max +u +∥ρ−1 +i (f (C)CE(ρiu) − g(C)CE(ρiu))∥ += +1 +|Ai|! +� +ρi∈Gi +max +u +∥f (C)CE(ρiu) − g(C)CE(ρiu)∥ += +1 +|Ai|! +� +ρi∈Gi +max +u +∥f (C)CE(u) − g(C)CE(u)∥ +=l∞(f (C)CE, g(C)CE) +Since Q = Q1 ◦ · · · ◦ Qn, we have +ℓ∞(Qf (C)CE, Qg(C)CE) ≤ ℓ∞(f (C)CE, g(C)CE). +(10) +Combing Lemma A.7, Equation (8), Equation (9) and Equation (10), we finish the proof. +A.9 +Proof of Theorem 4.8 +We first introduce a useful lemma. It is about the property of Ei(σ, u) +Lemma A.8. Ei(σ, u) is +1. Linear on σi, i.e. +pEi((σ1 +i , σ−i), u) + (1 − p)Ei((σ2 +i , σ−i), u) = Ei((pσ1 +i + (1 − p)σ2 +i , σ−i), u), ∀p ∈ [0, 1] +2. Convex on σj, i.e. +pEi((σ1 +j , σ−j), u) + (1 − p)Ei((σ2 +j , σ−j), u) ≥ Ei((pσ1 +j + (1 − p)σ2 +j , σ−j), u), ∀p ∈ [0, 1], j ̸= i +Proof. We recall the definition Ei(σ, u) = maxai∈Ai ui(ai, σ−i) − ui(σ). Notice that ui(σ) is linear on +σk for all k ∈ [n], thus both ui(ai, σ−i) and ui(σ) are linear on σk for any k ∈ [n]. Moreover, the +maximum operator on a set of linear functions will induce a convex function. +Proof of Theorem 4.8. We prove the theorem in two steps. +Step 1 +First, we show that +Eu∼D[Ei((Pif NE)(u), u)] = Eu∼D[Ei(f NE(u), u)], +∀f NE ∈ FNE +22 + +By definition, +Eu∼D[Ei(Pif NE(u), u)] +=Eu∼D[Ei(( +1 +|Ai|! +� +ρi∈Gi +ρ−1 +i f(ρiu)i, f(u)−i), u)] += +1 +|Ai|! +� +ρi∈Gi +Eu∼D[Ei((ρ−1 +i +f(ρiu)i, f(u)−i), u)] +, by linearity of Ei(σ, u) on σi += +1 +|Ai|! +� +ρi∈Gi +Ev∼D[Ei((ρ−1 +i +f(v)i, f(ρ−1 +i v)−i), ρ−1 +i v)] +, let v = ρiu and use the invariance of D += +1 +|Ai|! +� +ρi∈Gi +Ev∼D[Ei((ρ−1 +i +f(v)i, f(v)−i), ρ−1 +i v)] +, OPI of f += +1 +|Ai|! +� +ρi∈Gi +Eu∼D[Ei((f(u)i, f(u)−i), u)] +, invariance of Ei(σ, u) under ρ−1 +i +∈ Gi +=Eu∼D[Ei(f NE(u), u)] +Step 2 +Then we show that +Eu∼D[Ej((Pif NE)(u), u)] ≤ Eu∼D[Ej(f NE(u), u)], +∀f NE ∈ FNE, j ̸= i +Eu∼D[Ej((Pif NE)(u), u)] +=Eu∼D[Ej(( +1 +|Ai|! +� +ρi∈Gi +ρ−1 +i f(ρiu)i, f(u)−i), u)] +≤ +1 +|Ai|! +� +ρi∈Gi +Eu∼D[Ej((ρ−1 +i +f(ρiu)i, f(u)−i), u)] +, by convexity of Ej(σ, u) on σi += +1 +|Ai|! +� +ρi∈Gi +Ev∼D[Ej((ρ−1 +i +f(v)i, f(ρ−1 +i v)−i), ρ−1 +i v)] +, let v = ρiu and use the invariance of D += +1 +|Ai|! +� +ρi∈Gi +Ev∼D[Ej((ρ−1 +i +f(v)i, f(v)−i), ρ−1 +i v)] +, OPI of f += +1 +|Ai|! +� +ρi∈Gi +Eu∼D[Ej((f(u)i, f(u)−i), u)] +, invariance of Ej(σ, u) under ρ−1 +i +∈ Gi +=Eu∼D[Ej(f NE(u), u)] +Since P = ◦iPi and E = maxi Ei, we have +Eu∼D[E((Pf NE)(u), u)] ≤ Eu∼D[E(f NE(u), u)] +A.10 +Proof of Theorem 4.6 +Similar to the proof of Theorem 4.8, we first prove a lemma about the property of Ei(π, u) and +ECE +i +(π, u). +Lemma A.9. Ei(π, u) and ECE +i +(π, u) are convex on π, i.e. +pE(CE) +i +(π1, u) + (1 − p)E(CE) +i +(π2, u) ≥ E(CE) +i +(pπ1 + (1 − p)π2, u), +∀p ∈ [0, 1] +23 + +Proof. We recall the definition Ei(π, u) = maxai∈Ai ui(ai, π−i) − ui(π) for CCE approximator and +ECE +i +(π, u) = maxφi∈Ai→Ai +� +a π(a)ui(φi(ai), a−i) − ui(π) for CE approximator. ui(ai, π−i) is linear +on π. +Given φ, � +a π(a)ui(φi(ai), a−i) is also linear on π. Moreover, the maximum operator on a set of +linear functions will induce a convex function. +Proof of Theorem 4.6. For f ∈ F(C)CE and ∀i, j ∈ [n], +Eu∼D[E(CE) +i +(Qjf(u), u)] =Eu∼D[E(CE) +i +( +1 +|Aj|! +� +ρj∈Gj +ρ−1 +j f(ρju), u)] +, by definition +≤ +1 +|Aj|! +� +ρj∈Gj +Eu∼D[E(CE) +i +(ρ−1 +j f(ρju), u)] +, by convexity += +1 +|Aj|! +� +ρj∈Gj +Ev∼D[E(CE) +i +(ρ−1 +j f(v), ρ−1 +j v)] +, let v = ρju += +1 +|Aj|! +� +ρj∈Gj +Ev∼D[E(CE) +i +(f(v), v)] +, invariance of E(CE) +i +(π, u) under ρ−1 +j +∈ Gj +=Eu∼D[E(CE) +i +(f(u), u)] +Since Q = ◦iQi and E = maxi Ei, we have +Eu∼D[E(Qf(u), u)] ≤ Eu∼D[E(f(u), u)] +A.11 +Proof of Theorem 4.9 +We prove the theorem in two steps, similar to the proof of Theorem 4.8. +Step 1 +First we show that for player i ∈ {1, 2}, let {j} = {1, 2}\{i}, +Eu∼D[Ei((Oif NE)(u), u)] ≤ Eu∼D[Ei(f NE(u), u)] +This is because +Eu∼D[Ei((Oif NE)(u), u)] =Eu∼D[Ei((f NE(u)i, +1 +|Ai|! +� +ρi∈Gi +f NE(ρiu)j), u)] +≤ +1 +|Ai|! +� +ρi∈Gi +Eu∼D[Ei((f NE(u)i, f NE(ρiu)j), u)] +, by convexity of Ei on σj += +1 +|Ai|! +� +ρi∈Gi +Ev∼D[Ei((f NE(ρ−1 +i v)i, f NE(v)j), ρ−1 +i +v)] +, let v = ρiu += +1 +|Ai|! +� +ρi∈Gi +Ev∼D[Ei((ρ−1 +i +f NE(v)i, f NE(v)j), ρ−1 +i +v)] +, by PPE of f NE += +1 +|Ai|! +� +ρi∈Gi +Ev∼D[Ei((f NE(v)i, f NE(v)j), v)] +, invariance of Ei(σ, u) under ρ−1 +i +∈ G +=Eu∼D[Ei((f NE)(u), u)] +Step 2 +Then we show that if j ̸= i and {i, j} = {1, 2} +Eu∼D[Ej((Oif NE)(u), u)] = Eu∼D[Ej(f NE(u), u)] +24 + +This is because +Eu∼D[Ej((Oif NE)(u), u)] =Eu∼D[Ej((f NE(u)i, +1 +|Ai|! +� +ρi∈Gi +f NE(ρiu)j), u)] += +1 +|Ai|! +� +ρi∈Gi +Eu∼D[Ej((f NE(u)i, f NE(ρiu)j), u)] +, by linearity of Ej on σj += +1 +|Ai|! +� +ρi∈Gi +Ev∼D[Ej((f NE(ρ−1 +i v)i, f NE(v)j), ρ−1 +i v)] +, let v = ρiu += +1 +|Ai|! +� +ρi∈Gi +Ev∼D[Ej((ρ−1 +i f NE(v)i, f NE(v)j), ρ−1 +i v)] +, by PPE of f NE += +1 +|Ai|! +� +ρi∈Gi +Ev∼D[Ej((f NE(v)i, f NE(v)j), v)] +, invariance of Ej(σ, u) under ρ−1 +i +∈ Gi +=Eu∼D[Ej(f NE(u), u)] +Since O = ◦iOi and E = maxi Ei, we have +Eu∼D[E(Of NE(u), u)] ≤ Eu∼D[E(f NE(u), u)] +A.12 +Proof of Theorem 4.7 +We only prove for the P-projected case; the proof for O-projected case is similar and therefore +omitted. +Recall +Ei(σ, u) = max +ai∈Ai ui(ai, σ−i) − ui(σ) +Denote u1(σ) + u2(σ) ≡ c, then +� +i +Ei(σ, u) = +max +a1∈A1,a2∈A2 u1(a1, σ2) + u2(a2, σ1) − c +Then we have +Eu∼D[ +� +i +Ei((Pf NE)(u), u)] =Eu∼D[max +a1,a2 u1(a1, (Pf NE)(u)2) + u2(a2, (Pf NE)(u)1) − c] +=Eu∼D[max +a1 u1(a1, (Pf NE)(u)2)] + Eu∼D[max +a2 u2(a2, (Pf NE)(u)1)] − c +For the first term, +Eu∼D[max +a1 u1(a1, (Pf NE)(u)2)] =Eu∼D[max +a1 u1(a1, +1 +|A2|! +� +ρ2∈G2 +ρ−1 +2 f NE(ρ2u)2)] +≤ +1 +|A2|! +� +ρ2∈G2 +Eu∼D[max +a1 u1(a1, ρ−1 +2 f NE(ρ2u)2)] += +1 +|A2|! +� +ρ2∈G2 +Ev∼D[max +a1 (ρ−1 +2 v)1(a1, ρ−1 +2 f NE(v)2)] += +1 +|A2|! +� +ρ2∈G2 +Ev∼D[max +a1 v1(a1, f NE(v)2)] +=Eu∼D[max +a1 u1(a1, f NE(u)2)] +Similarly, for the second term, +Eu∼D[max +a2 u2(a2, (Pf NE)(u)1)] ≤ Eu∼D[max +a2 u2(a2, f NE(u)1)] +25 + +Above all, +Eu∼D[ +� +i +Ei((Pf NE)(u), u)] =Eu∼D[max +a1 u1(a1, (Pf NE)(u)2)] + Eu∼D[max +a2 u2(a2, (Pf NE)(u)1)] − c +≤Eu∼D[max +a1 u1(a1, f NE(u)2)] + Eu∼D[max +a2 u2(a2, f NE(u)1)] − c +=Eu∼D[ +� +i +Ei(f NE(u), u)] +A.13 +Proof of Theorem 5.3 +Let f be a PPE and OPI NE approximator. Denote f(u) = (σi)i∈[n]. For player k that a∗ +k ∈ V (ρk), +we get +σk = f(u)k +(a) += f(ρu)k +(b) += f(ρku)k +(c) += ρkf(u)k = ρkσk, +(11) +where (a) holds since u is permutable w.r.t. ρ, (b) holds by OPI of f, and (c) holds by PPE of f. +If a∗ can be found by f, we will have 1 = σk(a∗ +k) +(d) += ρkσk(a∗ +k) = σk(ρ−1 +k (a∗ +k)), where (d) holds by +Equation (11). However, such result leads to a contradiction, because a∗ +k ̸= ρ−1 +k (ak) but σk(a∗ +k) = +σ(ρ−1 +k (a∗ +k)) = 1. +Let f be a PE (C)CE approximator. Denote f(u) = π, we have +π = f(u) +(a) += f(ρu) +(b) += ρf(u) = ρπ +(12) +where (a) holds since u is permutable w.r.t. ρ, (b) holds by PE of f. If a∗ can be found by f, we +will have 1 = π(a∗) +(c) += ρπ(a∗) = π(ρ−1a∗) = π(ρ−1 +1 a∗ +1, · · · , ρ−1 +n a∗ +n), where (c) holds by Equation (12). +However, from a∗ +k ∈ V (ρk) we know ρ−1 +k (a∗ +k) ̸= a∗ +k, then ρ−1a∗ ̸= a∗, but π(a∗) = π(ρ−1a∗) = 1. +A.14 +Proof of Theorem 5.6 +Proof. Assume f ∈ F(C)CE +general is an (C)CE approximator that always finds the strategy that maximizes +the social welfare. Afterward, we construct another f0 that satisfies PE and always finds the strategy +that maximizes social welfare. f0 is constructed by orbit averaging: +f0(u) = Qf(u), +thus f0 is PE. +Denote D as an arbitrary payoff distribution of u such that D is invariant under permutation and +the cardinality of its support is finite. We have +Eu∼DSW(Qif(u), u) =Eu∼DSW( +1 +|Ai|! +� +ρi∈Gi +ρ−1 +i +f(ρiu), u) +=Eu∼D +n +� +i=1 +ui( +1 +|Ai|! +� +ρi∈Gi +ρ−1 +i f(ρiu)) += +1 +|Ai|! +� +ρi∈Gi +Eu∼D +n +� +i=1 +ui(ρ−1 +i +f(ρiu)) += +1 +|Ai|! +� +ρi∈Gi +Ev∼D +n +� +i=1 +(ρ−1 +i +v)i(ρ−1 +i f(v)) +, let v = ρiu += +1 +|Ai|! +� +ρi∈Gi +Ev∼D +n +� +i=1 +vi(f(v)) +=Eu∼D +n +� +i=1 +ui(f(u)) +=Eu∼DSW(f(u), u) +26 + +Due to that Q = Q1 ◦ · · · ◦ Qn, we have +Eu∼DSW(f0(u), u) = Eu∼DSW(f(u), u) +Due to the arbitrariness of D, we know that f0 maximizes the social welfare w.r.t. any u. +From above, we immediately know +SWRN,M(F(C)CE +PE +, F(C)CE +general) = 1 +A.15 +Proof of Theorem 5.7 +A.15.1 +Proof of Equation (1) and Equation (3) +We first prove the theorem with respect to FNE +OPI and FNE +both +Step 1 +On the one part, we prove +SWRN,M(FNE +OPI, FNE +general) +SWRN,M(FNE +both, FNE +general) +� +≤ +1 +M N−1 +We prove this by construction. +Consider a game with N player and Ai = [M] for i ∈ [N]. ∀a ∈ A, i ∈ [N], define the payoff ¯u as +follows: +¯ui(a) = +� +1 +, if a1 = a2 = · · · = aN +0 +, otherwise +Define U = {u′|u′ = ◦iρi¯u, ρi ∈ Gi} and D as a uniform distribution on U. Easy to certify that D is a +permutation-invariant distribution. +Let ˜f ∈ ˜FNE +general be the NE oracle that ˜f(¯u)i = 1 and for any u′ = ◦iρi¯u ∈ U, ˜f(u′)i = ρi(1). +Intuitively, the oracle will choose the action that will provide all players with revenue 1, leading to a +social welfare of N. Since each player has got her maximum possible utility, we have +max +f∈F NE +general +Eu∼DSW(f(u), u) = +max +˜f∈ � +F NE +general +Eu∼DSW( ˜f(u), u) = N. +(13) +For any j1, j2 ∈ [M] and j1 < j2, let ρ(j1,j2) +i += (1, . . . , j2, . . . , j1, . . . , M) for all player i ∈ [N] be +the swap permutation that swaps actions j1 and j2 and keeps other actions still. Then ◦i̸=jρ(j1,j2) +i +¯u = +ρ(j1,j2) +j +¯u for player j. For f ∈ FNE +OPI, we have f(¯u)j = f(◦i̸=jρ(j1,j2) +i +¯u)j = f(ρ(j1,j2) +j +¯u)j for arbitrary swap +permutation ρ(j1,j2) +j +. Since any permutation can be achieved by composition of swap permutations, +we have ∀ρj ∈ Gj, f(¯u)j = f(ρj ¯u)j. +Based on that, and by OPI of f, ∀ρ = ◦i∈[N]ρi we have +f(¯u)j = f(ρ¯u)j, i.e. f is a constant function on U. Without loss of generality, we denote f(u) ≡ σ for +all u ∈ U. Then +Eu∼DSW(f(u), u) = +1 +|U| +� +u′∈U +SW(σ, u′) = +1 +(M!)N−1 SW(σ, +� +u′∈U +u′). +Additionally, we have (� +u′∈U u′)(a) = ((M − 1)!)N−1 for any a ∈ A. Based on that, we have +max +f∈F NE +OPI +Eu∼DSW(f(u), u) = +1 +(M!)N−1 · N((M − 1)!)N−1 = +N +M N−1 . +(14) +Combining Equation (13) and Equation (14), we have +SWRN,M(FNE +OPI, FNE +general) ≤ +1 +M N−1 . +Due to FNE +both ⊆ FNE +OPI, we immediately know +SWRN,M(FNE +both, FNE +general) ≤ +1 +M N−1 +27 + +Step 2 +On the other part, we prove +SWRN,M(FNE +OPI, FNE +general) +SWRN,M(FNE +both, FNE +general) +� +≥ 1/M N−1 +Define the maximum possible utility (MPU) for player i with respect to utility ui and action ai as +MPU(ui, ai) := +max +a−i∈A−i ui(ai, a−i) +(15) +Define the set of maximum possible utility best response for player i w.r.t. ui as +Bi(ui) := {ai ∈ Ai : MPU(ui, ai) = max +a′ +i∈Ai MPU(ui, a′ +i)} +We first conduct some simplification to the target. +SWRN,M(FNE +both, FNE +general) = inf +D +maxf∈F NE +both Eu∼DSW(f(u), u) +maxf∈F NE +general Eu∼DSW(f(u), u) ≥ inf +D +maxf∈F NE +both Eu∼DSW(f(u), u) +Eu∼D maxσ SW(σ, u) +Then we constrain u to be a cooperation game. For a normal form game Γu, we define ˜u = (˜ui)i∈[n] +in which ˜ui = 1 +n +�n +i=1 ui. Then we have SW(σ, u) = SW(σ, ˜u), which means that constraining u to be +a cooperation game will induce the same social welfare. Then +inf +D +maxf∈F NE +both Eu∼DSW(f(u), u) +Eu∼D maxσ SW(σ, u) += inf +D +maxf∈F NE +both Eu∼DSW(f(u), ˜u) +Eu∼D maxσ SW(σ, ˜u) +Denote f0 be the approximator that always outputs uniform strategy on Bi(˜ui) for player i. It’s +obvious that f0 is both OPI and PPE because the operations from u to f0(u) are all permutation- +equivariant. Then, +inf +D +maxf∈F NE +both Eu∼DSW(f(u), ˜u) +Eu∼D maxσ SW(σ, ˜u) +≥ inf +D +Eu∼DSW(f0(u), ˜u) +Eu∼D maxσ SW(σ, ˜u) +Ignore the infimum and the expectation operator, consider +SW(f0(u),˜u) +maxσ SW(σ,˜u) for arbitrary ˜u, denote b +be the maximum element appeared in ˜u, then the denominator equals Nb. But for the numerator, +for player i, no matter what action ai ∈ Bi(˜ui) she chooses, she always has probability at least +� +j̸=i +1 +|Bj| ≥ +1 +MN−1 to achieve revenue b, therefore inducing SW(f0(u), ˜u) ≥ +Nb +MN−1 . +Then, +SW(f0(u),˜u) +maxσ SW(σ,˜u) ≥ +1 +MN−1 , so as infD +Eu∼DSW(f0(u),˜u) +Eu∼D maxσ SW(σ,˜u), SWRN,M(FNE +both) and SWRN,M(FNE +OPI). +Above all, +SWRN,M(FNE +OPI, FNE +general) +SWRN,M(FNE +both, FNE +general) +� += +1 +M N−1 +A.15.2 +Proof of Equation (2) +We next prove the theorem with respect to FNE +PPEthat +SWRN,M(FNE +PPE, FNE +general) ≤ 1 +M +Consider a bimatrix game and Ai = [M] for i ∈ [2]. ∀a ∈ A, i ∈ [2], define the payoff ¯u as follows: +¯ui(a) = +� +1 +, if a1 = a2 +0 +, otherwise +Define U := {u′|u′ = ρ1ρ2¯u, ρi ∈ Gi} and D as a uniform distribution on U. Easy to certify that +U = {u′|u′ = ρ1¯u, ρ1 ∈ G1} = {u′|u′ = ρ2¯u, ρ2 ∈ G2} and D is a permutation-invariant distribution. +28 + +Let ˜f ∈ ˜FNE +general be the NE oracle that ˜f(¯u)i = 1 and for any u′ = ◦iρi¯u ∈ U, ˜f(u′)i = ρi(1). +Intuitively, the oracle will choose the action that will provide all players with revenue of 1, leading to +a social welfare of 2. +For a permutation ̺ on [M], let P̺ ∈ {0, 1}M×M be the corresponding permutation matrix. Denote +P as the set of all permutation matrice. As a result, ∀u ∈ U, ∀ρ1 ∈ G1, ρ1u = (Pρ1u1, Pρ1u2) =: Pρ1u +and ∀ρ2 ∈ G2, ρ2u = (u1P T +ρ2, u2P T +ρ2) =: uP T +ρ2. Specially, we have P̺¯uP T +̺ = ¯u. For f ∈ FNE +PPE, Denote +f(¯u) = σ = (σ1, σ2). For permutation ̺ in [M] and payoff u′ = P̺¯u = ¯u(P T +̺ )−1, by PPE of f, we have +f(u′)1 = f(P̺¯u)1 = P̺σ1 = ̺σ1, and f(u′)2 = f(¯u(P T +̺ )−1)2 = (P̺)−1σ2 = ̺−1σ2. Then we have +SW(f(u′), u′) = +� +i +(P̺¯u)i(̺σ1, ̺−1σ2) = +� +i +¯ui(σ1, ̺−1σ2) = +� +i +(¯uP T +̺ )i(σ1, σ2) = SW(f(¯u), ¯uP T +̺ ) +Therefore +Eu∼DSW(f(u), u) = +1 +|U| +� +u′∈U +SW(f(u′), u′) += +1 +|U| +� +P̺∈P +SW(f(¯u), ¯uP T +̺ ) += +1 +|U| +� +u=¯u(P T +̺ )∈U +SW(f(¯u), u) += +1 +|U|SW(σ, +� +u′∈U +u′). +Since |U| = +1 +M! and � +u′∈U u′ is a tensor with all elements equal to (M −1)!. Thus Eu∼DSW(f(u), u) = +2 +M and +SWRN,M(FNE +PPE, FNE +general) ≤ 1 +M +A.15.3 +Proof of Equation (4) +Consider a 3 × 3 game as follows, where ǫ ∈ (0, 1 +2): +u = + + +1, 1 +0, 0 +0, 1 +2 + ε +0, 0 +1, 1 +0, 1 +2 + ε +1 +2 + ε, 0 +1 +2 + ε, 0 +ε, ε + + +It is obvious that maxσ∗⊆NE(Γu) SW(σ∗, u) = 2, and the corresponding strategy has been bolded. +However, for NE oracles with both PPE and OPI, it can only output a unique NE with a pure strategy +that induces utility (ε, ε). +Let ρ1 = ρ2 = (2, 1, 3), we have ρ1ρ2u = u. From the analysis above we know if f NE ∈ � +FNE +both and +f NE(u) = (σ1, σ2), then σ1(1) = σ1(2), σ2(1) = σ2(2). We integrate the first two actions of player 1 +and player 2 into a new action that will choose randomly between the first two actions, then we form +the utility matrix below: +u = +� +1 +2, 1 +2 +0, 1 +2 + ε +1 +2 + ε, 0 +ε, ε +� +There is a unique NE in this Prisoner’s Dilemma, which has been bolded. The game u is the +same with the game u under the assumption that σ1(1) = σ1(2) and σ2(1) = σ2(2) in u. +Then +maxf∈ � +F NE +both SW(f(u), u) = 2ε. Since ε can be arbitrarily small, we have SWR2,3( � +FNE +both, �FNE +general) = 0. +As a result, we have SWRN,M( �FNE +both, �FNE +general) = 0 for all N ≥ 2 and M ≥ 3. +29 + diff --git a/VdFJT4oBgHgl3EQfOSwV/content/tmp_files/load_file.txt b/VdFJT4oBgHgl3EQfOSwV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..12c54efbe7eba65355aa2b5b9372de5bd539a195 --- /dev/null +++ b/VdFJT4oBgHgl3EQfOSwV/content/tmp_files/load_file.txt @@ -0,0 +1,992 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf,len=991 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='11481v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='GT] 27 Jan 2023 Are Equivariant Equilibrium Approximators Beneficial?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Zhijian Duan1, Yunxuan Ma1, Xiaotie Deng1,2 1Center on Frontiers of Computing Studies, Peking University 2Center for Multi-Agent Research, Institute for AI, Peking University {zjduan,charmingmyx,xiaotie}@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='cn Abstract Recently, remarkable progress has been made by approximating Nash equilibrium (NE), corre- lated equilibrium (CE), and coarse correlated equilibrium (CCE) through function approximation that trains a neural network to predict equilibria from game representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Furthermore, equiv- ariant architectures are widely adopted in designing such equilibrium approximators in normal- form games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' In this paper, we theoretically characterize benefits and limitations of equivariant equilibrium approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For the benefits, we show that they enjoy better generalizability than general ones and can achieve better approximations when the payoff distribution is permutation- invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For the limitations, we discuss their drawbacks in terms of equilibrium selection and social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Together, our results help to understand the role of equivariance in equilibrium approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 1 Introduction The equivariant equilibrium property states that, given a Nash Equilibrium (NE) solution of a game, the permuted solution is also an NE for the game whose actions of representation are permuted in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The same property also holds in correlated equilibrium (CE) and coarse correlated equilibrium (CCE), as well as the approximate solutions for all three solution concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' In this paper, we are interested in understanding the equivariant equilibrium property in designing neural networks that predict equilibria from game payoffs, following such recent approaches in de- signing equivariant equilibrium approximators [Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2021, Marris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2022] in normal-form games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Informally, such equivariant approximators keep the same permutation of the output strate- gies (represented as vectors or tensors) when the input game representations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', the game payoff tensors) are permuted 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' While equivariant approximators achieved empirical success, little work has theoretically discussed whether they are beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We theoretically characterize benefits and limitations of equivariant NE, CE and CCE approx- imators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For the benefits, we first show that equivariant approximators enjoy better generalizabil- ity, where we evaluate the approximators using the maximum exploitability [Lockhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2019, Goktas and Greenwald, 2022] over all players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' To get such a result, we derive the generalization bounds and the sample complexities of the NE, CE, and CCE approximators: The generalization bounds offer confidence intervals on the expected testing approximations based on the empirical training approxi- mations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The sample complexities describe how many training samples the equilibrium approximators need to achieve desirable generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The generalization bounds and sample complexities include the covering numbers [Shalev-Shwartz and Ben-David, 2014], which measure the representativeness of the approximators’ function classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Afterward, we prove that the equivariant approximators have lower covering numbers than the general models, therefore have lower generalization bounds and sam- ple complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We then show that the equivariant approximators can achieve better approximation when the payoff distribution is permutation-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' As for the limitations, we find the equivariant approximators unable to find all the equilibria of some normal-form games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Such a result is caused by the limited representativeness of the equivariant approximators’ function class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Besides, we find that the equivariant NE approximator may lose social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Specifically, in an example we constructed, while the maximum NE social welfare is large, the maximum social welfare of NEs that the equivariant NE approximators could find can be arbitrary 1We will provide a formal definition of equivariance equilibrium approximators in Section 3 1 close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Such a negative result inspires us to balance generalizability and social welfare when we design the approximators’ architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='1 Further Related Work Solving (approximate) NE, CE, and CCE for a single game are well studied [Fudenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 1998, Cesa-Bianchi and Lugosi, 2006].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' However, many similar games often need to be solved [Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2022] , both in practice and in some multi-agent learning algorithms [Marris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2021, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For instance, in repeated traffic routing games [Sessa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2020], the payoffs of games de- pend on the capacity of the underlying network, which can vary with time and weather condi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' In repeated sponsored search auctions, advertisers value different keywords based on the current marketing environment [Nekipelov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' In many multi-agent learning algorithms such as Nash Q-learning [Hu and Wellman, 2003], Correlated-Q learning [Greenwald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2003], V- learning [Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2022] and PSRO [Lanctot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2017], an NE, CE or CCE of a normal-form game need to be solved in every update step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' In these settings, it is preferred to accelerate the speed of game solving by function approximation: Marris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' [2022] introduces a neural equilibrium approximator to approximate CE and CCE for n- player normal-form games;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' [2021] proposes a neural NE approximator in PSRO [Lanctot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2017];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Wu and Lisser [2022] designs a CNN-based NE approximator for zero-sum bimatrix games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Dif- ferentiable approximators have also been developed to learn QREs [Ling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2018], NE in chance- constrained games [Wu and Lisser, 2023], and opponent’s strategy [Hartford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Equivariance is an ideal property of the equilibrium approximator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We will discuss the literates of equivariant approximators after formally defining them in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 2 Preliminary In this section, we introduce the preliminary and notations of our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We also provide a brief introduction to equilibrium approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='1 Game Theory Normal-Form Game Let a normal-form game with joint payoff u be Γu = (n, A, u), in which n ∈ N≥2 is the number of players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Each player is represented by the index i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' A = ×i∈[n]Ai is the product action space of all players, where Ai = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', mi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For player i ∈ [n], let ai ∈ Ai be a specific action of i (An action is also referred to as a pure strategy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' A joint action a = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' , an) ∈ A represents one play of the game in which the player i takes action ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The action space A is a Cartesian product that contains all possible joint actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We have |A| = � i∈[n] |Ai| = � i∈[n] mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u = (ui)i∈[n] is the joint payoff or utility of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ui is an n-dimensional tensor (or matrix if n = 2) describing player i’s payoff on each joint action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' In our paper, following previous literatures [Tsaknakis and Spirakis, 2007, Deligkas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2022], we normalize all the elements of payoff into [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' A joint (mixed) strategy is a distribution over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Let σ = ×i∈[n]σi be a product strategy and π ∈ ∆A be a joint (possibly correlated) strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Denote πi as the marginal strategy of player i in π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The expected utility of player i under π is ui(π) = Ea∼π[ui(a)] = � a∈A π(a)ui(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Besides, on behalf of player i, the other players’ joint strategy is denoted as π−i, so as a−i and σ−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 2 Nash Equilibrium (NE) We say a product strategy σ∗ = (σ∗ 1, σ∗ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' , σ∗ n) is a NE if each player’s strategy is the best response given the strategies of others, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', ui(σi, σ∗ −i) ≤ ui(σ∗ i , σ∗ −i), ∀i ∈ [n], σi ∈ ∆Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (NE) Computing NE for even general 2-player or 3-player games is PPAD-hard [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2009, Daskalakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2009], which leads to research on approximate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For arbitrary ǫ > 0, we say a product strat- egy ˆσ is an ǫ-approximate Nash equilibrium (ǫ-NE) if no one can achieve more than ǫ utility gain by deviating from her current strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Formally, ui(σi, ˆσ−i) ≤ ui(ˆσi, ˆσ−i) + ǫ, ∀i ∈ [n], σi ∈ ∆Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (ǫ-NE) The definition of ǫ-NE reflects the idea that players might not be willing to deviate from their strategies when the amount of utility they could gain by doing so is tiny (not more than ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Coarse Correlated Equilibrium (CCE) We say a joint (possibly correlated) strategy π∗ is a CCE if no player can receive a higher payoff by acting independently, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', ui(σi, π∗ −i) ≤ ui(π∗), ∀i ∈ [n], σi ∈ ∆Ai, (CCE) and we say ˆπ is an ǫ-approximate coarse correlated equilibrium (ǫ-CCE) for ǫ > 0 if ui(σi, ˆπ−i) ≤ ui(ˆπ) + ǫ, ∀i ∈ [n], σi ∈ ∆Ai, (ǫ-CCE) The difference between NE and CCE is that in an NE, players execute their strategy individu- ally in a decentralized way, while in a CCE, players’ strategies are possibly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' A stan- dard technique to correlate the strategy is sending each player a signal from a centralized controller [Shoham and Leyton-Brown, 2008].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Correlated Equilibrium (CE) CE is similar to CCE, except that in a CE, each player can observe her recommended action before she acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Thus, player i deviates her strategy through strategy mod- ification φi : Ai → Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' φi maps actions in Ai to possibly different actions in Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Based on strategy modification, we say a joint (possibly correlated) strategy π∗ is a CE if � a∈A π∗(a)ui(φi(ai), a−i) ≤ ui(π∗), ∀i, ∀φi, (CE) and a joint strategy ˆπ is an ǫ-approximate correlated equilibrium (ǫ-CE) for ǫ > 0 if � a∈A ˆπ(a)ui(φi(ai), a−i) ≤ ui(ˆπ) + ǫ, ∀i, ∀φi, (ǫ-CE) Note that for a finite n-player normal-form game, at least one NE, CE, and CCE must exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' This is because NE always exists [Nash et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 1950] and NE ⊆ CE ⊆ CCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Equilibrium Approximation To evaluate the quality of a joint strategy to approximate an equilib- rium, we define approximation based on exploitability [Lockhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2019, Goktas and Greenwald, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='1 (Exploitability and Approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Given a joint strategy π, the exploitability (or regret) Ei(π, u) of player i is the maximum payoff gain of i by deviating from her current strategy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', Ei(π, u) := max σ′ i ui(σ′ i, π−i) − ui(π) = max a′ i ui(a′ i, π−i) − ui(π) and the exploitability under strategy modification ECE i (π, u) of player i is the maximum payoff gain of i by deviating through strategy modification, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', ECE i (π, u) := max φi � a∈A π(a)ui(φi(ai), a−i) − ui(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 3 Algorithm 1 Example: learning NE/CCE approximator via minibatch SGD 1: Input: Training set S 2: Parameters: Number of iterations T > 0, batch size B > 0, learning rate η > 0, initial parameters w0 ∈ Rd of the approximator model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 3: for t = 0 to T do 4: Receive minibatch St = {u(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' , u(B)} ⊂ S 5: Compute the empirical average approximation of St: 6: LSt(f wt) ← 1 B �B i=1 E(f wt(u(i)), u(i)) 7: Update model parameters: 8: wt+1 ← wt − η∇wtLSt(f wt) 9: end for The equilibrium approximation is defined as the maximum exploitability over all players 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', E(π, u) := � maxi∈[n] Ei(π, u) , for NE and CCE maxi∈[n] ECE i (π, u) , for CE Based on approximation, we can restate the definition of solution concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' A product strategy σ is an NE of game Γu if E(σ, u) = 0 and is an ǫ-NE if E(σ, u) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' A joint strategy π is a (C)CE of Γu if E(π, u) = 0 and is an ǫ-(C)CE if E(π, u) ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='2 Equilibrium Approximator The equilibrium approximators, including NE, CE, and CCE approximators, aim to predict solution concepts from game representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' In our paper, we fix the number of players n and action space A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We denote by U the set of all the possible game payoffs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The NE approximator f NE : U → ×i∈[n]∆Ai maps a game payoff to a product strategy, where f NE(u)i ∈ ∆Ai is player i’s predicted strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We define FNE as the function class of the NE approximator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Similarly, we define (C)CE approximator as f (C)CE : U → ∆A and (C)CE approximator class as F(C)CE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' An equilibrium approximator can be learned through machine learning paradigms based on empir- ical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For instance, Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' [2021] learn the NE approximator using the game payoffs generated in the learning process of PSRO, and optimize the approximator by gradient descent in exploitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Marris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' [2022] learn the CE and CCE approximators using the games i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' generated from a manually designed distribution, and optimize the approximators using maximum welfare minimum relative entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Such a loss balances the equilibrium approximation, the social welfare, and the relative entropy of the joint strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Additionally, another simple way to learn NE or CCE equilibrium approximator is to apply minibatch stochastic gradient descent (SGD) on approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Specifically, we denote w ∈ Rd as the d-dimensional parameter of the approximator, such as the weights of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We can optimize w by the standard minibatch SGD algorithm on approximation (See Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 3 Equivariant Equilibrium Approximator In this section, we introduce the equivariance of the equilibrium approximators and the way how we apply orbit averaging [Elesedy and Zaidi, 2021] to construct equivariant approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Equiv- ariant approximator has been developed in many literatures [Hartford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2016, Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2021, Marris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2022, Wu and Lisser, 2022], which we will discuss latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We first define the permutation of a game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Let ρi : Ai → Ai be a permutation function of player i, which is a bijection from Ai to Ai itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Define Gi ∋ ρi as the class of permutation function for player i, which forms an Abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='1 (Permutation of a game).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For a normal-form game Γu = (n, u, A), we define the ρi-permutation of payoff tensor u as ρiu = (ρiuj)j∈[n], in which (ρiuj)(ai, a−i) = uj(ρ−1 i (ai), a−i), ∀a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 2A similar metric of equilibrium approximation is called Nikaido-Isoda function [Nikaidˆo and Isoda, 1955] or Nash- Conv [Lockhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2019], which is the sum of exploitability over all players, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', � i∈[n] Ei(π, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 4 We also define the ρi-permutation of joint strategy π as ρiπ, where (ρiπ)(ai, a−i) = π(ρ−1 i (ai), a−i), ∀a ∈ A, and the ρi-permutation of product strategy σ as ρiσ = (ρiσj)j∈[n], where ∀aj ∈ Aj, ρiσj(aj) = � σj(aj) , if j ̸= i σi(ρ−1 i ai) , if j = i Equivariant NE Approximator Considering the relationship of ρi-permuted game and ρi-permuted product strategy, we have the following result for NE: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' In a normal-form game Γu = (n, u, A), for arbitrary player i ∈ [n] and any (ǫ-)NE strategy σ = (σi, σ−i), ρiσ = (ρiσi, σ−i) is also an (ǫ-)NE for the ρi-permuted game Γρiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='2 tells the unimportance of action order with respect to NE approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Based on it, we can summarize two ideal properties for NE approximators, which are defined as follows: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='3 (Player-Permutation-Equivariance, PPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We say an NE approximator f NE satisfies player i permutation-equivariant (i-PE) if for arbitrary permutation function ρi ∈ Gi we have f NE(ρiu)i = ρif NE(u)i, (i-PE) Moreover, we say f NE is player-permutation-equivariant (PPE) if f NE satisfies i-PE for all player i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='4 (Opponent-Permutation-Invariance, OPI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We say an NE approximator f NE is oppo- nent i permutation-invariant (i-PI) if for any other player j ∈ [n] − {i} and arbitrary permutation function ρi ∈ Gi we have f NE(ρiu)j = f NE(u)j, ∀j ̸= i (i-PI) and we say f NE is opponent-permutation-invariant (OPI) if f NE satisfies i-PI for all player i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Equivariant (C)CE approximator Considering the relationship of ρi-permuted game and ρi- permuted joint strategy, we have a similar result for CE and CCE: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' In a normal-form game Γu = (n, u, A), for an arbitrary player i ∈ [n] and any (ε-)CE or (ǫ-)CCE strategy π, ρiπ is also an (ε-)CE or (ǫ-)CCE for the ρi-permuted game Γρiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Inspired by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5, we can also summarize an ideal property for CE and CCE approximators defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='6 (Permutation-Equivariance,PE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We say an (C)CE approximator f (C)CE is player i permutation-equivariant (i-PE) if for permutation function ρi ∈ Gi we have f (C)CE(ρiu) = ρif (C)CE(u), and we say f (C)CE is permutation-equivariant (PE) if f (C)CE satisfies i-PE for all player i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Equivariant Approximators in Literature For two-player games, Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' [2021] propose an MLP-based NE approximator that satisfies both PPE and OPI for zero-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Additionally, they also design a Conv1d-based NE approximator that satisfies PPE only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Hartford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' [2016] give a PPE approximator to predict players’ strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The traditional algorithms Tsaknakis and Spirakis [2007] and Deligkas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' [2022], which approximate NE by optimization, are also PPE and OPI to payoff and the initial strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For n-player general games, Marris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' [2022] provide a permutation- equivariant approximator to approximate CE and CCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Equivariant architectures are also adopted in optimal auction design [Rahme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2021, Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2022, Ivanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2022], and Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' [2022] theoretically characterize the benefits of permutation-equivariant in auction mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We follow the rough idea of Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' [2022] when we analyze the benefits of equivariant equilibrium approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='1 Orbit Averaging Orbit averaging is a well-known method to enforce equivariance or invariance for a function [Schulz-Mirbach, 1994].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' It averages the inputs of a function over the orbit of a group (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', the permutation group in our paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Orbit Averaging for NE Approximator For an NE approximator f NE and any player i ∈ [n], we can construct a i-PI or i-PE NE approximator by averaging with respect to all the permutations of player i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Specifically, we construct an i-PI NE approximator by operator Oi with (Oif NE)(u)j = � f NE(u)i , if j = i 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi f NE(ρiu)j , otherwise and we construct an i-PE NE approximator by operator Pi with: (Pif NE)(u)j = � 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 i f NE(ρiu)i , if j = i f NE(u)j , otherwise Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Oif NE is i-PI and Pif NE is i-PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Specially, if f NE is already i-PI or i-PE, then we have Oif NE = f NE or Pif NE = f NE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' To construct a PPE or OPI NE approximator, we composite the operators with respect to all players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Let O = O1 ◦ O2 ◦ · · · ◦ On and P = P1 ◦ P2 ◦ · · · ◦ Pn, we get the following corollary: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Of NE is OPI and Pf NE is PPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' If f NE is already OPI or PPE, we have Of NE = f NE or Pf NE = f NE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Furthermore, we can also compose P ◦O to construct a NE approximator with both PPE and OPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Orbit Averaging for (C)CE Approximator For CE or CCE approximator f, we define Qi- project for player i ∈ [n] to construct an i-PE approximator, which averages with respect to all the permutations of player i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (Qif (C)CE)(u) = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 i f (C)CE(ρiu) Similarly, we define Q = Q1 ◦ Q2 ◦ · · · ◦ Qn as the composite operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Qif (C)CE is i-PE and Qf (C)CE is PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Specifically, If f (C)CE is already i-PE or PE, then we have Qif (C)CE = f (C)CE or Qf (C)CE = f (C)CE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Combined with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='7, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='8 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='9, we can have the following corollary directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' O2 = O, P2 = P, Q2 = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The benefit of using orbit averaging is shown in the following lemma: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Denote X as an idempotent operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' X 2 = X (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' O, P or Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For function class F of NE, CE or CCE approximator, let FX be any subset of F that is closed under X, then XFX is the largest subset of FX that is invariant under X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' According to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='8, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='9 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='11, OFNE(or PFNE/QF(C)CE) is the largest subset of FNE(or FNE/F(C)CE) with the corresponding property (OPI, PPE, or PE) if FNE(or FNE/F(C)CE) is closed operator under O(or P/Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The result tells that the orbit averaging oper- ators, while enforcing the operated function to be equivariance or invariance, keep as large capacity of the function class as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Therefore, we believe that orbit averaging is an ideal approach to constructing equivariant or invariant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 6 4 Theoretical Analysis of Benefits In this section, we theoretically analyze the benefits of equivariant approximators with respect to generalizability and approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='1 Benefits for Generalization We first derive the generalization bound and sample complexity for general approximator classes, and then we show the benefits of equivariant approximators by applying orbit averaging to the ap- proximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The representativeness of an approximator class is measured by the covering numbers [Shalev-Shwartz and Ben-David, 2014] under ℓ∞-distance, which are defined as follows: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='1 (ℓ∞-distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The ℓ∞-distance between two equilibrium approximators f, g is: ℓ∞(f, g) = max u∈U ∥f(u) − g(u)∥, where we define the distance of two product strategies σ and σ′ as ∥σ1 − σ2∥ = max i∈[n] � ai∈Ai |σ1 i (ai) − σ2 i (ai)| and the distance of two joint strategy π and π′ as ∥π1 − π2∥ = � a∈A |π1(a) − π2(a)| Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='2 (r-covering number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For r > 0, we say function class Fr r-covers another function class F under ℓ∞-distance if for all function f ∈ F, there exists fr ∈ Fr such that ∥f − fr∥∞ ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The r-covering number N∞(F, r) of F is the cardinality of the smallest function class Fr that r-covers F under ℓ∞-distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Based on covering numbers, we provide the generalization bounds of NE, CE and CCE approxima- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The bounds describe the difference between the expected testing approximation and empirical training approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='3 (Generalization bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For function class F of NE, CE or CCE approximator, with probability at least 1 − δ over draw of the training set S (with size m) from payoff distribution D, for all approximator f ∈ F we have Eu∼D[E(f(u), u)] − 1 m � u∈S E(f(u), u) ≤ 2 · inf r>0{ � 2 ln N∞(F, r) m + Lr} + 4 � 2 ln(4/δ) m , where L = 2n for NE approximator, and L = 2 for CE and CCE approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' To get the theorem, we first show that all three equilibrium approximations are Lipschitz continuous with respect to strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Afterward, we derive the Rademacher complexity [Bartlett and Mendelson, 2002] of the expected approximation based on the Lipschitz continuity and covering numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='6 for the detailed proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We can see from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='3 that, with a large enough training set, the generalization gaps of equilibrium approximators go to zero if the covering number N∞(F, r) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' As a result, we can estimate the expected testing performance through the empirical training performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We can also derive the sample complexities of equilibrium approximators to achieve the desirable generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='4 (Sample complexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For ǫ, δ ∈ (0, 1), function class F of NE, CE or CCE approxi- mator and distribution D, with probability at least 1 − δ over draw of the training set S with m ≥ 9 2ǫ2 � ln 2 δ + ln N∞(F, ǫ 3L) � 7 games sampled from D, ∀f ∈ F we have Eu∼D[E(f(u), u)] ≤ 1 m � u∈S E(f(u), u) + ǫ, where L = 2n for NE approximators, and L = 2 for CE and CCE approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The proof is based on the Lipschitz continuity of approximation, uniform bound, and concentration inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='7 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='4 is also called the uniform convergence of function class F, which is a sufficient condition for agnostic PAC learnable [Shalev-Shwartz and Ben-David, 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' As for the benefits of equivariant approximators for generalizability, the following result indicates that the projected equilibrium approximators have smaller covering numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The O-projected, P-projected and Q-projected approximator classes have smaller cov- ering numbers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', ∀r > 0 we have N∞(OFNE, r) ≤ N∞(FNE, r), N∞(PFNE, r) ≤ N∞(FNE, r), N∞(QF(C)CE, r) ≤ N∞(F(C)CE, r) The proof is done by showing all the operators are contraction mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='8 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Both the generalization bounds in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='3 and the sample complexities in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='4 decrease with the decrease of covering numbers N∞(F, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Thus, we can see from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5 that both PPE and OPI can improve the generalizability of NE approximators, and PE can improve the generalizability of CE and CCE approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='2 Benefits for Approximation We then show the benefits of equivariance for approximation when the payoff distribution is invari- ant under permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The permutation-invariant distribution holds when the action is anonymous or indifferent or when we pre-train the equilibrium approximators using a manually designed distribu- tion [Marris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (C)CE Approximator The following theorem tells the benefit of permutation-equivariance in de- creasing the exploitability of (C)CE approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' When the payoff distribution D is invariant under the permutation of payoffs, the Q-projected (C)CE approximator has a smaller expected equilibrium approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Formally, for all f (C)CE ∈ F(C)CE and permutation-invariant distribution D, we have Eu∼D[E(Qf (C)CE(u), u)] ≤ Eu∼D[E(f (C)CE(u), u)], The proof is done by using the convexity of approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='10 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We can see from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='6 that, when payoff distribution is invariant under permutation, it is beneficial to use equivariant architecture as the CE or CCE approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' NE Approximator As for NE approximator, we have similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For bimatrix constant-sum games, when the payoff distribution D is invariant under the permutation of payoffs, then the X-projected (X ∈ {O, P}) NE approximator has a smaller expected exploitability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Formally, for all f NE ∈ FNE and permutation-invariant distribution D for bimatrix constant-sum games, we have Eu∼D[ � i Ei((Xf NE)(u), u)] ≤ Eu∼D[ � i Ei(f NE(u), u)] 8 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' When the payoff distribution D is invariant under the permutation of payoffs, and f NE satisfies OPI, then the P-projected NE approximator has a smaller expected NE approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Formally, for all f NE ∈ FNE that is OPI and permutation-invariant distribution D, we have Eu∼D[E((Pf NE)(u), u)] ≤ Eu∼D[E(f NE(u), u)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For bimatrix games, when the payoff distribution D is invariant under the permutation of payoffs, and f NE satisfies PPE, then the O-projected NE approximator has a smaller expected NE approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Formally, for all f NE ∈ FNE that is PPE and permutation-invariant distribution D of bimatrix games, we have Eu∼D[E((Of NE)(u), u)] ≤ Eu∼D[E(f NE(u), u)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='8 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='9 tell that PPE and OPI approximators can achieve better approxi- mation than ones with only PPE or OPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Meanwhile, we can see from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='7 that for bimatrix constant-sum games (such as zero-sum games), it can be preferred to introduce PPE or OPI to the architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 5 Theoretical Analysis of Limitations As we discussed in Section 4, equivariant approximators enjoy better generalizability and better approximation sometimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' However, as we will show, they have some limitations regarding equilibrium selection and social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Such limitations attribute to the limited representativeness caused by equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='1 Equilibrium Selection We first show that there may be equilibria points that equivariant approximators will never find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We illustrate such limitation in permutation-invariant games, which is defined as follows: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='1 (Permutation-ρ-Invariant Game).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We say a game Γu is permutation-ρ-invariant, where ρ = ◦i∈[n]ρi, if the payoff u is permutation-invariant with respect to ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' That is, ρu = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Permutation-ρ-invariance indicates that one cannot distinguish joint action a from ρa using only the payoff u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We’d like to provide an example to show more insight of permutation-ρ-invariant games: Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For a 2-player game Γu = (2, u = (u1, u2), A = ([m1], [m2])) , Let ρi = (mi, mi − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' , 1) and ρ = ρ1 ◦ ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' If one of the following conditions holds, then u is permutation-ρ-invariant: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u1 and u2 are symmetric and persymmetric (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', symmetric with respect to the northeast-to- southwest diagonal) squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Both u1 and u2 are centrosymmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', ui(x, y) = ui(m1 +1−x, m2 +1−y) for i ∈ {1, 2}, x ∈ [m1] and y ∈ [m2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For permutation ρ = (◦i∈[n]ρi) and player k ∈ [n], we denote the set of non-fixed actions of player k under ρk as V (ρk) := {ak|ak ∈ Ak, ρk(ak) ̸= ak}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Based on V (ρk), we find some equilibria points of permutation-ρ-invariant games that any equivariant approximators will never find.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For a permutation-ρ-invariant game Γu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' if there is a pure NE a∗ = (a∗ i )i∈[n] and at least one player k ∈ [n] such that a∗ k ∈ V (ρk), then a∗ will never be found by any NE approximator with both PPE and OPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Besides, a∗ (as a pure CE or CCE) will also never be found by any CE or CCE approximator with PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We illustrate Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='3 by the following example: 9 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Consider a bimatrix game with identity utility u = � 1, 1 0, 0 0, 0 1, 1 � There are two pure NE (bolded in the above matrix) and one mixed NE of σ1 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5) and σ2 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Let ρi be the unique permute function (except for identity function) of player i ∈ [2], and ρ = ρ1 ◦ ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The game is permutation-ρ-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Case 1: Let f be a permutation-equivariant CE or CCE approximator, and denote π = f(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We have π = f(u) (a) = f(ρu) (b) = ρf(u), where (a) holds by permutation-ρ-invariance of u, and (b) holds by PE of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Thus, we have π1,1 = π2,2 ∈ [0, 1 2] and π1,2 = π2,1 ∈ [0, 1 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' As a result, the two pure (C)CEs cannot be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Case 2: Let f be a NE approximator that holds PPE and OPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Denote f(u) = (σ1, σ2), where σ1 = (p1, 1 − p1) and σ2 = (p2, 1 − p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' By PPE and OPI of f, we have f(u)1 = (p1, 1 − p1) (a) = f(ρ1ρ2u)1 (b) = ρ1f(ρ2u)1 (c) = ρ1f(u)1 = (1 − p1, p1), where (a) holds by permutaion-ρ-invariance of u, (b) holds by PPE of f, and (c) holds by OPI of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' As a result, the only NE that f could find is the mixed NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' As we can see from the example and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='3, the equivariance, while introducing inductive bias to the approximator architecture, is also a strong constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Such a constraint is why the equivariant approximators cannot find all the equilibria points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='2 Social Welfare The social welfare of a joint strategy π is defined as the sum of all players’ utilities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', SW(π, u) = � i∈[n] ui(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The equilibrium with higher social welfare is usually preferred [Marris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' To analyze the social welfare of equivariant approximators, we define the worst social welfare ratio as follows: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For any N, M ≥ 2 and two NE (or CE/CCE) approximator classes F1, F2 that target on games with number of players n ≤ N and |Ai| ≤ M, we define the worst social welfare ratio of F1 over F2 as: SWRN,M(F1, F2) := inf D maxf1∈F1 Eu∼DSW(f1(u), u) maxf2∈F2 Eu∼DSW(f2(u), u) SWRN,M(F1, F2) measures the relative representativeness of F1 over F2 in terms of social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Based on that, we have the following result for equivariant CE and CCE approximator classes: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Given N, M ≥ 2, let F(C)CE PE be the function class (target on games with number of players n ≤ N and |Ai| ≤ M) of all the (C)CE approximators with PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Denote by F(C)CE general the function class of all the (C)CE approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Then we have SWRN,M(F(C)CE PE , F(C)CE general) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='6 tells that, while the permutation-equivariant (C)CE approximator class may not be able to find all the (C)CE in a game, it can keep the social welfare of the output solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' However, when considering equivariant NE approximators, we have the following negative result: 10 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Given N, M ≥ 2, let FNE OPI, FNE PPE and FNE both be the function classes (target on games with number of players n ≤ N and |Ai| ≤ M) of all the NE approximators with OPI, PPE and both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Denote the function class of all the NE approximators as FNE general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Then we have SWRN,M(FNE OPI, FNE general) = 1 M N−1 , (1) SWRN,M(FNE PPE, FNE general) ≤ 1 M , (2) SWRN,M(FNE both, FNE general) = 1 M N−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (3) Additionally, when M ≥ 3, denote by �FNE both the function class of all the NE oracles (functions that always output exact NE solutions of the input games) with both PPE and OPI, and by � FNE general the function class of all the NE oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Then we have SWRN,M( �FNE both, �FNE general) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (4) The proof is done by construction (See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='15 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' As an illustration of Equa- tion (4), consider a bimatrix game with the following payoff: u = \uf8ee \uf8f0 1, 1 0, 0 0, 1 2 + ε 0, 0 1, 1 0, 1 2 + ε 1 2 + ε, 0 1 2 + ε, 0 ε, ε \uf8f9 \uf8fb for ǫ ∈ (0, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The maximum NE (the upper-left corner of u) social welfare is 2, which can be found by at least one NE oracle in �FNE general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' However, the only NE (the lower-right corner of u) that the NE oracles in �FNE both could find only has a social welfare of 2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' As a result, SWR2,3( �FNE both, �FNE general) ≤ 2ǫ 2 = ǫ, which goes to zero as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Recall that we always have SWRN,M ≥ 0, thus Equation (4) holds when N = 2 and M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='7 tells that equivariant NE approximators may lose some social welfare while enjoying better generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Such a result inspires us to balance generalizability and social welfare when designing the NE approximator architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 6 Conclusion and Future Work In this paper, we theoretically analyze the benefits and limitations of equivariant equilibrium approximators, including player-permutation-equivariant (PPE) and opponent-permutation-invariant (OPI) NE approximator, and permutation-equivariant (PE) CE and CCE approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For the benefits, we first show that these equivariant approximators enjoy better generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' To get the result, we derive the generalization bounds and sample complexities based on covering numbers, and then we prove that the symmetric approximators have lower covering numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We then show that the equivariant approximators can decrease the exploitability when the payoff distribution is invariant under permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For the limitations, we find the equivariant approximators may fail to find some equilibria points due to their limited representativeness caused by equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Besides, while equiv- ariant (C)CE approximators can keep the social welfare, the equivariant NE approximators reach a small worst social welfare ratio comparing to the general approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Such a result indicates that equivariance may reduce social welfare;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' therefore, we’d better balance the generalizability and social welfare when we design the architectures of NE approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' As for future work, since in our paper we assume the training and testing payoff distribution are the same, an interesting topic is to study the benefits of equivariant approximators under the payoff distribution shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Moreover, since we consider fixed and discrete action space, another interesting future direction is to analyze the benefits of equivariant approximators in varying or continuous action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 11 References Peter L Bartlett and Shahar Mendelson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Rademacher and gaussian complexities: Risk bounds and structural results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Journal of Machine Learning Research, 3(Nov):463–482, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Nicolo Cesa-Bianchi and G´abor Lugosi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Prediction, learning, and games.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 3-Conference C: Signal Processing (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 94CH3440-5), volume 2, pages 387–390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' IEEE, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Pier Giuseppe Sessa, Ilija Bogunovic, Andreas Krause, and Maryam Kamgarpour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Contextual games: Multi-agent learning with side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 33:21912–21922, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Shai Shalev-Shwartz and Shai Ben-David.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Understanding machine learning: From theory to algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Cambridge university press, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Yoav Shoham and Kevin Leyton-Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Multiagent systems: Algorithmic, game-theoretic, and logical foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Cambridge University Press, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Haralampos Tsaknakis and Paul G Spirakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' An optimization approach for approximate Nash equilib- ria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' In International Workshop on Web and Internet Economics, pages 42–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Springer, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Dawen Wu and Abdel Lisser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Using CNN for solving two-player zero-sum games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Expert Systems with Applications, page 117545, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Dawen Wu and Abdel Lisser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' CCGnet: A deep learning approach to predict Nash equilibrium of chance-constrained games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Information Sciences, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 13 A Omitted Proof A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='1 Useful Lemma We first introduce a lemma, which will be frequently used in the following proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ∀i, j ∈ [n], ρi ∈ Gi we have (ρiu)j(σi, σ−i) = uj(ρ−1 i σi, σ−i) and (ρiu)j(π) = uj(ρ−1 i π) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Define �ai := ρ−1 i ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For product strategy σ = (σi)i∈[n],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (ρiu)j(σi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' σ−i) = � ai∈Ai � a−i∈A−i (ρiu)j(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) · σi(ai) · σ−i(a−i) = � ai∈Ai � a−i∈A−i uj(ρ−1 i ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) · σi(ai) · σ−i(a−i) = � ai∈Ai � a−i∈A−i uj(ρ−1 i ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) · (ρ−1 i σi)(ρ−1 i ai) · σ−i(a−i) = � �ai∈Ai � a−i∈A−i uj(�ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) · (ρ−1 i σi)(�ai) · σ−i(a−i) =uj(ρ−1 i σi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' σ−i) For joint strategy π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (ρiu)j(π) = � ai∈Ai � a−i∈A−i (ρiuj)(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) · π(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) = � ai∈Ai � a−i∈A−i uj(ρ−1 i ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) · π(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) = � ai∈Ai � a−i∈A−i uj(ρ−1 i ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) · (ρ−1 i π)(ρ−1 i ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) = � �ai∈Ai � a−i∈A−i uj(�ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) · (ρ−1 i π)(�ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) =uj(ρ−1 i π) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='2 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For player i, we have Ei(ρiσ, ρiu) = max ai∈Ai ρiui(ai, ρiσ−i) − ρiui(ρiσ) = max ai∈Ai ρiui(ai, σ−i) − ρiui(ρiσi, σ−i) = max ai∈Ai ui(ρ−1 i ai, σ−i) − ui(ρ−1 i ρiσi, σ−i) (a) = max ai∈Ai ui(ai, σ−i) − ui(σi, σ−i) = Ei(σ, u), where (a) holds since ρi is a bijection on Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For player j ̸= i, we have Ej(ρiσ, ρiu) = max aj∈A ρiuj(aj, ρiσ−j) − ρiuj(ρiσ) = max aj∈Aj uj(aj, ρ−1 i ρiσ−j) − uj(ρ−1 i ρiσ) = max aj∈Aj uj(aj, σ−j) − uj(σ) = Ej(σ, u) From above, we have E(ρiσ, ρiu) = E(σ, u), thus if σ is a ε-NE of Γu, then ρiσ must be a ε-NE of Γρiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='3 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5 CCE For player i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' we have Ei(ρiπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ρiu) = max ai∈Ai(ρiui)(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (ρiπ)−i) − (ρiui)(ρiπi) = max ai∈Ai(ρiui)(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (ρiπ)−i) − ui(ρ−1 i ρiπi) = max ai∈Ai(ρiui)(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (ρiπ)−i) − ui(πi) = max ai∈Ai � b∈A (ρiui)(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' b−i) · (ρiπ)(b) − ui(πi) = max ai∈Ai � bi∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='b−i∈A−i ui(ρ−1 i ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' b−i) · π(ρ−1 i bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' b−i) − ui(πi) = max ai∈Ai � bi∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='b−i∈A−i ui(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' b−i) · π(bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' b−i) − ui(πi) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ρi is a bijection on Ai =Ei(π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u) For player j ̸= i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' we have Ej(ρiπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ρiu) = max aj∈Aj(ρiuj)(aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (ρiπ)−j) − (ρiuj)(ρiπj) = max aj∈Aj(ρiuj)(aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (ρiπ)−j) − uj(ρ−1 i ρiπj) = max aj∈Aj(ρiuj)(aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (ρiπ)−j) − uj(πj) = max aj∈Aj � b∈A (ρiuj)(aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' b−j) · (ρiπ)(b) − uj(πj) = max aj∈Aj � bi∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='b−i∈A−i uj(aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (b−j)−i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ρ−1 i bi) · π(ρ−1 i bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' b−i) − uj(πj) = max aj∈Aj � bi∈Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='b−i∈A−i uj(aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (b−j)−i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' bi) · π(bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' b−i) − uj(πj) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ρi is a bijection on Ai =Ej(π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u) Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' we have E(ρiπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ρiu) = E(π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Thus, if π is a ε-CCE of Γu, then ρiπ must be a ε-CCE of Γρiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' CE For player j ̸= i, we have ECE j (ρiπ, ρiu) = max φj:Aj→Aj � a∈A (ρiπ)(a) · (ρiuj)(φj(aj), a−j) − (ρiuj)(ρiπ) = max φj:Aj→Aj � a∈A π(ρ−1 i ai, a−i) · uj(φj(aj), a−i,j, ρ−1 i ai) − uj(π) = max φj:Aj→Aj � a∈A π(ai, a−i) · uj(φj(aj), a−i,j, ai) − uj(π) , ρi is a bijection on Ai =ECE j (π, u) For player i, we define operator ¯ρi as (¯ρiφi)(ai) = ρ−1 i φi(ρiai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We can verify that ¯ρi is a bijection on {φi : Ai → Ai}, because ¯· is a homomorphism in the sense that ρ1 i ◦ ρ2 i = ρ2 i ρ1 i and ¯· maps the identity mapping of Ai to the identity mapping of {Ai → Ai}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Specifically, ρ1 i ◦ ρ2 i φi(ai) = (ρ1 i )−1(ρ2 i φi)(ρ1 i ai) = (ρ1 i )−1(ρ2 i )−1φi(ρ2 i ρ1 i ai) = ρ2 i ρ1 i φi(ai), and eiφi(ai) = e−1 i φi(eiai) = φi(ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 15 Based on ¯ρi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' we have ECE i (ρiπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ρiu) = max φi:Ai→Ai � a∈A (ρiπ)(a) · (ρiui)(φi(ai),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) − ui(π) = max φi:Ai→Ai � a∈A π(ρ−1 i ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i)ui(ρ−1 i φi(ai),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) − ui(π) = max φi:Ai→Ai � a∈A π(ρ−1 i ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i)ui(ρ−1 i φi(ρi(ρ−1 i ai)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) − ui(π) = max φi:Ai→Ai � a∈A π(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i)ui(ρ−1 i φi(ρiai),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) − ui(π) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ρi is a bijection on Ai = max φi:Ai→Ai � a∈A π(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i)ui((¯ρiφi)(ai),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) − ui(π) = max φi:Ai→Ai � a∈A π(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i)ui(φi(ai),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−i) − ui(π) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ¯ρi is a bijection on {Ai → Ai} =ECE i (π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u) Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' we have E(ρiπ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ρiu) = E(π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' thus if π is a ε-CE of Γu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' then ρiπ must be a ε-CE of Γρiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='4 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='7 to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='9 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ∀j ̸= i, ρ0 ∈ Gi, for operator Oi we have (Oif NE)(ρ0u)j = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi f NE(ρiρ0u)j (a) = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � �ρi∈Gi f NE(�ρiu)j = (Oif NE)(u)j where in (a) we define �ρi = ρiρ0, and (a) holds since ρ0 is a bijection on Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' As a result, Oif NE is i-PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For operator Pi we have (Pif NE)(ρ0u)i = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 i f NE(ρiρ0u)j = ρ0 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 0 ρ−1 i f NE(ρiρ0u)j =ρ0 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � �ρi∈Gi �ρ−1 i f NE(�ρiu)j = ρ0(Pif NE)(u)i, therefore Pif NE is i-PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' If f NE is already i-PI, ∀j ̸= i we have Oif NE(u)j = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi f NE(ρiu)j = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi f NE(u)j = f NE(u)j, and Oif NE(u)i = f NE(u)i according to definition of Oi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Therefore, Oif NE = f NE for i-PI f NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' If f NE is already i-PE, we have Pif NE(u)i = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 i f NE(ρiu)i = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 i ρif NE(u)i = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi f NE(u)i = f NE(u)i, and ∀j ̸= i, Pif NE(u)j = f NE(u)j according to definition of Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Therefore, Pif NE = f NE for i-PE f NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' A direct inference from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='7 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ∀ρ0 ∈ Gi, we have 16 (Qif (C)CE)(ρ0u) = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 i f (C)CE(ρiρ0u) = ρ0 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 0 ρ−1 i f (C)CE(ρiρ0u) =ρ0 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � �ρi∈Gi �ρ−1 i f (C)CE(�ρiu) = ρ0(Qif (C)CE)(u) If f (C)CE is already i-PE, we have Qif (C)CE(u) = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 i f (C)CE(ρiu) = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 i ρif (C)CE(u) = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi f (C)CE(u) = f (C)CE(u) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5 Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='11 We prove the three claims below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' XFX ⊆ FX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' X 2FX = XFX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' If XY = Y ⊆ FX , then Y ⊆ XFX The first claim holds because FX is closed under X, and the second claim holds because X is idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For the third claim, from Y ⊆ FX we know XY ⊆ XFX , then Y = XY ⊆ XFX .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We immediately know XFX is the largest subset of FX that is invariant under X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='6 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='3 Some of the techniques come from D¨utting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' [2019] and Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' [2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We first introduce some useful lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Denote ℓ : F × U → R as the loss function (such as ℓ(f, u) := E(f(u), u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We measure the capacity of the composite function class ℓ ◦ F using the empirical Rademacher complex- ity [Bartlett and Mendelson, 2002] on the training set S, which is defined as: RS(ℓ ◦ F) := 1 mEx∼{+1,−1}m � sup f∈F m � i=1 xi · ℓ(f, u(i)) � , where x is distributed i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' according to uniform distribution in {+1, −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We have Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='2 (Shalev-Shwartz and Ben-David [2014]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Let S be a training set of size m drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' from distribution D over U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Then with probability at least 1 − δ over draw of S from D, for all f ∈ F, Eu∼D[ℓ(f, u)] − 1 m � u∈S ℓ(l, u) ≤ 2RS(ℓ ◦ F) + 4 � 2 ln(4/δ) m Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' If |ℓ(·)| ≤ c for constant c > 0 and ∀f, f ′ ∈ F, |ℓ(f, u) − ℓ(f ′, u)| ≤ L∥f − f ′∥∞, then we have Eu∼D[ℓ(f, u)] − 1 m � u∈S ℓ(l, u) ≤ 2 inf r>0 � c � 2 ln N∞(F, r) m + Lr � + 4 � 2 ln(4/δ) m Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For function class F, let Fr with |Fr| = N∞(F, r) be the function class that r-covers F for 17 some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Similarly, ∀f ∈ F, denote fr ∈ Fr be the function that r-covers f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We have RS(ℓ ◦ F) = 1 mEx � sup f∈F m � i=1 xi · ℓ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u(i)) � = 1 mEx � sup f∈F m � i=1 xi · � ℓ(fr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u(i)) + ℓ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u(i)) − ℓ(fr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u(i)) �� ≤ 1 mEx � sup fr∈Fr m � i=1 xi · ℓ(fr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u(i)) � + 1 mEx � sup f∈F m � i=1 |xi · Lr| � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' |ℓ(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u) − ℓ(fr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u)| ≤ L∥f − fr∥∞ = Lr ≤ sup fr∈Fr � � � � m � i=1 ℓ2(fr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u(i)) · � 2 ln N∞(F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' r) m + Lr m Ex∥x∥ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' the first term holds by Massart’s lemma ≤ √ c2m · � 2 ln N∞(F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' r) m + Lr m Ex∥x∥ ≤c � 2 ln N∞(F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' r) m + Lr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (5) Combining Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='2 and Equation (5), we get Eu∼D[ℓ(f, u)] − 1 m � u∈S ℓ(l, u) ≤ 2 inf r>0 � c � 2 ln N∞(F, r) m + Lr � + 4 � 2 ln(4/δ) m NE Approximator Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For arbitrary product mixed strategy σ and σ′, we have |E(σ, u) − E(σ′, u)| ≤ 2n∥σ − σ′∥, Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ∀σ, σ′, we define y−j := (σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' , σj−1, σ′ j+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' , σ′ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Then, ∀i ∈ [n] we have |ui(σ) − ui(σ′)| =|ui(σ1, σ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' , σn) − ui(σ′, σ′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' , σ′ n)| = ��� n � j=1 � ui(σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' , σj, σ′ j+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' , σ′ n) − ui(σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' , σ′ j, σ′ j+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' σ′ n) ���� = ��� n � j=1 � ui(σj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' y−j) − ui(σ′ j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' y−j) ���� = ��� n � j=1 � aj (σj(aj) − σ′ j(aj)) � a−j ui(aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−j)y−j(a−j) ��� ≤ n � j=1 � aj ���σj(aj) − σ′ j(aj) ��� � a−j ui(aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' a−j)y−j(a−j) ≤ n � j=1 � aj ���σj(aj) − σ′ j(aj) ��� � a−j y−j(a−j) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ui(·) ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 1] ≤ n � j=1 � aj∈Aj ���σj(aj) − σ′ j(aj) ��� ≤ n max j∈[n] � aj∈Aj ���σj(aj) − σ′ j(aj) ��� =n∥σ − σ′∥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ∀ai ∈ Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ui(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' σ−i) − ui(σ) =ui(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' σ−i) − ui(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' σ′ −i) + ui(ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' σ′ −i) − ui(σ′) + ui(σ′) − ui(σ) ≤n∥σ − σ′∥ + E(σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u) + n∥σ − σ′∥ =E(σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' u) + 2n∥σ − σ′∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 18 Based on that, we get E(σ, u) = max i∈N,ai∈Ai[ui(ai, σ−i) − ui(σ)] ≤ E(σ′, u) + 2n∥σ − σ′∥ Similarly, we also have E(σ′, u) ≤ E(σ, u) + 2n∥σ − σ′∥ Based on Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='4, ∀f, f ′ ∈ FNE, we have E(f(u), u) − E(f ′(u), u) ≤ 2∥f(u) − f ′(u)∥ ≤ 2∥f − f ′∥∞ Considering that |E(·)| ≤ 1, according to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='3, we have: Eu∼D[E(f NE(u), u)] − 1 m � u∈S E(f NE(u), u) ≤ 2 · inf r>0 �� 2 ln N∞(FNE, r) m + 2nr � + 4 � 2 ln(4/δ) m CCE Approximator Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For arbitrary joint mixed strategy π and π′, we have |E(π, u) − E(π′, u)| ≤ 2∥π − π′∥, Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ∀π, π′, ∀i ∈ [n] we have |ui(π) − ui(π′)| = � a∈A (π(a) − π′(a))ui(a) (a) ≤ � a∈A |π(a) − π′(a)| = ∥π − π′∥ (6) where (a) holds since ui(·) ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Therefore, ∀ai ∈ Ai, ui(ai, π−i) − ui(π) =ui(ai, π−i) − ui(ai, π′ −i) + ui(ai, π′ −i) − ui(π′) + ui(π′) − ui(π) ≤∥π − π′∥ + E(π′, u) + ∥π − π′∥ =E(π′, u) + 2∥π − π′∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Based on that, we get E(π, u) = max i∈N,ai∈Ai[ui(ai, π−i) − ui(π)] ≤ E(π′, u) + 2∥π − π′∥ Similarly, we also have E(π′, u) ≤ E(π, u) + 2∥π − π′∥ Based on Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5, ∀f, f ′ ∈ FCCE, we have E(f(u), u) − E(f ′(u), u) ≤ 2∥f(u) − f ′(u)∥ ≤ 2∥f − f ′∥∞ Considering that |E(·)| ≤ 1, according to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='3, we have: Eu∼D[E(f CCE(u), u)] − 1 m � u∈S E(f CCE(u), u) ≤ 2 · inf r>0 �� 2 ln N∞(FCCE, r) m + 2r � + 4 � 2 ln(4/δ) m 19 CE Approximator Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For arbitrary joint mixed strategy π and π′, we have |ECE(π, u) − ECE(π′, u)| ≤ 2∥π − π′∥, Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ∀ai ∈ Ai, ∀φi, we have � a∈A π(a)ui(φ(ai), a−i) − ui(π) = � a∈A π(a)ui(φ(ai), a−i) − � a∈A π′(a)ui(φ(ai), a−i) + � a∈A π′(a)ui(φ(ai), a−i) − ui(π′) + ui(π′) − ui(π) ≤∥π − π′∥ + ECE(π′, u) + ∥π − π′∥ =ECE(π′, u) + 2∥π − π′∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Based on that, we get ECE(π, u) = max i∈N max φi � a∈A π(a)ui(φ(ai), a−i) − ui(π) ≤ ECE(π′, u) + 2∥π − π′∥ Similarly, we also have ECE(π′, u) ≤ ECE(π, u) + 2∥π − π′∥ Based on Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5, ∀f, f ′ ∈ FCE, we have ECE(f(u), u) − ECE(f ′(u), u) ≤ 2∥f(u) − f ′(u)∥ ≤ 2∥f − f ′∥∞ Considering that |E(·)| ≤ 1, according to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='3, we have: Eu∼D[ECE(f CE(u), u)] − 1 m � u∈S ECE(f CE(u), u) ≤ 2 · inf r>0 �� 2 ln N∞(FCE, r) m + 2r � + 4 � 2 ln(4/δ) m A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='7 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='4 For function class F of NE, CE or CCE approximators, according to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='4, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='6, ∀f, g ∈ F we have E(CE)(f(u), u) − E(CE)(g(u), u) ≤ L∥f(u) − g(u)∥ ≤ L∥f − g∥∞, (7) where L = 2n for NE approximators, and L = 2 for CE and CCE approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For simplicity, we denote LS(f) = 1 m � u∈S E(CE)(f(u), u) and LD(f) = Eu∼D[E(CE)(f(u), u)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' let Fr with |Fr| = N∞(F, r) be the function class that r-covers F for some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ∀ǫ ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' by setting r = ǫ 3L we have PS∼Dm � ∃f ∈ F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ��LS(f) − LD(f) �� > ǫ � ≤PS∼Dm � ∃f ∈ F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ��LS(f) − LS(fr) �� + ��LS(fr) − LD(fr) �� + ��LD(fr) − LD(f) �� > ǫ � (a) ≤PS∼Dm � ∃f ∈ F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Lr + ��LS(fr) − LD(fr) �� + Lr > ǫ � ≤PS∼Dm � ∃fr ∈ Fr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ��LS(fr) − LD(fr) �� > ǫ − 2Lr � (b) ≤N∞(F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' r)PS∼Dm ���LS(f) − LD(f) �� > ǫ − 2Lr � (c) ≤2N∞(F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' r) exp(−2m(ǫ − 2Lr)2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' =2N∞(F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ǫ 3L) exp(−2 9mǫ2) where (a) holds by Equation (7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (b) holds by union bound,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' and (c) holds by Hoeffding inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' As a result, when m ≥ 9 2ǫ2 � ln 2 δ + ln N∞(F, ǫ 3L) � , we have PS∼Dm � ∃f ∈ F, ���LS(f) − LD(f) ��� > ǫ � < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='8 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='5 We first provide an auxiliary lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For function class F and orbit averaging operator X, if ∀f, g ∈ F, ℓ∞(Xf, Xg) ≤ ℓ∞(f, g), then N∞(XF, r) ≤ N∞(F, r) for any r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ∀r > 0, Denote Fr as the smallest r-covering set that covers F with size N∞(F, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ∀f ∈ F, let fr ∈ Fr be the function that r-covers f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We have ℓ∞(Xfr, Xf) ≤ ℓ∞(fr, f) ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Therefore, XFr is a r-covering set of XF, and we have N∞(XF, r) ≤ |XFr| ≤ |Fr| = N∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For player i ∈ [n] and ∀f NE, gNE ∈ FNE, assuming U is closed under any ρi ∈ Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For Oi, l∞(Oif NE, OigNE) = max u∈U ∥Oif NE(u) − OigNE(u)∥ = max j∈[n] max u∈U ∥(Oif NE)(u)j − (OigNE)(u)j∥ = max � max u∈U ∥f NE(u)i − gNE(u)i∥, max j̸=i max u∈U ∥ 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi (f NE(ρiu)j − gNE(ρiu)j)∥ � ≤ max � max u∈U ∥f NE(u)i − gNE(u)i∥, max j̸=i 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi max u∈U ∥f NE(ρiu)j − gNE(ρiu)j∥ � = max � max u∈U ∥f NE(u)i − gNE(u)i∥, max j̸=i 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi max u∈U ∥f NE(u)j − gNE(u)j∥ � = max � max u∈U ∥f NE(u)i − gNE(u)i∥, max j̸=i max u ∥f NE(u)j − gNE(u)j∥ � =l∞(f NE, gNE) Since O = O1 ◦ · · · ◦ On, we have ℓ∞(Of NE, OgNE) ≤ ℓ∞(f NE, gNE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (8) For Pi, l∞(Pif NE, PigNE) = max u∈U max j∈[n] ∥(Pif NE)(u)j − (PigNE)(u)j∥ = max � max j̸=i max u ∥f NE(u)j − gNE(u)j∥, max u ∥ 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 i (f NE(ρiu)i − gNE(ρiu)i)∥ � = max � max j̸=i max u ∥f NE(u)j − gNE(u)j∥, max u ∥ 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi (f NE(ρiu)i − gNE(ρiu)i)∥ � ≤ max � max j̸=i max u ∥f NE(u)j − gNE(u)j∥, 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi max u ∥f NE(ρiu)i − gNE(ρiu)i∥ � = max � max j̸=i max u ∥f NE(u)j − gNE(u)j∥, 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi max u ∥f NE(u)i − gNE(u)i∥ � = max � max j̸=i max u ∥f NE(u)j − gNE(u)j∥, max u ∥f NE(u)i − gNE(u)i∥ � =l∞(f NE, gNE) Since P = P1 ◦ · · · ◦ Pn, we have ℓ∞(Pf NE, PgNE) ≤ ℓ∞(f NE, gNE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (9) 21 For CE or CCE approximator f (C)CE ∈ F(C)CE and Qi, we have l∞(Qif (C)CE, Qig(C)CE) = max u∈U ∥(Qif (C)CE)(u) − (Qig(C)CE)(u)∥ = max u ∥ 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 i (f (C)CE(ρiu) − g(C)CE(ρiu))∥ ≤ max u 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ∥ρ−1 i (f (C)CE(ρiu) − g(C)CE(ρiu))∥ ≤ 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi max u ∥ρ−1 i (f (C)CE(ρiu) − g(C)CE(ρiu))∥ = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi max u ∥f (C)CE(ρiu) − g(C)CE(ρiu)∥ = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi max u ∥f (C)CE(u) − g(C)CE(u)∥ =l∞(f (C)CE, g(C)CE) Since Q = Q1 ◦ · · · ◦ Qn, we have ℓ∞(Qf (C)CE, Qg(C)CE) ≤ ℓ∞(f (C)CE, g(C)CE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (10) Combing Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='7, Equation (8), Equation (9) and Equation (10), we finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='9 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='8 We first introduce a useful lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' It is about the property of Ei(σ, u) Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Ei(σ, u) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Linear on σi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' pEi((σ1 i , σ−i), u) + (1 − p)Ei((σ2 i , σ−i), u) = Ei((pσ1 i + (1 − p)σ2 i , σ−i), u), ∀p ∈ [0, 1] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Convex on σj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' pEi((σ1 j , σ−j), u) + (1 − p)Ei((σ2 j , σ−j), u) ≥ Ei((pσ1 j + (1 − p)σ2 j , σ−j), u), ∀p ∈ [0, 1], j ̸= i Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We recall the definition Ei(σ, u) = maxai∈Ai ui(ai, σ−i) − ui(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Notice that ui(σ) is linear on σk for all k ∈ [n], thus both ui(ai, σ−i) and ui(σ) are linear on σk for any k ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Moreover, the maximum operator on a set of linear functions will induce a convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We prove the theorem in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Step 1 First, we show that Eu∼D[Ei((Pif NE)(u), u)] = Eu∼D[Ei(f NE(u), u)], ∀f NE ∈ FNE 22 By definition, Eu∼D[Ei(Pif NE(u), u)] =Eu∼D[Ei(( 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 i f(ρiu)i, f(u)−i), u)] = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Eu∼D[Ei((ρ−1 i f(ρiu)i, f(u)−i), u)] , by linearity of Ei(σ, u) on σi = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Ev∼D[Ei((ρ−1 i f(v)i, f(ρ−1 i v)−i), ρ−1 i v)] , let v = ρiu and use the invariance of D = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Ev∼D[Ei((ρ−1 i f(v)i, f(v)−i), ρ−1 i v)] , OPI of f = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Eu∼D[Ei((f(u)i, f(u)−i), u)] , invariance of Ei(σ, u) under ρ−1 i ∈ Gi =Eu∼D[Ei(f NE(u), u)] Step 2 Then we show that Eu∼D[Ej((Pif NE)(u), u)] ≤ Eu∼D[Ej(f NE(u), u)], ∀f NE ∈ FNE, j ̸= i Eu∼D[Ej((Pif NE)(u), u)] =Eu∼D[Ej(( 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 i f(ρiu)i, f(u)−i), u)] ≤ 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Eu∼D[Ej((ρ−1 i f(ρiu)i, f(u)−i), u)] , by convexity of Ej(σ, u) on σi = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Ev∼D[Ej((ρ−1 i f(v)i, f(ρ−1 i v)−i), ρ−1 i v)] , let v = ρiu and use the invariance of D = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Ev∼D[Ej((ρ−1 i f(v)i, f(v)−i), ρ−1 i v)] , OPI of f = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Eu∼D[Ej((f(u)i, f(u)−i), u)] , invariance of Ej(σ, u) under ρ−1 i ∈ Gi =Eu∼D[Ej(f NE(u), u)] Since P = ◦iPi and E = maxi Ei, we have Eu∼D[E((Pf NE)(u), u)] ≤ Eu∼D[E(f NE(u), u)] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='10 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='6 Similar to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='8, we first prove a lemma about the property of Ei(π, u) and ECE i (π, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Ei(π, u) and ECE i (π, u) are convex on π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' pE(CE) i (π1, u) + (1 − p)E(CE) i (π2, u) ≥ E(CE) i (pπ1 + (1 − p)π2, u), ∀p ∈ [0, 1] 23 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We recall the definition Ei(π, u) = maxai∈Ai ui(ai, π−i) − ui(π) for CCE approximator and ECE i (π, u) = maxφi∈Ai→Ai � a π(a)ui(φi(ai), a−i) − ui(π) for CE approximator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ui(ai, π−i) is linear on π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Given φ, � a π(a)ui(φi(ai), a−i) is also linear on π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Moreover, the maximum operator on a set of linear functions will induce a convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For f ∈ F(C)CE and ∀i, j ∈ [n], Eu∼D[E(CE) i (Qjf(u), u)] =Eu∼D[E(CE) i ( 1 |Aj|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρj∈Gj ρ−1 j f(ρju), u)] , by definition ≤ 1 |Aj|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρj∈Gj Eu∼D[E(CE) i (ρ−1 j f(ρju), u)] , by convexity = 1 |Aj|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρj∈Gj Ev∼D[E(CE) i (ρ−1 j f(v), ρ−1 j v)] , let v = ρju = 1 |Aj|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρj∈Gj Ev∼D[E(CE) i (f(v), v)] , invariance of E(CE) i (π, u) under ρ−1 j ∈ Gj =Eu∼D[E(CE) i (f(u), u)] Since Q = ◦iQi and E = maxi Ei, we have Eu∼D[E(Qf(u), u)] ≤ Eu∼D[E(f(u), u)] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='11 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='9 We prove the theorem in two steps, similar to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Step 1 First we show that for player i ∈ {1, 2}, let {j} = {1, 2}\\{i}, Eu∼D[Ei((Oif NE)(u), u)] ≤ Eu∼D[Ei(f NE(u), u)] This is because Eu∼D[Ei((Oif NE)(u), u)] =Eu∼D[Ei((f NE(u)i, 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi f NE(ρiu)j), u)] ≤ 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Eu∼D[Ei((f NE(u)i, f NE(ρiu)j), u)] , by convexity of Ei on σj = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Ev∼D[Ei((f NE(ρ−1 i v)i, f NE(v)j), ρ−1 i v)] , let v = ρiu = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Ev∼D[Ei((ρ−1 i f NE(v)i, f NE(v)j), ρ−1 i v)] , by PPE of f NE = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Ev∼D[Ei((f NE(v)i, f NE(v)j), v)] , invariance of Ei(σ, u) under ρ−1 i ∈ G =Eu∼D[Ei((f NE)(u), u)] Step 2 Then we show that if j ̸= i and {i, j} = {1, 2} Eu∼D[Ej((Oif NE)(u), u)] = Eu∼D[Ej(f NE(u), u)] 24 This is because Eu∼D[Ej((Oif NE)(u), u)] =Eu∼D[Ej((f NE(u)i, 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi f NE(ρiu)j), u)] = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Eu∼D[Ej((f NE(u)i, f NE(ρiu)j), u)] , by linearity of Ej on σj = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Ev∼D[Ej((f NE(ρ−1 i v)i, f NE(v)j), ρ−1 i v)] , let v = ρiu = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Ev∼D[Ej((ρ−1 i f NE(v)i, f NE(v)j), ρ−1 i v)] , by PPE of f NE = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Ev∼D[Ej((f NE(v)i, f NE(v)j), v)] , invariance of Ej(σ, u) under ρ−1 i ∈ Gi =Eu∼D[Ej(f NE(u), u)] Since O = ◦iOi and E = maxi Ei, we have Eu∼D[E(Of NE(u), u)] ≤ Eu∼D[E(f NE(u), u)] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='12 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='7 We only prove for the P-projected case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' the proof for O-projected case is similar and therefore omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Recall Ei(σ, u) = max ai∈Ai ui(ai, σ−i) − ui(σ) Denote u1(σ) + u2(σ) ≡ c, then � i Ei(σ, u) = max a1∈A1,a2∈A2 u1(a1, σ2) + u2(a2, σ1) − c Then we have Eu∼D[ � i Ei((Pf NE)(u), u)] =Eu∼D[max a1,a2 u1(a1, (Pf NE)(u)2) + u2(a2, (Pf NE)(u)1) − c] =Eu∼D[max a1 u1(a1, (Pf NE)(u)2)] + Eu∼D[max a2 u2(a2, (Pf NE)(u)1)] − c For the first term, Eu∼D[max a1 u1(a1, (Pf NE)(u)2)] =Eu∼D[max a1 u1(a1, 1 |A2|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρ2∈G2 ρ−1 2 f NE(ρ2u)2)] ≤ 1 |A2|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρ2∈G2 Eu∼D[max a1 u1(a1, ρ−1 2 f NE(ρ2u)2)] = 1 |A2|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρ2∈G2 Ev∼D[max a1 (ρ−1 2 v)1(a1, ρ−1 2 f NE(v)2)] = 1 |A2|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρ2∈G2 Ev∼D[max a1 v1(a1, f NE(v)2)] =Eu∼D[max a1 u1(a1, f NE(u)2)] Similarly, for the second term, Eu∼D[max a2 u2(a2, (Pf NE)(u)1)] ≤ Eu∼D[max a2 u2(a2, f NE(u)1)] 25 Above all, Eu∼D[ � i Ei((Pf NE)(u), u)] =Eu∼D[max a1 u1(a1, (Pf NE)(u)2)] + Eu∼D[max a2 u2(a2, (Pf NE)(u)1)] − c ≤Eu∼D[max a1 u1(a1, f NE(u)2)] + Eu∼D[max a2 u2(a2, f NE(u)1)] − c =Eu∼D[ � i Ei(f NE(u), u)] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='13 Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='3 Let f be a PPE and OPI NE approximator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Denote f(u) = (σi)i∈[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For player k that a∗ k ∈ V (ρk), we get σk = f(u)k (a) = f(ρu)k (b) = f(ρku)k (c) = ρkf(u)k = ρkσk, (11) where (a) holds since u is permutable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ρ, (b) holds by OPI of f, and (c) holds by PPE of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' If a∗ can be found by f, we will have 1 = σk(a∗ k) (d) = ρkσk(a∗ k) = σk(ρ−1 k (a∗ k)), where (d) holds by Equation (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' However, such result leads to a contradiction, because a∗ k ̸= ρ−1 k (ak) but σk(a∗ k) = σ(ρ−1 k (a∗ k)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Let f be a PE (C)CE approximator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Denote f(u) = π, we have π = f(u) (a) = f(ρu) (b) = ρf(u) = ρπ (12) where (a) holds since u is permutable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ρ, (b) holds by PE of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' If a∗ can be found by f, we will have 1 = π(a∗) (c) = ρπ(a∗) = π(ρ−1a∗) = π(ρ−1 1 a∗ 1, · · · , ρ−1 n a∗ n), where (c) holds by Equation (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' However, from a∗ k ∈ V (ρk) we know ρ−1 k (a∗ k) ̸= a∗ k, then ρ−1a∗ ̸= a∗, but π(a∗) = π(ρ−1a∗) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='14 Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Assume f ∈ F(C)CE general is an (C)CE approximator that always finds the strategy that maximizes the social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Afterward, we construct another f0 that satisfies PE and always finds the strategy that maximizes social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' f0 is constructed by orbit averaging: f0(u) = Qf(u), thus f0 is PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Denote D as an arbitrary payoff distribution of u such that D is invariant under permutation and the cardinality of its support is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We have Eu∼DSW(Qif(u), u) =Eu∼DSW( 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 i f(ρiu), u) =Eu∼D n � i=1 ui( 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi ρ−1 i f(ρiu)) = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Eu∼D n � i=1 ui(ρ−1 i f(ρiu)) = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Ev∼D n � i=1 (ρ−1 i v)i(ρ−1 i f(v)) , let v = ρiu = 1 |Ai|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' � ρi∈Gi Ev∼D n � i=1 vi(f(v)) =Eu∼D n � i=1 ui(f(u)) =Eu∼DSW(f(u), u) 26 Due to that Q = Q1 ◦ · · · ◦ Qn, we have Eu∼DSW(f0(u), u) = Eu∼DSW(f(u), u) Due to the arbitrariness of D, we know that f0 maximizes the social welfare w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' any u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' From above, we immediately know SWRN,M(F(C)CE PE , F(C)CE general) = 1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='15 Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='7 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='1 Proof of Equation (1) and Equation (3) We first prove the theorem with respect to FNE OPI and FNE both Step 1 On the one part, we prove SWRN,M(FNE OPI, FNE general) SWRN,M(FNE both, FNE general) � ≤ 1 M N−1 We prove this by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Consider a game with N player and Ai = [M] for i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ∀a ∈ A, i ∈ [N], define the payoff ¯u as follows: ¯ui(a) = � 1 , if a1 = a2 = · · · = aN 0 , otherwise Define U = {u′|u′ = ◦iρi¯u, ρi ∈ Gi} and D as a uniform distribution on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Easy to certify that D is a permutation-invariant distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Let ˜f ∈ ˜FNE general be the NE oracle that ˜f(¯u)i = 1 and for any u′ = ◦iρi¯u ∈ U, ˜f(u′)i = ρi(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Intuitively, the oracle will choose the action that will provide all players with revenue 1, leading to a social welfare of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Since each player has got her maximum possible utility, we have max f∈F NE general Eu∼DSW(f(u), u) = max ˜f∈ � F NE general Eu∼DSW( ˜f(u), u) = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (13) For any j1, j2 ∈ [M] and j1 < j2, let ρ(j1,j2) i = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' , j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' , j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' , M) for all player i ∈ [N] be the swap permutation that swaps actions j1 and j2 and keeps other actions still.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Then ◦i̸=jρ(j1,j2) i ¯u = ρ(j1,j2) j ¯u for player j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For f ∈ FNE OPI, we have f(¯u)j = f(◦i̸=jρ(j1,j2) i ¯u)j = f(ρ(j1,j2) j ¯u)j for arbitrary swap permutation ρ(j1,j2) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Since any permutation can be achieved by composition of swap permutations, we have ∀ρj ∈ Gj, f(¯u)j = f(ρj ¯u)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Based on that, and by OPI of f, ∀ρ = ◦i∈[N]ρi we have f(¯u)j = f(ρ¯u)j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' f is a constant function on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Without loss of generality, we denote f(u) ≡ σ for all u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Then Eu∼DSW(f(u), u) = 1 |U| � u′∈U SW(σ, u′) = 1 (M!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' )N−1 SW(σ, � u′∈U u′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Additionally, we have (� u′∈U u′)(a) = ((M − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' )N−1 for any a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Based on that, we have max f∈F NE OPI Eu∼DSW(f(u), u) = 1 (M!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' )N−1 · N((M − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' )N−1 = N M N−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' (14) Combining Equation (13) and Equation (14), we have SWRN,M(FNE OPI, FNE general) ≤ 1 M N−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Due to FNE both ⊆ FNE OPI, we immediately know SWRN,M(FNE both, FNE general) ≤ 1 M N−1 27 Step 2 On the other part, we prove SWRN,M(FNE OPI, FNE general) SWRN,M(FNE both, FNE general) � ≥ 1/M N−1 Define the maximum possible utility (MPU) for player i with respect to utility ui and action ai as MPU(ui, ai) := max a−i∈A−i ui(ai, a−i) (15) Define the set of maximum possible utility best response for player i w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ui as Bi(ui) := {ai ∈ Ai : MPU(ui, ai) = max a′ i∈Ai MPU(ui, a′ i)} We first conduct some simplification to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' SWRN,M(FNE both, FNE general) = inf D maxf∈F NE both Eu∼DSW(f(u), u) maxf∈F NE general Eu∼DSW(f(u), u) ≥ inf D maxf∈F NE both Eu∼DSW(f(u), u) Eu∼D maxσ SW(σ, u) Then we constrain u to be a cooperation game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For a normal form game Γu, we define ˜u = (˜ui)i∈[n] in which ˜ui = 1 n �n i=1 ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Then we have SW(σ, u) = SW(σ, ˜u), which means that constraining u to be a cooperation game will induce the same social welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Then inf D maxf∈F NE both Eu∼DSW(f(u), u) Eu∼D maxσ SW(σ, u) = inf D maxf∈F NE both Eu∼DSW(f(u), ˜u) Eu∼D maxσ SW(σ, ˜u) Denote f0 be the approximator that always outputs uniform strategy on Bi(˜ui) for player i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' It’s obvious that f0 is both OPI and PPE because the operations from u to f0(u) are all permutation- equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Then, inf D maxf∈F NE both Eu∼DSW(f(u), ˜u) Eu∼D maxσ SW(σ, ˜u) ≥ inf D Eu∼DSW(f0(u), ˜u) Eu∼D maxσ SW(σ, ˜u) Ignore the infimum and the expectation operator, consider SW(f0(u),˜u) maxσ SW(σ,˜u) for arbitrary ˜u, denote b be the maximum element appeared in ˜u, then the denominator equals Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' But for the numerator, for player i, no matter what action ai ∈ Bi(˜ui) she chooses, she always has probability at least � j̸=i 1 |Bj| ≥ 1 MN−1 to achieve revenue b, therefore inducing SW(f0(u), ˜u) ≥ Nb MN−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Then, SW(f0(u),˜u) maxσ SW(σ,˜u) ≥ 1 MN−1 , so as infD Eu∼DSW(f0(u),˜u) Eu∼D maxσ SW(σ,˜u), SWRN,M(FNE both) and SWRN,M(FNE OPI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Above all, SWRN,M(FNE OPI, FNE general) SWRN,M(FNE both, FNE general) � = 1 M N−1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='2 Proof of Equation (2) We next prove the theorem with respect to FNE PPEthat SWRN,M(FNE PPE, FNE general) ≤ 1 M Consider a bimatrix game and Ai = [M] for i ∈ [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' ∀a ∈ A, i ∈ [2], define the payoff ¯u as follows: ¯ui(a) = � 1 , if a1 = a2 0 , otherwise Define U := {u′|u′ = ρ1ρ2¯u, ρi ∈ Gi} and D as a uniform distribution on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Easy to certify that U = {u′|u′ = ρ1¯u, ρ1 ∈ G1} = {u′|u′ = ρ2¯u, ρ2 ∈ G2} and D is a permutation-invariant distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 28 Let ˜f ∈ ˜FNE general be the NE oracle that ˜f(¯u)i = 1 and for any u′ = ◦iρi¯u ∈ U, ˜f(u′)i = ρi(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Intuitively, the oracle will choose the action that will provide all players with revenue of 1, leading to a social welfare of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For a permutation ̺ on [M], let P̺ ∈ {0, 1}M×M be the corresponding permutation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Denote P as the set of all permutation matrice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' As a result, ∀u ∈ U, ∀ρ1 ∈ G1, ρ1u = (Pρ1u1, Pρ1u2) =: Pρ1u and ∀ρ2 ∈ G2, ρ2u = (u1P T ρ2, u2P T ρ2) =: uP T ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Specially, we have P̺¯uP T ̺ = ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For f ∈ FNE PPE, Denote f(¯u) = σ = (σ1, σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' For permutation ̺ in [M] and payoff u′ = P̺¯u = ¯u(P T ̺ )−1, by PPE of f, we have f(u′)1 = f(P̺¯u)1 = P̺σ1 = ̺σ1, and f(u′)2 = f(¯u(P T ̺ )−1)2 = (P̺)−1σ2 = ̺−1σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Then we have SW(f(u′), u′) = � i (P̺¯u)i(̺σ1, ̺−1σ2) = � i ¯ui(σ1, ̺−1σ2) = � i (¯uP T ̺ )i(σ1, σ2) = SW(f(¯u), ¯uP T ̺ ) Therefore Eu∼DSW(f(u), u) = 1 |U| � u′∈U SW(f(u′), u′) = 1 |U| � P̺∈P SW(f(¯u), ¯uP T ̺ ) = 1 |U| � u=¯u(P T ̺ )∈U SW(f(¯u), u) = 1 |U|SW(σ, � u′∈U u′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Since |U| = 1 M!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' and � u′∈U u′ is a tensor with all elements equal to (M −1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='. Thus Eu∼DSW(f(u), u) = 2 M and SWRN,M(FNE PPE, FNE general) ≤ 1 M A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content='3 Proof of Equation (4) Consider a 3 × 3 game as follows, where ǫ ∈ (0, 1 2): u = \uf8ee \uf8f0 1, 1 0, 0 0, 1 2 + ε 0, 0 1, 1 0, 1 2 + ε 1 2 + ε, 0 1 2 + ε, 0 ε, ε \uf8f9 \uf8fb It is obvious that maxσ∗⊆NE(Γu) SW(σ∗, u) = 2, and the corresponding strategy has been bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' However, for NE oracles with both PPE and OPI, it can only output a unique NE with a pure strategy that induces utility (ε, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Let ρ1 = ρ2 = (2, 1, 3), we have ρ1ρ2u = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' From the analysis above we know if f NE ∈ � FNE both and f NE(u) = (σ1, σ2), then σ1(1) = σ1(2), σ2(1) = σ2(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' We integrate the first two actions of player 1 and player 2 into a new action that will choose randomly between the first two actions, then we form the utility matrix below: u = � 1 2, 1 2 0, 1 2 + ε 1 2 + ε, 0 ε, ε � There is a unique NE in this Prisoner’s Dilemma, which has been bolded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' The game u is the same with the game u under the assumption that σ1(1) = σ1(2) and σ2(1) = σ2(2) in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Then maxf∈ � F NE both SW(f(u), u) = 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' Since ε can be arbitrarily small, we have SWR2,3( � FNE both, �FNE general) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' As a result, we have SWRN,M( �FNE both, �FNE general) = 0 for all N ≥ 2 and M ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} +page_content=' 29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VdFJT4oBgHgl3EQfOSwV/content/2301.11481v1.pdf'} diff --git a/VtE4T4oBgHgl3EQfMgzk/content/tmp_files/2301.04949v1.pdf.txt b/VtE4T4oBgHgl3EQfMgzk/content/tmp_files/2301.04949v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5f69c925babb6092d5d930e092524df21f0ef83f --- /dev/null +++ b/VtE4T4oBgHgl3EQfMgzk/content/tmp_files/2301.04949v1.pdf.txt @@ -0,0 +1,3642 @@ +arXiv:2301.04949v1 [math.OC] 12 Jan 2023 +A FORMAL POWER SERIES APPROACH TO MULTIPLICATIVE +DYNAMIC FEEDBACK CONNECTION +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +Abstract. The goal of the paper is multi-fold. The first of which is to derive an explicit +formula to compute the generating series of a closed-loop system when a plant, given in a +Chen–Fliess series description is in multiplicative output feedback connection with another +system given in Chen–Fliess series description. Further, the objective extends in showing +that the multiplicative dynamic output feedback connection has a natural interpretation as +a transformation group acting on the plant. A computational framework for computing the +generating series for multiplicative dynamic output feedback is devised utilizing the dual +Hopf algebras corresponding to the shuffle group and the multiplicative feedback group. +Contents +1. +Introduction +2 +2. +Preliminaries: Formal Power Series +2 +2.1. +Shuffle Product +3 +3. +Bialgebra and Hopf algebra: Preliminaries +4 +3.1. +Algebra +4 +3.2. +Coalgebra +5 +3.3. +Bialgebra +6 +3.4. +Hopf Algebra +7 +4. +Unshuffle Hopf algebra and its Coaction +8 +4.1. +Unshuffle Hopf Algebra +8 +4.2. +Gradation of Bialgebra H +10 +4.3. +Coaction of H +11 +5. +Chen–Fliess Series and its Interconnections +13 +5.1. +Chen–Fliess Series +13 +5.2. +Interconnections of Chen–Fliess Series: Parallel and Cascade Connections +14 +5.3. +Cascading of Chen–Fliess with Multiplicative Feedforward of Input +15 +5.4. +Multiplicative Dynamic Output Feedback Group +16 +6. +Chen–Fliess Series Under Multiplicative Dynamic Output Feedback +18 +7. +Invariance of Class and Relative Degree under multiplicative dynamic feedback +connection +20 +8. +Computational Framework for Multiplicative Mixed Composition & Dynamic +Feedback Product +24 +8.1. +Hopf Algebra Corresponding to the Multiplicative Dynamic Feedback Subgroup 24 +8.2. +Coaction of Hopf algebra H on Algebra of Coordinate Map +25 +8.3. +Coaction of Hopf algbera H on the Hopf algebra H +27 +8.4. +Coproduct, Antipode Computations and Grading of Hopf algebra H +29 +9. +Conclusions and Future work +35 +References +35 + +2 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +1. Introduction +The objective of the document is two fold and works with the Chen–Fliess functional +series [Fliess(1981)]. There is no need that these input-output systems have a state space +realization and thus, the results presented here are independent of any state space embed- +ding when a realization is possible [Fliess(1983)]. Firstly, let Fc and Fd be two nonlinear +input-output systems represented by Chen–Fliess series. It was shown in [Gray & Li(2005)] +that the additive feedback interconnection of two such systems result in a Chen–Fliess se- +ries description for the closed-loop system. The convergence of the closed-loop system was +characterized in [Thitsa & Gray(2012)]. An efficient computation of the generating series for +closed-loop system is facilitated through a combinatorial Hopf algebra [Gray, et al.(2014a), +Foissy(2015), Duffaut Espinosa, et al.(2016)]. The feedback product formula and its com- +putation were used to solve system inversion problems [Gray, et al.(2014b)] and trajectory +generation problems [Duffaut Espinosa & Gray(2017)]. +However, when the nature of interconnection becomes multiplicative feedback, the similar +set of questions persist in general. It is known that, in single-input single-output (SISO) +setting, the closed-loop system in the affine feedback case (of which multiplicative feedback +is a special case) has a Chen–Fliess series description and the computation of feedback for- +mula is facilitated through a combinatorial Hopf algebra [Gray & Ebrahimi-Fard(2017)]. The +present document, in one part, shows that even in multi-input multi-output (MIMO) setting +the closed-loop system under multiplicative feedback has a Chen–Fliess series representation +and provides an explicit expression of the closed-loop generating series which will be called +as multiplicative dynamic feedback product . Furthermore, it will be shown that this feedback +product has a natural interpretation as a transformation group acting on the plant. The +algorithmic framework for the computation of the multiplicative dynamic feedback product +formula for a general MIMO case is devised using the dual Hopf algebras corresponding to +the shuffle product and to the multiplicative dynamic output feedback group. The charac- +terization of convergence of the Chen–Fliess series for the closed-loop system is deferred for +future work. +The paper is organized as follows. The next section provides a summary of the concepts +related to non-commutative formal power series, Hopf algebra, Chen–Fliess series and their +interconnections. The Section 5.4 builds the pivotal multiplicative dynamic output feedback +group. The Hopf algebra construction corresponding to the shuffle group is drafted in Sec- +tion 4. Section 6 is where the multiplicative dynamic feedback connection is analyzed. The +invariance of relative degree under multiplicative output feedback is asserted in Section 7. +The framework for computing the feedback product is devised in Section 8 and is demon- +strated using examples. The conclusions of the paper and directions for future work is given +in the last section. +2. Preliminaries: Formal Power Series +A finite nonempty set of noncommuting symbols X = {x0, x1, . . . , xm} is called an alphabet. +Each element of X is called a letter. Any finite sequence, η = xi1 · · · xik, of letters from X +is called a word over X and its length is |η| = k. The set X∗ of all words includes the +empty word, denoted ∅ ∈ X∗ and X+ := X∗\∅, and forms a monoid under catenation. +Any mapping c : X∗ → Rℓ is called a formal power series. The value of c at η ∈ X∗ is +denoted by (c, η) and called the coefficient of η in c. Normally, c is written as a formal sum +c = � +η∈X∗(c, η)η. A series c is proper when the coefficient (c, ∅) = 0 else it is a non-proper +series. The support of c is the set supp(c) containing all words having nonzero coefficients. +The order of c, denoted ord(c), is the length of the minimal length word in its support. The + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +3 +collection of all formal power series over X is denoted by Rℓ⟨⟨X⟩⟩. The ith component of a +vector v ∈ Rℓ is denoted by vi and consequently the ith component of a series c ∈ Rℓ⟨⟨X⟩⟩ +is denoted by ci viz. (ci, η) = (c, η)i. +A series c′ ∈ Rℓ⟨⟨X⟩⟩ is called a subseries of c ∈ Rℓ⟨⟨X⟩⟩ if there exists another series +c′′ ∈ Rℓ⟨⟨X⟩⟩ such that the intersection supp (c′) ∩ supp (c′′) is empty and the series c can +be decomposed as c = c′ + c′′. +Definition 2.1. Let c ∈ Rℓ⟨⟨X⟩⟩, then the natural part of the series c is the subseries +denoted by cN such that c = cN + cF and supp (cF) ⊆ X∗ \ {xk +0 : k ∈ N0}. The subseries cF +is called as forced part of the series c. +Definition 2.1 asserts that the forced part cF of a series c should not contain any word +formed by the letter x0 alone, including the empty word ∅. For the remainder of the docu- +ment, Rℓ is given the structure of a unital commutative ring under Hadamard or pointwise +product viz. (xy)i = xiyi with ll = [1 1 · · ·1]t ∈ Rℓ being the multiplicative unit. Formal +power series Rℓ⟨⟨X⟩⟩ form a Rℓ-module and the submodule of all proper series in Rℓ⟨⟨X⟩⟩ +is denoted by Rℓ +p ⟨⟨X⟩⟩, while the subset of non-proper series is denoted by Rℓ +np ⟨⟨X⟩⟩. +Definition 2.2. A series c ∈ Rℓ⟨⟨X⟩⟩ is called purely improper if ci is non-proper ∀i = +1, . . . , ℓ. The subset of all purely improper series in Rℓ⟨⟨X⟩⟩ is denoted by Rℓ +pi ⟨⟨X⟩⟩. +Observe that Rℓ +pi ⟨⟨X⟩⟩ ⊊ Rℓ +np ⟨⟨X⟩⟩ if ℓ > 1, otherwise Rpi ⟨⟨X⟩⟩ = Rnp ⟨⟨X⟩⟩. +2.1. Shuffle Product. The shuffle product α +β of two words is a bilinear product on the +linear span of words, which can be uniquely specified iteratively +(xiη) +(xjξ) := xi(η +(xjξ)) + xj((xiη) +ξ), +where η, ξ ∈ X∗ and xi, xj ∈ X. See for instance [Fliess(1981)]. The shuffle product of two +series, (c, d) �→ c +d is defined as +(c +d, η) = +� +ζ,ν∈X∗ +η∈supp(ζ +ν) +(c, ζ) (d, ν) . +We define for any xi, xj ∈ X and any word η ∈ X∗ +x−1 +i (xjη) := +�η, +i = j +0, +else +The following proposition is vital in understanding the bialgebra and Hopf algebra devised +in Sections 4.1 and 4.3. +Proposition 2.1. If c, d ∈ Rℓ⟨⟨X⟩⟩, then ∀xi ∈ X +x−1 +i +(c +d) = +� +x−1 +i +(c) +d +� ++ +� +c +x−1 +i +(d) +� +. +Note that Rℓ⟨⟨X⟩⟩ forms an associative and commutative Rℓ-algebra under the shuffle +product. If d ∈ Rℓ +pi ⟨⟨X⟩⟩, then shuffle inverse of d, denoted by d +−1 is defined as +d +−1 +i += (di, ∅)−1 +�� +k∈N0 +(d′ +i) +k +� +, +where d′ +i = 1 − (di/ (di, ∅)). Hence, Rℓ +pi ⟨⟨X⟩⟩ forms an Abelian group under the shuffle +product with ll as the identity element. + +4 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +Example 2.1. Let X = {x0, x1} and c ∈ R⟨⟨X⟩⟩ described as c = 1 − x1. Then the shuffle +inverse is computed as: +c +−1 = +� +k∈N0 +(1 − (1 − x1)) +k += +� +k∈N0 +x +k +1 += +� +k∈N0 +k!xk +1. +Therefore, c +−1 = 1 + x1 + 2x2 +1 + 6x3 +1 + · · · + n!xn +1 + · · · . +Observe that (c +d, ∅) = (c, ∅) (d, ∅). Hence, the set +M += { ll + c : c ∈ Rn +p ⟨⟨X⟩⟩}, +where c is a proper series in Rn⟨⟨X⟩⟩, forms a subgroup of the shuffle group. The group +M +is vital in the design of a computational framework of multiplicative dynamic feedback +product as explained in Section 8. +The set Rℓ⟨⟨X⟩⟩ is endowed with ultrametric structure where the metric κ is defined as +κ(c, d) = σord(c−d), +for c, d ∈ Rℓ⟨⟨X⟩⟩ and σ ∈]0, 1[. For brevity, κ(c, 0) is written as κ(c), and κ(c, d) = κ(c−d). +The ultrametric space (Rℓ⟨⟨X⟩⟩, κ) is Cauchy complete [Berstel & Reutenauer(1988)]. The +following definition of contraction maps between metric spaces will be useful. +Definition 2.3. Given metric spaces (E, d) and (E′, d′), a map f : E −→ E′ is said to be a +strong contraction map if ∀s, t ∈ E, it satisfies the condition d′(f(s), f(t)) ≤ αd(s, t) where +α ∈ [0, 1[. If α = 1, then the map f is said to be a weak contraction map or a non-expansive +map. +3. Bialgebra and Hopf algebra: Preliminaries +The goal is to provide the definitions of algebraic structures such as algebra, coalgebra, +bialgebra and Hopf algebra [Abe(2004), Sweedler(1969)]. We let K be a commutative ring +with identity 1K. +3.1. Algebra. The definition of an algebra can be facilitated through the category of mod- +ules. It allows to define the concept of a coalgebra (the dual notion) with ease. +Definition 3.1. An algebra over K is a K-module A along with the morphisms of K- +modules m : A ⊗ A −→ A , called the multiplication or product map, and η : K −→ A , +called the unit map, such that the following diagrams are commutative. +(1) +A ⊗ A ⊗ A +m⊗idA +� +idA ⊗m +� +A ⊗ A +m +� +A ⊗ A +m +� A +K ⊗ A +η⊗idA +� +∼ += +�▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +A ⊗ A +m +� +A +A ⊗ K +∼ += +�r +r +r +r +r +r +r +r +r +r +r +r +r +r +r +r +r +r +r +idA ⊗η +� A ⊗ A +m +� + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +5 +The tuple (A , m, η) is called a K-algebra. +The commutative diagrams (1) mean that a K-algebra A must satisfy the following prop- +erties: +(1) The product map m must be associative. +(2) The scalar multiplication through the η map must have a unit. +The concept of a K-algebra morphism is defined next. +Definition 3.2. Let (A , m, η), (A ′, m′, η′) be K-algebras. A map f : A −→ A ′ is called +a K-algebra morphism provided the following diagrams commute. +A ⊗ A +m +� +f⊗f +� +A +f +� +A ′ ⊗ A ′ +m′ +� A ′ +K +η +� +η′ +�❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +A +f +�①①①①①①①①①①①①①①①① +A ′ +Definition 3.3. Let P and Q be modules over K. The twisting morphism τ of K-modules +is τ : P ⊗ Q −→ Q ⊗ P with +τ(p ⊗ q) = q ⊗ p +∀ q ∈ Q, p ∈ P. +A K-algebra A is commutative if and only if the following diagram commutes. +A ⊗ A +τ +� +m +�■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +A ⊗ A +m +� A +A K-algebra A is a graded algebra if the underlying K-module structure is graded +viz. A = � +n∈N0 An, where An is a K-module for all n ∈ N0 such that m (Am ⊗ An) ⊆ +Am+n, for all m, n ∈ N0. The graded K-algebra is connected if η : K −→ A0 is a K-algebra +isomorphism. +3.2. Coalgebra. The notion of a K-coalgebra is a categorical structure dual to that of a +K-algebra. +Definition 3.4. A K-coalgebra C is a K-module with the K-module morphisms ∆ : C −→ +C ⊗ C , called the comultiplication or coproduct map, and ǫ : C −→ K, called the counit +map, such that the following diagrams commute. +(2) +C +∆ +� +∆ +� +C ⊗ C +∆⊗idC +� +C ⊗ C +idC ⊗∆ +� C ⊗ C ⊗ C +C ⊗ C +ǫ⊗idC +� K ⊗ C +∼ += +� +C +∆ +�❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ +∆ +�sssssssssssssssssss +C ⊗ C +idC ⊗ǫ +� C ⊗ K +∼ += +� + +6 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +The tuple (C , ∆, ǫ) is called a K-coalgebra. +The commutative diagrams (2) imply that a K-coalgebra C must satisfy the following +properties: +(1) The coproduct map ∆ must be coassociative. +(2) The counit map ǫ is the categorical dual to the unit map η for a K-algebra. +The coalgebra C is called cocommutative if the following diagram commutes, +C +∆ +� +∆ +�❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +C ⊗ C +τ +� C ⊗ C +where τ is the twisting morphism given in Definition 3.3. Sweedler’s notation is very useful +in representing the coproduct map and is adopted in Sections 4 and 8. +Definition 3.5. [Sweedler(1969)]. Given the K-coalgebra tuple (C , ∇, ǫ) and an element +c ∈ C , then the Sweedler notation for the coproduct +∆(c) = +� +(c) +c(1) ⊗ c(2), +where c(1), c(2) ∈ C are the components of the tensors resulting from the coproduct of c. +Next, the definition of a K-coalgebra morphism is given. +Definition 3.6. Let (C , ∆, ǫ), (C ′, ∆′, ǫ′) be K-coalgebras. A map f : C −→ C ′ is called a +K-coalgebra morphism provided the following diagrams commute. +C +∆ +� +f +� +C ⊗ C +f⊗f +� +C ′ +∆′ +� C ′ ⊗ C ′ +C +ǫ +� +f +�❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +K +C ′ +ǫ′ +�② +② +② +② +② +② +② +② +② +② +② +② +② +② +② +② +3.3. Bialgebra. The bialgebra structure over a commutative ring is fundamental for defining +a Hopf algebra. A bialgebra is an amalgamation of the algebra and coalgebra structures such +that both are compatible with each other. +Definition 3.7. A bialgebra H over K is a tuple (H, m, η, ∆, ǫ) such that +(1) H is a K-module. +(2) (H, m, η) is a K-algebra, where m and η are the product and unit maps, respectively. +(3) (H, ∆, ǫ) is a K-coalgebra, where ∆ and ǫ are the coproduct and counit maps, respec- +tively. +such that the following diagrams commute. +(3) +H ⊗ H +m +� +∆⊗∆ +� +H +∆ +� H ⊗ H +H ⊗ H ⊗ H ⊗ H +idH⊗τ⊗idH +� H ⊗ H ⊗ H ⊗ H +m⊗m +� + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +7 +(4) +H ⊗ H +m +� +ǫ⊗ǫ +�▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +H +ǫ +� +K ∼= K ⊗ K +η⊗η +�qqqqqqqqqqqqqqqqqqqqq +η +� +H ⊗ H +H +∆ +� +(5) +H +ǫ +�❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +K +η +�② +② +② +② +② +② +② +② +② +② +② +② +② +② +② +② +idK +� K +The diagrams (3) and (4) state that the product map m and the unit map η are K- +coalgebra morphisms, while the coproduct map ∆ and the counit map ǫ are K-algebra +morphisms. Diagram (5) describes that the unit map η is a section of the counit map ǫ in +the category of K-modules. +3.4. Hopf Algebra. Hopf algebras are an important class of bialgebras. A Hopf algebra is +a bialgebra equipped with a particular K-linear map called antipode. +Definition 3.8. A Hopf algebra H over K is a tuple (H, m, η, ∆, ǫ, S) such that the following +conditions are satisfied: +(1) (H, m, η, ∆, ǫ) is a K-bialgebra. +(2) S : H −→ H is a K-linear map such that the following diagram commutes. +(6) +H ⊗ H +idH⊗S +� H ⊗ H +m +�❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +H +ǫ +� +∆ +�✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +∆ +�❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +K +η +� H +H ⊗ H +S⊗idH +� H ⊗ H +m +�✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +An element a ∈ H is called group-like if ∆(a) = a ⊗ a and thus a̸∈ker(ǫ), where ker(.) +represents the kernel of a K-module map. A graded Hopf algebra H = � +n∈N0 Hn is connected +if and only if H0 ∼= Kη(1K) as K-modules. +Equivalently, a graded Hopf algebra H is +connected if and only if H+ := � +k≥1 Hk is isomorphic to ker(ǫ) as K-modules viz. η◦ǫ = idH0 +and zero otherwise. For simplicity denote m (a, b) := ab, for all a, b, ∈ H. Using Sweedler’s + +8 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +notation, diagram (6) implies that for all c ∈ H, +� +(c) +S +� +c(1) +� +c(2) = +� +(c) +c(1)S +� +c(2) +� += ǫ (c) 1H , +where 1H is the multiplicative unit of the Hopf algebra H. The computation of the antipode +of an element c becomes easier when the algebra structure of H is graded and connected. +Theorem 3.1. If the Hopf algebra H is graded and connected, then the antipode can be +computed for any a ∈ H+ := � +k≥1 Hk as +S(a) = −a − +� +a′ +(1)S(a′ +(2)), +where the summation is taken over all components of the reduced coproduct ∆′ defined as: +∆′ (a) := ∆ (a) − a ⊗ η (1K) − η (1K) ⊗ a. +4. Unshuffle Hopf algebra and its Coaction +The goal of this section is to explain and illustrate the computational framework to +compute the shuffle product of two series and the shuffle inverse using the coordinate +maps of the series. The framework is well-developed in the literature [Foissy(2015)] and +was utilized in study of interconnections of Chen–Fliess series [Venkatesh & Gray(2022), +Venkatesh & Gray(2021), Venkatesh & Gray (2020), Gray, et al.(2014b), Gray, et al.(2014a)]. +4.1. Unshuffle Hopf Algebra. We construct a dual Hopf algebra reflecting the group +structure of M +as defined in Section 2. The antipode constructed in the Hopf algebra +provides a framework for computing the shuffle inverse of purely improper series c. +Let the set Wb ⊂ Rm⟨⟨X⟩⟩∗ (dual module of Rm⟨⟨X⟩⟩) be defined as the collection of +coordinate maps: +Wb = {aη : aη(c) = (c, η), η ∈ X∗, c ∈ Rm⟨⟨X⟩⟩}. +Define W to be the free Rm-module spanned by the set Wb. Let H +denote the reduced +symmetric algebra generated by the module W. The Rm-algebra H +can equivalently be +seen as the polynomial algebra of coordinate maps (corresponding to non-empty words) of +Rm⟨⟨X⟩⟩. The unit map ξ : Rm −→ H +is defined by ξ( ll) = a∅. Observe that a∅ : c �→ ll, +for all c ∈ M +. By construction, H +is an Rm-associative, commutative and unital algebra +with addition and scalar multiplication defined, respectively, as +(aη + aζ)(c) = aη(c) + aζ(c) +(kaη)(c) = k(aη(c)), +where c ∈ Rm⟨⟨X⟩⟩ and k ∈ Rm, and product +m(aη, aζ)(c) = aη(c).aζ(c), +for c ∈ M +. Then H +is equipped with a coproduct ˆ∆ +: H +−→ H +� H +such that +ˆ∆ +aη(c, d) = (c +d, η), for all c, d ∈ M +and η ∈ X∗. The counit map ǫ : H +−→ Rm is +defined as +ǫ(h) = +� ll : h = a∅ +0 : otherwise. +Since the shuffle product is associative and commutative, thus dually the coproduct ˆ∆ +is +coassociative and cocommutative. Therefore, (H +, m, ξ, ˆ∆ +, ǫ) forms a Rm-bialgebra. The + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +9 +following lemma is vital in the framework for computing both shuffle product and dynamic +feedback group product. Define a collection of linear endomorphisms {θi}m +i=0 on W +θi : W −→ W +aη �−→ axiη, +for all xi ∈ X, η ∈ X∗. Thus θi (aη) (c) = aη +� +x−1 +i +(c) +� +. +The coproduct ˆ∆ +can be recursively constructed as defined in the following proposition. +Proposition 4.1. [Foissy(2015)] On the module W +ˆ∆ +◦ θk = (θk ⊗ id + id ⊗ θk) ◦ ˆ∆ +, +for all i = 1, 2, . . . , m and k = 0, 1, . . . , m with base case being ˆ∆ +a∅ = a∅ ⊗ a∅. +Proposition 4.1 infers that the maps θi, for i = 1, 2, . . . , m, are coderivations on the +underlying coalgebra of H +. +We note that the unshuffle coproduct ˆ∆ +was utilized in the design of an algorithmic +framework for computation of Wiener-Fliess composition product and subsequently additive +static feedback product [Venkatesh & Gray(2021), Venkatesh & Gray(2022), Venkatesh(2021)] +and also in the computation of shuffle-rational series from its representation [Venkatesh & Gray (2020), +Venkatesh(2021)]. Moreover, the unshuffle coproduct was also crucial in the computational +framework for the multivariate additive output feedback [Gray, et al.(2014a), Gray, et al.(2014b)] +and for SISO affine output feedback [Gray & Ebrahimi-Fard(2017)]. +Let {πi}m +i=1 be the collection of co-ordinate projection maps on the module W defined as +ai +η(c) := πi(aη)(c) = (c, η)i = (ci, η), +for all η ∈ X∗. Thus, define the following notation +ˆ∆j ai +η := (πi ⊗ πj) ◦ ˆ∆ +aη. +Note that the projection maps {πi}m +i=1 commute with the maps {θj}m +j=0 viz. θi +� +aj +η +� += aj +xiη. +The significance of these notations are well-reflected in the computational framework in +Section 8. The following example is to demonstrate the result of Proposition 4.1 for a few +words. +Example 4.1. A few examples of the computation of deshuffle coproduct ˆ∆ +on W (akin +to Example 4.3) using Proposition 4.1 are given as follows(indices i = 1, 2, . . . , m and k, s = +0, 1, . . . , m): +ˆ∆j ai +xk = ai +xk ⊗ aj +∅ + ai +∅ ⊗ aj +xk. +ˆ∆j ai +xkxk = ai +xkxk ⊗ aj +∅ + 2ai +xk ⊗ aj +xk + ai +∅ ⊗ aj +xkxk. +ˆ∆j ai +xkxs = ai +xkxs ⊗ aj +∅ + ai +xk ⊗ aj +xs + ai +xs ⊗ aj +xk + ai +∅ ⊗ aj +xkxs. +The connected Rm-bialgebra H +is endowed with an antipode map S +given as: +S +: H +−→ H +aη �→ S +aη +such that S +aη (c) = (c +−1, η), for η ∈ X∗, c ∈ M +. + +10 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +4.2. Gradation of Bialgebra H +. The Hopf algebra H +can be equipped with a grading +such that it is connected and all its homogeneous components are finite-dimensional. +Definition 4.1. Given η ∈ X+, define the degree of aη as deg (aη) = |η|. +(1) Define gradation on the Rm-module W viz. +W = +� +k≥1 +Wk, +where Wk is the free Rm-module spanned by the aη of deg (aη) = k. +(2) The gradation on the module W induces a graded structure on the algebra H +as +H += +� +n∈N0 +ˆHn, +with ˆH0 ∼= Rm in the category of Rm-modules. +The following proposition asserts that the above gradation is connected and all its homo- +geneous components are finite-dimensional. +Proposition 4.2. Given the gradation for the Hopf algebra H +, +(1) H +is a graded and connected Hopf algebra viz. +ˆ∆ +� +ˆHn +� +⊆ +� +i+j=n +i,j≥0 +ˆHi ⊗ ˆHj. +(2) For all k: define wk = dim (Wk) and FW = � +k≥1 wkZk is the geometric series given +by +FW = +1 +1 − mZ , +where m = |X| and for all k ≥ 1: +wk = dim (Wk) = mk. +(3) Define F ˆ +H = � +n≥1 hnZn where hn = dim( ˆHn) then +F ˆ +H = +∞ +� +k=1 +1 +(1 − Zk)wk . +Proof: +(1) The Hopf algebra H +follows from the fact that if γ(̸= η, ζ) ∈ supp(η +ζ) then +deg (γ) = |γ| = |η| + |ζ| = deg (η) + deg (ζ) , +for all η, ζ, γ ∈ X∗. +(2) Define the formal power series +F(Z0, Z1, . . . , Zm) = +� +k≥1 +� +i0,i1,...,im≥0 +i0+i1+···+im=k +#{η : |η|xj = ij ∀ j = 0, 1, 2, . . . , m}Zi0 +0 Zi1 +1 · · · Zim +m += +(Z0 + Z1 + · · · + Zm) +1 − (Z0 + Z1 + · · · + Zm). + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +11 +Since each letter contributes equally to the degree (viz. length), thus +FW = F(Z, Z, . . ., Z) = +mZ +1 − mZ . +(3) The proposition follows from the item 2 as ˆH is the symmetric algebra generated by +the Rm-module W. +Table 1. Dimensions of the homogeneous components of module W and H +(when m = 2) +k +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +dim (Wk) +1 +2 +4 +8 +16 +32 +64 +128 +256 +512 +1024 +dim( ˆHk) +1 +2 +7 +20 +59 +162 +449 +1200 +3194 +8348 +21646 +. . . +Example 4.2. The dimensions of the homogeneous components of the graded module W +(up to k = 10) and the graded algebra H +for m = 2 viz when X = {x0, x1} is tabulated in +Table 1. +The sequence {dim( ˆHk)}k∈N0 is the sequence A034899 in [OEIS(2022)] which corresponds +to the number of multisets of binary words of total length n. +4.3. Coaction of H +. The subsection explains the coaction of the Hopf algebra H +(4.1) +on the algebra of coordinate functions. It is utilized subsequently to develop an algorithm to +compute the multiplicative mixed composition product explained in Section 5.2 and dynamic +feedback product as defined in Theorem 6.2. Let W to be the Rm-module as described in +Section 4.1. Let S+ (W) denote the reduced symmetric algebra generated by the module W. +The non-unital Rm-algebra S+(W) are equivalently the polynomials without constant term +of coordinate maps of Rm⟨⟨X⟩⟩. By construction S+(W) has a non-unital Rm-associative, +commutative algebra structure with addition, scalar multiplication and product defined, +respectively, as +(aη + aζ)(c) = aη(c) + aζ(c) +(kaη)(c) = k(aη(c)) +where c ∈ Rm⟨⟨X⟩⟩, and +m(aη, aζ)(c) = aη(c).aζ(c), +where c ∈ M +. The Rm-algebra S+(W) is isomorphic to the algebra structure of H +with +forgetting of the unit map ξ. The right coaction map ρ +: S+ (W) −→ S+ (W) ⊗ H +is +recursively defined on the module V as given by the following proposition. +Proposition 4.3. For all i = 0, 1, 2, . . . , m : +ρ +◦ θi = (θi ⊗ id + id ⊗ θi) ◦ ρ +, +with base case being ρ +a∅ = a∅ ⊗ a∅. +Proposition 4.3 might appear as repetition of Proposition 4.1. It is vital to note that +Proposition 4.1 is for defining the coproduct of Hopf algebra H +, where a∅ is the unit +element. Observe that, +ρ +ai +η(c, d) = ai +η(c +d), + +12 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +where c ∈ R⟨⟨X⟩⟩ (not necessarily in M +) and d ∈ M +. +The coaction ρ +thus is a +corepresentation of the Hopf algebra H +on the algebra S+ (W) or equivalently, ρ +makes +S+ (W), a H +-algebra. Let {πi}m +i=1 be the collection of co-ordinate projection maps on the +module W defined as +ai +η(c) := πi(aη)(c) = (c, η)i = (ci, η), +for all η ∈ X∗ and thus the following notation is well-defined, +ρj ai +η := (πi ⊗ πj) ◦ ρ +aη. +These notations are very much utilized in developing computational framework for the +multiplicative mixed composition product as discussed in Section 8. +Corollary 4.1. If n ∈ N0, then for all i = 0, 1, 2, . . ., m and j, k = 1, 2, . . . , m (defining +x0 +j := ∅): +ρj ak +xin = +n +� +r=0 +�n +r +� +ak +xir ⊗ aj +xin−r . +Proof: The statement is proved by induction on n ∈ N0. The base case (n = 0) follows from +Proposition 4.3. Assume the statement is true for n = p − 1, then +ρj ak +xip = ρj ◦ θiak +xip−1 += (θi ⊗ id + id ⊗ θi) ◦ ∆j ak +xip−1. +Using the induction hypothesis, +ρj ak +xip = (θi ⊗ id + id ⊗ θi) +�p−1 +� +r=0 +�p − 1 +r +� +ak +xir ⊗ aj +xip−1−r +� += +p +� +r=1 +�p − 1 +r − 1 +� +ak +xir ⊗ aj +xip−r + +p−1 +� +r=0 +�p − 1 +r +� +ak +xir ⊗ aj +xip−r. += +p +� +r=0 +�n +r +� +ak +xir ⊗ aj +xip−r. +Since the S+ (W) and H +are isomorphic as Rm-modules, the following lemma states the +coaction of H +on S+ (W) and the unshuffle coproduct coincide when the evaluation of +coordinate maps are restricted to the group M +. +Lemma 4.1. Given c, d ∈ M +, η ∈ X∗ and i = 1, 2, . . . , m, +ˆ∆ +aη (c, d) = (c +d, η) = ρ +aη (c, d) , +where c, d ∈ M +and ˆ∆i +is the coproduct from the bialgebra H +constructed in Section 4.3. +Example 4.3. A few examples of the computation of the coaction map ρ +on W using +Proposition 4.3 are given as follows(indices i, j = 1, 2, . . . , m and k, s = 0, 1, . . . , m): +∆j ai +∅ = ai +∅ ⊗ aj +∅. +∆j ai +xk = ai +xi ⊗ aj +∅ + ai +∅ ⊗ aj +xi. +∆j ai +xkxk = ai +xkxk ⊗ aj +∅ + 2ai +xk ⊗ aj +xk + ai +∅ ⊗ aj +xkxk. + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +13 +∆j ai +xkxs = ai +xkxs ⊗ aj +∅ + ai +xk ⊗ aj +xs + ai +xs ⊗ aj +xk + ai +∅ ⊗ aj +xkxs. +The following example illustrates the application of the deshuffle coproduct ∆ +in the +computation of the shuffle product of two series. +Example 4.4. Let X = {x0, x1} and c, d ∈ R2⟨⟨X⟩⟩ described as +c = +� +1 + x1 + x2 +1 + x3 +1 + · · · +x0 + x0x1 + x100 +1 +� +& +d = +� +1 + x2 +0 + exp (x1) +1 + x2 +0x1 +� +, +where exp(.) is the standard exponential function expressed in its Taylor series. Note that +c ̸∈ M +but d ∈ M +. The coefficient of x0x2 +1 in series c2 +d1 can be computed as: +� +c2 +d1, x0x2 +1 +� += ∆1 a2 +x0x2 +1 (c, d) = (π2 ⊗ π1) ◦ ∆ +ax0x2 +1 (c, d) += ∆1 ◦ θ0ax2 +1 (c, d) . +Using Proposition 4.3, +� +c2 +d1, x0x2 +1 +� += (θ0 ⊗ id + id ⊗ θ0) ◦ ∆1 a2 +x2 +1 (c, d) . +Using Corollary 4.1, +� +c2 +d1, x0x2 +1 +� += (θ0 ⊗ id + id ⊗ θ0) ◦ +� +a2 +x12 ⊗ a1 +∅ + 2a2 +x1 ⊗ a1 +x1 + a2 +∅ ⊗ a1 +x12 +� +(c, d) += +� +a2 +x0x12 ⊗ a1 +∅ + 2a2 +x0x1 ⊗ a1 +x1 + a2 +x0 ⊗ a1 +x12 + a2 +x12 ⊗ a1 +x0+ +2a2 +x1 ⊗ a1 +x0x1 + a2 +∅ ⊗ a1 +x0x12 +� +(c, d) += (0)(1) + 2(1)(1) + (1)(0.5) + (0)(0) + 2(0)(0) + (0)(0) = 2.5. +Therefore (c2 +d1, x0x2 +1) = 2.5. +5. Chen–Fliess Series and its Interconnections +The objective of the section is to describe Chen–Fliess series and the necessary non- +recursive interconnections of Chen–Fliess series to understand the results about the multi- +plicative dynamic feedback product in Section 6. +5.1. Chen–Fliess Series. Let p ≥ 1 and t0 < t1 be given. For a Lebesgue measurable +function u : [t0, t1] → Rm, define ∥u∥p = max{∥ui∥p : +1 ≤ i ≤ m}, where ∥ui∥p is the +usual Lp-norm for a measurable real-valued function, ui, defined on [t0, t1]. Let Lm +p [t0, t1] +denote the set of all measurable functions defined on [t0, t1] having a finite ∥ · ∥p norm +and Bm +p (R)[t0, t1] := {u ∈ Lm +p [t0, t1] : ∥u∥p ≤ R}. +Given any series c ∈ Rℓ⟨⟨X⟩⟩, the +corresponding Chen–Fliess series is +(7) +Fc[u](t) = +� +η∈X∗ +(c, η) Fη[u](t, t0), +where E∅[u] = 1 and +Fxi¯η[u](t, t0) = +� t +t0 +ui(τ)F¯η[u](τ, t0) dτ +with xi ∈ X, ¯η ∈ X∗, and u0 = 1 [Fliess(1981)]. If there exist constants K, M > 0 such that +|(ci, η)| ≤ KM|η||η|!, ∀η ∈ X∗, ∀i = 1, . . . , ℓ , +(8) + +14 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +then Fc constitutes a well-defined mapping from Bm +p (R)[t0, t0 + T] into Bℓ +q(S)[t0, t0 + T] +for sufficiently small R, T > 0, where the numbers p, q ∈ [1, ∞] are conjugate exponents, +i.e., 1/p + 1/q = 1 [Gray & Wang(2002)]. +This map is referred to as a Fliess operator. +A series c ∈ Rℓ⟨⟨X⟩⟩ obeying the growth condition in (8) is called a locally convergent +generating series. The set of all locally convergent generating series is denoted by Rℓ +LC⟨⟨X⟩⟩. +The supremum of the set of all max{R, T} for which a Fliess operator Fc is a well-defined +mapping from Bm +p (R)[t0, t0 + T] into Bℓ +q(S)[t0, t0 + T] is called the radius of convergence +of the Fliess operator Fc and is denoted by ρ (Fc). A Fliess operator Fc is called locally +convergent if ρ (Fc) > 0. If there exist constants K, M > 0 and γ ∈ [0, 1[ such that +|(ci, η)| ≤ KM|η| (|η|!)γ , ∀η ∈ X∗, ∀i = 1, . . . , ℓ , +(9) +then Fc constitutes a well defined mapping from Bm +p (R)[t0, t0 + T] into Bℓ +q(S)[t0, t0 + T] +for all R, T > 0 [Winter-Arboleda(2019), Winter-Arboleda, et al.(2015)]. The infimum of all +the γ ∈ [0, 1[ such that (9) is satisfied for a series c ∈ Rℓ⟨⟨X⟩⟩ is called the Gevrey order of +the series c. +A series c ∈ Rℓ⟨⟨X⟩⟩ obeying the growth condition in (9) is called a globally convergent +series. The set of all globally convergent series in Rℓ⟨⟨X⟩⟩ is denoted as Rℓ +GC⟨⟨X⟩⟩. A Fliess +operator Fc is globally convergent if and only if there exists no real number M > 0 such +that ρ (Fc) < M. Observe that a noncommutative polynomial R⟨X⟩ is a globally convergent +series with Gevrey degree 0. As described above, a series c ∈ Rℓ +GC⟨⟨X⟩⟩ is only a sufficient +condition for the corresponding Fliess operator Fc to be globally convergent. +Necessary +conditions are well-detailed in the literature [Winter-Arboleda(2019), Venkatesh(2021)]. In +the absence of any convergence criterion, (7) only defines an operator in a formal sense. +5.2. Interconnections of Chen–Fliess Series: Parallel and Cascade Connections. +Given Chen–Fliess series Fc and Fd, where c, d ∈ Rℓ⟨⟨X⟩⟩, the parallel and product connec- +tions satisfy Fc + Fd = Fc+d and FcFd = Fc +d, respectively [Ree(1958), Fliess(1981)]. The +parallel and product connections preserve local convergence and hence the interconnected +systems has a Fliess operator representation [Thitsa & Gray(2012), Venkatesh(2021)]. When +Chen–Fliess series Fc and Fd with c ∈ Rk⟨⟨X′⟩⟩ and d ∈ Rℓ⟨⟨X⟩⟩ are interconnected in a +cascade fashion, where |X′| = ℓ + 1, the composite system Fc ◦ Fd has a Chen–Fliess series +representation Fc◦d, where the composition product of c and d is given by +(10) +c ◦ d = +� +η∈X′∗ +(c, η) ψd(η)(1) +[Ferfera(1979), Ferfera(1980)]. Here 1 denotes the monomial 1∅, and ψd is the continuous +(in the ultrametric sense) algebra homomorphism from R⟨⟨X′⟩⟩ to the set of vector space +endomorphisms on R⟨⟨X⟩⟩, End (R⟨⟨X⟩⟩), uniquely specified by +ψd(x′ +iη) = ψd(x′ +i) ◦ ψd(η) +with ψd(x′ +i)(e) = x0(di +e), i = 0, 1, . . . , m for any e ∈ R⟨⟨X⟩⟩, and where di is the i-th +component series of d (d0 := 1). By definition, ψd(∅) is the identity map on R⟨⟨X⟩⟩. The +cascade interconnection preserves local convergence and thus the composite has a Fliess +operator representation [Thitsa & Gray(2012)]. The linearity of the composition product in +the left argument is evident form the definition. However, the following theorem states that +the composition product distributes over the shuffle product from the right. +Theorem 5.1. [Gray & Li(2005)] Let c, d ∈ Rk⟨⟨X′⟩⟩ and e ∈ Rℓ⟨⟨X⟩⟩, such that |X′| = +ℓ + 1, then (c +d) ◦ e = (c ◦ e) +(d ◦ e). + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +15 +Given a series e ∈ Rℓ⟨⟨X⟩⟩, define a map Υe : Rk⟨⟨X′⟩⟩ −→ Rk⟨⟨X⟩⟩ defined as c �→ +c ◦ e. Theorem 5.1 infers that Υe is an R-algebra homomorphism from the shuffle algebra of +Rk⟨⟨X′⟩⟩ to the shuffle algebra of Rℓ⟨⟨X⟩⟩. The composition product preserves the purely +improper property of the left argument which is stated in the following theorem. +Theorem 5.2. If c ∈ Rk⟨⟨X′⟩⟩ and d ∈ Rℓ⟨⟨X⟩⟩ such that |X′| = ℓ + 1, then (c ◦ d, ∅) = +(c, ∅). Hence, if c ∈ Rk +pi ⟨⟨X′⟩⟩ then c ◦ d ∈ Rk +pi ⟨⟨X⟩⟩ and vice-versa. Similarly if c is a +proper series then c ◦ d is also a proper series and vice-versa. +Proof: The proof follows immediately from (10). +The composition product is a strong contraction map with respect to its right argument +in the ultrametric topology and is stated in the following theorem. +Theorem 5.3. [Gray & Li(2005)] Let c ∈ Rk⟨⟨X′⟩⟩ and d, e ∈ Rℓ⟨⟨X⟩⟩, such that |X′| = +ℓ + 1, then κ (c ◦ d, c ◦ e) ≤ σκ (d, e) where σ ∈ [0, 1[. +5.3. Cascading of Chen–Fliess with Multiplicative Feedforward of Input. The cas- +cade interconnection of a Chen–Fliess series Fc and Fd along with the multiplicative feed- +forward of the input, as shown in Figure 1, arises primarily in the analysis of multiplicative +feedback interconnection discussed in Section 6. A semblance of such an interconnection +has appeared in Definition 3.1 of [Gray & Ebrahimi-Fard(2017)], without being explicit and +limited to the SISO case. With respect to Figure 1, the map u �→ y viz. y = Fc[u.Fd[u]] has +Chen–Fliess series representation denoted by Fc↶d, where c ↶ d denotes the multiplicative +mixed composition product of c ∈ Rp⟨⟨X⟩⟩ and d ∈ Rm⟨⟨X⟩⟩ defined as +c ↶ d = +� +η∈X∗ +(c, η) η ↶ d := +� +η∈X∗ +(c, η) ¯φd (η) (1) . +(11) +Here, ¯φd : R⟨⟨X⟩⟩ −→ End (R⟨⟨X⟩⟩) is an R-algebra homomorphism such that +¯φd(x0)(e) = x0e +and +¯φd(xi)(e) = xi(di +e). +Recall that R⟨⟨X⟩⟩ is an R-algebra under Cauchy product and End (R⟨⟨X⟩⟩). The multi- +plicative mixed composition defined in (11) asserts that, for all η ∈ X∗ and d ∈ Rm⟨⟨X⟩⟩, +∅ ↶ d = ∅ +x0η ↶ d = x0 (η ↶ d) +xiη ↶ d = xi (di +(η ↶ d)) +∀ i = 1, 2, . . . , m. +For later reference, we summarise the properties of (11) in the following +Theorem 5.4. The multiplicative mixed composition product (11) is linear in its left argu- +ment and (c ↶ d, ∅) = (c, ∅), for all c ∈ Rp⟨⟨X⟩⟩ and d ∈ Rm⟨⟨X⟩⟩. +The following results are already known in the single-input single-output (SISO) setting. +However, their multi-input multi-output (MIMO) extensions are straightforward and to avoid +reiteration of the proofs, only the statements are provided in this document. The foremost +of the theorems asserts that the multiplicative mixed composition product distributes over +shuffle product from the right. +Theorem 5.5. [Gray & Ebrahimi-Fard(2017)] Let c, d ∈ Rp⟨⟨X⟩⟩ and e ∈ Rm⟨⟨X⟩⟩, then +(c +d) ↶ e = (c ↶ e) +(d ↶ e). + +16 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +Fd +Fc +u +y +Figure 1. Cascade connection of Chen–Fliess Fd with Fc along with multi- +plicative feedforward of input +The inference of Theorem 5.5 is that for any e ∈ Rm⟨⟨X⟩⟩, the map Γe : Rp⟨⟨X⟩⟩ −→ +Rp⟨⟨X⟩⟩ given by d �→ d ↶ e is an R-algebra endomorphism on the shuffle algebra Rp⟨⟨X⟩⟩. +The next lemma is essential in proving that multiplicative mixed composition product is a +strong contraction map in its right argument in the ultrametric topology. +Lemma 5.1. [Gray & Ebrahimi-Fard(2017)] Let η ∈ X∗ and d, e ∈ Rm⟨⟨X⟩⟩, then +κ (η ↶ d, η ↶ e) ≤ σ|η|κ (d, e) where σ ∈ [0, 1[. +The following theorem states the strong contraction property of the multiplicative mixed +composition product which is an essential result in Section 6. +Theorem 5.6. [Gray & Ebrahimi-Fard(2017)] Let d, e ∈ Rm⟨⟨X⟩⟩ and c ∈ Rp⟨⟨X⟩⟩, then +κ (c ↶ d, c ↶ e) ≤ σord(c′)κ (d, e), where c′ = c − (c, ∅), the proper part of c. +Since ord (c′) ≥ 1 and σ ∈]0, 1[, then from Theorem 5.6, the map ¯Γc : e �→ c ↶ e is a strong +contraction map in the ultrametric topology. The following lemma is essential in proving +the mixed associativity of the composition and multiplicative mixed composition product. +The result, along with Theorem 5.7 can be inferred in the SISO setting from Lemma 3.6 in +[Gray & Ebrahimi-Fard(2017)], and its extension to the MIMO case is straightforward. +Lemma 5.2. [Gray & Ebrahimi-Fard(2017)] Let X′ = {x′ +0, . . . , x′ +p} and η ∈ X′∗. Let d ∈ +Rp⟨⟨X⟩⟩ and e ∈ Rm⟨⟨X⟩⟩, then η ◦ (d ↶ e) = (η ◦ d) ↶ e. +The following theorem states that the composition product and multiplicative mixed com- +position product are associative in combination. +Theorem 5.7. [Gray & Ebrahimi-Fard(2017)] Let X′ = {x′ +0, . . . , x′ +p} and c ∈ Rq⟨⟨X′⟩⟩. Let +d ∈ Rp⟨⟨X⟩⟩ and e ∈ Rm⟨⟨X⟩⟩, then c ◦ (d ↶ e) = (c ◦ d) ↶ e. +5.4. Multiplicative Dynamic Output Feedback Group. The dynamic multiplicative +feedback group plays a vital role in computation of the multiplicative dynamic feedback +formula, as well as in assessing the feedback as a group action in Section 6. Indeed, consider +the cascade interconnection of two Chen–Fliess series Fc and Fd along with their multiplica- +tive feedforward of inputs displayed in Figure 2, where c, d ∈ Rm⟨⟨X⟩⟩. The input-output +relation of the composite system, u �→ y is u.Fd[u]Fc[u.Fd[u]] and can be represented by +Chen–Fliess series as follows. Consider +u.Fc⋆d[u] := u.Fd[u]Fc[u.Fd[u]], +where the multiplicative composition product of c and d is defined as +c ⋆ d = d +(c ↶ d) . +(12) +The following theorems appeared in [Gray & Ebrahimi-Fard(2017)] in the SISO setting. +We underline that the latter restriction is not essential, that is, the statements along with +the proofs naturally extend to the MIMO setting. + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +17 +Figure 2. Cascade connection of Chen–Fliess Fd with Fc along with multi- +plicative feedforward of their inputs. +Theorem 5.8. [Gray & Ebrahimi-Fard(2017)] Let c, d, e ∈ Rm⟨⟨X⟩⟩, then, (c ⋆ d) ⋆ e = +c ⋆ (d ⋆ e). +Observe that (12) and Theorem 5.8 infer that Rm⟨⟨X⟩⟩ forms a non-commutative monoid +under multiplicative composition product, with the identity element ll. The following theo- +rem states that the multiplicative mixed composition product is a right action on Rq⟨⟨X⟩⟩ +by the monoid (Rm⟨⟨X⟩⟩, ⋆, ll). +Theorem 5.9. [Gray & Ebrahimi-Fard(2017)] Let c ∈ Rq⟨⟨X⟩⟩ and d, e ∈ Rm⟨⟨X⟩⟩, then +(c ↶ d) ↶ e = c ↶ (d ⋆ e). +The prominent question is to find the invertible elements of the monoid (Rm⟨⟨X⟩⟩, ⋆) and +the motivation to find the unit elements of the monoid shall be evident in Section 6. Let +d, e ∈ Rm +pi ⟨⟨X⟩⟩ and suppose +d ⋆ e = ll. +Observe that d ∈ Rm +pi ⟨⟨X⟩⟩ implies (d ↶ e) ∈ Rm +pi ⟨⟨X⟩⟩ and using Theorem 5.5, +e = (d ↶ e) +−1 = d +−1 ↶ e. +Hence, for e to be right inverse of d, the purely improper series e has to satisfy the fixed +point equation +e = d +−1 ↶ e +(13) +Observe from Theorem 5.6 that the map e �→ d +−1 ↶ e is a strong contraction in the +ultrametric space inferring that (13) has a unique fixed point. Suppose e is the left inverse +of d viz. e ⋆ d, then a similar procedure shows that e has to satisfy the equation +d = e +−1 ↶ d +(14) +Note that if e is a solution of (13), then e satisfies (14) and also the converse holds true. +Hence, e is the unique inverse of d and is given the notation d⋆−1 for d ∈ Rm +pi ⟨⟨X⟩⟩. Thus, +Rm +pi ⟨⟨X⟩⟩ forms a group under multiplicative composition product, ⋆, and is termed as the +multiplicative dynamic output feedback group and is formally stated in the following theorem. +Theorem 5.10. +� +Rm +pi ⟨⟨X⟩⟩, ⋆ +� +forms a group with the identity element ll. +It is worth noting that [Gray & Ebrahimi-Fard(2017)] proved Theorem 5.10 for one- +dimensional case viz. m = 1. In light of Theorem 5.10, Theorem 5.5 and (12) one obtains +the following relations for c ∈ Rm +pi ⟨⟨X⟩⟩: +c⋆−1 = c +−1 ↶ c⋆−1 +(15) +� +c⋆−1� +−1 = c ↶ c⋆−1. +The following lemma is essential in defining a subgroup of the multiplicative dynamic out- +put feedback group upon which the computational framework for the multiplicative feedback +products is discussed in Section 8. + +F +F +n18 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +Lemma 5.3. Let c, d ∈ Rm +pi ⟨⟨X⟩⟩, then (c ⋆ d, ∅) = (c, ∅) (d, ∅). +Proof: Observe from (12) that, +(c ⋆ d, ∅) = (d +(c ↶ d) , ∅) += (c ↶ d, ∅) (d, ∅) +Since (c ↶ d, ∅) = (c, ∅), +(c ⋆ d, ∅) = (c, ∅) (d, ∅) . +Lemma 5.3 thus proves that the set of all series which are of the form ll + c, where c is +a proper series, forms a subgroup of the multiplicative dynamic feedback group, which is +stated in the following theorem. +Theorem 5.11. Let M = { ll + c : c ∈ Rm +p ⟨⟨X⟩⟩}, then (M, ⋆, ll) forms a subgroup of the +multiplicative dynamic feedback group. +The algorithmic framework for the computation of multiplicative feedback products is +fundamentally based on the subgroup M as asserted in Theorem 5.11. The group M is +isomorphic to the character group of the Hopf algebra H which is used for computation of +feedback and the framework is explained in detail in Section 8. +6. Chen–Fliess Series Under Multiplicative Dynamic Output Feedback +Let Fc be a Chen–Fliess series with a generating series c ∈ Rq⟨⟨X⟩⟩. Assume it is intercon- +nected with a Chen–Fliess series Fd with a purely improper generating series d ∈ Rm +pi ⟨⟨X′⟩⟩, +as shown in Figure 3. Note that, |X| = m + 1 and |X′| = q + 1. The primary goal of this +section is to show that the closed-loop system has a Chen–Fliess series representation, say +y = Fe[v], where e ∈ Rq⟨⟨X⟩⟩. If this is the case, then necessarily +y = Fe[v] = Fc[u] = Fc[vFd[y]] += Fc[vFd[Fe[v]]] = Fc[vFd◦e[v]] += Fc↶(d◦e)[v] +for any admissible input v. Therefore, the series e has to satisfy the fixed point equation +e = c ↶ (d ◦ e) . +(16) +Observe that, in light of Theorem 5.3 and Theorem 5.6 the map e �→ c ↶ (d ◦ e) is a +strong contraction map in the ultrametric space and thus (16) has a unique fixed point. The +following thoerem establishes the first main result of this section, which follows immediately. +Theorem 6.1. The series c ↶ (d +−1 ◦ c)⋆−1 ∈ Rq⟨⟨X⟩⟩ is the unique fixed point of the map +e �→ c ↶ (d ◦ e). +Proof: If e := c ↶ (d +−1 ◦ c)⋆−1, then +c ↶ (d ◦ e) = c ↶ +� +d ◦ +� +c ↶ +� +d +−1 ◦ c +�⋆−1�� +Using Theorem 5.7 and then Theorem 5.5, +c ↶ (d ◦ e) = c ↶ +� +(d ◦ c) ↶ +� +d +−1 ◦ c +�⋆−1� + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +19 +Fc +v +Fd +y +u +Figure 3. Chen–Fliess series Fc in multiplicative output feedback with Chen- +Flies series Fd += c ↶ +� +(d ◦ c) +−1 ↶ +� +d +−1 ◦ c +�⋆−1� +−1 +. +Using Theorem 5.1, +c ↶ (d ◦ e) = c ↶ +�� +d +−1 ◦ c +� +↶ +� +d +−1 ◦ c +�⋆−1� +−1 +. +Using the relations (15), +c ↶ (d ◦ e) = c ↶ +��� +d +−1 ◦ c +�⋆−1� +−1� +−1 += c ↶ +� +d +−1 ◦ c +�⋆−1 = e. +Theorem 6.2. Given a series c ∈ Rq⟨⟨X⟩⟩ and a purely improper series d ∈ Rm +pi ⟨⟨X′⟩⟩ (such +that |X| = m + 1 and |X′| = q + 1), then the generating series for the closed-loop system in +Figure 3 is given by the multiplicative dynamic feedback product cˇ@d := c ↶ (d +−1 ◦ c)⋆−1. +The notion that feedback can described mathematically as a transformation group acting +on the plant is well established in control theory [Brockett(1978)]. The following theorem +describes the situation in the present context. +Theorem 6.3. The multiplicative dynamic feedback product is a right group action by the +multiplicative group +� +Rm +pi ⟨⟨X′⟩⟩, +, ll +� +on the set Rq⟨⟨X⟩⟩, where |X| = m + 1 and |X′| = +q + 1. +Proof: Let c ∈ Rq⟨⟨X⟩⟩. Observe that from Theorem 6.2, +cˇ@ ll = c ↶ +� +ll +−1 ◦ c +�⋆−1 += c ↶ ll = c. +Let d1, d2 ∈ Rm +pi ⟨⟨X′⟩⟩. It needs to be proven that +� +cˇ@d1 +� ˇ@d2 = cˇ@ (d1 +d2). From Theo- +rem 6.2, observe that +� +cˇ@d1 +� ˇ@d2 = +� +cˇ@d1 +� +↶ +� +d +−1 +2 +◦ +� +cˇ@d1 +��⋆−1 += +� +c ↶ +� +d +−1 +1 +◦ c +�⋆−1� +↶ +� +d +−1 +2 +◦ +� +c ↶ +� +d +−1 +1 +◦ c +�⋆−1��⋆−1 +. +Applying Theorem 5.7, +� +cˇ@d1 +� ˇ@d2 = +� +c ↶ +� +d +−1 +1 +◦ c +�⋆−1� +↶ +�� +d +−1 +2 +◦ c +� +↶ +� +d +−1 +1 +◦ c +�⋆−1�⋆−1 +. + +20 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +Applying Theorem 5.9 and fact that the group inverse is anti-homomorphism with respect +to the group product, +� +cˇ@d1 +� ˇ@d2 = c ↶ +� � +d +−1 +1 +◦ c +�⋆−1 ⋆ +�� +d +−1 +2 +◦ c +� +↶ +� +d +−1 +1 +◦ c +�⋆−1�⋆−1 � += c ↶ +� �� +d +−1 +2 +◦ c +� +↶ +� +d +−1 +1 +◦ c +�⋆−1� +⋆ +� +d +−1 +1 +◦ c +� �⋆−1 +. +Applying (12), +� +cˇ@d1 +� ˇ@d2 = c ↶ +� +� +d +−1 +1 +◦ c +� +�� � +d +−1 +2 +◦ c +� +↶ +� +d +−1 +1 +◦ c +�⋆−1 � +↶ +� +d +−1 +1 +◦ c +� +��⋆−1 +. +Using Theorem 5.9, +� +cˇ@d1 +� ˇ@d2 = c ↶ +� +� +d +−1 +1 +◦ c +� +� +� +d +−1 +2 +◦ c +� +↶ +�� +d +−1 +1 +◦ c +�⋆−1 ⋆ +� +d +−1 +1 +◦ c +�� ��⋆−1 += c ↶ +�� +d +−1 +1 +◦ c +� +�� +d +−1 +2 +◦ c +� +↶ ll +��⋆−1 += c ↶ +�� +d +−1 +1 +◦ c +� +� +d +−1 +2 +◦ c +��⋆−1 . +In light of Theorem 5.1, +� +cˇ@d1 +� ˇ@d2 = c ↶ +�� +d +−1 +1 +d +−1 +2 +� +◦ c +�⋆−1 += c ↶ +� +(d1 +d2) +−1 ◦ c +�⋆−1 . +Therefore, +� +cˇ@d1 +� ˇ@d2 = cˇ@ (d1 +d2) . +It is worth noting that for the additive dynamic feedback product the transformation group +is the additive group (Rm⟨⟨X′⟩⟩, +, 0) while here (Rm +pi ⟨⟨X′⟩⟩, +, ll) plays the role. +7. Invariance of Class and Relative Degree under multiplicative dynamic +feedback connection +The notion of relative degree of a plant is very essential and prime in the studies of +feedback linearization [Isidori(1995)], flatness and system inversion etc. The existence and +quantification of relative degree of a interconnection of systems is vital in systems theory. +The notion of class and relative degree of a SISO Chen–Fliess series is equivalently char- +acterized by the notion of relative degree of its generating series and the definition was +furnished in [Gray, et al.(2014b), Gray & Venkatesh(2019)] and the existence and quantifi- +cation of relative degree of interconnected system of Chen–Fliess series was described in +[Gray & Venkatesh(2019), Venkatesh(2021)]. In addition, this definition of relative degree is +consistent with the classical definition whenever y = Fc[u] has an input-affine analytic state +space realization [Gray, et al.(2014b), Gray & Ebrahimi-Fard(2017)]. Let X = {x0, x1} and +the following definition explains the concept of a class, a weaker notion than the relative +degree of a series in R⟨⟨X⟩⟩. +Definition 7.1. +[Gray & Venkatesh(2019)] A series c ∈ R⟨⟨X⟩⟩ is said to be of r-class, +denoted by C (c) = r, if supp(cF) ⊆ xr−1 +0 +X+ and supp(cF) ⊈ xr +0X+. +By definition, let +C (c) = ∞ if cF = 0. +The notion of class is universal and is versed in the following theorem. + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +21 +Lemma 7.1. [Gray & Venkatesh(2019)] Every series c ∈ R⟨⟨X⟩⟩ has a class. +Definition 7.1 of class is illustrated in the following example. +Example 7.1. Let c = 1 + x0x2 +1 + x2 +0x1, so that cF = x0x2 +1 + x2 +0x1. Observe that supp(cF) ⊆ +x0X+ but supp(cF) ⊈ x2 +0X+. Thus, C (c) = 2. +The following lemma is essential in the proof of quantification of class for the multiplicative +mixed composition product. +Lemma 7.2. Let c, c′, d ∈ Rm⟨⟨X⟩⟩ such that supp (c′) ̸⊆ x0X∗. Then the following state- +ments are true: +(1) xk +0 ↶ d = xk +0 ∀k ∈ N0. +(2) cN ↶ d = cN where cN is the natural part of the series c. +(3) supp (c′ ↶ d) ̸⊆ x0X∗. +Proof: +(1) The proof is by induction on k ∈ N0. The base case being k = 0 is true viz ∅ ↶ d = ∅ +from (11). Assume the proposition is true for k = n − 1, then using (11) +xn +0 ↶ d = x0 +� +xn−1 +0 +↶ d +� += x0 +� +xn−1 +0 +� += xn +0. +Hence proved by induction on N0. +(2) Observe that from Definition 2.1, supp (cN) ⊆ {xk +0 : k ∈ N0}. Thus, using the previ- +ous statement (1) and Theorem 5.4 it follows that cN ↶ d = cN. +(3) Since supp (c′) ̸⊆ x0X∗, there exists a word xiη ∈ supp (c′) where xi ̸= x0 and η ∈ X∗. +Using (11), +xiη ↶ d = xi (di +(η ↶ d)) . +Thus, supp (xiη ↶ d) ⊆ xiX∗, where xi ̸= x0. Therefore, supp (c′ ↶ d) ̸⊆ x0X∗. +The following theorem quantifies that class is invariant under the multiplicative mixed +composition product +Theorem 7.1. Let c, d ∈ R⟨⟨X⟩⟩, then C (c ↶ d) = C (c). +Proof: Suppose the series c ∈ R⟨⟨X⟩⟩ is of r-class, then the series c can be written as: +c = cN + xr−1 +0 +c′, +where c′ is a proper series such that supp (c′) ̸⊆ x0X∗. Hence by Theorem 5.4, +c ↶ d = (cN ↶ d) + +� +xr−1 +0 +c′ ↶ d +� +. +Using (11), +c ↶ d = (cN ↶ d) + xr−1 +0 +(c′ ↶ d) . +Since supp (c′) ̸⊆ x0X∗, then by applying Lemma 7.2, +c ↶ d = cN + xr−1 +0 +(c′ ↶ d) , + +22 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +with supp (c′ ↶ d) ̸⊆ x0X∗. Given that c′ ∈ Rp ⟨⟨X⟩⟩, whence supp (c ↶ d)F ⊆ xr−1 +0 +X+ and +supp (c ↶ d)F ̸⊆ xr +0X+. Therefore, C (c ↶ d) = r = C (c). +Example 7.2. Consider the series c in Example 7.1, given by c = 1 + x2 +0x1 + x0x2 +1 and +d = 1 + x1 ∈ R⟨⟨X⟩⟩. Using (11), the multiplicative mixed composition product of c and d +is computed as: +c ↶ d = 1 + x0x2 +1 + 3x0x3 +1 + 3x0x4 +1 + x2 +0x1 + x2 +0x2 +1. +Observe that C (c ↶ d) = 2 = C (c), as in Example 7.1. +The following theorem asserts that class of a series is preserved under the multiplicative +dynamic feedback product which is one of the prime goal of this subsection. +Theorem 7.2. If c ∈ R⟨⟨X⟩⟩ with C (c) = r, and d ∈ Rpi ⟨⟨X⟩⟩, then C +� +cˇ@d +� += r = C (c). +Proof: From Theorem 6.2, +cˇ@d = c ↶ +� +d +−1 ◦ c +�⋆−1 . +Since C (c) = r, whence applying Theorem 7.1, +C +� +cˇ@d +� += C +� +c ↶ +� +d +−1 ◦ c +�⋆−1� += r = C (c) . +The preservation of class under the multiplicative dynamic feedback connections as as- +serted in Theorem 7.2 is further illustrated in the following example. +Example 7.3. Let c, d ∈ R⟨⟨X⟩⟩ c = x1 and d = 1 + � +k∈N k!xk +1. Note that the class of +series C (c) = 1. Using Theorem 6.2 the multiplicative feedback product is computed as: +cˇ@d = x1 + x1x0x1 + 3x1x0x1x0x1 + 4x1x2 +0x2 +1 + · · · . +Infer from Definition 7.1 that C +� +cˇ@d +� += C (c) = 1. +Finally, the main definition of the section details the concept of relative degree in the +context of Chen–Fliess series which is characterized on its generating series. +Definition 7.2. +[Gray & Venkatesh(2019)] A series c ∈ R⟨⟨X⟩⟩ has relative degree r if +C (c) = r and the word xr−1 +0 +x1 ∈ supp(cF). Otherwise, c does not have relative degree. +The following theorem asserts the quantification of relative degree under multiplicative +mixed composition product. +Theorem 7.3. If c ∈ R⟨⟨X⟩⟩ with relative degree rc and d ∈ R⟨⟨X⟩⟩ be non-proper, then +c ↶ d has relative degree rc. +Proof: From Theorem 7.1, C (c ↶ d) = rc. It remains to prove that xrc−1 +0 +x1 ∈ supp (c ↶ d). +Given that c ∈ R⟨⟨X⟩⟩ has relative degree rc, then c can be decomposed as: +c = cN + λxrc−1 +0 +x1 + xrc−1 +0 +c′, +where λ ̸= 0 and c′ is a proper series such that x1 ̸∈ supp (c′). Then, +c ↶ d = +� +cN + λxrc−1 +0 +x1 + xrc−1 +0 +c′� +↶ d. +Applying Theorem 5.4, +c ↶ d = (cN ↶ d) + λ +� +xrc−1 +0 +x1 ↶ d +� ++ +� +xrc−1 +0 +c′ ↶ d +� +. + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +23 +Using (11) and Lemma 7.2, +c ↶ d = cN + λxrc−1 +0 +x1d + xrc−1 +0 +(c′ ↶ d) . +Since d ∈ Rpi ⟨⟨X⟩⟩ −→ d = α + d′, where α ̸= 0 and d′ is a proper series. Hence, +c ↶ d = cN + λαxrc−1 +0 +x1 + xrc−1 +0 +x1d′ + xrc−1 +0 +(c′ ↶ d) . +Observe from (11), x1 ̸∈ supp (c′) =⇒ x1 ̸∈ supp (c′ ↶ d) and also that αλ ̸= 0. +Therefore xrc−1 +0 +x1 ∈ supp (c ↶ d), whence the relative degree of c ↶ d is rc, when d is a +non-proper series. +The following example illustrates the statement from Theorem 7.3. +Example 7.4. Let c = 1 + x2 +0 + x0x1 + x2 +0x1 and d = 1 + x1. Observe that by Definition 7.2, +the relative degree of c is rc = 2 and also that d is non-proper. The multiplicative mixed +composition product of c and d to computed as: +c ↶ d = 1 + x2 +0 + x0x1 + x0x2 +1 + x2 +0x1 + x2 +0x2 +1. +Using Definition 7.2, note that the relative degree of c ↶ d is 2 = rc. +The following theorem is the prime objective of this section stating that the relative degree +of a series remains invariant under multiplicative dynamic feedback product. +Theorem 7.4. If c ∈ R⟨⟨X⟩⟩ with relative degree rc and d ∈ Rpi ⟨⟨X⟩⟩, then the relative +degree of +� +cˇ@d +� +is rc. +Proof: Since c ∈ R⟨⟨X⟩⟩ and d ∈ Rpi ⟨⟨X⟩⟩, then by Theorem 6.2, +cˇ@d = c ↶ +� +d +−1 ◦ c +�⋆−1 . +Observe that d ∈ Rpi ⟨⟨X⟩⟩ ⇔ d +−1 ∈ Rpi ⟨⟨X⟩⟩. +Then by Theorem 5.2 (d +−1 ◦ c) ∈ +Rpi ⟨⟨X⟩⟩. As per Theorem 5.10, the group inverse +� +d +−1 ◦ c +�⋆−1 ∈ Rpi ⟨⟨X⟩⟩. +Hence by Theorem 7.3, +cˇ@d = c ↶ +� +d +−1 ◦ c +�⋆−1 . +has relative degree rc. +The invariance of the relative degree of a Chen–Fliess series under multiplicative dynamic +feedback connections as stated in Theorem 7.4 is illustrated through the following example. +Example 7.5. Consider the Example 7.3 again where c = x1 and d = 1 + � +k∈N k!xk +1. +Observe that by Definition 7.2, the relative degree of c is rc = 1. The multiplicative feedback +product is computed as: +cˇ@d = x1 + x1x0x1 + 3x1x0x1x0x1 + 4x1x2 +0x2 +1 + · · · +Infer that the relative degree of cˇ@d = 1 = rc as stated in Theorem 7.4. + +24 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +8. Computational Framework for Multiplicative Mixed Composition & +Dynamic Feedback Product +The goal of this section is to describe the computational framework for multiplicative +dynamic feedback product as explained in Section 6. +The section further illustrates the +framework with examples but prior to that it is imperative to understand the dual bialgebra +and Hopf algebra constructions corresponding to the multiplicative dynamic output feedback +group. +8.1. Hopf Algebra Corresponding to the Multiplicative Dynamic Feedback Sub- +group. The goal of the subsection is to construct a dual Hopf algebra reflecting the group +structure of the multiplicative dynamic feedback subgroup M as asserted in Theorem 5.11. +The group inverse is computed the antipode of the constructed Hopf algebra and thus pro- +vides a computational framework to compute the multiplicative dynamic feedback group +inverse. As a recall, the group M is defined as +M = { ll + d : d ∈ Rm +p ⟨⟨X⟩⟩}, +where ll = [1 · · ·1 1]T ∈ Rm. In light of Theorem 5.11, (M, ⋆) forms a subgroup of the +multiplicative dynamic feedback group. The algebra structure is same as the algebra of H +in Section 4.1. Let the set Wb ⊂ Rm⟨⟨X⟩⟩∗ (dual module of Rm⟨⟨X⟩⟩) be defined as the +collection of coordinate maps defined as: +Wb = {aη : aη(c) = (c, η) : η ∈ X∗}, +where c ∈ Rm⟨⟨X⟩⟩. +Define W to be the free Rm-module spanned by the set Wb. +Let +H denote the reduced symmetric algebra generated by the module W. The unit map ξ : +Rm −→ W is defined by ξ( ll) = a∅. Note that a∅ (c) = ll ∀c ∈ M. By construction H is +an Rm-associative, commutative and unital algebra with addition, scalar multiplication and +product defined, respectively, as +(aη + aζ)(c) = aη(c) + aζ(c) +(kaη)(c) = k(ai +η(c)) +m(aη, aζ)(c) = aη(c)aζ(c), +where c ∈ Rm⟨⟨X⟩⟩. Then H is given a coproduct ∆H : H −→ H � H such that for all +c, d ∈ M: ∆Hai +η(c, d) = ai +η(c ⋆ d) = ((c ⋆ d)i , η) ∀η ∈ X+. The counit map ǫ : H −→ R is +defined as +ǫ(h) = +� ll : h = a∅ +0 : otherwise. +Since ◦ is associative (from Theorem 5.8), thus by the dual the coproduct ∆H is coasso- +ciative. Therefore, (H, m, ξ, ∆H, ǫ) forms a Rm-bialgebra. Owing to the group structure of +(M, ◦), the bialgebra H is equipped with antipode S defined as: +Saη (c) = aη +� +c⋆−1� += +� +c⋆−1, η +� +, +for all i = 1, 2, . . . , m and η ∈ X+. Hence H is a Rm-Hopf algebra. The computation of +coproduct ∆H is well-understood through the right coaction of Hopf algebra H on the Hopf +algebra H +. Prior to that, it is imperative to understand the right coaction of Hopf algebra +H on the non-unital algebra of coordinate functions. + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +25 +8.2. Coaction of Hopf algebra H on Algebra of Coordinate Map. The subsection +explains the coaction of the Hopf algebra H defined in Section 8.1 on the algebra of coordinate +functions. The results in this subsection are utilized subsequently to explain the coaction of +H on the bialgebra H +, particularly in proofs in Section 8.3. The right coaction of the Hopf +algebra H is on Rm-algebra of coordinate maps S+ (W) constructed in Section 4.3. +The right coaction map ˜∆ : S+ (W) −→ S+ (W) � H is defined such that for all c ∈ +Rm⟨⟨X⟩⟩, d ∈ M and η ∈ X∗, +˜∆aη (c, d) = (c ↶ d, η) . +(17) +The map ˜∆ being a right coaction map is a reflection of Theorem 5.9. It remains to +show how the coaction map ˜∆ is computed on S+(W), for which it is sufficient to define its +computation on the module W. Observe that for all aη ∈ W, +˜∆aη = +� +˜∆ ◦ π1 ˜∆ ◦ π2 · · · ˜∆ ◦ πm�t +aη. +On the dual side, the above statement infers that for all c ∈ Rm⟨⟨X⟩⟩, d ∈ M and η ∈ X∗, +(c ↶ d, η) = [((c ↶ d)1 , η) · · · ((c ↶ d)m , η)]t . +Hence, the notation ˜∆ai +η := ˜∆ ◦ πiaη for all η ∈ X∗ and i = 1, 2, . . . , m. The following +proposition provides a recursive definition to compute ˜∆ on the module V viz to compute +the ˜∆ +� +aj +η +� +∀η ∈ X∗ and j = 1, 2, . . . , m. +Proposition 8.1. For all i = 1, . . . , m: +(1) ˜∆ai +∅ = ai +∅ ⊗ ai +∅. +(2) ˜∆ ◦ θ0 = (θ0 ⊗ id) ◦ ˜∆. +(3) ˜∆ ◦ θi = (θi ⊗ m) ◦ +� +˜∆ ⊗ id +� +◦ ρi , +where ρ +is the coaction map of Hopf algebra H +on S+ (W) as defined in Section 4.3. +Proof: Observe that ∀c ∈ Rm⟨⟨X⟩⟩ and d ∈ M, +c = (c, ∅) + +m +� +j=0 +xj +� +x−1 +j +(c) +� +. +Hence by Theorem 5.4, +c ↶ d = (c, ∅) + x0 +� +x−1 +0 (c) ↶ d +� ++ +m +� +j=1 +xj +� +dj +� +x−1 +j +(c) ↶ d +�� +. +(18) +The proof for each of the statement as follows: +(1) Let c, d ∈ Rm⟨⟨X⟩⟩. From (17) and (18), +˜∆ai +∅ (c, d) = ((c ↶ d)i , ∅) += (ci ↶ d, ∅) = (ci, ∅) .1 = ai +∅ ⊗ ai +∅ (c, d) . +Therefore, ˜∆ai +∅ = ai +∅ ⊗ ai +∅. + +26 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +(2) Let c, d ∈ Rm⟨⟨X⟩⟩, η ∈ X∗ and ∀ j = 1, 2, . . . , m. Then, +� +˜∆ ◦ θ0 +� +aj +η (c, d) = +� +(c ↶ d)j , x0η +� += +� +x−1 +0 (c ↶ d)j , η +� +From (18), +� +˜∆ ◦ θ0 +� +aj +η (c, d) = +� +x−1 +0 (cj) ↶ d, η +� += ˜∆aj +η +� +x−1 +0 (c) , d +� += (θ0 ⊗ id) ◦ ˜∆aη (c, d) . +Therefore, ˜∆ ◦ θ0 = (θ0 ⊗ id) ◦ ˜∆. +(3) Let c, d ∈ Rm⟨⟨X⟩⟩ and η ∈ X∗. Then ∀ i, j = 1, 2, . . . , m, +� +˜∆ ◦ θi +� +aj +η (c, d) = +� +(c ↶ d)j , xiη +� += +� +x−1 +i +(c ↶ d)j , η +� +From (18), +� +˜∆ ◦ θi +� +aj +η (c, d) = +� +di +x−1 +i +(cj) ↶ d, η +� += ρi aj +η +� +x−1 +i +(c) ↶ d, d +� += ρi aj +η +� +x−1 +i +(c) ↶ d +� += ρi aj +η +� +x−1 +i +(c) ↶ d, d +� += +� +˜∆ ⊗ id +� +◦ ρi aj +η +� +x−1 +i +(c) , d, d +� += (θi ⊗ m) ◦ +� +˜∆ ⊗ id +� +◦ ρi aj +η (c, d) . +Therefore, ˜∆ ◦ θi = (θi ⊗ m) ◦ +� +˜∆ ⊗ id +� +◦ ρi +∀i = 1, 2, . . . , m. +Example 8.1. A few examples of the computation of ˜∆ on V using Proposition 8.1 are +given as follows(indices i, j, k = 1, 2, . . . , m.): +˜∆ai +∅ = ai +∅ ⊗ ai +∅. +˜∆ai +x0 = ai +x0 ⊗ ai +∅. +˜∆aj +xi = aj +xi ⊗ ai +∅. +˜∆ai +x2 +0 = ai +x2 +0 ⊗ ai +∅. +˜∆aj +x0xi = aj +x0xi ⊗ ai +∅. +˜∆aj +xix0 = +� +aj +xix0 ⊗ ai +∅ +� ++ +� +aj +xi ⊗ ai +x0 +� +. + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +27 +˜∆ak +xixj = +� +ak +xixj ⊗ aj +∅ai +∅ +� ++ +� +ak +xi ⊗ ai +xj +� +. +The coaction map ˜∆ thus provides a framework to compute the multiplicative mixed com- +position product and multiplicative dynamic feedback group product whenever c ∈ Rm⟨⟨X⟩⟩ +and d ∈ M ⊊ Rm⟨⟨X⟩⟩. For computing the multiplicative mixed composition product for +c ∈ Rp⟨⟨X⟩⟩ and d ∈ M ⊊ Rm⟨⟨X⟩⟩ where p = m, +(1) If p < m, then define ˇc ∈ Rm⟨⟨X⟩⟩ such that ˇci = ci ∀ i = 1, 2, . . . , p and ˇci = 0 ∀ i = +p + 1, p + 2, . . . , m. Then for all η ∈ X∗, +((c ↶ d)i , η) = ˜∆ai +η (ˇc, d) +∀i = 1, 2, . . . , p. +Note that (ˇc ↶ d)j = 0 ∀j = p + 1, p + 2, . . . , m. +(2) If p > m, then this can be reduced to Case 1 by performing computations component +wise viz computing ci ↶ d for all i = 1, 2, . . . , p. +Thus the computational framework to compute the multiplicative mixed composition prod- +uct of c ∈ Rp⟨⟨X⟩⟩ and d ∈ M, denoted by c ↶ d for arbitrary p and m is well-defined via +the coaction map ˜∆. The computations of the coproduct ∆H and antipode S (defined in +Section 8.1) are well-understood once the right coaction of Hopf algebra H on Hopf algebra +H +. +8.3. Coaction of Hopf algbera H on the Hopf algebra H +. The objective of the +subsection is to define the right coaction map of Hopf algebra H on the unshuffle Hopf algebra +H +defined in Section 4.1. The right coaction is pivotal in computation of the coproduct +and antipode of Hopf algebra H which in turn are essential to compute the multiplicative +dynamic feedback product. +The right coaction map of H on H +is defined to be ˜∆H : H +−→ H +� H such that +for all c, d ∈ M (the underlying sets of M and M +are identical) and η ∈ X∗, +˜∆Haη (c, d) = (c ↶ d, η) . +(19) +Observe that the algebra of coordinate functions S+(W) and H +are isomorphic as Rm- +modules. Thus it is vital to understand the relationship between the operator ˜∆ operating +on the module S+(W) and operator ˜∆H operating on H +, which is stated in the following +lemma. +Lemma 8.1. If c, d ∈ M, then for all η ∈ X∗ +˜∆Haη (c, d) = ˜∆aη (c, d) . +Proof: If c, d ∈ M and η ∈ X+, +˜∆Haη (c, d) = (c ↶ d, η) = ˜∆aη (c, d) . +Despite the statement of Lemma 8.1, it is vital to understand the difference between the +coaction maps ˜∆ and ˜∆H. +The coaction map ˜∆H is compatible with the Hopf algebra +structure of H +viz. +m1,3,24 ◦ +� +˜∆H ⊗ ˜∆H +� +◦ ∆ += (∆ +⊗ id) ◦ ˜∆H, +˜∆H ◦ S = (S +⊗ id) ◦ ˜∆H, +where m1,3,24 = (m ⊗ m) ◦ (id ⊗ τ ⊗ id). + +28 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +Thus the coaction map ˜∆H makes H +a comodule-Hopf algebra over H. Equivalently, +the coaction map ˜∆H is a corepresentation of Hopf algebra H over unshuffle Hopf algebra +H +. Similar to Section 8.2, for all aη ∈ W, +˜∆Haη = +� +˜∆H ◦ π1 ˜∆H ◦ π2 · · · ˜∆H ◦ πm� +aη. +The map to compute the ˜∆H +� +aj +η +� +∀η ∈ X∗ and j = 1, 2, . . . , m is g module W is stated +in the following proposition. +Proposition 8.2. For all i, j = 1, 2 . . . , m and η ∈ X∗: +(1) ˜∆Hai +∅ = ai +∅ ⊗ ai +∅. +(2) ˜∆H ◦ θ0aj +η = (θ0 ⊗ id) ◦ ˜∆Haj +η. +(3) +� +˜∆H ◦ θi +� +aj +η = (θi ⊗ m) ◦ +� +˜∆H ⊗ id +� +◦ ∆i aj +η, +where ∆ +is the unshuffle coproduct defined in Section 4.1. +Proof: Observe that ∀c ∈ M, +c = ll + +m +� +j=0 +xj +� +x−1 +j +(c) +� +. +Hence by Theorem 5.4, +c ↶ d = ll + x0 +� +x−1 +0 (c) ↶ d +� ++ +m +� +j=1 +xj +� +dj +� +x−1 +j +(c) ↶ d +�� +. +(20) +The proof for each of the statement as follows: +(1) Let c, d ∈ M. From (19) and (20), +˜∆Hai +∅ (c, d) = ((c ↶ d)i , ∅) += (ci ↶ d, ∅) = 1 = (ci, ∅)(di, ∅) += ai +∅ ⊗ ai +∅(c, d). +Therefore, ˜∆Hai +∅ = ai +∅ ⊗ ai +∅. +(2) Let c, d ∈ M, η ∈ X∗ and ∀ j = 1, 2, . . . , m. Then, +� +˜∆H ◦ θ0 +� +aj +η (c, d) = +� +(c ↶ d)j , x0η +� += +� +x−1 +0 (c ↶ d)j , η +� +Observe that x−1 +0 (c) may not belong to M and from (20), +� +˜∆H ◦ θ0 +� +aj +η (c, d) = +� +x−1 +0 (cj) ↶ d, η +� += ˜∆aj +η +� +x−1 +0 (c) , d +� += (θ0 ⊗ id) ◦ ˜∆aη (c, d) . + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +29 +Since η ∈ X+ and c, d ∈ M, then by Lemma 8.1 +� +˜∆H ◦ θ0 +� +aj +η (c, d) = (θ0 ⊗ id) ◦ ˜∆Haη (c, d) . +Therefore, ˜∆H ◦ θ0 = (θ0 ⊗ id) ◦ ˜∆H. +(3) Let c, d ∈ M and η ∈ X∗. Then ∀ i, j = 1, 2, . . . , m, +� +˜∆H ◦ θi +� +aj +η (c, d) = +� +(c ↶ d)j , xiη +� += +� +x−1 +i +(c ↶ d)j , η +� +From (20), +� +˜∆H ◦ θi +� +aj +η (c, d) = +� +di +x−1 +i +(cj) ↶ d, η +� +. +Since x−1 +i +(c) may not belong to group M (also M +), += ρi aj +η +� +x−1 +i +(c) ↶ d, d +� += ρi aj +η +� +x−1 +i +(c) ↶ d +� += ρi aj +η +� +x−1 +i +(c) ↶ d, d +� += +� +˜∆ ⊗ id +� +◦ ρi aj +η +� +x−1 +i +(c) , d, d +� += (θi ⊗ m) ◦ +� +˜∆ ⊗ id +� +◦ ρi aj +η (c, d) . +Since η ∈ X+ and c, d ∈ M, then by Lemma 8.1 and Lemma 4.1, +� +˜∆H ◦ θi +� +aj +η (c, d) = (θi ⊗ m) ◦ +� +˜∆H ⊗ id +� +◦ ∆i aj +η. +Therefore, +� +˜∆H ◦ θi +� += (θi ⊗ m) ◦ +� +˜∆H ⊗ id +� +◦ ∆i +for all i = 1, 2, . . . , m. +8.4. Coproduct, Antipode Computations and Grading of Hopf algebra H. The +objective of this subsection is to define and illustrate the computation of coproduct ∆H of +the bialgebra H. Further, a graded and connected structure is endowed with the bialgebra +owing to which the antipode computation is possible owing to Theorem 3.1. The following +proposition asserts the essential reason behind the definition of ˜∆H. +Proposition 8.3. For all η ∈ X∗ and i = 1, 2, . . . , m, +∆Hai +η = (id ⊗ m) ◦ +� +˜∆H ⊗ id +� +◦ ˆ∆i ai +η. +Proof: Proof: Observe that for all c, d ∈ M and η ∈ X∗, +∆ai +η (c, d) = ((c ⋆ d)i , η) +∀ i = 1, 2, . . . , m. +Using (12), +∆Hai +η (c, d) = (di +ci ↶ d, η) + +30 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD += ˆ∆i ai +η(c ↶ d, d) += +� +˜∆H ⊗ id +� +◦ ˆ∆i ai +η (c, d, d) += (id ⊗ m) ◦ +� +˜∆H ⊗ id +� +◦ ˆ∆i ai +η (c, d) . +Proposition 8.3 asserts that the computation of coproduct ∆H on the module W (sub- +sequently on the algebra H) can be carried out post the computation of the operator ˜∆H +on W. The computation of the coproduct ∆H for the some coordinate maps are given as +follows: +∆Hai +∅ = ai +∅ ⊗ ai +∅. +∆Hai +x0 = +� +ai +x0 ⊗ ai +∅ +� ++ +� +ai +∅ ⊗ ai +x0 +� +. +∆Haj +xi = +� +aj +xi ⊗ ai +∅aj +∅ +� ++ +� +aj +∅ ⊗ aj +xi +� +. +∆Hai +x2 +0 = +� +ai +x2 +0 ⊗ ai +∅ +� ++ 2 +� +ai +x0 ⊗ ai +x0 +� ++ +� +ai +∅ ⊗ ai +x2 +0 +� +. +∆Haj +x0xi = +� +aj +x0xi ⊗ aj +∅ +� ++ +� +aj +x0 ⊗ aj +xi +� ++ +� +aj +xi ⊗ ai +∅aj +x0 +� ++ +� +aj +∅ ⊗ aj +x0xi +� +. +∆Haj +xix0 = +� +aj +xix0 ⊗ ai +∅aj +∅ +� ++ +� +aj +xi ⊗ ai +x0aj +∅ +� ++ +� +aj +xi ⊗ ai +∅aj +x0 +� ++ +� +aj +x0 ⊗ aj +xi +� ++ +� +aj +∅ ⊗ aj +xix0 +� +. +∆Hak +xixj = +� +ak +xixj ⊗ aj +∅ai +∅ak +∅ +� ++ +� +ak +xi ⊗ ai +xjak +∅ +� ++ +� +ak +xi ⊗ ai +∅ak +xj +� ++ +� +ak +xj ⊗ aj +∅ak +xi +� ++ +� +ak +∅ ⊗ ak +xixj +� +. +If m = 2 (two input-two output MIMO case) viz. X = {x0, x1, x2}, then from above +computations +∆Hax1x2 = + + +� +a1 +x1x2 ⊗ +� +a1 +∅ +�2 a2 +∅ +� ++ +� +a1 +x1 ⊗ a1 +x2a1 +∅ +� ++ +� +a1 +x1 ⊗ a1 +∅a1 +x2 +� ++ +� +a1 +x2 ⊗ a2 +∅a1 +x1 +� ++ +� +a1 +∅ ⊗ a1 +x1x2 +� +� +a2 +x1x2 ⊗ a1 +∅ +� +a2 +∅ +�2� ++ +� +a2 +x1 ⊗ a1 +x2a2 +∅ +� ++ +� +a2 +x1 ⊗ a1 +∅a2 +x2 +� ++ +� +a2 +x2 ⊗ a2 +∅a2 +x1 +� ++ +� +a2 +∅ ⊗ a2 +x1x2 +� + + +which can be rewritten as +∆Hax1x2 = +� +ax1x2 ⊗ (a1 +∅a2 +∅ ll)a∅ +� ++ +� +ax1 ⊗ (a1 +x2 ll)a∅ +� ++ +� +ax1 ⊗ (a1 +∅ ll)ax2 +� ++ +� +ax2 ⊗ (a2 +∅ ll)ax1 +� ++ (a∅ ⊗ ax1x2) , +where ll = [1 1]t. It is vital to observe that the term +� +ax1x2 ⊗ (a1 +∅a2 +∅ ll)a∅ +� +is a primitive +term of the coproduct as a1 +∅a2 +∅ ll ∼= ll since a∅ is the unit of H. +The following corollary is resultant of the Proposition 8.2 to the words of the form xn +0 for +all n ≥ 0. +Corollary 8.1. If n ∈ N0, then for all i = 1, 2, . . . , m (defining x0 +0 := ∅): +˜∆Hai +xn +0 = ai +xn +0 ⊗ ai +∅. +∆Hai +xn +0 = +n +� +k=0 +�n +k +� +ai +xk +0 ⊗ ai +∅ai +xn−k +0 +. +Proof: The proof is by induction on n ∈ N0. The base case (n = 0) : +˜∆Hai +∅ = ai +∅ ⊗ ai +∅, + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +31 +is proved in Proposition 8.1. Assume the statement is true for n = k, then +˜∆ai +xk+1 +0 += +� +˜∆ ◦ θ0 +� +ai +xk +0. +Using Proposition 8.1, +˜∆ai +xk+1 +0 += (θ0 ⊗ id) ◦ ˜∆ai +xk +0 += (θ0 ⊗ id) {ai +xk +0 ⊗ ai +∅} += ai +xk+1 +0 +⊗ ai +∅. +Hence proved by induction on n ∈ N0 that: ˜∆ai +xn +0 = ai +xn +0 ⊗1. Observe that from Proposition ?? +∆ai +xn +0 = (id ⊗ m) ◦ +� +˜∆ ⊗ id +� +◦ ∆i ai +xn +0 . +Using Corollary 4.1, +∆ai +xn +0 = (id ⊗ m) ◦ +� +˜∆ ⊗ id +� � n +� +k=0 +�n +k +� +ai +xk +0 ⊗ ai +xn−k +0 +� += (id ⊗ m) +� n +� +k=0 +�n +k +� +˜∆ai +xk +0 ⊗ ai +xn−k +0 +� += (id ⊗ m) +� n +� +k=0 +�n +k +� +ai +xk +0 ⊗ ai +∅ ⊗ ai +xn−k +0 +� += +n +� +k=0 +�n +k +� +ai +xk +0 ⊗ ai +∅ai +xn−k +0 +. +Proposition 8.3 asserted that the calculation of coproduct ∆H is carried out post the +computation of ˜∆H. However the converse is also true viz. the computation of ˜∆H can be +carried out if the evaluation fo the coproduct ∆H is known a priori which is well-asserted in +the following proposition. +Proposition 8.4. For all η ∈ X+ and for all i = 1, 2, . . . , m, +˜∆Hai +η (c, d) = (id ⊗ m) ◦ (∆H ⊗ S +) ◦ ˆ∆i ai +η. +Proof: Given c, d ∈ M, by Theorem (12) +(c ⋆ d) = (d +(c ↶ d)) . +Observe that d ∈ M implies that d is shuffle invertible. Thus for any η ∈ X+, +((c ↶ d)i , η) = +� +d +−1 +i +(c ⋆ d)i , η +� +, +for all i = 1, 2, . . . , m. Hence, +((c ↶ d)i , η) = ˜∆Hai +η (c, d) += ˆ∆i ai +η +� +c ⋆ d, d +−1� +. += (∆H ⊗ S +) ◦ ˆ∆i ai +η (c, d, d) . += (id ⊗ m) ◦ (∆H ⊗ S +) ◦ ˆ∆i ai +η (c, d) . + +32 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +The key point of the Proposition 8.4 is the shuffle-invertibility of a series c ∈ M +The goal of this subsection is to provide a graded structure on the R-module W and +consequently on the underlying R-module structure of the Hopf algebra H such that H is +connected and the homogeneous components of H are finite-dimensional. +Definition 8.1. Given a word η ∈ X+, denote the degree of the word as deg (η) and define +deg (η) = |η| and for all k ≥ 1: +Xk := {aη : deg (η) = k}. +(1) Define gradation on the R-module W viz. +W = +� +k≥1 +Wk, +where Wk is the free R-module spanned by Xk. +(2) The gradation on the module W induces a graded structure on the algebra H as +H = +� +n∈N0 +Hn, +with H0 ∼= R in the category of R-modules. +The following lemma aids in proving that the gradation in Definition 8.1 makes the Hopf +algebra H is well-defined. +Lemma 8.2. If η ∈ X∗ such that deg (η) = n then +˜∆H (aη) ∈ +� +i+j=n +Wi ⊗ Hj, +for all k = 1, 2, . . . , m. +Proof: The following observations will help in proving the lemma. +(1) The map {θi}m +i=0 is a homogeneous operator of degree 1 on the module W. If deg (η) = +|η| = n for some η ∈ X∗, then |xiη| = n + 1 for all i = 0, 1, . . . , m. Hence, +θi : Wn −→ Wn+1 +for all i = 0, 1, . . . , m and n ≥ 1. +(2) Observe that if η, ζ, γ ∈ X∗ such that |γ| = n and γ ∈ supp (η +ζ) then |γ| = n = +|ζ|+|η|. Thus, the reduced coproduct ˆ∆ +: W −→ W ⊗W is homogeneous operator +of degree 0 viz. +ˆ∆ +: Wn −→ (W ⊗ W)n . +Let us prove the statement ot the lemma by induction on degree (equivalently length) n +of the word η ∈ X∗. The base case is n = 0 ⇔ η = ∅. From Proposition 8.2, +˜∆Ha∅ = a∅ ⊗ a∅ +∈ W0 ⊗ H0, +Thus the statement holds true for the base case. Assume that the statement of theorem +holds true for all η ∈ X∗ such that deg (η) ≤ k. Let η′ such that deg (η′) = k + 1. Then two +cases can occur. +(1) Let η′ = x0η where |η| = k. Then +˜∆Haη′ = ˜∆H ◦ θ0aj +η. + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +33 +By Proposition 8.2, +˜∆Haη′ = (θ0 ⊗ id) ◦ ˜∆Haη. +Since aη ∈ Wk, then by the induction hypothesis ˜∆H (aη) ⊆ � +i+j=k Wi ⊗ Hj. Then, +(θ0 ⊗ id) +� � +i+j=k +Wi ⊗ Hj +� +⊆ +� +i+j=k +Wi+1 ⊗ Hj +⊆ +� +i+j=k+1 +Wi ⊗ Hj. +Thus, ˜∆Haη′ ∈ � +i+j=k+1 Wi ⊗ Hj where |η′| = k + 1. +(2) Let η′ = xiη where |η| = k and xi ̸= x0. Then from Proposition 8.2, +� +˜∆H ◦ πj +� +aη′ = (θi ⊗ m) ◦ +� +˜∆H ⊗ id +� +◦ (πj ⊗ πi) ◦ ˜∆ +aη. += (θi ⊗ m) ◦ +� +( ˜∆H ◦ πj) ⊗ πi +� +◦ ˜∆ +aη +Thus, +˜∆Haη′ = (θi ⊗ m) ◦ +� +˜∆H ⊗ ll.πi +� +˜∆ +aη, +where ll.πi = [πi πi · · · πi]t. Since deg(η) = k, +˜∆ +aη ⊆ (W ⊗ W)k. +Note that ll.πi(aη)(c) = [ai +η ai +η · · · ai +η](c) = aη[ci ci · · · ci]. Thus ll.πiaη ∈ W and then +applying the induction hypothesis ˜∆HWn ⊆ (W ⊗ H)n for n ≤ k, +� +˜∆H ⊗ ll.πi +� +(W ⊗ W)k ⊆ (W ⊗ H ⊗ W)k . +Finally, +(θi ⊗ m) (W ⊗ H ⊗ W)k ⊆ (W ⊗ H)k+1, +as θi is homogeneous operator of degree 1. Thus, ˜∆Haη′ ∈ � +i+j=k+1 Wi ⊗ Hj where +|η′| = k + 1. +Hence proved by induction that for all n ≥ 0: ˜∆H (aη) ∈ � +i+j=n Wi ⊗ Hj where |η| = n. +The following proposition asserts that the grading on H in Definition 8.1 is compatible +with bialgebraic structure of H. +Proposition 8.5. With the grading on the Hopf algebra H as in Definition 8.1, +∆H (Hn) ⊆ +� +i+j=n +Hi ⊗ Hj +for all n ≥ 0. +Proof: Observe that the statement is true for n = 0. Prior to the proving the statement for +n ≥ 1, the following statement needs to be proved: +∆H (Wn) ⊆ +� +i+j=n +Wi ⊗ Hj +∀ n ≥ 0. + +34 +VENKATESH G. S. AND KURUSCH EBRAHIMI-FARD +Observe that from Proposition 8.2, +∆H ◦ πi = (id ⊗ m) ◦ +� +˜∆H ⊗ id +� +◦ (πi ⊗ πi) ◦ ˜∆ +. +Thus (by grouping them along the coordinate i), +∆H = (id ⊗ m) ◦ +� +˜∆ ◦ id +� +◦ ˜∆ +. +Hence, +∆H(Wn) = (id ⊗ m) ◦ +� +˜∆ ◦ id +� +◦ ˜∆ +(Wn) +⊆ (id ⊗ m) ◦ +� +˜∆ ◦ id +� +(W ⊗ W)n. +Using Proposition 8.2, +∆H(Wn) ⊆ (id ⊗ m) (W ⊗ H ⊗ W)n +⊆ (W ⊗ H)n. +Therefore, the intermediate statement holds true viz. +∆H (Wn) ⊆ +� +i+j=n +Wi ⊗ Hj +∀ n ≥ 0. +The statement of the theorem then holds true as ∆ is an Rn-algebra morphism from H to +H ⊗ H. +Thus Proposition 8.5 asserts that the grading defined on the Hopf algebra H in Defini- +tion 8.1 is well-defined and connected. The homogeneous components are finite-dimensional +and dimensions respect the Proposition 4.2 since the bialgebras H and H +are isomorphic +with respect to the underlying graded algebraic structures. +The following example is rework of the Example 4.10 in [Gray & Ebrahimi-Fard(2017)] +acting as a check for the computation of feedback group inverse in one-dimensional case. +Example 8.2. Let c = 1−x1 ∈ R⟨⟨X⟩⟩. The series c◦−1 = 1+· · ·+· · · . Using the recursive +computation formula for antipode as in Theorem 3.1 +ax1(c◦−1) = Sax1 (c) = −ax1(c) = 1. +Observe that +∆′ +Hax2 +1 = 3ax1 ⊗ ax1. +Thus, +ax2 +1 +� +c◦−1� += Sax2 +1 (c) += −ax2 +1 − 3ax1.Sax1 = −ax2 +1 + 3a2 +x1. +Therefore, ax2 +1 (c◦−1) = 0 + 3(1)2 = 3. In similar fashion the reduced coproduct of a3 +x1 is +∆′ +Hax3 +1 = 4ax1 ⊗ ax2 +1 + 6ax2 +1 ⊗ ax1 + 3ax1 ⊗ a2 +x1. +Thus, +ax3 +1 +� +c◦−1� += +� +−ax3 +1 − 4ax1.Sax2 +1 − 6ax2 +1.Sax1 − 3ax1. (Sax1)2� +(c) += 0 − 4(−1)(3) − 6(0)(−1) − 3(−1)(1)2 = 15. +Therefore c◦−1 = 1 + x1 + 3x2 +1 + 15x3 +1 + 105x4 +1 + · · · . +The result matches exactly with that of Example 4.10 in [Gray & Ebrahimi-Fard(2017)]. + +FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION +35 +9. Conclusions and Future work +It was shown that the closed-loop system of a plant in Chen–Fliess series description +in multiplicative output feedback with another system, given by Chen–Fliess series, has a +Chen–Fliess series representation. An explicit expression of the closed-loop generating series +was derived and the multiplicative dynamic feedback connection has a natural interpretation +as a transformation group acting on the plant. A computational framework has been devised +utilizing the dual Hopf algebras corresponding to the shuffle group and multiplicative output +dynamic feedback group. 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M., On Analytic Nonlinear Input-output Systems: Expanded +Global Convergence and System Interconnections, Ph.D. Dissertation, Old Dominion University, 2019. +[Winter-Arboleda, et al.(2015)] Winter-Arboleda, I. M., Gray, W. S. and Duffaut Espinosa, L. A., Frac- +tional Fliess Operators: Two Approaches, Proc. 49th Conference on Information Sciences and Systems, +Baltimore, MD, 2015, pp. 1–6 +Department of Mathematical Sciences, Norwegian University of Science and Technology +(NTNU), 7491 Trondheim, Norway +Email address: subbarao.v.guggilam@ntnu.no +Department of Mathematical Sciences, Norwegian University of Science and Technology +(NTNU), 7491 Trondheim, Norway +Email address: kurusch.ebrahimi-fard@ntnu.no +URL: https://folk.ntnu.no/kurusche/ + diff --git a/VtE4T4oBgHgl3EQfMgzk/content/tmp_files/load_file.txt b/VtE4T4oBgHgl3EQfMgzk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..745835021b3b9bf204f86f424a355cdc98049d34 --- /dev/null +++ b/VtE4T4oBgHgl3EQfMgzk/content/tmp_files/load_file.txt @@ -0,0 +1,1512 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf,len=1511 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='04949v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='OC] 12 Jan 2023 A FORMAL POWER SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The goal of the paper is multi-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The first of which is to derive an explicit formula to compute the generating series of a closed-loop system when a plant, given in a Chen–Fliess series description is in multiplicative output feedback connection with another system given in Chen–Fliess series description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Further, the objective extends in showing that the multiplicative dynamic output feedback connection has a natural interpretation as a transformation group acting on the plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A computational framework for computing the generating series for multiplicative dynamic output feedback is devised utilizing the dual Hopf algebras corresponding to the shuffle group and the multiplicative feedback group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Introduction 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Preliminaries: Formal Power Series 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Shuffle Product 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Bialgebra and Hopf algebra: Preliminaries 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Algebra 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Coalgebra 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Bialgebra 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hopf Algebra 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Unshuffle Hopf algebra and its Coaction 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Unshuffle Hopf Algebra 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Gradation of Bialgebra H 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Coaction of H 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Chen–Fliess Series and its Interconnections 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Chen–Fliess Series 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Interconnections of Chen–Fliess Series: Parallel and Cascade Connections 14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Cascading of Chen–Fliess with Multiplicative Feedforward of Input 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Multiplicative Dynamic Output Feedback Group 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Chen–Fliess Series Under Multiplicative Dynamic Output Feedback 18 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Invariance of Class and Relative Degree under multiplicative dynamic feedback connection 20 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Computational Framework for Multiplicative Mixed Composition & Dynamic Feedback Product 24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hopf Algebra Corresponding to the Multiplicative Dynamic Feedback Subgroup 24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Coaction of Hopf algebra H on Algebra of Coordinate Map 25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Coaction of Hopf algbera H on the Hopf algebra H 27 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Coproduct, Antipode Computations and Grading of Hopf algebra H 29 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Conclusions and Future work 35 References 35 2 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Introduction The objective of the document is two fold and works with the Chen–Fliess functional series [Fliess(1981)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' There is no need that these input-output systems have a state space realization and thus, the results presented here are independent of any state space embed- ding when a realization is possible [Fliess(1983)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Firstly, let Fc and Fd be two nonlinear input-output systems represented by Chen–Fliess series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' It was shown in [Gray & Li(2005)] that the additive feedback interconnection of two such systems result in a Chen–Fliess se- ries description for the closed-loop system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The convergence of the closed-loop system was characterized in [Thitsa & Gray(2012)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' An efficient computation of the generating series for closed-loop system is facilitated through a combinatorial Hopf algebra [Gray, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2014a), Foissy(2015), Duffaut Espinosa, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='(2016)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The feedback product formula and its com- putation were used to solve system inversion problems [Gray, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2014b)] and trajectory generation problems [Duffaut Espinosa & Gray(2017)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' However, when the nature of interconnection becomes multiplicative feedback, the similar set of questions persist in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' It is known that, in single-input single-output (SISO) setting, the closed-loop system in the affine feedback case (of which multiplicative feedback is a special case) has a Chen–Fliess series description and the computation of feedback for- mula is facilitated through a combinatorial Hopf algebra [Gray & Ebrahimi-Fard(2017)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The present document, in one part, shows that even in multi-input multi-output (MIMO) setting the closed-loop system under multiplicative feedback has a Chen–Fliess series representation and provides an explicit expression of the closed-loop generating series which will be called as multiplicative dynamic feedback product .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Furthermore, it will be shown that this feedback product has a natural interpretation as a transformation group acting on the plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The algorithmic framework for the computation of the multiplicative dynamic feedback product formula for a general MIMO case is devised using the dual Hopf algebras corresponding to the shuffle product and to the multiplicative dynamic output feedback group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The charac- terization of convergence of the Chen–Fliess series for the closed-loop system is deferred for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The next section provides a summary of the concepts related to non-commutative formal power series, Hopf algebra, Chen–Fliess series and their interconnections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4 builds the pivotal multiplicative dynamic output feedback group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The Hopf algebra construction corresponding to the shuffle group is drafted in Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Section 6 is where the multiplicative dynamic feedback connection is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The invariance of relative degree under multiplicative output feedback is asserted in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The framework for computing the feedback product is devised in Section 8 and is demon- strated using examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The conclusions of the paper and directions for future work is given in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Preliminaries: Formal Power Series A finite nonempty set of noncommuting symbols X = {x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , xm} is called an alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Each element of X is called a letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Any finite sequence, η = xi1 · · · xik, of letters from X is called a word over X and its length is |η| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The set X∗ of all words includes the empty word, denoted ∅ ∈ X∗ and X+ := X∗\\∅, and forms a monoid under catenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Any mapping c : X∗ → Rℓ is called a formal power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The value of c at η ∈ X∗ is denoted by (c, η) and called the coefficient of η in c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Normally, c is written as a formal sum c = � η∈X∗(c, η)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A series c is proper when the coefficient (c, ∅) = 0 else it is a non-proper series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The support of c is the set supp(c) containing all words having nonzero coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The order of c, denoted ord(c), is the length of the minimal length word in its support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 3 collection of all formal power series over X is denoted by Rℓ⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The ith component of a vector v ∈ Rℓ is denoted by vi and consequently the ith component of a series c ∈ Rℓ⟨⟨X⟩⟩ is denoted by ci viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (ci, η) = (c, η)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A series c′ ∈ Rℓ⟨⟨X⟩⟩ is called a subseries of c ∈ Rℓ⟨⟨X⟩⟩ if there exists another series c′′ ∈ Rℓ⟨⟨X⟩⟩ such that the intersection supp (c′) ∩ supp (c′′) is empty and the series c can be decomposed as c = c′ + c′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let c ∈ Rℓ⟨⟨X⟩⟩, then the natural part of the series c is the subseries denoted by cN such that c = cN + cF and supp (cF) ⊆ X∗ \\ {xk 0 : k ∈ N0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The subseries cF is called as forced part of the series c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 asserts that the forced part cF of a series c should not contain any word formed by the letter x0 alone, including the empty word ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' For the remainder of the docu- ment, Rℓ is given the structure of a unital commutative ring under Hadamard or pointwise product viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (xy)i = xiyi with ll = [1 1 · · ·1]t ∈ Rℓ being the multiplicative unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Formal power series Rℓ⟨⟨X⟩⟩ form a Rℓ-module and the submodule of all proper series in Rℓ⟨⟨X⟩⟩ is denoted by Rℓ p ⟨⟨X⟩⟩, while the subset of non-proper series is denoted by Rℓ np ⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A series c ∈ Rℓ⟨⟨X⟩⟩ is called purely improper if ci is non-proper ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The subset of all purely improper series in Rℓ⟨⟨X⟩⟩ is denoted by Rℓ pi ⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that Rℓ pi ⟨⟨X⟩⟩ ⊊ Rℓ np ⟨⟨X⟩⟩ if ℓ > 1, otherwise Rpi ⟨⟨X⟩⟩ = Rnp ⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Shuffle Product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The shuffle product α β of two words is a bilinear product on the linear span of words, which can be uniquely specified iteratively (xiη) (xjξ) := xi(η (xjξ)) + xj((xiη) ξ), where η, ξ ∈ X∗ and xi, xj ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' See for instance [Fliess(1981)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The shuffle product of two series, (c, d) �→ c d is defined as (c d, η) = � ζ,ν∈X∗ η∈supp(ζ ν) (c, ζ) (d, ν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' We define for any xi, xj ∈ X and any word η ∈ X∗ x−1 i (xjη) := �η, i = j 0, else The following proposition is vital in understanding the bialgebra and Hopf algebra devised in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If c, d ∈ Rℓ⟨⟨X⟩⟩, then ∀xi ∈ X x−1 i (c d) = � x−1 i (c) d � + � c x−1 i (d) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Note that Rℓ⟨⟨X⟩⟩ forms an associative and commutative Rℓ-algebra under the shuffle product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If d ∈ Rℓ pi ⟨⟨X⟩⟩, then shuffle inverse of d, denoted by d −1 is defined as d −1 i = (di, ∅)−1 �� k∈N0 (d′ i) k � , where d′ i = 1 − (di/ (di, ∅)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence, Rℓ pi ⟨⟨X⟩⟩ forms an Abelian group under the shuffle product with ll as the identity element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 4 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let X = {x0, x1} and c ∈ R⟨⟨X⟩⟩ described as c = 1 − x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Then the shuffle inverse is computed as: c −1 = � k∈N0 (1 − (1 − x1)) k = � k∈N0 x k 1 = � k∈N0 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='xk 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore, c −1 = 1 + x1 + 2x2 1 + 6x3 1 + · · · + n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='xn 1 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that (c d, ∅) = (c, ∅) (d, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence, the set M = { ll + c : c ∈ Rn p ⟨⟨X⟩⟩}, where c is a proper series in Rn⟨⟨X⟩⟩, forms a subgroup of the shuffle group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The group M is vital in the design of a computational framework of multiplicative dynamic feedback product as explained in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The set Rℓ⟨⟨X⟩⟩ is endowed with ultrametric structure where the metric κ is defined as κ(c, d) = σord(c−d), for c, d ∈ Rℓ⟨⟨X⟩⟩ and σ ∈]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' For brevity, κ(c, 0) is written as κ(c), and κ(c, d) = κ(c−d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The ultrametric space (Rℓ⟨⟨X⟩⟩, κ) is Cauchy complete [Berstel & Reutenauer(1988)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following definition of contraction maps between metric spaces will be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Given metric spaces (E, d) and (E′, d′), a map f : E −→ E′ is said to be a strong contraction map if ∀s, t ∈ E, it satisfies the condition d′(f(s), f(t)) ≤ αd(s, t) where α ∈ [0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If α = 1, then the map f is said to be a weak contraction map or a non-expansive map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Bialgebra and Hopf algebra: Preliminaries The goal is to provide the definitions of algebraic structures such as algebra, coalgebra, bialgebra and Hopf algebra [Abe(2004), Sweedler(1969)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' We let K be a commutative ring with identity 1K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The definition of an algebra can be facilitated through the category of mod- ules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' It allows to define the concept of a coalgebra (the dual notion) with ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' An algebra over K is a K-module A along with the morphisms of K- modules m : A ⊗ A −→ A , called the multiplication or product map, and η : K −→ A , called the unit map, such that the following diagrams are commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (1) A ⊗ A ⊗ A m⊗idA � idA ⊗m � A ⊗ A m � A ⊗ A m � A K ⊗ A η⊗idA � ∼ = �▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ A ⊗ A m � A A ⊗ K ∼ = �r r r r r r r r r r r r r r r r r r r idA ⊗η � A ⊗ A m � FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 5 The tuple (A , m, η) is called a K-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The commutative diagrams (1) mean that a K-algebra A must satisfy the following prop- erties: (1) The product map m must be associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) The scalar multiplication through the η map must have a unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The concept of a K-algebra morphism is defined next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let (A , m, η), (A ′, m′, η′) be K-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A map f : A −→ A ′ is called a K-algebra morphism provided the following diagrams commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A ⊗ A m � f⊗f � A f � A ′ ⊗ A ′ m′ � A ′ K η � η′ �❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ A f �①①①①①①①①①①①①①①①① A ′ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let P and Q be modules over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The twisting morphism τ of K-modules is τ : P ⊗ Q −→ Q ⊗ P with τ(p ⊗ q) = q ⊗ p ∀ q ∈ Q, p ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A K-algebra A is commutative if and only if the following diagram commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A ⊗ A τ � m �■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ A ⊗ A m � A A K-algebra A is a graded algebra if the underlying K-module structure is graded viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A = � n∈N0 An, where An is a K-module for all n ∈ N0 such that m (Am ⊗ An) ⊆ Am+n, for all m, n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The graded K-algebra is connected if η : K −→ A0 is a K-algebra isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Coalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The notion of a K-coalgebra is a categorical structure dual to that of a K-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A K-coalgebra C is a K-module with the K-module morphisms ∆ : C −→ C ⊗ C , called the comultiplication or coproduct map, and ǫ : C −→ K, called the counit map, such that the following diagrams commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) C ∆ � ∆ � C ⊗ C ∆⊗idC � C ⊗ C idC ⊗∆ � C ⊗ C ⊗ C C ⊗ C ǫ⊗idC � K ⊗ C ∼ = � C ∆ �❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ ∆ �sssssssssssssssssss C ⊗ C idC ⊗ǫ � C ⊗ K ∼ = � 6 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD The tuple (C , ∆, ǫ) is called a K-coalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The commutative diagrams (2) imply that a K-coalgebra C must satisfy the following properties: (1) The coproduct map ∆ must be coassociative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) The counit map ǫ is the categorical dual to the unit map η for a K-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The coalgebra C is called cocommutative if the following diagram commutes, C ∆ � ∆ �❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ C ⊗ C τ � C ⊗ C where τ is the twisting morphism given in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Sweedler’s notation is very useful in representing the coproduct map and is adopted in Sections 4 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Sweedler(1969)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Given the K-coalgebra tuple (C , ∇, ǫ) and an element c ∈ C , then the Sweedler notation for the coproduct ∆(c) = � (c) c(1) ⊗ c(2), where c(1), c(2) ∈ C are the components of the tensors resulting from the coproduct of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Next, the definition of a K-coalgebra morphism is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let (C , ∆, ǫ), (C ′, ∆′, ǫ′) be K-coalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A map f : C −→ C ′ is called a K-coalgebra morphism provided the following diagrams commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' C ∆ � f � C ⊗ C f⊗f � C ′ ∆′ � C ′ ⊗ C ′ C ǫ � f �❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ K C ′ ǫ′ �② ② ② ② ② ② ② ② ② ② ② ② ② ② ② ② 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The bialgebra structure over a commutative ring is fundamental for defining a Hopf algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A bialgebra is an amalgamation of the algebra and coalgebra structures such that both are compatible with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A bialgebra H over K is a tuple (H, m, η, ∆, ǫ) such that (1) H is a K-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) (H, m, η) is a K-algebra, where m and η are the product and unit maps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (3) (H, ∆, ǫ) is a K-coalgebra, where ∆ and ǫ are the coproduct and counit maps, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' such that the following diagrams commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='(3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='H ⊗ H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='∆⊗∆ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='② ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='② ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='idK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='� K ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='The diagrams (3) and (4) state that the product map m and the unit map η are K- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='coalgebra morphisms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' while the coproduct map ∆ and the counit map ǫ are K-algebra morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Diagram (5) describes that the unit map η is a section of the counit map ǫ in the category of K-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hopf Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hopf algebras are an important class of bialgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A Hopf algebra is a bialgebra equipped with a particular K-linear map called antipode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A Hopf algebra H over K is a tuple (H, m, η, ∆, ǫ, S) such that the following conditions are satisfied: (1) (H, m, η, ∆, ǫ) is a K-bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) S : H −→ H is a K-linear map such that the following diagram commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (6) H ⊗ H idH⊗S � H ⊗ H m �❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ H ǫ � ∆ �✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ∆ �❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ K η � H H ⊗ H S⊗idH � H ⊗ H m �✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ An element a ∈ H is called group-like if ∆(a) = a ⊗ a and thus a̸∈ker(ǫ), where ker(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=') represents the kernel of a K-module map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A graded Hopf algebra H = � n∈N0 Hn is connected if and only if H0 ∼= Kη(1K) as K-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Equivalently, a graded Hopf algebra H is connected if and only if H+ := � k≥1 Hk is isomorphic to ker(ǫ) as K-modules viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' η◦ǫ = idH0 and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' For simplicity denote m (a, b) := ab, for all a, b, ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using Sweedler’s 8 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD notation, diagram (6) implies that for all c ∈ H, � (c) S � c(1) � c(2) = � (c) c(1)S � c(2) � = ǫ (c) 1H , where 1H is the multiplicative unit of the Hopf algebra H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The computation of the antipode of an element c becomes easier when the algebra structure of H is graded and connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If the Hopf algebra H is graded and connected, then the antipode can be computed for any a ∈ H+ := � k≥1 Hk as S(a) = −a − � a′ (1)S(a′ (2)), where the summation is taken over all components of the reduced coproduct ∆′ defined as: ∆′ (a) := ∆ (a) − a ⊗ η (1K) − η (1K) ⊗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Unshuffle Hopf algebra and its Coaction The goal of this section is to explain and illustrate the computational framework to compute the shuffle product of two series and the shuffle inverse using the coordinate maps of the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The framework is well-developed in the literature [Foissy(2015)] and was utilized in study of interconnections of Chen–Fliess series [Venkatesh & Gray(2022), Venkatesh & Gray(2021), Venkatesh & Gray (2020), Gray, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2014b), Gray, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2014a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Unshuffle Hopf Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' We construct a dual Hopf algebra reflecting the group structure of M as defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The antipode constructed in the Hopf algebra provides a framework for computing the shuffle inverse of purely improper series c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let the set Wb ⊂ Rm⟨⟨X⟩⟩∗ (dual module of Rm⟨⟨X⟩⟩) be defined as the collection of coordinate maps: Wb = {aη : aη(c) = (c, η), η ∈ X∗, c ∈ Rm⟨⟨X⟩⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Define W to be the free Rm-module spanned by the set Wb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let H denote the reduced symmetric algebra generated by the module W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The Rm-algebra H can equivalently be seen as the polynomial algebra of coordinate maps (corresponding to non-empty words) of Rm⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The unit map ξ : Rm −→ H is defined by ξ( ll) = a∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that a∅ : c �→ ll, for all c ∈ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' By construction, H is an Rm-associative, commutative and unital algebra with addition and scalar multiplication defined, respectively, as (aη + aζ)(c) = aη(c) + aζ(c) (kaη)(c) = k(aη(c)), where c ∈ Rm⟨⟨X⟩⟩ and k ∈ Rm, and product m(aη, aζ)(c) = aη(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='aζ(c), for c ∈ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Then H is equipped with a coproduct ˆ∆ : H −→ H � H such that ˆ∆ aη(c, d) = (c d, η), for all c, d ∈ M and η ∈ X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The counit map ǫ : H −→ Rm is defined as ǫ(h) = � ll : h = a∅ 0 : otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Since the shuffle product is associative and commutative, thus dually the coproduct ˆ∆ is coassociative and cocommutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore, (H , m, ξ, ˆ∆ , ǫ) forms a Rm-bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 9 following lemma is vital in the framework for computing both shuffle product and dynamic feedback group product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Define a collection of linear endomorphisms {θi}m i=0 on W θi : W −→ W aη �−→ axiη, for all xi ∈ X, η ∈ X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus θi (aη) (c) = aη � x−1 i (c) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The coproduct ˆ∆ can be recursively constructed as defined in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Foissy(2015)] On the module W ˆ∆ θk = (θk ⊗ id + id ⊗ θk) ◦ ˆ∆ , for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m and k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m with base case being ˆ∆ a∅ = a∅ ⊗ a∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 infers that the maps θi, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m, are coderivations on the underlying coalgebra of H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' We note that the unshuffle coproduct ˆ∆ was utilized in the design of an algorithmic framework for computation of Wiener-Fliess composition product and subsequently additive static feedback product [Venkatesh & Gray(2021), Venkatesh & Gray(2022), Venkatesh(2021)] and also in the computation of shuffle-rational series from its representation [Venkatesh & Gray (2020), Venkatesh(2021)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Moreover, the unshuffle coproduct was also crucial in the computational framework for the multivariate additive output feedback [Gray, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2014a), Gray, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2014b)] and for SISO affine output feedback [Gray & Ebrahimi-Fard(2017)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let {πi}m i=1 be the collection of co-ordinate projection maps on the module W defined as ai η(c) := πi(aη)(c) = (c, η)i = (ci, η), for all η ∈ X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus, define the following notation ˆ∆j ai η := (πi ⊗ πj) ◦ ˆ∆ aη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Note that the projection maps {πi}m i=1 commute with the maps {θj}m j=0 viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' θi � aj η � = aj xiη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The significance of these notations are well-reflected in the computational framework in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following example is to demonstrate the result of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 for a few words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A few examples of the computation of deshuffle coproduct ˆ∆ on W (akin to Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3) using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 are given as follows(indices i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m and k, s = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m): ˆ∆j ai xk = ai xk ⊗ aj ∅ + ai ∅ ⊗ aj xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ˆ∆j ai xkxk = ai xkxk ⊗ aj ∅ + 2ai xk ⊗ aj xk + ai ∅ ⊗ aj xkxk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ˆ∆j ai xkxs = ai xkxs ⊗ aj ∅ + ai xk ⊗ aj xs + ai xs ⊗ aj xk + ai ∅ ⊗ aj xkxs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The connected Rm-bialgebra H is endowed with an antipode map S given as: S : H −→ H aη �→ S aη such that S aη (c) = (c −1, η), for η ∈ X∗, c ∈ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 10 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Gradation of Bialgebra H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The Hopf algebra H can be equipped with a grading such that it is connected and all its homogeneous components are finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Given η ∈ X+, define the degree of aη as deg (aη) = |η|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (1) Define gradation on the Rm-module W viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' W = � k≥1 Wk, where Wk is the free Rm-module spanned by the aη of deg (aη) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) The gradation on the module W induces a graded structure on the algebra H as H = � n∈N0 ˆHn, with ˆH0 ∼= Rm in the category of Rm-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following proposition asserts that the above gradation is connected and all its homo- geneous components are finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Given the gradation for the Hopf algebra H , (1) H is a graded and connected Hopf algebra viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ˆ∆ � ˆHn � ⊆ � i+j=n i,j≥0 ˆHi ⊗ ˆHj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) For all k: define wk = dim (Wk) and FW = � k≥1 wkZk is the geometric series given by FW = 1 1 − mZ , where m = |X| and for all k ≥ 1: wk = dim (Wk) = mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (3) Define F ˆ H = � n≥1 hnZn where hn = dim( ˆHn) then F ˆ H = ∞ � k=1 1 (1 − Zk)wk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: (1) The Hopf algebra H follows from the fact that if γ(̸= η, ζ) ∈ supp(η ζ) then deg (γ) = |γ| = |η| + |ζ| = deg (η) + deg (ζ) , for all η, ζ, γ ∈ X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) Define the formal power series F(Z0, Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , Zm) = � k≥1 � i0,i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=',im≥0 i0+i1+···+im=k #{η : |η|xj = ij ∀ j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m}Zi0 0 Zi1 1 · · · Zim m = (Z0 + Z1 + · · · + Zm) 1 − (Z0 + Z1 + · · · + Zm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 11 Since each letter contributes equally to the degree (viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' length), thus FW = F(Z, Z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=', Z) = mZ 1 − mZ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (3) The proposition follows from the item 2 as ˆH is the symmetric algebra generated by the Rm-module W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Dimensions of the homogeneous components of module W and H (when m = 2) k 0 1 2 3 4 5 6 7 8 9 10 dim (Wk) 1 2 4 8 16 32 64 128 256 512 1024 dim( ˆHk) 1 2 7 20 59 162 449 1200 3194 8348 21646 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The dimensions of the homogeneous components of the graded module W (up to k = 10) and the graded algebra H for m = 2 viz when X = {x0, x1} is tabulated in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The sequence {dim( ˆHk)}k∈N0 is the sequence A034899 in [OEIS(2022)] which corresponds to the number of multisets of binary words of total length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Coaction of H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The subsection explains the coaction of the Hopf algebra H (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1) on the algebra of coordinate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' It is utilized subsequently to develop an algorithm to compute the multiplicative mixed composition product explained in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2 and dynamic feedback product as defined in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let W to be the Rm-module as described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let S+ (W) denote the reduced symmetric algebra generated by the module W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The non-unital Rm-algebra S+(W) are equivalently the polynomials without constant term of coordinate maps of Rm⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' By construction S+(W) has a non-unital Rm-associative, commutative algebra structure with addition, scalar multiplication and product defined, respectively, as (aη + aζ)(c) = aη(c) + aζ(c) (kaη)(c) = k(aη(c)) where c ∈ Rm⟨⟨X⟩⟩, and m(aη, aζ)(c) = aη(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='aζ(c), where c ∈ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The Rm-algebra S+(W) is isomorphic to the algebra structure of H with forgetting of the unit map ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The right coaction map ρ : S+ (W) −→ S+ (W) ⊗ H is recursively defined on the module V as given by the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' For all i = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m : ρ θi = (θi ⊗ id + id ⊗ θi) ◦ ρ , with base case being ρ a∅ = a∅ ⊗ a∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3 might appear as repetition of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' It is vital to note that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 is for defining the coproduct of Hopf algebra H , where a∅ is the unit element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that, ρ ai η(c, d) = ai η(c d), 12 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD where c ∈ R⟨⟨X⟩⟩ (not necessarily in M ) and d ∈ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The coaction ρ thus is a corepresentation of the Hopf algebra H on the algebra S+ (W) or equivalently, ρ makes S+ (W), a H algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let {πi}m i=1 be the collection of co-ordinate projection maps on the module W defined as ai η(c) := πi(aη)(c) = (c, η)i = (ci, η), for all η ∈ X∗ and thus the following notation is well-defined, ρj ai η := (πi ⊗ πj) ◦ ρ aη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' These notations are very much utilized in developing computational framework for the multiplicative mixed composition product as discussed in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If n ∈ N0, then for all i = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=', m and j, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m (defining x0 j := ∅): ρj ak xin = n � r=0 �n r � ak xir ⊗ aj xin−r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: The statement is proved by induction on n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The base case (n = 0) follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Assume the statement is true for n = p − 1, then ρj ak xip = ρj ◦ θiak xip−1 = (θi ⊗ id + id ⊗ θi) ◦ ∆j ak xip−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using the induction hypothesis, ρj ak xip = (θi ⊗ id + id ⊗ θi) �p−1 � r=0 �p − 1 r � ak xir ⊗ aj xip−1−r � = p � r=1 �p − 1 r − 1 � ak xir ⊗ aj xip−r + p−1 � r=0 �p − 1 r � ak xir ⊗ aj xip−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' = p � r=0 �n r � ak xir ⊗ aj xip−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Since the S+ (W) and H are isomorphic as Rm-modules, the following lemma states the coaction of H on S+ (W) and the unshuffle coproduct coincide when the evaluation of coordinate maps are restricted to the group M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Given c, d ∈ M , η ∈ X∗ and i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m, ˆ∆ aη (c, d) = (c d, η) = ρ aη (c, d) , where c, d ∈ M and ˆ∆i is the coproduct from the bialgebra H constructed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A few examples of the computation of the coaction map ρ on W using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3 are given as follows(indices i, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m and k, s = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m): ∆j ai ∅ = ai ∅ ⊗ aj ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ∆j ai xk = ai xi ⊗ aj ∅ + ai ∅ ⊗ aj xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ∆j ai xkxk = ai xkxk ⊗ aj ∅ + 2ai xk ⊗ aj xk + ai ∅ ⊗ aj xkxk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 13 ∆j ai xkxs = ai xkxs ⊗ aj ∅ + ai xk ⊗ aj xs + ai xs ⊗ aj xk + ai ∅ ⊗ aj xkxs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following example illustrates the application of the deshuffle coproduct ∆ in the computation of the shuffle product of two series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let X = {x0, x1} and c, d ∈ R2⟨⟨X⟩⟩ described as c = � 1 + x1 + x2 1 + x3 1 + · · · x0 + x0x1 + x100 1 � & d = � 1 + x2 0 + exp (x1) 1 + x2 0x1 � , where exp(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=') is the standard exponential function expressed in its Taylor series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Note that c ̸∈ M but d ∈ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The coefficient of x0x2 1 in series c2 d1 can be computed as: � c2 d1, x0x2 1 � = ∆1 a2 x0x2 1 (c, d) = (π2 ⊗ π1) ◦ ∆ ax0x2 1 (c, d) = ∆1 ◦ θ0ax2 1 (c, d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3, � c2 d1, x0x2 1 � = (θ0 ⊗ id + id ⊗ θ0) ◦ ∆1 a2 x2 1 (c, d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1, � c2 d1, x0x2 1 � = (θ0 ⊗ id + id ⊗ θ0) ◦ � a2 x12 ⊗ a1 ∅ + 2a2 x1 ⊗ a1 x1 + a2 ∅ ⊗ a1 x12 � (c, d) = � a2 x0x12 ⊗ a1 ∅ + 2a2 x0x1 ⊗ a1 x1 + a2 x0 ⊗ a1 x12 + a2 x12 ⊗ a1 x0+ 2a2 x1 ⊗ a1 x0x1 + a2 ∅ ⊗ a1 x0x12 � (c, d) = (0)(1) + 2(1)(1) + (1)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='5) + (0)(0) + 2(0)(0) + (0)(0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore (c2 d1, x0x2 1) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Chen–Fliess Series and its Interconnections The objective of the section is to describe Chen–Fliess series and the necessary non- recursive interconnections of Chen–Fliess series to understand the results about the multi- plicative dynamic feedback product in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Chen–Fliess Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let p ≥ 1 and t0 < t1 be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' For a Lebesgue measurable function u : [t0, t1] → Rm, define ∥u∥p = max{∥ui∥p : 1 ≤ i ≤ m}, where ∥ui∥p is the usual Lp-norm for a measurable real-valued function, ui, defined on [t0, t1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let Lm p [t0, t1] denote the set of all measurable functions defined on [t0, t1] having a finite ∥ · ∥p norm and Bm p (R)[t0, t1] := {u ∈ Lm p [t0, t1] : ∥u∥p ≤ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Given any series c ∈ Rℓ⟨⟨X⟩⟩, the corresponding Chen–Fliess series is (7) Fc[u](t) = � η∈X∗ (c, η) Fη[u](t, t0), where E∅[u] = 1 and Fxi¯η[u](t, t0) = � t t0 ui(τ)F¯η[u](τ, t0) dτ with xi ∈ X, ¯η ∈ X∗, and u0 = 1 [Fliess(1981)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If there exist constants K, M > 0 such that |(ci, η)| ≤ KM|η||η|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=', ∀η ∈ X∗, ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , ℓ , (8) 14 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD then Fc constitutes a well-defined mapping from Bm p (R)[t0, t0 + T] into Bℓ q(S)[t0, t0 + T] for sufficiently small R, T > 0, where the numbers p, q ∈ [1, ∞] are conjugate exponents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=', 1/p + 1/q = 1 [Gray & Wang(2002)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' This map is referred to as a Fliess operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A series c ∈ Rℓ⟨⟨X⟩⟩ obeying the growth condition in (8) is called a locally convergent generating series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The set of all locally convergent generating series is denoted by Rℓ LC⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The supremum of the set of all max{R, T} for which a Fliess operator Fc is a well-defined mapping from Bm p (R)[t0, t0 + T] into Bℓ q(S)[t0, t0 + T] is called the radius of convergence of the Fliess operator Fc and is denoted by ρ (Fc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A Fliess operator Fc is called locally convergent if ρ (Fc) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If there exist constants K, M > 0 and γ ∈ [0, 1[ such that |(ci, η)| ≤ KM|η| (|η|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' )γ , ∀η ∈ X∗, ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , ℓ , (9) then Fc constitutes a well defined mapping from Bm p (R)[t0, t0 + T] into Bℓ q(S)[t0, t0 + T] for all R, T > 0 [Winter-Arboleda(2019), Winter-Arboleda, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='(2015)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The infimum of all the γ ∈ [0, 1[ such that (9) is satisfied for a series c ∈ Rℓ⟨⟨X⟩⟩ is called the Gevrey order of the series c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A series c ∈ Rℓ⟨⟨X⟩⟩ obeying the growth condition in (9) is called a globally convergent series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The set of all globally convergent series in Rℓ⟨⟨X⟩⟩ is denoted as Rℓ GC⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A Fliess operator Fc is globally convergent if and only if there exists no real number M > 0 such that ρ (Fc) < M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that a noncommutative polynomial R⟨X⟩ is a globally convergent series with Gevrey degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' As described above, a series c ∈ Rℓ GC⟨⟨X⟩⟩ is only a sufficient condition for the corresponding Fliess operator Fc to be globally convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Necessary conditions are well-detailed in the literature [Winter-Arboleda(2019), Venkatesh(2021)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' In the absence of any convergence criterion, (7) only defines an operator in a formal sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Interconnections of Chen–Fliess Series: Parallel and Cascade Connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Given Chen–Fliess series Fc and Fd, where c, d ∈ Rℓ⟨⟨X⟩⟩, the parallel and product connec- tions satisfy Fc + Fd = Fc+d and FcFd = Fc d, respectively [Ree(1958), Fliess(1981)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The parallel and product connections preserve local convergence and hence the interconnected systems has a Fliess operator representation [Thitsa & Gray(2012), Venkatesh(2021)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' When Chen–Fliess series Fc and Fd with c ∈ Rk⟨⟨X′⟩⟩ and d ∈ Rℓ⟨⟨X⟩⟩ are interconnected in a cascade fashion, where |X′| = ℓ + 1, the composite system Fc ◦ Fd has a Chen–Fliess series representation Fc◦d, where the composition product of c and d is given by (10) c ◦ d = � η∈X′∗ (c, η) ψd(η)(1) [Ferfera(1979), Ferfera(1980)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Here 1 denotes the monomial 1∅, and ψd is the continuous (in the ultrametric sense) algebra homomorphism from R⟨⟨X′⟩⟩ to the set of vector space endomorphisms on R⟨⟨X⟩⟩, End (R⟨⟨X⟩⟩), uniquely specified by ψd(x′ iη) = ψd(x′ i) ◦ ψd(η) with ψd(x′ i)(e) = x0(di e), i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m for any e ∈ R⟨⟨X⟩⟩, and where di is the i-th component series of d (d0 := 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' By definition, ψd(∅) is the identity map on R⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The cascade interconnection preserves local convergence and thus the composite has a Fliess operator representation [Thitsa & Gray(2012)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The linearity of the composition product in the left argument is evident form the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' However, the following theorem states that the composition product distributes over the shuffle product from the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Gray & Li(2005)] Let c, d ∈ Rk⟨⟨X′⟩⟩ and e ∈ Rℓ⟨⟨X⟩⟩, such that |X′| = ℓ + 1, then (c d) ◦ e = (c ◦ e) (d ◦ e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 15 Given a series e ∈ Rℓ⟨⟨X⟩⟩, define a map Υe : Rk⟨⟨X′⟩⟩ −→ Rk⟨⟨X⟩⟩ defined as c �→ c ◦ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 infers that Υe is an R-algebra homomorphism from the shuffle algebra of Rk⟨⟨X′⟩⟩ to the shuffle algebra of Rℓ⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The composition product preserves the purely improper property of the left argument which is stated in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If c ∈ Rk⟨⟨X′⟩⟩ and d ∈ Rℓ⟨⟨X⟩⟩ such that |X′| = ℓ + 1, then (c ◦ d, ∅) = (c, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence, if c ∈ Rk pi ⟨⟨X′⟩⟩ then c ◦ d ∈ Rk pi ⟨⟨X⟩⟩ and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Similarly if c is a proper series then c ◦ d is also a proper series and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: The proof follows immediately from (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The composition product is a strong contraction map with respect to its right argument in the ultrametric topology and is stated in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Gray & Li(2005)] Let c ∈ Rk⟨⟨X′⟩⟩ and d, e ∈ Rℓ⟨⟨X⟩⟩, such that |X′| = ℓ + 1, then κ (c ◦ d, c ◦ e) ≤ σκ (d, e) where σ ∈ [0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Cascading of Chen–Fliess with Multiplicative Feedforward of Input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The cas- cade interconnection of a Chen–Fliess series Fc and Fd along with the multiplicative feed- forward of the input, as shown in Figure 1, arises primarily in the analysis of multiplicative feedback interconnection discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A semblance of such an interconnection has appeared in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 of [Gray & Ebrahimi-Fard(2017)], without being explicit and limited to the SISO case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' With respect to Figure 1, the map u �→ y viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' y = Fc[u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='Fd[u]] has Chen–Fliess series representation denoted by Fc↶d, where c ↶ d denotes the multiplicative mixed composition product of c ∈ Rp⟨⟨X⟩⟩ and d ∈ Rm⟨⟨X⟩⟩ defined as c ↶ d = � η∈X∗ (c, η) η ↶ d := � η∈X∗ (c, η) ¯φd (η) (1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (11) Here, ¯φd : R⟨⟨X⟩⟩ −→ End (R⟨⟨X⟩⟩) is an R-algebra homomorphism such that ¯φd(x0)(e) = x0e and ¯φd(xi)(e) = xi(di e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Recall that R⟨⟨X⟩⟩ is an R-algebra under Cauchy product and End (R⟨⟨X⟩⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The multi- plicative mixed composition defined in (11) asserts that, for all η ∈ X∗ and d ∈ Rm⟨⟨X⟩⟩, ∅ ↶ d = ∅ x0η ↶ d = x0 (η ↶ d) xiη ↶ d = xi (di (η ↶ d)) ∀ i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' For later reference, we summarise the properties of (11) in the following Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The multiplicative mixed composition product (11) is linear in its left argu- ment and (c ↶ d, ∅) = (c, ∅), for all c ∈ Rp⟨⟨X⟩⟩ and d ∈ Rm⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following results are already known in the single-input single-output (SISO) setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' However, their multi-input multi-output (MIMO) extensions are straightforward and to avoid reiteration of the proofs, only the statements are provided in this document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The foremost of the theorems asserts that the multiplicative mixed composition product distributes over shuffle product from the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Gray & Ebrahimi-Fard(2017)] Let c, d ∈ Rp⟨⟨X⟩⟩ and e ∈ Rm⟨⟨X⟩⟩, then (c d) ↶ e = (c ↶ e) (d ↶ e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 16 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD Fd Fc u y Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Cascade connection of Chen–Fliess Fd with Fc along with multi- plicative feedforward of input The inference of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='5 is that for any e ∈ Rm⟨⟨X⟩⟩, the map Γe : Rp⟨⟨X⟩⟩ −→ Rp⟨⟨X⟩⟩ given by d �→ d ↶ e is an R-algebra endomorphism on the shuffle algebra Rp⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The next lemma is essential in proving that multiplicative mixed composition product is a strong contraction map in its right argument in the ultrametric topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Gray & Ebrahimi-Fard(2017)] Let η ∈ X∗ and d, e ∈ Rm⟨⟨X⟩⟩, then κ (η ↶ d, η ↶ e) ≤ σ|η|κ (d, e) where σ ∈ [0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following theorem states the strong contraction property of the multiplicative mixed composition product which is an essential result in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Gray & Ebrahimi-Fard(2017)] Let d, e ∈ Rm⟨⟨X⟩⟩ and c ∈ Rp⟨⟨X⟩⟩, then κ (c ↶ d, c ↶ e) ≤ σord(c′)κ (d, e), where c′ = c − (c, ∅), the proper part of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Since ord (c′) ≥ 1 and σ ∈]0, 1[, then from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='6, the map ¯Γc : e �→ c ↶ e is a strong contraction map in the ultrametric topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following lemma is essential in proving the mixed associativity of the composition and multiplicative mixed composition product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The result, along with Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='7 can be inferred in the SISO setting from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='6 in [Gray & Ebrahimi-Fard(2017)], and its extension to the MIMO case is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Gray & Ebrahimi-Fard(2017)] Let X′ = {x′ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , x′ p} and η ∈ X′∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let d ∈ Rp⟨⟨X⟩⟩ and e ∈ Rm⟨⟨X⟩⟩, then η ◦ (d ↶ e) = (η ◦ d) ↶ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following theorem states that the composition product and multiplicative mixed com- position product are associative in combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Gray & Ebrahimi-Fard(2017)] Let X′ = {x′ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , x′ p} and c ∈ Rq⟨⟨X′⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let d ∈ Rp⟨⟨X⟩⟩ and e ∈ Rm⟨⟨X⟩⟩, then c ◦ (d ↶ e) = (c ◦ d) ↶ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Multiplicative Dynamic Output Feedback Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The dynamic multiplicative feedback group plays a vital role in computation of the multiplicative dynamic feedback formula, as well as in assessing the feedback as a group action in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Indeed, consider the cascade interconnection of two Chen–Fliess series Fc and Fd along with their multiplica- tive feedforward of inputs displayed in Figure 2, where c, d ∈ Rm⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The input-output relation of the composite system, u �→ y is u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='Fd[u]Fc[u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='Fd[u]] and can be represented by Chen–Fliess series as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Consider u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='Fc⋆d[u] := u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='Fd[u]Fc[u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='Fd[u]], where the multiplicative composition product of c and d is defined as c ⋆ d = d (c ↶ d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (12) The following theorems appeared in [Gray & Ebrahimi-Fard(2017)] in the SISO setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' We underline that the latter restriction is not essential, that is, the statements along with the proofs naturally extend to the MIMO setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 17 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Cascade connection of Chen–Fliess Fd with Fc along with multi- plicative feedforward of their inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Gray & Ebrahimi-Fard(2017)] Let c, d, e ∈ Rm⟨⟨X⟩⟩, then, (c ⋆ d) ⋆ e = c ⋆ (d ⋆ e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that (12) and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='8 infer that Rm⟨⟨X⟩⟩ forms a non-commutative monoid under multiplicative composition product, with the identity element ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following theo- rem states that the multiplicative mixed composition product is a right action on Rq⟨⟨X⟩⟩ by the monoid (Rm⟨⟨X⟩⟩, ⋆, ll).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Gray & Ebrahimi-Fard(2017)] Let c ∈ Rq⟨⟨X⟩⟩ and d, e ∈ Rm⟨⟨X⟩⟩, then (c ↶ d) ↶ e = c ↶ (d ⋆ e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The prominent question is to find the invertible elements of the monoid (Rm⟨⟨X⟩⟩, ⋆) and the motivation to find the unit elements of the monoid shall be evident in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let d, e ∈ Rm pi ⟨⟨X⟩⟩ and suppose d ⋆ e = ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that d ∈ Rm pi ⟨⟨X⟩⟩ implies (d ↶ e) ∈ Rm pi ⟨⟨X⟩⟩ and using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='5, e = (d ↶ e) −1 = d −1 ↶ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence, for e to be right inverse of d, the purely improper series e has to satisfy the fixed point equation e = d −1 ↶ e (13) Observe from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='6 that the map e �→ d −1 ↶ e is a strong contraction in the ultrametric space inferring that (13) has a unique fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Suppose e is the left inverse of d viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' e ⋆ d, then a similar procedure shows that e has to satisfy the equation d = e −1 ↶ d (14) Note that if e is a solution of (13), then e satisfies (14) and also the converse holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence, e is the unique inverse of d and is given the notation d⋆−1 for d ∈ Rm pi ⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus, Rm pi ⟨⟨X⟩⟩ forms a group under multiplicative composition product, ⋆, and is termed as the multiplicative dynamic output feedback group and is formally stated in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' � Rm pi ⟨⟨X⟩⟩, ⋆ � forms a group with the identity element ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' It is worth noting that [Gray & Ebrahimi-Fard(2017)] proved Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='10 for one- dimensional case viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' In light of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='10, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='5 and (12) one obtains the following relations for c ∈ Rm pi ⟨⟨X⟩⟩: c⋆−1 = c −1 ↶ c⋆−1 (15) � c⋆−1� −1 = c ↶ c⋆−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following lemma is essential in defining a subgroup of the multiplicative dynamic out- put feedback group upon which the computational framework for the multiplicative feedback products is discussed in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' F F n18 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let c, d ∈ Rm pi ⟨⟨X⟩⟩, then (c ⋆ d, ∅) = (c, ∅) (d, ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: Observe from (12) that, (c ⋆ d, ∅) = (d (c ↶ d) , ∅) = (c ↶ d, ∅) (d, ∅) Since (c ↶ d, ∅) = (c, ∅), (c ⋆ d, ∅) = (c, ∅) (d, ∅) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3 thus proves that the set of all series which are of the form ll + c, where c is a proper series, forms a subgroup of the multiplicative dynamic feedback group, which is stated in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let M = { ll + c : c ∈ Rm p ⟨⟨X⟩⟩}, then (M, ⋆, ll) forms a subgroup of the multiplicative dynamic feedback group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The algorithmic framework for the computation of multiplicative feedback products is fundamentally based on the subgroup M as asserted in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The group M is isomorphic to the character group of the Hopf algebra H which is used for computation of feedback and the framework is explained in detail in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Chen–Fliess Series Under Multiplicative Dynamic Output Feedback Let Fc be a Chen–Fliess series with a generating series c ∈ Rq⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Assume it is intercon- nected with a Chen–Fliess series Fd with a purely improper generating series d ∈ Rm pi ⟨⟨X′⟩⟩, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Note that, |X| = m + 1 and |X′| = q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The primary goal of this section is to show that the closed-loop system has a Chen–Fliess series representation, say y = Fe[v], where e ∈ Rq⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If this is the case, then necessarily y = Fe[v] = Fc[u] = Fc[vFd[y]] = Fc[vFd[Fe[v]]] = Fc[vFd◦e[v]] = Fc↶(d◦e)[v] for any admissible input v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore, the series e has to satisfy the fixed point equation e = c ↶ (d ◦ e) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (16) Observe that, in light of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='6 the map e �→ c ↶ (d ◦ e) is a strong contraction map in the ultrametric space and thus (16) has a unique fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following thoerem establishes the first main result of this section, which follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The series c ↶ (d −1 ◦ c)⋆−1 ∈ Rq⟨⟨X⟩⟩ is the unique fixed point of the map e �→ c ↶ (d ◦ e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: If e := c ↶ (d −1 ◦ c)⋆−1, then c ↶ (d ◦ e) = c ↶ � d ◦ � c ↶ � d −1 ◦ c �⋆−1�� Using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='7 and then Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='5, c ↶ (d ◦ e) = c ↶ � (d ◦ c) ↶ � d −1 ◦ c �⋆−1� FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 19 Fc v Fd y u Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Chen–Fliess series Fc in multiplicative output feedback with Chen- Flies series Fd = c ↶ � (d ◦ c) −1 ↶ � d −1 ◦ c �⋆−1� −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1, c ↶ (d ◦ e) = c ↶ �� d −1 ◦ c � ↶ � d −1 ◦ c �⋆−1� −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using the relations (15), c ↶ (d ◦ e) = c ↶ ��� d −1 ◦ c �⋆−1� −1� −1 = c ↶ � d −1 ◦ c �⋆−1 = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Given a series c ∈ Rq⟨⟨X⟩⟩ and a purely improper series d ∈ Rm pi ⟨⟨X′⟩⟩ (such that |X| = m + 1 and |X′| = q + 1), then the generating series for the closed-loop system in Figure 3 is given by the multiplicative dynamic feedback product cˇ@d := c ↶ (d −1 ◦ c)⋆−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The notion that feedback can described mathematically as a transformation group acting on the plant is well established in control theory [Brockett(1978)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following theorem describes the situation in the present context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The multiplicative dynamic feedback product is a right group action by the multiplicative group � Rm pi ⟨⟨X′⟩⟩, , ll � on the set Rq⟨⟨X⟩⟩, where |X| = m + 1 and |X′| = q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: Let c ∈ Rq⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2, cˇ@ ll = c ↶ � ll −1 ◦ c �⋆−1 = c ↶ ll = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let d1, d2 ∈ Rm pi ⟨⟨X′⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' It needs to be proven that � cˇ@d1 � ˇ@d2 = cˇ@ (d1 d2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' From Theo- rem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2, observe that � cˇ@d1 � ˇ@d2 = � cˇ@d1 � ↶ � d −1 2 � cˇ@d1 ��⋆−1 = � c ↶ � d −1 1 c �⋆−1� ↶ � d −1 2 � c ↶ � d −1 1 c �⋆−1��⋆−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Applying Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='7, � cˇ@d1 � ˇ@d2 = � c ↶ � d −1 1 c �⋆−1� ↶ �� d −1 2 c � ↶ � d −1 1 c �⋆−1�⋆−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 20 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD Applying Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='9 and fact that the group inverse is anti-homomorphism with respect to the group product, � cˇ@d1 � ˇ@d2 = c ↶ � � d −1 1 c �⋆−1 ⋆ �� d −1 2 c � ↶ � d −1 1 c �⋆−1�⋆−1 � = c ↶ � �� d −1 2 c � ↶ � d −1 1 c �⋆−1� ⋆ � d −1 1 c � �⋆−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Applying (12), � cˇ@d1 � ˇ@d2 = c ↶ � � d −1 1 c � �� � d −1 2 c � ↶ � d −1 1 c �⋆−1 � ↶ � d −1 1 c � ��⋆−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='9, � cˇ@d1 � ˇ@d2 = c ↶ � � d −1 1 c � � � d −1 2 c � ↶ �� d −1 1 c �⋆−1 ⋆ � d −1 1 c �� ��⋆−1 = c ↶ �� d −1 1 c � �� d −1 2 c � ↶ ll ��⋆−1 = c ↶ �� d −1 1 c � � d −1 2 c ��⋆−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' In light of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1, � cˇ@d1 � ˇ@d2 = c ↶ �� d −1 1 d −1 2 � c �⋆−1 = c ↶ � (d1 d2) −1 ◦ c �⋆−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore, � cˇ@d1 � ˇ@d2 = cˇ@ (d1 d2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' It is worth noting that for the additive dynamic feedback product the transformation group is the additive group (Rm⟨⟨X′⟩⟩, +, 0) while here (Rm pi ⟨⟨X′⟩⟩, , ll) plays the role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Invariance of Class and Relative Degree under multiplicative dynamic feedback connection The notion of relative degree of a plant is very essential and prime in the studies of feedback linearization [Isidori(1995)], flatness and system inversion etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The existence and quantification of relative degree of a interconnection of systems is vital in systems theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The notion of class and relative degree of a SISO Chen–Fliess series is equivalently char- acterized by the notion of relative degree of its generating series and the definition was furnished in [Gray, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2014b), Gray & Venkatesh(2019)] and the existence and quantifi- cation of relative degree of interconnected system of Chen–Fliess series was described in [Gray & Venkatesh(2019), Venkatesh(2021)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' In addition, this definition of relative degree is consistent with the classical definition whenever y = Fc[u] has an input-affine analytic state space realization [Gray, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2014b), Gray & Ebrahimi-Fard(2017)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let X = {x0, x1} and the following definition explains the concept of a class, a weaker notion than the relative degree of a series in R⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Gray & Venkatesh(2019)] A series c ∈ R⟨⟨X⟩⟩ is said to be of r-class, denoted by C (c) = r, if supp(cF) ⊆ xr−1 0 X+ and supp(cF) ⊈ xr 0X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' By definition, let C (c) = ∞ if cF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The notion of class is universal and is versed in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 21 Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Gray & Venkatesh(2019)] Every series c ∈ R⟨⟨X⟩⟩ has a class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 of class is illustrated in the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let c = 1 + x0x2 1 + x2 0x1, so that cF = x0x2 1 + x2 0x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that supp(cF) ⊆ x0X+ but supp(cF) ⊈ x2 0X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus, C (c) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following lemma is essential in the proof of quantification of class for the multiplicative mixed composition product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let c, c′, d ∈ Rm⟨⟨X⟩⟩ such that supp (c′) ̸⊆ x0X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Then the following state- ments are true: (1) xk 0 ↶ d = xk 0 ∀k ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) cN ↶ d = cN where cN is the natural part of the series c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (3) supp (c′ ↶ d) ̸⊆ x0X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: (1) The proof is by induction on k ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The base case being k = 0 is true viz ∅ ↶ d = ∅ from (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Assume the proposition is true for k = n − 1, then using (11) xn 0 ↶ d = x0 � xn−1 0 ↶ d � = x0 � xn−1 0 � = xn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence proved by induction on N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) Observe that from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1, supp (cN) ⊆ {xk 0 : k ∈ N0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus, using the previ- ous statement (1) and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4 it follows that cN ↶ d = cN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (3) Since supp (c′) ̸⊆ x0X∗, there exists a word xiη ∈ supp (c′) where xi ̸= x0 and η ∈ X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using (11), xiη ↶ d = xi (di (η ↶ d)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus, supp (xiη ↶ d) ⊆ xiX∗, where xi ̸= x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore, supp (c′ ↶ d) ̸⊆ x0X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following theorem quantifies that class is invariant under the multiplicative mixed composition product Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let c, d ∈ R⟨⟨X⟩⟩, then C (c ↶ d) = C (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: Suppose the series c ∈ R⟨⟨X⟩⟩ is of r-class, then the series c can be written as: c = cN + xr−1 0 c′, where c′ is a proper series such that supp (c′) ̸⊆ x0X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4, c ↶ d = (cN ↶ d) + � xr−1 0 c′ ↶ d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using (11), c ↶ d = (cN ↶ d) + xr−1 0 (c′ ↶ d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Since supp (c′) ̸⊆ x0X∗, then by applying Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2, c ↶ d = cN + xr−1 0 (c′ ↶ d) , 22 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD with supp (c′ ↶ d) ̸⊆ x0X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Given that c′ ∈ Rp ⟨⟨X⟩⟩, whence supp (c ↶ d)F ⊆ xr−1 0 X+ and supp (c ↶ d)F ̸⊆ xr 0X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore, C (c ↶ d) = r = C (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Consider the series c in Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1, given by c = 1 + x2 0x1 + x0x2 1 and d = 1 + x1 ∈ R⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using (11), the multiplicative mixed composition product of c and d is computed as: c ↶ d = 1 + x0x2 1 + 3x0x3 1 + 3x0x4 1 + x2 0x1 + x2 0x2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that C (c ↶ d) = 2 = C (c), as in Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following theorem asserts that class of a series is preserved under the multiplicative dynamic feedback product which is one of the prime goal of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If c ∈ R⟨⟨X⟩⟩ with C (c) = r, and d ∈ Rpi ⟨⟨X⟩⟩, then C � cˇ@d � = r = C (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: From Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2, cˇ@d = c ↶ � d −1 ◦ c �⋆−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Since C (c) = r, whence applying Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1, C � cˇ@d � = C � c ↶ � d −1 ◦ c �⋆−1� = r = C (c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The preservation of class under the multiplicative dynamic feedback connections as as- serted in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2 is further illustrated in the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let c, d ∈ R⟨⟨X⟩⟩ c = x1 and d = 1 + � k∈N k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='xk 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Note that the class of series C (c) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2 the multiplicative feedback product is computed as: cˇ@d = x1 + x1x0x1 + 3x1x0x1x0x1 + 4x1x2 0x2 1 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Infer from Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 that C � cˇ@d � = C (c) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Finally, the main definition of the section details the concept of relative degree in the context of Chen–Fliess series which is characterized on its generating series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Gray & Venkatesh(2019)] A series c ∈ R⟨⟨X⟩⟩ has relative degree r if C (c) = r and the word xr−1 0 x1 ∈ supp(cF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Otherwise, c does not have relative degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following theorem asserts the quantification of relative degree under multiplicative mixed composition product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If c ∈ R⟨⟨X⟩⟩ with relative degree rc and d ∈ R⟨⟨X⟩⟩ be non-proper, then c ↶ d has relative degree rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: From Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1, C (c ↶ d) = rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' It remains to prove that xrc−1 0 x1 ∈ supp (c ↶ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Given that c ∈ R⟨⟨X⟩⟩ has relative degree rc, then c can be decomposed as: c = cN + λxrc−1 0 x1 + xrc−1 0 c′, where λ ̸= 0 and c′ is a proper series such that x1 ̸∈ supp (c′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Then, c ↶ d = � cN + λxrc−1 0 x1 + xrc−1 0 c′� ↶ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Applying Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4, c ↶ d = (cN ↶ d) + λ � xrc−1 0 x1 ↶ d � + � xrc−1 0 c′ ↶ d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 23 Using (11) and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2, c ↶ d = cN + λxrc−1 0 x1d + xrc−1 0 (c′ ↶ d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Since d ∈ Rpi ⟨⟨X⟩⟩ −→ d = α + d′, where α ̸= 0 and d′ is a proper series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence, c ↶ d = cN + λαxrc−1 0 x1 + xrc−1 0 x1d′ + xrc−1 0 (c′ ↶ d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe from (11), x1 ̸∈ supp (c′) =⇒ x1 ̸∈ supp (c′ ↶ d) and also that αλ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore xrc−1 0 x1 ∈ supp (c ↶ d), whence the relative degree of c ↶ d is rc, when d is a non-proper series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following example illustrates the statement from Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let c = 1 + x2 0 + x0x1 + x2 0x1 and d = 1 + x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that by Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2, the relative degree of c is rc = 2 and also that d is non-proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The multiplicative mixed composition product of c and d to computed as: c ↶ d = 1 + x2 0 + x0x1 + x0x2 1 + x2 0x1 + x2 0x2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2, note that the relative degree of c ↶ d is 2 = rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following theorem is the prime objective of this section stating that the relative degree of a series remains invariant under multiplicative dynamic feedback product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If c ∈ R⟨⟨X⟩⟩ with relative degree rc and d ∈ Rpi ⟨⟨X⟩⟩, then the relative degree of � cˇ@d � is rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: Since c ∈ R⟨⟨X⟩⟩ and d ∈ Rpi ⟨⟨X⟩⟩, then by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2, cˇ@d = c ↶ � d −1 ◦ c �⋆−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that d ∈ Rpi ⟨⟨X⟩⟩ ⇔ d −1 ∈ Rpi ⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Then by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2 (d −1 ◦ c) ∈ Rpi ⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' As per Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='10, the group inverse � d −1 ◦ c �⋆−1 ∈ Rpi ⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3, cˇ@d = c ↶ � d −1 ◦ c �⋆−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' has relative degree rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The invariance of the relative degree of a Chen–Fliess series under multiplicative dynamic feedback connections as stated in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4 is illustrated through the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Consider the Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3 again where c = x1 and d = 1 + � k∈N k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='xk 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that by Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2, the relative degree of c is rc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The multiplicative feedback product is computed as: cˇ@d = x1 + x1x0x1 + 3x1x0x1x0x1 + 4x1x2 0x2 1 + · · · Infer that the relative degree of cˇ@d = 1 = rc as stated in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 24 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Computational Framework for Multiplicative Mixed Composition & Dynamic Feedback Product The goal of this section is to describe the computational framework for multiplicative dynamic feedback product as explained in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The section further illustrates the framework with examples but prior to that it is imperative to understand the dual bialgebra and Hopf algebra constructions corresponding to the multiplicative dynamic output feedback group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hopf Algebra Corresponding to the Multiplicative Dynamic Feedback Sub- group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The goal of the subsection is to construct a dual Hopf algebra reflecting the group structure of the multiplicative dynamic feedback subgroup M as asserted in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The group inverse is computed the antipode of the constructed Hopf algebra and thus pro- vides a computational framework to compute the multiplicative dynamic feedback group inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' As a recall, the group M is defined as M = { ll + d : d ∈ Rm p ⟨⟨X⟩⟩}, where ll = [1 · · ·1 1]T ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' In light of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='11, (M, ⋆) forms a subgroup of the multiplicative dynamic feedback group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The algebra structure is same as the algebra of H in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let the set Wb ⊂ Rm⟨⟨X⟩⟩∗ (dual module of Rm⟨⟨X⟩⟩) be defined as the collection of coordinate maps defined as: Wb = {aη : aη(c) = (c, η) : η ∈ X∗}, where c ∈ Rm⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Define W to be the free Rm-module spanned by the set Wb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let H denote the reduced symmetric algebra generated by the module W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The unit map ξ : Rm −→ W is defined by ξ( ll) = a∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Note that a∅ (c) = ll ∀c ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' By construction H is an Rm-associative, commutative and unital algebra with addition, scalar multiplication and product defined, respectively, as (aη + aζ)(c) = aη(c) + aζ(c) (kaη)(c) = k(ai η(c)) m(aη, aζ)(c) = aη(c)aζ(c), where c ∈ Rm⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Then H is given a coproduct ∆H : H −→ H � H such that for all c, d ∈ M: ∆Hai η(c, d) = ai η(c ⋆ d) = ((c ⋆ d)i , η) ∀η ∈ X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The counit map ǫ : H −→ R is defined as ǫ(h) = � ll : h = a∅ 0 : otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Since ◦ is associative (from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='8), thus by the dual the coproduct ∆H is coasso- ciative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore, (H, m, ξ, ∆H, ǫ) forms a Rm-bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Owing to the group structure of (M, ◦), the bialgebra H is equipped with antipode S defined as: Saη (c) = aη � c⋆−1� = � c⋆−1, η � , for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m and η ∈ X+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence H is a Rm-Hopf algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The computation of coproduct ∆H is well-understood through the right coaction of Hopf algebra H on the Hopf algebra H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Prior to that, it is imperative to understand the right coaction of Hopf algebra H on the non-unital algebra of coordinate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Coaction of Hopf algebra H on Algebra of Coordinate Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The subsection explains the coaction of the Hopf algebra H defined in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 on the algebra of coordinate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The results in this subsection are utilized subsequently to explain the coaction of H on the bialgebra H , particularly in proofs in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The right coaction of the Hopf algebra H is on Rm-algebra of coordinate maps S+ (W) constructed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The right coaction map ˜∆ : S+ (W) −→ S+ (W) � H is defined such that for all c ∈ Rm⟨⟨X⟩⟩, d ∈ M and η ∈ X∗, ˜∆aη (c, d) = (c ↶ d, η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (17) The map ˜∆ being a right coaction map is a reflection of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' It remains to show how the coaction map ˜∆ is computed on S+(W), for which it is sufficient to define its computation on the module W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that for all aη ∈ W, ˜∆aη = � ˜∆ ◦ π1 ˜∆ ◦ π2 · · · ˜∆ ◦ πm�t aη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' On the dual side, the above statement infers that for all c ∈ Rm⟨⟨X⟩⟩, d ∈ M and η ∈ X∗, (c ↶ d, η) = [((c ↶ d)1 , η) · · · ((c ↶ d)m , η)]t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence, the notation ˜∆ai η := ˜∆ ◦ πiaη for all η ∈ X∗ and i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following proposition provides a recursive definition to compute ˜∆ on the module V viz to compute the ˜∆ � aj η � ∀η ∈ X∗ and j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' For all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m: (1) ˜∆ai ∅ = ai ∅ ⊗ ai ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) ˜∆ ◦ θ0 = (θ0 ⊗ id) ◦ ˜∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (3) ˜∆ ◦ θi = (θi ⊗ m) ◦ � ˜∆ ⊗ id � ρi , where ρ is the coaction map of Hopf algebra H on S+ (W) as defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: Observe that ∀c ∈ Rm⟨⟨X⟩⟩ and d ∈ M, c = (c, ∅) + m � j=0 xj � x−1 j (c) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4, c ↶ d = (c, ∅) + x0 � x−1 0 (c) ↶ d � + m � j=1 xj � dj � x−1 j (c) ↶ d �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (18) The proof for each of the statement as follows: (1) Let c, d ∈ Rm⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' From (17) and (18), ˜∆ai ∅ (c, d) = ((c ↶ d)i , ∅) = (ci ↶ d, ∅) = (ci, ∅) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 = ai ∅ ⊗ ai ∅ (c, d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore, ˜∆ai ∅ = ai ∅ ⊗ ai ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 26 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD (2) Let c, d ∈ Rm⟨⟨X⟩⟩, η ∈ X∗ and ∀ j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Then, � ˜∆ ◦ θ0 � aj η (c, d) = � (c ↶ d)j , x0η � = � x−1 0 (c ↶ d)j , η � From (18), � ˜∆ ◦ θ0 � aj η (c, d) = � x−1 0 (cj) ↶ d, η � = ˜∆aj η � x−1 0 (c) , d � = (θ0 ⊗ id) ◦ ˜∆aη (c, d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore, ˜∆ ◦ θ0 = (θ0 ⊗ id) ◦ ˜∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (3) Let c, d ∈ Rm⟨⟨X⟩⟩ and η ∈ X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Then ∀ i, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m, � ˜∆ ◦ θi � aj η (c, d) = � (c ↶ d)j , xiη � = � x−1 i (c ↶ d)j , η � From (18), � ˜∆ ◦ θi � aj η (c, d) = � di x−1 i (cj) ↶ d, η � = ρi aj η � x−1 i (c) ↶ d, d � = ρi aj η � x−1 i (c) ↶ d � = ρi aj η � x−1 i (c) ↶ d, d � = � ˜∆ ⊗ id � ρi aj η � x−1 i (c) , d, d � = (θi ⊗ m) ◦ � ˜∆ ⊗ id � ρi aj η (c, d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore, ˜∆ ◦ θi = (θi ⊗ m) ◦ � ˜∆ ⊗ id � ρi ∀i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A few examples of the computation of ˜∆ on V using Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 are given as follows(indices i, j, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='): ˜∆ai ∅ = ai ∅ ⊗ ai ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ˜∆ai x0 = ai x0 ⊗ ai ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ˜∆aj xi = aj xi ⊗ ai ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ˜∆ai x2 0 = ai x2 0 ⊗ ai ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ˜∆aj x0xi = aj x0xi ⊗ ai ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ˜∆aj xix0 = � aj xix0 ⊗ ai ∅ � + � aj xi ⊗ ai x0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 27 ˜∆ak xixj = � ak xixj ⊗ aj ∅ai ∅ � + � ak xi ⊗ ai xj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The coaction map ˜∆ thus provides a framework to compute the multiplicative mixed com- position product and multiplicative dynamic feedback group product whenever c ∈ Rm⟨⟨X⟩⟩ and d ∈ M ⊊ Rm⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' For computing the multiplicative mixed composition product for c ∈ Rp⟨⟨X⟩⟩ and d ∈ M ⊊ Rm⟨⟨X⟩⟩ where p = m, (1) If p < m, then define ˇc ∈ Rm⟨⟨X⟩⟩ such that ˇci = ci ∀ i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , p and ˇci = 0 ∀ i = p + 1, p + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Then for all η ∈ X∗, ((c ↶ d)i , η) = ˜∆ai η (ˇc, d) ∀i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Note that (ˇc ↶ d)j = 0 ∀j = p + 1, p + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) If p > m, then this can be reduced to Case 1 by performing computations component wise viz computing ci ↶ d for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus the computational framework to compute the multiplicative mixed composition prod- uct of c ∈ Rp⟨⟨X⟩⟩ and d ∈ M, denoted by c ↶ d for arbitrary p and m is well-defined via the coaction map ˜∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The computations of the coproduct ∆H and antipode S (defined in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1) are well-understood once the right coaction of Hopf algebra H on Hopf algebra H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Coaction of Hopf algbera H on the Hopf algebra H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The objective of the subsection is to define the right coaction map of Hopf algebra H on the unshuffle Hopf algebra H defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The right coaction is pivotal in computation of the coproduct and antipode of Hopf algebra H which in turn are essential to compute the multiplicative dynamic feedback product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The right coaction map of H on H is defined to be ˜∆H : H −→ H � H such that for all c, d ∈ M (the underlying sets of M and M are identical) and η ∈ X∗, ˜∆Haη (c, d) = (c ↶ d, η) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (19) Observe that the algebra of coordinate functions S+(W) and H are isomorphic as Rm- modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus it is vital to understand the relationship between the operator ˜∆ operating on the module S+(W) and operator ˜∆H operating on H , which is stated in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If c, d ∈ M, then for all η ∈ X∗ ˜∆Haη (c, d) = ˜∆aη (c, d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: If c, d ∈ M and η ∈ X+, ˜∆Haη (c, d) = (c ↶ d, η) = ˜∆aη (c, d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Despite the statement of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1, it is vital to understand the difference between the coaction maps ˜∆ and ˜∆H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The coaction map ˜∆H is compatible with the Hopf algebra structure of H viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' m1,3,24 ◦ � ˜∆H ⊗ ˜∆H � ∆ = (∆ ⊗ id) ◦ ˜∆H, ˜∆H ◦ S = (S ⊗ id) ◦ ˜∆H, where m1,3,24 = (m ⊗ m) ◦ (id ⊗ τ ⊗ id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 28 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD Thus the coaction map ˜∆H makes H a comodule-Hopf algebra over H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Equivalently, the coaction map ˜∆H is a corepresentation of Hopf algebra H over unshuffle Hopf algebra H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Similar to Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2, for all aη ∈ W, ˜∆Haη = � ˜∆H ◦ π1 ˜∆H ◦ π2 · · · ˜∆H ◦ πm� aη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The map to compute the ˜∆H � aj η � ∀η ∈ X∗ and j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m is g module W is stated in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' For all i, j = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m and η ∈ X∗: (1) ˜∆Hai ∅ = ai ∅ ⊗ ai ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) ˜∆H ◦ θ0aj η = (θ0 ⊗ id) ◦ ˜∆Haj η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (3) � ˜∆H ◦ θi � aj η = (θi ⊗ m) ◦ � ˜∆H ⊗ id � ∆i aj η, where ∆ is the unshuffle coproduct defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: Observe that ∀c ∈ M, c = ll + m � j=0 xj � x−1 j (c) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4, c ↶ d = ll + x0 � x−1 0 (c) ↶ d � + m � j=1 xj � dj � x−1 j (c) ↶ d �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (20) The proof for each of the statement as follows: (1) Let c, d ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' From (19) and (20), ˜∆Hai ∅ (c, d) = ((c ↶ d)i , ∅) = (ci ↶ d, ∅) = 1 = (ci, ∅)(di, ∅) = ai ∅ ⊗ ai ∅(c, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore, ˜∆Hai ∅ = ai ∅ ⊗ ai ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) Let c, d ∈ M, η ∈ X∗ and ∀ j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Then, � ˜∆H ◦ θ0 � aj η (c, d) = � (c ↶ d)j , x0η � = � x−1 0 (c ↶ d)j , η � Observe that x−1 0 (c) may not belong to M and from (20), � ˜∆H ◦ θ0 � aj η (c, d) = � x−1 0 (cj) ↶ d, η � = ˜∆aj η � x−1 0 (c) , d � = (θ0 ⊗ id) ◦ ˜∆aη (c, d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 29 Since η ∈ X+ and c, d ∈ M, then by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 � ˜∆H ◦ θ0 � aj η (c, d) = (θ0 ⊗ id) ◦ ˜∆Haη (c, d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore, ˜∆H ◦ θ0 = (θ0 ⊗ id) ◦ ˜∆H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (3) Let c, d ∈ M and η ∈ X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Then ∀ i, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m, � ˜∆H ◦ θi � aj η (c, d) = � (c ↶ d)j , xiη � = � x−1 i (c ↶ d)j , η � From (20), � ˜∆H ◦ θi � aj η (c, d) = � di x−1 i (cj) ↶ d, η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Since x−1 i (c) may not belong to group M (also M ), = ρi aj η � x−1 i (c) ↶ d, d � = ρi aj η � x−1 i (c) ↶ d � = ρi aj η � x−1 i (c) ↶ d, d � = � ˜∆ ⊗ id � ρi aj η � x−1 i (c) , d, d � = (θi ⊗ m) ◦ � ˜∆ ⊗ id � ρi aj η (c, d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Since η ∈ X+ and c, d ∈ M, then by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1, � ˜∆H ◦ θi � aj η (c, d) = (θi ⊗ m) ◦ � ˜∆H ⊗ id � ∆i aj η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore, � ˜∆H ◦ θi � = (θi ⊗ m) ◦ � ˜∆H ⊗ id � ∆i for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Coproduct, Antipode Computations and Grading of Hopf algebra H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The objective of this subsection is to define and illustrate the computation of coproduct ∆H of the bialgebra H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Further, a graded and connected structure is endowed with the bialgebra owing to which the antipode computation is possible owing to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following proposition asserts the essential reason behind the definition of ˜∆H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' For all η ∈ X∗ and i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m, ∆Hai η = (id ⊗ m) ◦ � ˜∆H ⊗ id � ˆ∆i ai η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: Proof: Observe that for all c, d ∈ M and η ∈ X∗, ∆ai η (c, d) = ((c ⋆ d)i , η) ∀ i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using (12), ∆Hai η (c, d) = (di ci ↶ d, η) 30 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD = ˆ∆i ai η(c ↶ d, d) = � ˜∆H ⊗ id � ˆ∆i ai η (c, d, d) = (id ⊗ m) ◦ � ˜∆H ⊗ id � ˆ∆i ai η (c, d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3 asserts that the computation of coproduct ∆H on the module W (sub- sequently on the algebra H) can be carried out post the computation of the operator ˜∆H on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The computation of the coproduct ∆H for the some coordinate maps are given as follows: ∆Hai ∅ = ai ∅ ⊗ ai ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ∆Hai x0 = � ai x0 ⊗ ai ∅ � + � ai ∅ ⊗ ai x0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ∆Haj xi = � aj xi ⊗ ai ∅aj ∅ � + � aj ∅ ⊗ aj xi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ∆Hai x2 0 = � ai x2 0 ⊗ ai ∅ � + 2 � ai x0 ⊗ ai x0 � + � ai ∅ ⊗ ai x2 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ∆Haj x0xi = � aj x0xi ⊗ aj ∅ � + � aj x0 ⊗ aj xi � + � aj xi ⊗ ai ∅aj x0 � + � aj ∅ ⊗ aj x0xi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ∆Haj xix0 = � aj xix0 ⊗ ai ∅aj ∅ � + � aj xi ⊗ ai x0aj ∅ � + � aj xi ⊗ ai ∅aj x0 � + � aj x0 ⊗ aj xi � + � aj ∅ ⊗ aj xix0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ∆Hak xixj = � ak xixj ⊗ aj ∅ai ∅ak ∅ � + � ak xi ⊗ ai xjak ∅ � + � ak xi ⊗ ai ∅ak xj � + � ak xj ⊗ aj ∅ak xi � + � ak ∅ ⊗ ak xixj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If m = 2 (two input-two output MIMO case) viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' X = {x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' x2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' then from above ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='computations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='∆Hax1x2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='\uf8ee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='\uf8ef\uf8ef\uf8ef\uf8f0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='x1x2 ⊗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='� ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='∆Hax1x2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='ax1x2 ⊗ (a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='∅a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='∅ ll)a∅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='ax1 ⊗ (a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='x2 ll)a∅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='ax1 ⊗ (a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='∅ ll)ax2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='ax2 ⊗ (a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='∅ ll)ax1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='+ (a∅ ⊗ ax1x2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' where ll = [1 1]t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' It is vital to observe that the term � ax1x2 ⊗ (a1 ∅a2 ∅ ll)a∅ � is a primitive term of the coproduct as a1 ∅a2 ∅ ll ∼= ll since a∅ is the unit of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following corollary is resultant of the Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2 to the words of the form xn 0 for all n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If n ∈ N0, then for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m (defining x0 0 := ∅): ˜∆Hai xn 0 = ai xn 0 ⊗ ai ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ∆Hai xn 0 = n � k=0 �n k � ai xk 0 ⊗ ai ∅ai xn−k 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: The proof is by induction on n ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The base case (n = 0) : ˜∆Hai ∅ = ai ∅ ⊗ ai ∅, FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 31 is proved in Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Assume the statement is true for n = k, then ˜∆ai xk+1 0 = � ˜∆ ◦ θ0 � ai xk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1, ˜∆ai xk+1 0 = (θ0 ⊗ id) ◦ ˜∆ai xk 0 = (θ0 ⊗ id) {ai xk 0 ⊗ ai ∅} = ai xk+1 0 ⊗ ai ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence proved by induction on n ∈ N0 that: ˜∆ai xn 0 = ai xn 0 ⊗1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that from Proposition ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ∆ai xn 0 = (id ⊗ m) ◦ � ˜∆ ⊗ id � ∆i ai xn 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1, ∆ai xn 0 = (id ⊗ m) ◦ � ˜∆ ⊗ id � � n � k=0 �n k � ai xk 0 ⊗ ai xn−k 0 � = (id ⊗ m) � n � k=0 �n k � ˜∆ai xk 0 ⊗ ai xn−k 0 � = (id ⊗ m) � n � k=0 �n k � ai xk 0 ⊗ ai ∅ ⊗ ai xn−k 0 � = n � k=0 �n k � ai xk 0 ⊗ ai ∅ai xn−k 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='3 asserted that the calculation of coproduct ∆H is carried out post the computation of ˜∆H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' However the converse is also true viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' the computation of ˜∆H can be carried out if the evaluation fo the coproduct ∆H is known a priori which is well-asserted in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' For all η ∈ X+ and for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m, ˜∆Hai η (c, d) = (id ⊗ m) ◦ (∆H ⊗ S ) ◦ ˆ∆i ai η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: Given c, d ∈ M, by Theorem (12) (c ⋆ d) = (d (c ↶ d)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that d ∈ M implies that d is shuffle invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus for any η ∈ X+, ((c ↶ d)i , η) = � d −1 i (c ⋆ d)i , η � , for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence, ((c ↶ d)i , η) = ˜∆Hai η (c, d) = ˆ∆i ai η � c ⋆ d, d −1� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' = (∆H ⊗ S ) ◦ ˆ∆i ai η (c, d, d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' = (id ⊗ m) ◦ (∆H ⊗ S ) ◦ ˆ∆i ai η (c, d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 32 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD The key point of the Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='4 is the shuffle-invertibility of a series c ∈ M The goal of this subsection is to provide a graded structure on the R-module W and consequently on the underlying R-module structure of the Hopf algebra H such that H is connected and the homogeneous components of H are finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Given a word η ∈ X+, denote the degree of the word as deg (η) and define deg (η) = |η| and for all k ≥ 1: Xk := {aη : deg (η) = k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (1) Define gradation on the R-module W viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' W = � k≥1 Wk, where Wk is the free R-module spanned by Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) The gradation on the module W induces a graded structure on the algebra H as H = � n∈N0 Hn, with H0 ∼= R in the category of R-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following lemma aids in proving that the gradation in Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 makes the Hopf algebra H is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If η ∈ X∗ such that deg (η) = n then ˜∆H (aη) ∈ � i+j=n Wi ⊗ Hj, for all k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: The following observations will help in proving the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (1) The map {θi}m i=0 is a homogeneous operator of degree 1 on the module W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' If deg (η) = |η| = n for some η ∈ X∗, then |xiη| = n + 1 for all i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence, θi : Wn −→ Wn+1 for all i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' , m and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) Observe that if η, ζ, γ ∈ X∗ such that |γ| = n and γ ∈ supp (η ζ) then |γ| = n = |ζ|+|η|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus, the reduced coproduct ˆ∆ : W −→ W ⊗W is homogeneous operator of degree 0 viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ˆ∆ : Wn −→ (W ⊗ W)n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let us prove the statement ot the lemma by induction on degree (equivalently length) n of the word η ∈ X∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The base case is n = 0 ⇔ η = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' From Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2, ˜∆Ha∅ = a∅ ⊗ a∅ ∈ W0 ⊗ H0, Thus the statement holds true for the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Assume that the statement of theorem holds true for all η ∈ X∗ such that deg (η) ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let η′ such that deg (η′) = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Then two cases can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (1) Let η′ = x0η where |η| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Then ˜∆Haη′ = ˜∆H ◦ θ0aj η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 33 By Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2, ˜∆Haη′ = (θ0 ⊗ id) ◦ ˜∆Haη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Since aη ∈ Wk, then by the induction hypothesis ˜∆H (aη) ⊆ � i+j=k Wi ⊗ Hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Then, (θ0 ⊗ id) � � i+j=k Wi ⊗ Hj � ⊆ � i+j=k Wi+1 ⊗ Hj ⊆ � i+j=k+1 Wi ⊗ Hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus, ˜∆Haη′ ∈ � i+j=k+1 Wi ⊗ Hj where |η′| = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (2) Let η′ = xiη where |η| = k and xi ̸= x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Then from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2, � ˜∆H ◦ πj � aη′ = (θi ⊗ m) ◦ � ˜∆H ⊗ id � (πj ⊗ πi) ◦ ˜∆ aη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' = (θi ⊗ m) ◦ � ( ˜∆H ◦ πj) ⊗ πi � ˜∆ aη Thus, ˜∆Haη′ = (θi ⊗ m) ◦ � ˜∆H ⊗ ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='πi � ˜∆ aη, where ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='πi = [πi πi · · · πi]t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Since deg(η) = k, ˜∆ aη ⊆ (W ⊗ W)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Note that ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='πi(aη)(c) = [ai η ai η · · · ai η](c) = aη[ci ci · · · ci].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='πiaη ∈ W and then applying the induction hypothesis ˜∆HWn ⊆ (W ⊗ H)n for n ≤ k, � ˜∆H ⊗ ll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='πi � (W ⊗ W)k ⊆ (W ⊗ H ⊗ W)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Finally, (θi ⊗ m) (W ⊗ H ⊗ W)k ⊆ (W ⊗ H)k+1, as θi is homogeneous operator of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus, ˜∆Haη′ ∈ � i+j=k+1 Wi ⊗ Hj where |η′| = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence proved by induction that for all n ≥ 0: ˜∆H (aη) ∈ � i+j=n Wi ⊗ Hj where |η| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following proposition asserts that the grading on H in Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 is compatible with bialgebraic structure of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' With the grading on the Hopf algebra H as in Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1, ∆H (Hn) ⊆ � i+j=n Hi ⊗ Hj for all n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Proof: Observe that the statement is true for n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Prior to the proving the statement for n ≥ 1, the following statement needs to be proved: ∆H (Wn) ⊆ � i+j=n Wi ⊗ Hj ∀ n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 34 VENKATESH G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' AND KURUSCH EBRAHIMI-FARD Observe that from Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2, ∆H ◦ πi = (id ⊗ m) ◦ � ˜∆H ⊗ id � (πi ⊗ πi) ◦ ˜∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus (by grouping them along the coordinate i), ∆H = (id ⊗ m) ◦ � ˜∆ ◦ id � ˜∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Hence, ∆H(Wn) = (id ⊗ m) ◦ � ˜∆ ◦ id � ˜∆ (Wn) ⊆ (id ⊗ m) ◦ � ˜∆ ◦ id � (W ⊗ W)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2, ∆H(Wn) ⊆ (id ⊗ m) (W ⊗ H ⊗ W)n ⊆ (W ⊗ H)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore, the intermediate statement holds true viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' ∆H (Wn) ⊆ � i+j=n Wi ⊗ Hj ∀ n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The statement of the theorem then holds true as ∆ is an Rn-algebra morphism from H to H ⊗ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='5 asserts that the grading defined on the Hopf algebra H in Defini- tion 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 is well-defined and connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The homogeneous components are finite-dimensional and dimensions respect the Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2 since the bialgebras H and H are isomorphic with respect to the underlying graded algebraic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The following example is rework of the Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='10 in [Gray & Ebrahimi-Fard(2017)] acting as a check for the computation of feedback group inverse in one-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Let c = 1−x1 ∈ R⟨⟨X⟩⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The series c◦−1 = 1+· · ·+· · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Using the recursive computation formula for antipode as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='1 ax1(c◦−1) = Sax1 (c) = −ax1(c) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Observe that ∆′ Hax2 1 = 3ax1 ⊗ ax1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus, ax2 1 � c◦−1� = Sax2 1 (c) = −ax2 1 − 3ax1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='Sax1 = −ax2 1 + 3a2 x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore, ax2 1 (c◦−1) = 0 + 3(1)2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' In similar fashion the reduced coproduct of a3 x1 is ∆′ Hax3 1 = 4ax1 ⊗ ax2 1 + 6ax2 1 ⊗ ax1 + 3ax1 ⊗ a2 x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Thus, ax3 1 � c◦−1� = � −ax3 1 − 4ax1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='Sax2 1 − 6ax2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='Sax1 − 3ax1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' (Sax1)2� (c) = 0 − 4(−1)(3) − 6(0)(−1) − 3(−1)(1)2 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Therefore c◦−1 = 1 + x1 + 3x2 1 + 15x3 1 + 105x4 1 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' The result matches exactly with that of Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='10 in [Gray & Ebrahimi-Fard(2017)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' FORMAL SERIES APPROACH TO MULTIPLICATIVE DYNAMIC FEEDBACK CONNECTION 35 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Conclusions and Future work It was shown that the closed-loop system of a plant in Chen–Fliess series description in multiplicative output feedback with another system, given by Chen–Fliess series, has a Chen–Fliess series representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' An explicit expression of the closed-loop generating series was derived and the multiplicative dynamic feedback connection has a natural interpretation as a transformation group acting on the plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A computational framework has been devised utilizing the dual Hopf algebras corresponding to the shuffle group and multiplicative output dynamic feedback group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Future work will be to address the solemn problem regarding the local convergence of the both multiplicative dynamic and static output feedback connections and to identify both the multiplicative dynamic and static feedback invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' References [Abe(2004)] Abe, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=', Hopf Algebras, Cambridge University Press, Cambridge, UK, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' Abramowitz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' and Stegun, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Berstel & Reutenauer(1988)] Berstel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' and Reutenauer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=', Rational Series and Their Languages, Springer-Verlag, Berlin, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' [Brockett(1978)] Brockett, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' W.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' and Duffaut Espinosa, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=', Frac- tional Fliess Operators: Two Approaches, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 49th Conference on Information Sciences and Systems, Baltimore, MD, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content=' 1–6 Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway Email address: subbarao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='guggilam@ntnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='no Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway Email address: kurusch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='ebrahimi-fard@ntnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='no URL: https://folk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='ntnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} +page_content='no/kurusche/' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE4T4oBgHgl3EQfMgzk/content/2301.04949v1.pdf'} diff --git a/X9AyT4oBgHgl3EQfWfd0/content/tmp_files/2301.00164v1.pdf.txt b/X9AyT4oBgHgl3EQfWfd0/content/tmp_files/2301.00164v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d09c66d88be47638ec9108f43e84b15c8655a84a --- /dev/null +++ b/X9AyT4oBgHgl3EQfWfd0/content/tmp_files/2301.00164v1.pdf.txt @@ -0,0 +1,2493 @@ +arXiv:2301.00164v1 [eess.SP] 31 Dec 2022 +1 +Design of a Multi-User Wireless Powered +Communication System Employing Either +Active IRS or AF Relay +Omid Rezaei, Maryam Masjedi, Ali Kanaani, Mohammad Mahdi Naghsh∗, +Saeed Gazor, and Mohammad Mahdi Nayebi +Abstract +In this paper, we optimize a Wireless Powered Communication (WPC) system including multiple +pair of users, where transmitters employ single-antenna to transmit their information and power to their +receivers with the help of one multiple-antennas Amplify-and-Forward (AF) relay or an active Intelligent +Reflecting Surface (IRS). We propose a joint Time Switching (TS) scheme in which transmitters, +receivers, and the relay/IRS are either in their energy or information transmission/reception modes. +The transmitted multi-carrier unmodulated and modulated waveforms are used for Energy Harvesting +(EH) and Information Decoding (ID) modes, respectively. In order to design an optimal fair system, we +maximize the minimum rate of all pairs for both relay and IRS systems through a unified framework. +This framework allows us to simultaneously design energy waveforms, find optimal relay/IRS amplifi- +cation/reflection matrices, allocate powers for information waveforms, and allocate time durations for +various phases. In addition, we take into account the non-linearity of the EH circuits in our problem. This +problem turns out to be non-convex. Thus, we propose an iterative algorithm by using the Minorization- +Maximization (MM) technique, which quickly converges to the optimal solution. Numerical examples +show that the proposed method improves the performance of the multi-pair WPC relay/IRS system +under various setups. +O. Rezaei and M. M. Nayebi are with the Department of Electrical Engineering, Sharif University of Technology, Tehran, +11155-4363, Iran. M. Masjedi, A. Kanaani, and M. M. Naghsh are with the Department of Electrical and Computer Engineering, +Isfahan University of Technology, Isfahan, 84156-83111, Iran. S. Gazor is with the Department of Electrical and Computer +Engineering, Queen’s University, Kingston, Ontario, K7L 3N6, Canada. +∗Please address all the correspondence to M. M. Naghsh, Phone: (+98) 31-33912450; Fax: (+98) 31-33912451; Email: +mm naghsh@cc.iut.ac.ir +January 3, 2023 +DRAFT + +2 +Index Terms +Fair Throughput Maximization, Intelligent Reflecting Surface (IRS), Minorization-Maximization +(MM), Relay Networks, Wireless Powered Communication (WPC). +I. INTRODUCTION +Wireless Power Transfer (WPT) technology is introduced to extend the lifetime of devices +in wireless networks in which the energy is emitted from the dedicated power sources to the +devices [1]. Interestingly, WPT enables Simultaneous Wireless Information and Power Transfer +(SWIPT) [2], where devices not only receive and decode information, but also harvest the +energy from Radio Frequency (RF) signals. The Time Switching (TS) and Power Splitting +(PS) schemes are two well-known implementing protocols of SWIPT [3]. Recently, SWIPT +models are designed to employ relays to further enhance the coverage and spectral efficiency +of wireless networks [4]–[6]. In [4], a Multiple-Input Multiple-Output (MIMO) two-way relay +system is introduced in which two transceivers exchange their information through a relay. The +authors in [5] designed the relay and source precoders by minimizing the bit error rate at the +destination for a full-duplex MIMO relay system and SWIPT-enabled users. A similar system +with a half-duplex two-way relay is designed in [6] by minimizing the mean square error at the +destination. +Another impactful technology that is currently emerging is to use of the Intelligent Reflecting +Surface (IRS) in wireless communication systems. This promising solution not only is capable +of improving energy delivery but also can enhance the spectral efficiency of future wireless +communication networks. [7]. An IRS is an array of large number of Reflecting Elements (REs) +designed to have controllable electromagnetic properties. Each RE introduces a phase shift on +the impinging signal, allowing beamforming/manipulation of the reflection waveforms. Precisely, +the IRS matrix allows controlling the reflected signal (amplify, attenuated, steer in the desired +direction, and so on) toward optimal desired directions by purposefully designing the phase +shift matrix. The IRS is exploited recently in SWIPT systems in [8]–[11]. The weighted sum +harvested power maximization problem was studied in [8] for an IRS-aided SWIPT model in +which a multiple-antennas access point serves multiple single-antenna users. In [9], the model +in [8] is extended for a more practical multi-objective optimization problem by taking into +account the trade-off between sum rate and sum harvested power maximization. In [10], the +total transmission power is minimized for a Multiple-Input Single-Output (MISO) SWIPT system +January 3, 2023 +DRAFT + +3 +employing multiple IRSs. In [11], the MISO SWIPT in [8]–[10] is extended to the MIMO case, +and the weighted sum rate is maximized in an IRS-assisted system. +The design of the energy waveform remarkably affects the performance of WPT-based systems. +Indeed, an efficient waveform leads to significant improvement in the efficacy of power delivery. +Experiments reveal that signals with high Peak to Average Power Ratio (PAPR) such as multi- +sine signals provide more DC power at Energy Harvesting (EH) circuits than constant envelope +signals with the same average RF power [12]. Based on this interesting observation, a multi-sine +waveform design for WPT has been examined in several works [13]–[21]. Waveform design +with a non-linear EH model was considered in [13] and [14] for MISO and Single-Input Single- +Output (SISO) systems, respectively. The authors of [15] proposed a low-complexity method for +a waveform design in a SISO WPT system. In [16], the transmit waveform was designed based +on limited Channel State Information (CSI) feedback WPT system. Then, the authors in [17] +studied waveform design for an IRS-aided SWIPT MISO system. The aforementioned methods +for design of single-user systems were extended to the multi-user case in [18]. Also, a waveform +design was performed in [19] for wireless powered backscatter communication networks, and this +work was extended to multi-user backscatter systems in [20]. The authors of [21] investigated +the waveform and transceiver design problem in a MISO SWIPT system and determined the +multi-sine waveforms for Information Decoding (ID) and EH phases. +In this paper, we optimize a multi-user wireless powered relay/IRS system using a multi-sine +waveform with the following main contributions: +• Relay model: To the best of our knowledge, a multi-sine waveform design for multiple +user pairs in a wireless powered relay system has not been addressed in the literature. In +this paper, we consider a multi-carrier Wireless Powered Communication (WPC) system +for multi-user relay channels. Precisely, in our proposed model, an amplify-and-forward +(AF) relay provides energy/information transmission from K transmitters to their receivers +by adopting the TS scheme in all nodes1. Herein, the aim is to design the multi-carrier +unmodulated energy waveforms and the allocated power for information waveforms at the +transmitters, the amplification matrices in a relay, and the time durations for the EH and ID +modes in order to maximize the minimum rate of the user pairs. In addition, we consider +1Note that the system in [22], where a TS scheme is only applied for a receiver, is considered as a special case of the proposed +joint TS scheme. +January 3, 2023 +DRAFT + +4 +the effect of the non-linearity of EH circuits in the design problem. +• IRS model: In the case of IRS-assisted communication, multi-pair WPC has not been +considered in the literature, and therefore, herein, we consider this type of IRS-assisted +systems. Precisely, in this case, an active IRS2 replaced with the AF relay in the proposed +system model mentioned in the above paragraph. Also, some comparisons are made between +relay and IRS models in terms of architecture and performance (see Remark 1-2). +• Unified consideration of relay and IRS: Both proposed AF relay and active IRS-aided +systems are modeled under a unified formulation, and we handle the resulting optimization +problems under a unified mathematical umbrella. We show that the problem is non-convex +and consequently, is hard to solve. To deal with this problem, we devise a method based on +the Minorization-Maximization (MM) technique. Interestingly, the proposed algorithm can +deal with relay and IRS systems by switching between Kronecker and Hadamard products +for parameters used in the algorithm (see Lemma 2). +• Sub-optimal methods: Some sub-optimal methods with lower signaling overhead and com- +putational complexity are proposed and then, their performance are compared. +• Numerical result: Simulation results are reported to illustrate the effectiveness of the pro- +posed method; particularly, the impact of the relay/IRS matrix and energy waveform design. +Also, numerical examples show that the minimum rate of users increases linearly/super- +linearly with the number of antennas/REs in relay/IRS systems. +The rest of this paper is organized as follows: The signal and system models are explained +in Section II. In Section III, the minimum rate maximization problem is formulated, and a +unified optimization framework is proposed for both relay and IRS models. Section IV presents +numerical examples to illustrate the effectiveness of the proposed method. Finally, conclusions +are drawn in Section V. +Notation: Bold lowercase (uppercase) letters are used for vectors (matrices). The notations +arg(·), E[·], ℜ{·}, ∥·∥2, (·)T, (·)H, (·)∗, tr{·}, λmax(·), vec(·), Diag(·), ∇xf(·) and ∇2 +xf(·) indicate +the phase of a complex number, statistical expectation, real-part, l2-norm of a vector, transpose, +Hermitian, complex conjugate, trace of a matrix, the principal eigenvalue of a matrix, stacking +of the column of a matrix, a diagonal matrix formed by the entries, the gradient of a function +with respect to (w.r.t.) x and the Hessian of a function w.r.t. x, respectively. The symbols ⊗ and +2Note that in an active IRS, REs can amplify the reflected signals using their reflection-type amplifiers [23]. +January 3, 2023 +DRAFT + +5 +T1 ... +τ +T − τ +Information +Transmitter +Energy +Transmitter +Tk ... +TK +AF Relay or Active IRS +R1 +... +Rk +... +τ +T − τ +Information +Decoder +Energy +Harvester +RK +Blockage +Fig. 1. Multi-user wireless powered relay/IRS system based on TS scheme with blocked direct path. +⊙ stand for the Kronecker and Hadamard products of two matrices. We denote CN (ω, Σ) as +a circularly symmetric complex Gaussian (CSCG) distribution with mean ω and covariance Σ. +The set R+ represents non-negative real numbers and CN×N and DN×N are the set of N × N +complex and complex diagonal matrices, respectively. The set of N × N positive (semi-)definite +and identity matrices are denoted by SN +++ ⊂ CN×N (SN ++ ⊂ CN×N) and IN, respectively. The +notation A ≻ B (A ⪰ B) means that A − B is positive (semi-)definite. +II. SYSTEM MODEL +We consider a multi-carrier wireless powered relay/IRS system with K user pairs {(Tk, Rk)}K +k=1 +as shown in Fig. 1, where the direct links between the transmitters and receivers are likely blocked +(see [24] and [25], [26] for similar models of multiple user pairs with blocked direct path for +relay and IRS systems, respectively). The single-antenna transmitter Tk communicates with its +receiver Rk through either an AF relay with MR antennas or an active IRS with MIRS REs. +We assume that Rk harvests a part of its required power, whereas Tk and the relay/IRS have +no energy concern [4], [5]. In each time duration T, the relay/IRS helps Rk not only harvest +energy from the signal of all {Tk}K +k=1, but also decode the information from its corresponding +transmitter Tk by using a joint TS scheme. Precisely, Tk, relay/IRS, and Rk switch simultaneously +at time t = τ from their energy delivery modes to their communication modes. We assume that all +nodes are perfectly synchronized as shown in Fig. 2 for this switching [22]. We consider a multi- +carrier system with a total bandwidth of Bt equally divided into N orthogonal subbands. We +also model all channels to have a frequency-selective block fading, i.e., the channel coefficients +January 3, 2023 +DRAFT + +6 +Relay: +IRS: +{Tk}K +k=1 → IRS → {Rk}K +k=1 +{Tk}K +k=1 → relay relay → {Rk}K +k=1 +{Tk}K +k=1 → IRS → {Rk}K +k=1 +{Tk}K +k=1 → relay +relay → {Rk}K +k=1 +τ +2 +τ (energy waveform) +τ +2 +T −τ +2 +T − τ (information waveform) +T −τ +2 +Fig. 2. The transmission, amplification/reflection, and reception timeline for the proposed relay/IRS model. +remain constant for at least T seconds. Let the complex random matrices HR +n ∈ CMR×K and +GR +n ∈ CK×MR denote the channels from transmitters to the relay and the channels from the +relay to the receivers for nth subband, respectively. The elements of HR +n and GR +n are zero mean +CSCG random variables in the case of Rayleigh fading. Similarly, we denote the channels from +transmitters to the IRS and the channels from the IRS to the receivers by HIRS +n +∈ CMIRS×K +and GIRS +n +∈ CK×MIRS, respectively. In the sequel, Hn and Gn refer to either HR +n and GR +n or +HIRS +n +and GIRS +n , depending on the case under discussion. In addition, we assume that we can +control the relay and IRS by collecting and using the CSI of all links [25]–[27]. For example, +a relay itself can act as a controller. The CSI may be estimated in various ways, e.g., by using +orthogonal pilot sequences ( see [28], [29] for more details). The CSI estimation is out of the +scope of this paper. Also, we propose two low-complexity implementation methods mentioned +in Remark 3 to reduce the signaling overhead in the controller node. +Each Tk transmits a multi-sine energy waveform xE,k(t) and a multi-carrier modulated infor- +mation waveform xI,k(t) to the relay/IRS during the first-hop transmission at the EH and ID +time slots, respectively, as follows +xE,k(t) = +N +� +n=1 +aE,k,ncos(2πfnt + φE,k,n), = ℜ +� N +� +n=1 +sE,k,nej2πfnt +� +, +(1) +xI,k(t) = +N +� +n=1 +aI,k,n(τ)cos(2πfnt + φI,k,n) = ℜ +� N +� +n=1 +sI,k,nej2πfnt +� +, +(2) +where sE,k,n = aE,k,nejφE,k,n and sI,k,n = aI,k,nejφI,k,n are the baseband complex signal represen- +tations for the energy and information waveforms, respectively. We assume that the baseband +information signals are i.i.d. CSCG random variable variables, i.e., sI,k,n ∼ CN (0, pI,k,n). The +transmitted energy by Tk is constrained by +τ +2ρ|sE,k,n|2 + T − τ +2ρ +pI,k,n ≤ Tprf +k,n, ∀k, n, +(3) +January 3, 2023 +DRAFT + +7 +where prf +k,n is the maximum power budget at Tk for the nth subband and ρ addresses both ρR = 2 +for relay and ρIRS = 1 for IRS system according to the proposed timeline in Fig. 2 (see also +Remark 1). By defining sE,n = [sE,1,n, · · · , sE,K,n]T and sI,n = [sI,1,n, · · · , sI,K,n]T, the received +signal at the relay/IRS is expressed as +r(t) = + + + + + + + +N� +n=1 +{HnsE,n + zn}, t ∈ TEH, for EH, +N� +n=1 +{HnsI,n + zn}, t ∈ TID, for ID, +(4) +where +TEH = + + + + + +0 ≤ t ≤ τ +2, for relay, +0 ≤ t ≤ τ, for IRS, +TID = + + + + + +τ ≤ t ≤ τ + T−τ +2 , for relay, +τ ≤ t ≤ T, for IRS, +(5) +and r denotes either rR or rIRS. Furthermore, the AWGN zn denotes either zR +n ∼ CN (0, σ2 +R,nIMR) +or zIRS +n +∼ CN (0, σ2 +IRS,nIMIRS) for relaying or reflecting modes. In contrast to the passive IRS, +an active IRS adds non-negligible noise (which is introduced by the active elements [23], [30]); +however, the added noise of an active IRS has considerably less impact compared to the relay +noise (which is introduced by RF chains), i.e., σ2 +IRS,n ≤ σ2 +R,n [31]. +In the second-hop transmission, the relay/IRS amplifies the energy and information signals of +Tk by amplification/reflection matrices and then forwards them to Rk. For AF relay system, the +amplification matrices is introduced as UR +E,n and UR +I,n ∈ CMR×MR, ∀n for energy and information +phases, respectively. In the case of IRS-aided system, the reflection matrices is defined as UIRS +E += +Diag(θE) and UIRS +I += Diag(θI) for energy and information time slots, respectively, where θE = +[ηE,1ejθE,1, ηE,2ejθE,2, · · · , ηE,MIRSejθE,MIRS]T and θI = [ηI,1ejθI,1, ηI,2ejθI,2, · · · , ηI,MIRSejθI,MIRS]T +with ηE,m, ηI,m ≥ 1 and θE,m, θI,m ∈ [0, 2π] respectively denote the reflection amplitude and +the phase shift at the mth RE3. +Remark 1. An active IRS amplifies the signal without any significant delay. However, in an +AF relay, the signal reception, amplification, and transmission at the RF chain cause a long +delay. Therefore, in practice, the AF relay requires twice time compared to the active IRS for +transmission one information symbol [23]. +3Note that passive and passive lossless IRS require ηE,m, ηI,m ∈ [0, 1] and ηE,m = ηI,m = 1, respectively. +January 3, 2023 +DRAFT + +8 +We define UE,n and UI,n to address both UR +E,n, UIRS +E +and UR +I,n, UIRS +I +, respectively. The +forwarded signal by the relay/IRS is given by +�r(t)= + + + + + +�N +n=1 UE,n (HnsE,n + zn) , t ∈ �TEH, for EH, +�N +n=1 UI,n (HnsI,n + zn) , t ∈ �TID, for ID, +where +�TEH = + + + + + +τ +2 ≤ t ≤ τ, for relay, +0 ≤ t ≤ τ, for IRS, +�TID = + + + + + +τ + T−τ +2 +≤ t ≤ T, for relay, +τ ≤ t ≤ T, for IRS, +(6) +and �r denotes either �rR or �rIRS for relay or IRS system, with a slight abuse of notation. +Then, the power of �r(t) from the relay/IRS is written as +E +� +∥�r(t)∥2 +2 +� += + + + + + + + +1 +2 +N� +n=1 +� +sH +E,nVE,nsE,n + σ2 +ntr +� +UE,nUH +E,n +�� +, for EH, +1 +2 +N� +n=1 +� +tr {QI,nVI,n} + σ2 +ntr +� +UI,nUH +I,n +�� +, for ID, +(7) +where σ2 +n addresses both σ2 +R,n and σ2 +IRS,n, QI,n = Diag(pI,1,n, pI,2,n, · · · , pI,K,n) and +VE,n = HH +n UH +E,nUE,nHn, ∀n, +VI,n = HH +n UH +I,nUI,nHn, ∀n. +(8) +Using (7), the total consumed energy is bounded at the relay/IRS in t ∈ [0, T] as +τ +2ρ +� +sH +E,nVE,nsE,n + σ2 +ntr{UE,nUH +E,n} +� ++ T − τ +2ρ +� +tr{QI,nVI,n} + σ2 +ntr{UI,nUH +I,n} +� +≤ Tprf +n , ∀n, (9) +where prf +n denotes either the maximum power budget at the relay prf +R,n or IRS prf +IRS. We can write +received signal at Rk as +yk(t) = + + + + + + + +N� +n=1 +� +gT +k,nUE,n (HnsE,n + zn) + zk,n +� +, t ∈ �TEH, ∀k, for EH, +N� +n=1 +� +gT +k,nUI,n (HnsI,n + zn) + zk,n + �zk,n +� +, t ∈ �TID, ∀k, for ID, +where gk,n is the kth column vector of GT +n, and zk,n as well as �zk,n are the AWGN from +the antenna and baseband processing noises at Rk, respectively, with zk,n ∼ CN(0, σ2 +k,n) and +�zk,n ∼ CN (0, δ2 +k,n). The information signals at Rk corresponding to the nth subband can be +expanded as +yk,n(t) =gT +k,nUI,nhk,nsI,k,n + gT +k,nUI,n +K +� +j=1,j̸=k +hj,nsI,j,n + gT +k,nUI,nzn + zk,n + �zk,n, ∀k, n, +(10) +January 3, 2023 +DRAFT + +9 +where hk,n and gk,n are the kth column vector of Hn and GT +n, respectively. By defining pI,n = +[pI,1,n, pI,2,n, · · · , pI,K,n]T, the SINR at the ID part for the nth subband is given by +γk,n(pI,n, UI,n) = +pI,k,nψk,k,n +�K +j=1,j̸=k pI,j,nψk,j,n + σ2n �ψk,n + δ2 +k,n + σ2 +k,n +, ∀k, n, +(11) +where ψk,j,n = gT +k,nUI,nhj,nhH +j,nUH +I,ng∗ +k,n and �ψk,n = gT +k,nUI,nUH +I,ng∗ +k,n. From Remark 1, we obtain +the achievable rate at the kth pair as follows +Rk +� +{pI,n}N +n=1, {UI,n}N +n=1, τ +� += T − τ +ρT +N +� +n=1 +log2 +� +1 + γk,n(pI,n, UI,n) +� +. +(12) +For the EH stream, we assume the noise power is negligible compared to the received signal +power. We take into account the rectifier non-linearity by employing the results from [32] where +the harvested energy at Rk is approximated by +Ek +� +{sE,n}N +n=1, {UE,n}N +n=1, τ +� += τ +ρexp +� +�alog2pE,k +� +p +�b +E,kexp�c, ∀k, +(13) +where �a, �b, and �c are the curve fitting constants and pE,k is the average input power to Rk’s +harvester as +pE,k +� +{sE,n}N +n=1, {UE,n}N +n=1 +� += 1 +2 +N +� +n=1 +sH +E,nΞk,nsE,n, ∀k, +(14) +with +Ξk,n = HH +n UH +E,ng∗ +k,ngT +k,nUE,nHn, ∀k, n. +(15) +Remark 2. Note that the reflection matrix cannot be designed separately for each subband in the +IRS system, while, thanks to the RF chain circuits in a relay, the amplification matrix design is +considered for each subband. We note that an active IRS is considerably less expensive than an +AF relay. This is because an AF relay requires massive integrated circuits (including analog-to- +digital/digital-to-analog converter, self-interference cancellation circuits, etc). The delay caused +by RF chain processing of an AF relay contributes to latency, leads to lower transmission time, +and requires more power for energy and information signals (see (3) and (9)). Therefore, a +relay-IRS trade-off exists in the system performance (see (12) and (13)). +Remark 3. An approach with lower implementation complexity is considered in which only one +amplification/reflection matrix needs to be designed for both energy and information time slots, +called the t-static approach. Also, one can consider another approach with only one amplification +matrix design in both time slots and all subbands, referred to as t-f-static in the relay system. +These design methodologies lead to a lower signaling overhead and system performance. +January 3, 2023 +DRAFT + +10 +III. THE PROPOSED MINIMUM RATE MAXIMIZATION METHOD +In this section, the aim is to maximize the minimum rate of the multi-user relay/IRS WPC +system w.r.t. multi-sine energy waveforms sE,n, allocated power pI,n, amplification/reflection +matrices UE,n, UI,n, and the time allocation parameter τ. The unified minimum rate maximization +problem for both relay and IRS systems is cast as +max +τ,{sE,n,pI,n,UE,n,UI,n}N +n=1 +min +1≤k≤K +Rk +(16) +s.t. +� +τ, {sE,n, pI,n, UE,n, UI,n}N +n=1 +� +∈ Ω, +where Ω = Ω0 ∩ Ωind with +Ω0 = +� +C1 : 0 ≤ τ ≤ T, C2 : (3), pI,k,n ≥ 0, ∀k, n, C3 : (9), C4 : Ek ≥ Emin,k, ∀k +� +, +(17) +Ωind = + + + + + +CR : UE,n, UI,n ∈ CMR×MR, ∀n, for relay, +CIRS : UE,n, UI,n ∈ DMIRS×MIRS, ∀n, |θE,m| ≥ 1, |θI,m| ≥ 1, ∀m, for IRS, +(18) +and Emin,k in C4 is the minimum required harvested energy for the kth user. +The problem in (16) is non-convex due to the coupled design variables in the objective function +and the constraints C2 − C4 and CIRS. To deal with this non-convex problem, we first solve +the problem w.r.t. {UE,n, UI,n} for fixed {sE,n, pI,n, τ}, then optimize {sE,n, pI,n} for given +{UE,n, UI,n, τ}, and finally, solve the problem w.r.t. τ via a closed-form solution. The procedure +is repeated until convergence. +A. Maximization over {UE,n, UI,n} +Here, we first consider the relay problem, and then the IRS problem is investigated. +1) Relay System: The problem in (16) for fixed {sE,n, pI,n} reduces to the following opti- +mization +max +{UE,UI}N +n=1 +min +1≤k≤K +N +� +n=1 +log2 (1 + γk,n(UI,n)) +(19) +s.t. +C3, C4, +January 3, 2023 +DRAFT + +11 +which is still a non-convex problem. To start solving the problem, first we need to reformulate +the obtained expressions for the relay power constraint (7), SINR (11), and the input power of +harvesters (14) from Section II. We can rewrite (7) as (see Appendix A for the derivation) +E +� +∥�r(t)∥2 +2 +� += + + + + + + + +1 +2 +N� +n=1 +uH +E,n �AR +E,nuE,n, 0 ≤ t ≤ τ, for EH, +1 +2 +N� +n=1 +uH +I,n �AR +I,nuI,n, τ ≤ t ≤ T, for ID, +(20) +where uE,n = vec(UE,n), uI,n = vec(UI,n), and +�AR +E,n = +� +HnsE,nsH +E,nHH +n +�T ⊗ IMR + σ2 +nIM2 +R, +�AR +I,n = +� +HnQI,nHH +n +�T ⊗ IMR + σ2 +nIM2 +R. +(21) +Therefore, we rewrite the relay power constraint in (9) by using (20) as +τ +2ρuH +E,n �AR +E,nuE,n + T − τ +2ρ +uH +I,n �AR +I,nuI,n ≤ Tprf +R,n, ∀n. +(22) +Next, we rewrite the SINR and the input power at Rk’s harvester in (11) and (14) as +γk,n(uI,n) = +uH +I,nAR +k,nuI,n +uH +I,n �AR +k,nuI,n + δ2 +k,n + σ2 +k,n +, ∀k, n, +(23) +pE,k +� +{uE,n}N +n=1 +� += 1 +2 +N +� +n=1 +uH +E,n ¯AR +k,nuE,n, ∀k, +(24) +where +AR +k,n = pI,k,n +� +hk,nhH +k,n +�T ⊗ g∗ +k,ngT +k,n, +(25) +�AR +k,n= +K +� +j=1,j̸=k +pI,j,n +� +hj,nhH +j,n +�T ⊗ g∗ +k,ngT +k,n + σ2 +nIMR ⊗ g∗ +k,ngT +k,n, +(26) +¯AR +k,n = +� +HnsE,nsH +E,nHH +n +�T ⊗ g∗ +k,ngT +k,n. +(27) +By using (22), (23), and (24) with an auxiliary variable αa the optimization problem in (19) can +be equivalently rewritten as +max +αa,{uE,uI}N +n=1 +αa +(28) +s.t. +C3 : (22), C4 : Ek +� +{uE,n}N +n=1 +� +≥ Emin,k, ∀k, +C5 : +N +� +n=1 +log2 +� +1 + +uH +I,nAR +k,nuI,n +uH +I,n �AR +k,nuI,n + ζk,n,a +� +≥ αa, ∀k, +where ζk,n,a = σ2 +k,n + δ2 +k,n. The constraint C5 can be equivalently rewritten as +C5 : +N +� +n=1 +� +log2 +� +uH +I,nBk,nuI,n + ζk,n,a +� +− log2 +� +uH +I,n �AR +k,nuI,n + ζk,n,a +� � +≥ αa, +(29) +January 3, 2023 +DRAFT + +12 +where Bk,n = �AR +k,n + AR +k,n. It is observed that this constraint is non-convex. Therefore, we +employ the MM technique to tackle its non-convexity. Precisely, we minorize the denominator +term − log2 +� +uH +I,n �AR +k,nuI,n +ζk,n,b +� +by the using the following inequality +log2(x) ≤ log2(x0) + log2 e +x0 +(x − x0). +(30) +By setting x = uH +I,n �AR +k,nuI,n + ζk,n,a and x0 = +� +u0 +I,n +�H �AR +k,nu0 +I,n + ζk,n,a in (30) we obtain +− log2 +� +uH +I,n �AR +k,nuI,n + ζk,n,a +� +≥ − log2 +�� +u0 +I,n +�H �AR +k,nu0 +I,n + ζk,n,a +� +(31) +− +log2 e +� +uH +I,n �AR +k,nuI,n − +� +u0 +I,n +�H �AR +k,nu0 +I,n +� +� +u0 +I,n +�H �AR +k,nu0 +I,n + ζk,n,a +. +Applying the above minorizer, the constraint C5 in (29) is rewritten at the ith iteration of the +MM technique as +N +� +n=1 +� +log2 +� +uH +I,nBk,nuI,n + ζk,n,a +� +− log2 +�� +u(i−1) +I,n +�H �AR +k,nu(i−1) +I,n ++ ζk,n,a +� +(32) +− +log2 e +� +u(i−1) +I,n +�H �AR +k,nu(i−1) +I,n ++ ζk,n,a +� +uH +I,n �AR +k,nuI,n − +� +u(i−1) +I,n +�H �AR +k,nu(i−1) +I,n +� � +≥ αa. +The following lemma lays the ground for dealing with the first non-concave logarithmic term +in (32) in light of the MM technique. +Lemma 1. Let s(x) = − log2 +� +xHTx + ν +� +and xHQx ≤ P for any positive-definite matrices +T, Q ∈ SN +++ and P ∈ R+. Then, s(x) is bounded for all x and x0 as follows +s(x) ≤ s(x0) + ℜ +� +bH(x − x0) +� ++ (x − x0)HD(x − x0), +where b = +−2 log2 e +xH +0 Tx0+νTx0, D = +4P +wH +1 Qw1IM2 +R, and w1 is the principal eigenvector of T and ǫ > 0. +Proof. See Appendix B. +Using Lemma 1 and noting that the term τ +2uH +E,n �AR +E,nuE,n in (22) is positive, we obtain the +following minorizer for the term log2(uH +I,nBk,nuI,n + ζk,n,a) in (32) at any given u0 +I,n +log2(uH +I,nBk,nuI,n + ζk,n,a) ≥ log2 +�� +u0 +I,n +�H Bk,nu0 +I,n + ζk,n,a +� +− ℜ +� +bH +k,n(uI,n − u0 +I,n) +� +(33) +− +� +uI,n − u0 +I,n +�H Dk,n(uI,n − u0 +I,n), +where +bk,n = +−2 log2 e +� +u0 +I,n +�H Bk,nu0 +I,n + ζk,n,a +Bk,nu0 +I,n, +Dk,n = +16T +T−τ prf +R,n +�wH +k,n �AR +I,n �wk,n +IM2 +R, +January 3, 2023 +DRAFT + +13 +and �wk,n denotes the principal eigenvector of Bk,n. Applying (33), the constraint in (32) is +restated as +− +N +� +n=1 +� +log2 e uH +I,n �AR +k,nuI,n +� +u(i−1) +I,n +�H �AR +k,nu(i−1) +I,n ++ ζk,n,a ++ uH +I,nDk,nuI,n + ℜ +�� +bk,n − 2Dk,nu(i−1) +I,n +�H +uI,n +� ++ d(i) +k,n +� +≥ αa, ∀k, +(34) +where +d(i) +k,n = log2 +� +u(i−1) +I,n +�H �AR +k,nu(i−1) +I,n ++ ζk,n,a +� +u(i−1) +I,n +�H +Bk,nu(i−1) +I,n ++ ζk,n,a +− ℜ +� +bH +k,nu(i−1) +I,n +� ++ +� +u(i−1) +I,n +�H +Dk,nu(i−1) +I,n +(35) +− +log2 e +� +u(i−1) +I,n +�H �AR +k,nu(i−1) +I,n +� +u(i−1) +I,n +�H �AR +k,nu(i−1) +I,n ++ ζk,n,a +. +Then, we can simplify constraint in (34) as +− +N +� +n=1 +� +uH +I,nF(i) +k,nuI,n + ℜ +� +(f(i) +k,n)HuI,n +� ++ d(i) +k,n +� +≥ αa, ∀k, +(36) +where +F(i) +k,n = +log2 e �AR +k,n +� +u(i−1) +I,n +�H �AR +k,nu(i−1) +I,n ++ ζk,n,a ++ Dk,n, +f(i) +k,n = bk,n − 2Dk,nu(i−1) +I,n +. +(37) +Finally, we focus on the constraint C4. From (13) and (24), we see that in the left-hand side +(LHS) of C4, Ek is neither convex nor concave w.r.t. uE,n. To apply the MM technique on LHS +of C4, we first define a parameter4 βk,n,a such that ∇2 +uE,nEk +� +{uE,n}N +n=1 +� ++βk,n,aIM2 +R ⪰ 0, ∀k, n, +and write Ek as the sum of a convex and a concave function as +Ek +� +{uE,n}N +n=1 +� +=Ek +� +{uE,n}N +n=1 +� ++ 1 +2 +N +� +n=1 +βk,n,auH +E,nuE,n − 1 +2 +N +� +n=1 +βk,n,auH +E,nuE,n, ∀k. +(38) +We now apply the MM technique to C4 and obtain a convex constraint. To do so, we keep the +concave part and minorize the convex part of (38) and rewrite C4 as +Ek +� +{u(i−1) +E,n }N +n=1 +� ++ 1 +2 +N +� +n=1 +βk,n,a +� +u(i−1) +E,n +�H +u(i−1) +E,n ++ +N +� +n=1 +ℜ +� +ϑ(i) +k,n,a +� +uE,n − u(i−1) +E,n +�� +(39) +− 1 +2 +N +� +n=1 +βk,n,auH +E,nuE,n ≥ Emin,k, ∀k, +4See Appendix C for a selection of βk,n,a. +January 3, 2023 +DRAFT + +14 +where +ϑ(i) +k,n,a =βk,n,a +� +u(i−1) +E,n +�H ++ τexp�c +2 +exp +� +�alog2ω(i) +k,a +� � +ω(i) +k,a +��b−1 � +2�a log ω(i) +k,a + �b +� � +u(i−1) +E,n +�H ¯AR +k,n, +with ω(i) +k,a = 1 +2 +�N +n=1 +� +u(i−1) +E,n +�H ¯AR +k,nu(i−1) +E,n . Therefore, the ith MM iteration for (19) is the solution +of the following convex problem +max +αa,{uE,n,uI,n}N +n=1 +αa +(40) +s.t. C3 : (22), C4 : (39), C5 : (36), +which can be solved efficiently. +2) IRS System: By considering UE,n = Diag(θE), UI,n = Diag(θI) and adding the constraint +CIRS in (18), the optimization problem in (19) is considered in this subsection. Since UE,n and +UI,n are diagonal matrices, the expressions in (22)-(24) are modified as +τ +2θH +E �AIRS +E,nθE + T − τ +2 +θH +I �AIRS +I,n θI ≤ Tprf +IRS, ∀n, +(41) +γk,n(θI) = +θH +I AIRS +k,n θI +θH +I �AIRS +k,n θI + δ2 +k,n + σ2 +k,n +, ∀k, n, +(42) +pE,k (θE) = 1 +2 +N +� +n=1 +θH +E ¯AIRS +k,n θE, ∀k, +(43) +where their parameters are defined in Lemma 2 below. +Lemma 2. The parameters �AIRS +E,n, �AIRS +I,n , AIRS +k,n , �AIRS +k,n , and ¯AIRS +k,n are expressed as follows: +�AIRS +E,n = +� +HnsE,nsH +E,nHH +n +�T ⊙ IMIRS + σ2 +nIMIRS, +(44) +�AIRS +I,n = +� +HnQI,nHH +n +�T ⊙ IMIRS + σ2 +nIMIRS, +(45) +AIRS +k,n = pI,k,n +� +hk,nhH +k,n +�T ⊙ g∗ +k,ngT +k,n, +(46) +�AIRS +k,n = +K +� +j=1,j̸=k +pI,j,n +� +hj,nhH +j,n +�T⊙ g∗ +k,ngT +k,n+ σ2 +nIMIRS⊙ g∗ +k,ngT +k,n, +(47) +¯AIRS +k,n = +� +HnsE,nsH +E,nHH +n +�T ⊙ g∗ +k,ngT +k,n. +(48) +It is worth pointing out that the only difference between the parameters above and their corre- +sponding expressions in (21) and (25)-(27), is the symbol of multiplication, i.e., ⊗ and ⊙, in a +January 3, 2023 +DRAFT + +15 +proper dimension. The proper dimension consideration means MR → MIRS for all of the above +parameters and IM2 +R → IMIRS for the second terms of �AIRS +E,n and �AIRS +I,n . +Proof. See Appendix D. +Next, we focus on constraint CIRS. First, let us introduce the following minorizer [33] +|x| ≥ ℜ +� +x∗ x0 +|x0| +� +. +(49) +Then, considering the above minorizer, the constraint CIRS is expressed as the ith iteration of +MM as +ℜ +� +θ∗ +E,m +θ(i−1) +E,m +|θ(i−1) +E,m | +� +≥ 1, ℜ +� +θ∗ +I,m +θ(i−1) +I,m +|θ(i−1) +I,m | +� +≥ 1, +∀m. +(50) +Therefore, the optimization problem in (28) is modified as +max +αa,θE,θI +αa +(51) +s.t. +C3 : (41), C4 : Ek (θE) ≥ Emin,k, ∀k, CIRS : (50), +C5 : +N +� +n=1 +log2 +� +1 + +θH +I AIRS +k,n θI +θH +I �AIRS +k,n θI + ζk,n,a +� +≥ αa, ∀k, +where the steps for constraints C3-C5 in Subsection III-A1 are used exactly here. +B. Maximization over {sE,n, pI,n} +By introducing an auxiliary variable αb, the relay/IRS problem in (16) for fixed {UE,n, UI,n, τ} +boils down to the following optimization: +max +αb,{sE,n,pI,n}N +n=1 +αb +(52) +s.t. +C2 : (3), pI,k,n ≥ 0, ∀k, n, C3 : (9), C4 : Ek +� +{sE}N +n=1 +� +≥ Emin,k, ∀k, +C5 : +N +� +n=1 +log2 (1 + γk,n(pI,n)) ≥ αb, ∀k. +The constraints C4 and C5 of this sub-problem are non-convex. We first rewrite the SINR +associated with the kth pair in (11) as +γk,n(pI,n) = +aT +k,npI,n +bT +k,npI,n + σ2n �ψk,n + δ2 +k,n + σ2 +k,n +, +(53) +where ak,n = ψk,k,nek, bk,n = [ψk,1,n, ψk,2,n, · · · , ψk,k−1,n, 0 , ψk,k+1,n, · · · , ψk,K,n]T, and ek is +the kth unit vector. Therefore, the LHS of C5 in (52) is written as +N +� +n=1 +� +log2 +� +qT +k,npI,n + ζk,n,b +� +− log2 +� +bT +k,npI,n + ζk,n,b +� � +, +(54) +January 3, 2023 +DRAFT + +16 +where qk,n = ak,n +bk,n and ζk,n,b = σ2 +n �ψk,n +σ2 +k,n +δ2 +k,n. Then, similar to the procedure in Sub- +section III-A for C5, we resort to the MM technique. Precisely, considering the inequality in (30), +the second term in (54) is minorized by setting x = bT +k,npI,n + ζk,n,b and x0 = bT +k,np0 +I,n + ζk,n,b. +By substituting the minorizer in (54), the constraint C5 at the ith iteration is obtained as +C5 : +N +� +n=1 +� +log2 +� +qT +k,npI,n + ζk,n,b +� +− log2(bT +k,np(i−1) +I,n ++ ζk,n,b) +(55) +− +log2 e +bT +k,np(i−1) +I,n ++ ζk,n,b +bT +k,n +� +pI,n − p(i−1) +I,n +� � +≥ αb. +Next, we consider the non-convex constraint C4. It is observed that the term Ek +� +{sE,n}N +n=1 +� +in +the LHS of the this constraint is neither convex nor concave w.r.t. sE,n. Therefore, similar to +the procedure in Subsection III-A, we apply the MM by selecting βk,n,b (see Appendix C) and +minorize C4 at the ith iteration by +Ek +�� +s(i−1) +E,n +�N +n=1 +� ++ 1 +2 +N +� +n=1 +βk,n,b +� +s(i−1) +E,n +�H +s(i−1) +E,n ++ +N +� +n=1 +ℜ +� +ϑ(i) +k,n,b +� +sE,n − s(i−1) +E,n +�� +(56) +− 1 +2 +N +� +n=1 +βk,n,bsH +E,nsE,n ≥ Emin,k, ∀k, +where we define +ϑ(i) +k,n,b =βk,n,b +� +s(i−1) +E,n +�H ++ τ +ρexp�cexp +� +�alog2ω(i) +k,b +� � +ω(i) +k,b +��b−1 � +2�a log ω(i) +k,b + �b +� � +s(i−1) +E,n +�H +Ξk,n, +with ω(i) +k,b = +1 +2 +�N +n=1 +� +s(i−1) +E,n +�H +Ξk,ns(i−1) +E,n . Consequently, the ith iteration of the MM update +for (52) is obtained easily as the interior point solution of the following convex problem +max +αb,{sE,n,pI,n}N +n=1 +αb +(57) +s.t. +C2, C3, C4 : (56), C5 : (55). +C. Maximization over τ +The optimization problem in (16) w.r.t. τ becomes +min +τ +τ +(58) +s.t. +C1 : 0 ≤ τ ≤ T, +C2 : τvk ≤ �vk, ∀k, +C3 : τ�v1 ≤ �v2, +C4 : τ ≥ ¯vk, ∀k, +where +vk = 1 +2ρ +N +� +n=1 +� +|sE,k,n|2 − pI,k,n +� +, ∀k, +�vk = T +N +� +n=1 +� +prf +k,n − pI,k,n +2ρ +� +, ∀k, +(59) +January 3, 2023 +DRAFT + +17 +Algorithm 1 The Proposed Method for Minimum Rate Maximization in Relay/IRS Systems +1. Relay: Initialize U(l) +E,n, U(l) +I,n ∈ CMR×MR, τ (l) ∈ R+, l ← 0. +1. IRS: Initialize θ(l) +E , θ(l) +I +∈ CMIRS, τ (l) ∈ R+, l ← 0. +repeat +2. Relay: Initialize U(i) +E,n and U(i) +I,n and set i = 0. +2. IRS: Initialize θ(i) +E , θ(i) +I +and set i = 0. +repeat +3. Relay: Solve (40) to obtain {UE,n, UI,n, αa}. +3. IRS: Solve (51) to obtain {θE, θI, αa}. +4. Update i ← i + 1. +until convergence +5. Relay/IRS: Initialize s(i) +E,n, p(i) +I,n and set i = 0. +repeat +6. Relay/IRS: Solve the convex problem in (57) to obtain {sE,n, pI,n, αb}. +7. Update i ← i + 1. +until convergence +8. Relay/IRS: Compute τ (l) via the closed-form solution in (63). +9. Update l ← l + 1. +until convergence +�v1 = 1 +2ρ +N +� +n=1 +� +sH +E,nVE,nsE,n + σ2 +ntr +� +UE,nUH +E,n +� +− tr{QI,nVI,n} − σ2 +ntr{UI,nUH +I,n} +� +, +(60) +�v2 = T +N +� +n=1 +� +prf +n − 1 +2ρ +� +tr{QI,nVI,n} + σ2 +ntr +� +UI,nUH +I,n +��� +, +(61) +¯vk = +ρEmin,k +exp +� +�alog2pE,k +� +p�b +E,kexp (�c) +, ∀k. +(62) +Therefore, a closed-form solution (for a non-empty feasible set5) can be obtained as +τopt = max{¯v1, ¯v2, ..., ¯vK}. +(63) +5The following conditions lead to a non-empty feasible set for the problem: +1) ¯vk ≤ T, ∀k, 2) �vk ≥ 0, ∀k, 3) �v2 ≥ 0, 4) +�vj +vj |K +j=1 ≥ ¯vk, ∀k (for vj ≥ 0, ∀j), 5) �v2 +�v1 ≥ ¯vk, ∀k (for v1 ≥ 0). +January 3, 2023 +DRAFT + +18 +TABLE I +THE COMPUTATIONAL COMPLEXITY ORDER (PER INNER ITERATIONS) FOR STEP 3 OF THE ALGORITHM 1. +Relay +O +�� +2NM 2 +R(1 + 2N)(1 + K) +�3.5� +Relay (t-static) +O +�� +NM 2 +R(1 + N)(1 + 2K) +�3.5� +Relay (t-f-static) +O +�� +2M 2 +R(1 + 2K) +�3.5� +IRS +O +� +(6MIRS(N + K + 1))3.5� +IRS (t-static) +O +� +(2MIRS(N + 2K + 1))3.5� +Algorithm 1 summarizes the discussions in Section III and represents the steps of the proposed +method for maximizing the minimum rate of all user pairs in relay/IRS WPC systems. Note that +similar mathematical derivations are used to develop t-f-static algorithm for relay system as well +as t-static algorithm for both relay and IRS systems. +Remark 4 (convergence). It has been shown that under some mild conditions, the MM technique +converges to the stationary points of the problem [34], [35]. +D. Complexity Analysis +The main computational burdens in Algorithm 1 are associated with steps 3, 6, and 8. At +each inner iteration in step 3, the convex problems in (40) and (51) are solved via interior +point methods for relay and IRS system design, respectively, with a computational complexity +of O +� +(2NM2 +R(1 + 2N)(1 + K))3.5� +and O +� +(6MIRS(N + K + 1))3.5� +[36]. Table I compares +the computational complexity of step 3 for other versions of relay/IRS models. Similar to step +3, the complexity (per inner iterations) for step 6 which solves (57) (e.g., by using the interior +point methods) is O +� +(KN(1 + 2N)(5 + 2K))3.5� +for all versions of relay/IRS models. In step +8, the closed-form expression in (63) must be calculated leading to the complexity of6 O(N3). +IV. NUMERICAL EXAMPLES +Here, we evaluate the proposed relay/IRS method in different scenarios. The channels from +transmitters to the relay and the channels from the relay to the receivers are modeled as Hn = +0.1 +� �d1 +d0 +� −�γ +2 �Hn and Gn = 0.1 +� �d2 +d0 +� −�γ +2 �Gn, respectively, where d0 = 1 m is a reference distance, +�d1 is the distance between Tk and the relay, �d2 is the distance between the relay and Rk, and +6This can be decreased to O(N 2.3) via finding the best order of matrix multiplications (see [37] for details). +January 3, 2023 +DRAFT + +19 +T1 +T2 +TK +Relay or IRS +rT +d3 +d1 +d2 +rR +R1 +... +... +R2 +RK +Fig. 3. Simulation setup for relay/IRS WPC systems with K user pairs. +TABLE II +THE BASELINE BENCHMARK METHODS +Baseline 1 +Baseline 2 +Information Power Allocation +✓ +✓ +Energy Waveform Design +✗ +✓ +Time Allocation +✓ +✓ +Energy/Information Relay Beamforming +✓ +✗ +�γ = 3 is the path-loss exponent. It is assumed that the elements of �Hn and �Gn are i.i.d. CSCG +random variables with zero mean and unit variance. As shown in Fig. 3, the transmitters and +receivers are distributed uniformly within a circle with radius rT and rR, respectively. We set +the distance parameters as d1 = d2 = d3 = 10 m and rT = rR = 5 m. The maximum power +budget for Tk, relay, and IRS are set to prf +k,n = prf +R,n = 28 dBm, prf +IRS = 20 dBm, ∀k, n, and +the noise power at the relay, IRS and receivers are supposed to be σ2 +R,n = σ2 +k,n = δ2 +k,n = −80 +dBm, σ2 +IRS,n = −100 dBm, ∀k, n. The total bandwidth is fixed to Bt = 1 MHz. We further +assume the total operation time T = 1. The curve fitting parameters for non-linear EH circuits +are equal to �a = −0.11, �b = −1.17, and �c = −12 [32]. Also, we set K = 5, N = 8, MR = 6, +and Emin,k = Emin = 10 µW, ∀k, unless otherwise specified. We solve the convex optimization +problems using CVX [38]. +A. Relay System +Here, we compare the results of the proposed algorithms with partially optimized methods +(referred to as baseline schemes in the sequel) listed in Table II. For the first baseline method, +the energy signals are not optimized, and in the second baseline method, there is no optimization +January 3, 2023 +DRAFT + +20 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +6 +Minimum Rate (bps/Hz) +Number of Outer Iterations +0.2 +0.8 +5.75 +5.8 +Different Initializations +(a) +1 +2 +3 +4 +5 +6 +7 +0 +1 +2 +3 +4 +Minimum Rate (bps/Hz) +Number of Inner Iterations +(b) +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +4 +4.2 +4.4 +4.6 +4.8 +5 +Number of Inner Iterations +Minimum Rate (bps/Hz) + + +(c) +Fig. 4. +Convergence behavior of the proposed method in Algorithm 1: (a) outer iterations for three random initial points, +(b) inner iterations associated with the sub-problem III-A1 in the first outer iteration, (c) inner iterations associated with the +sub-problem III-B in the first outer iteration. +for the relay beamformer; more precisely, the relay amplification matrices are assumed to be +identity matrices, i.e. UR +E,n = �αE,nIMR, ∀n, and UR +I,n = �αI,nIMR, ∀n, where the scalar parameters +�αE,n and �αI,n are employed to satisfy the feasible set Ω in (16). The convergence of the proposed +algorithm for inner and outer iterations (see Algorithm 1) are plotted in Fig. 4. This figure shows +that the proposed algorithm converges within a few outer iterations. Also, in this example, the +three different initializations lead to almost the same final value. +In Fig. 5.a, we illustrate the rate-energy region of the proposed method in comparison with the +first baseline method for different number of subbands. We can observe that the minimum rate +increases as N grows. The optimal time allocation parameter τopt w.r.t. the EH target is depicted +in Fig. 5.b. It is seen that the increased energy threshold Emin leads to a larger τ. As τ increases, +the duration of the ID phase decreases. Therefore, as we observe in Fig. 5.a, the minimum rate +reduces with increasing Emin. Also, the impact of the energy waveform design is evident in both +figures. In Fig. 6.a and Fig. 6.b, we compare the minimum rate of the proposed optimal and sub- +optimal approaches with baseline methods. As we can see in Fig. 6.a, increasing the number +of pairs results in lower minimum rate for all methods with MR = 9. Furthermore, Fig. 6.b +shows that a larger MR increases the minimum rate with an almost linear trend. The importance +of the energy waveform and relay beamforming design is observed through both figures. We +can see that the method with no relay beamforming has the worst performance compared to +other methods since without a relay amplification matrix design, inter-pair interference cannot +be managed. +January 3, 2023 +DRAFT + +21 +10 20 30 40 50 60 70 80 90 100110120130 +0 +1 +2 +3 +4 +5 +6 +7 +EH Target, Emin (µW ) +Minimum Rate (bps/Hz) + + +Proposed - N = 8 +Baseline 1 - N = 8 +Proposed - N = 9 +Baseline 1 - N = 9 +(a) the rate-energy region +10 20 30 40 50 60 70 80 90 100110120130 +0 +0.2 +0.4 +0.6 +0.8 +1 +EH Target, Emin (µW ) +Time Allocation Parameter (s) + + +Proposed - N = 8 +Baseline 1 - N = 8 +Proposed - N = 9 +Baseline 1 - N = 9 +(b) the time allocation parameter τopt +Fig. 5. Comparison of the proposed and baseline 1 methods for different number of subbands N = 8, 9. +3 +4 +5 +6 +7 +8 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Number of Pairs, K +Minimum Rate (bps/Hz) + + +Proposed +Baseline 1 +Proposed (t−static) +Proposed (t−f−static) +Baseline 2 +(a) minimum rate versus number of pairs K +5 +6 +7 +8 +9 +10 +2 +3 +4 +5 +6 +7 +8 +9 +Number of Antennas, MR +Minimum Rate (bps/Hz) + + +Proposed +Baseline 1 +Proposed (t−static) +Proposed (t−f−static) +Baseline 2 +(b) minimum rate versus number of antennas MR +Fig. 6. Comparison of the proposed optimal and sub-optimal methods with baseline methods. +B. IRS System +In this subsection, the performance of the proposed IRS-assisted WPC system is evaluated. +Since most of the studied scenarios for relay (i.e., Fig. 4, Fig. 5, and Fig. 6.a) have similar trends +for IRS, we only consider the scenario of Fig. 6.b, for the sake of brevity. As we can see from +Fig.7, in the case of IRS, the minimum rate has a super-linear ascent property versus increasing +MIRS. +V. CONCLUSION +In this paper, the max-min rate maximization in a multi-carrier relay/IRS WPC system with a +joint TS scheme was considered. A unified framework was proposed to maximize the minimum +rate of the user pairs in both relay and IRS systems by jointly designing the energy waveforms, +January 3, 2023 +DRAFT + +22 +7 +10 +13 +16 +19 +22 +3 +3.5 +4 +4.5 +5 +5.5 +Number of REs, MIRS +Minimum Rate (bps/Hz) + + +Proposed +Proposed (t−static) +Fig. 7. The effect of the number of REs MIRS for the proposed optimal and sub-optimal IRS methods. +power of information waveforms, amplification matrices, and the time allocation parameter. +The non-linearity in EH circuits was also considered in the design problem. The non-convex +problem was handled via the MM technique. Numerical results demonstrated the effectiveness of +the proposed algorithm in terms of the minimum rate. As a extended future work in this area, it +might be interesting to develop a distributed algorithm for design of multi-user relay/IRS WPC +systems. +APPENDIX A +THE DERIVATION OF THE EXPRESSIONS IN (21) AND (25)-(27) +Using tr +� +XHY +� += vec(X)Hvec(Y) and vec(XYZ) = (ZT ⊗ X)vec(Y), the power of the +relay signal for ID mode in (7) can be obtained as +E +� +∥�r(t)∥2 +2 +� += 1 +2 +N +� +n=1 +� +uH +I,nvec +� +UI,nHnQI,nHH +n +� ++ σ2 +nuH +I,nuI,n +� += 1 +2 +N +� +n=1 +� +uH +I,n +�� +HnQI,nHH +n +�T ⊗ IMR +� +uI,n + σ2 +nuH +I,nuI,n +� += 1 +2 +N +� +n=1 +uH +I,n �AR +I,nuI,n. +Similarly, we can derive the power of the relay signal for the EH mode in (20) and the expressions +in (25)–(27). +January 3, 2023 +DRAFT + +23 +APPENDIX B +PROOF OF LEMMA 1 +By defining a positive semi-definite matrix D such that ∇2 +xs(x) ⪯ D, we can write the +following majorizer for s(x) as [39] +s(x) ≤ s(x0) + ℜ +� +(∇xs(x))H |x=x0(x − x0) +� ++ (x − x0)HD(x − x0), +(64) +where the gradient and Hessian of s(x) are respectively expressed as +∇xs(x) = −2 log2 e +xHTx + ν Tx, +(65) +∇2 +xs(x) = +� +−2T +xHTx + ν + +4TxxHT +(xHTx + ν)2 +� +log2 e. +Since T ⪰ 0, the term +−2T +xHTx+ν is negative semi-definite, and thus we obtain ξ > 0 such that for +any ν ⩾ 0 +4TxxHT +(xHTx + ν)2 log2 e ⩽ 4TxxHT +(xHTx)2 ⩽ ξIM2 +R. +Also, as TxxHT is a rank-one matrix, we can choose ξ as ξ ⩾ 4φ, where φ is given as +φ = max +x +xHT2x +(xHTx)2. +(66) +Then by choosing a = VHx, where V is a full-rank matrix such that T = VVH, the following +optimization is equivalently obtained from (66) as +φ = max +a +aHVHVa +aHa +1 +aHa. +(67) +Using xHQx ≤ P and applying a similar procedure in [39, Appendix B], we can write +φ ≤ +Pλmax(T) +vH +1 V−1QV−Hv1 +, +where v1 is the principal eigenvector of VHV. Finally, from (64), (65), and ξ = 4φ, we obtain +b = ∇s(x)|x=x0 = +−2 log2 e +xH +0 Tx0+νTx0 and D = +4P +wH +1 Qw1IM2 +R, where w1 is the principal eigenvector of +T. +January 3, 2023 +DRAFT + +24 +APPENDIX C +A SELECTION OF βk,n,a AND βk,n,b +The value of βk,n,b should be selected such that ∇2 +sE,nEk +� +{sE,n}N +n=1 +� ++βk,n,bIK ⪰ 0. The term +∇2 +sE,nEk +� +{sE,n}N +n=1 +� +is straightforwardly calculated as +∇2 +sE,nEk +� +{sE,n}N +n=1 +� +=̺k +N +� +n=1 +Ξk,n + ηk +N +� +n=1 +N +� +n′=1 +Ξk,nsE,nsH +E,n′Ξk,n′, +(68) +where +̺k = τ +ρexp�c exp +� +�alog2pE,k +� +p +�b−1 +E,k +� +2�a log pE,k + �b +� +, +ηk = +τexp�c exp +� +�alog2pE,k +� +p +�b−2 +E,k +ρ +� +4�a2log2pE,k + +� +4�a�b − 2�a +� +log pE,k + �b2 −�b + 2�a +� +. +As �a < 0,�b < 0, Ξk,n ⪰ 0, and Ξk,nsE,nsH +E,nΞk,n ⪰ 0, it suffices to choose βk,n,b such that +βk,n,bIK ⪰ − �̺k +N +� +n=1 +Ξk,n − �ηk +N +� +n=1 +N +� +n′=1 +Ξk,nsE,nsH +E,n′Ξk,n′, +(69) +where +�̺k = τ +ρ +�b exp�c exp +� +�alog2pE,k +� +p +�b−1 +E,k , +�ηk =τ +ρexp�c exp +� +�alog2pE,k +� +p +�b−2 +E,k +� +log pE,k +� +4�a�b − 2�a +� ++ 2�a +� +. +Thus from (3), we can write +∥sE,n∥2 +2 ≤ 2ρT +τ +K +� +k=1 +prf +k,n. +(70) +Finally, using (16), (69), (70) and knowing that sH +E,nΞk,nsE,n ≤ ∥sE,n∥2 +2λmax (Ξk,n), we can select +βk,n,b > βt +k,n,b where +βt +k,n,b = − τ +ρexp�c exp +� +2�alog2T +N +� +n=1 +λmax (Ξk,n) +K +� +k=1 +prf +k,n +� +�f +�b−2 +k +� �� +4�a�b − 2�a +� +log �fk + 2�a +� +× +N +� +n=1 +N +� +n′=1 +λmax (Ξk,nΞk,n′) +K +� +k=1 +� +prf +k,nprf +k,n′ + �b �fk +N +� +n=1 +λmax (Ξk,n) +� +, +with �fk = exp +� +−�b− +� +�b2−4�a log +ρEmin,k +τexp�c +2�a +� +. We can take similar steps for selecting βt +k,n,a. +January 3, 2023 +DRAFT + +25 +APPENDIX D +PROOF OF LEMMA 2 +The ID part of the relay power constraint in (20) is uH +I,n �AR +I,nuI,n. Only (iMIRS + i+ 1)th, 0 ≤ +i ≤ MIRS − 1 entries of uI,n = vec(Diag(θI)) are non-zero for the IRS system. Thus, we can +rewrite uH +I,n �AR +I,nuI,n for IRS system as θH +I �AIRS +I,n θI, where �AIRS +I,n contains only the (kMIRS + k + +1, lMIRS +l+1)th, 0 ≤ k, l ≤ MIRS −1 entries of �AI,n which is the same as �AR +I,n with replacing +MR by MIRS. Therefore, from (21) and by using some matrix manipulations, we obtain +�AIRS +I,n = +� +HnQI,nHH +n +�T ⊙ IMIRS + σ2 +nIMIRS. +(71) +Other expressions in (44)-(48) are similarly obtained. +REFERENCES +[1] C. K. Ho and R. Zhang, “Optimal energy allocation for wireless communications with energy harvesting constraints,” +IEEE Transactions on Signal Processing, vol. 60, no. 9, pp. 4808–4818, Sept. 2012. +[2] T. D. P. Perera, D. N. K. Jayakody, S. K. Sharma, S. 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Ottersten, “Efficient sum-rate maximization for medium-scale +MIMO AF-relay networks,” IEEE Transactions on Wireless Communications, vol. 15, no. 9, pp. 6400–6411, Sept. 2016. +January 3, 2023 +DRAFT + diff --git a/X9AyT4oBgHgl3EQfWfd0/content/tmp_files/load_file.txt b/X9AyT4oBgHgl3EQfWfd0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..82569ed9ae64cc009b61d7e730c09f554907cda0 --- /dev/null +++ b/X9AyT4oBgHgl3EQfWfd0/content/tmp_files/load_file.txt @@ -0,0 +1,949 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf,len=948 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='00164v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='SP] 31 Dec 2022 1 Design of a Multi-User Wireless Powered Communication System Employing Either Active IRS or AF Relay Omid Rezaei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Maryam Masjedi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Ali Kanaani,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Mohammad Mahdi Naghsh∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Saeed Gazor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' and Mohammad Mahdi Nayebi Abstract In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' we optimize a Wireless Powered Communication (WPC) system including multiple pair of users,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' where transmitters employ single-antenna to transmit their information and power to their receivers with the help of one multiple-antennas Amplify-and-Forward (AF) relay or an active Intelligent Reflecting Surface (IRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We propose a joint Time Switching (TS) scheme in which transmitters, receivers, and the relay/IRS are either in their energy or information transmission/reception modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The transmitted multi-carrier unmodulated and modulated waveforms are used for Energy Harvesting (EH) and Information Decoding (ID) modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In order to design an optimal fair system, we maximize the minimum rate of all pairs for both relay and IRS systems through a unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' This framework allows us to simultaneously design energy waveforms, find optimal relay/IRS amplifi- cation/reflection matrices, allocate powers for information waveforms, and allocate time durations for various phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In addition, we take into account the non-linearity of the EH circuits in our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' This problem turns out to be non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Thus, we propose an iterative algorithm by using the Minorization- Maximization (MM) technique, which quickly converges to the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Numerical examples show that the proposed method improves the performance of the multi-pair WPC relay/IRS system under various setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Rezaei and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Nayebi are with the Department of Electrical Engineering, Sharif University of Technology, Tehran, 11155-4363, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Masjedi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Kanaani, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Naghsh are with the Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, 84156-83111, Iran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Gazor is with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, Ontario, K7L 3N6, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' ∗Please address all the correspondence to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Naghsh, Phone: (+98) 31-33912450;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Fax: (+98) 31-33912451;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Email: mm naghsh@cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='iut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='ir January 3, 2023 DRAFT 2 Index Terms Fair Throughput Maximization, Intelligent Reflecting Surface (IRS), Minorization-Maximization (MM), Relay Networks, Wireless Powered Communication (WPC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' INTRODUCTION Wireless Power Transfer (WPT) technology is introduced to extend the lifetime of devices in wireless networks in which the energy is emitted from the dedicated power sources to the devices [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Interestingly, WPT enables Simultaneous Wireless Information and Power Transfer (SWIPT) [2], where devices not only receive and decode information, but also harvest the energy from Radio Frequency (RF) signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The Time Switching (TS) and Power Splitting (PS) schemes are two well-known implementing protocols of SWIPT [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Recently, SWIPT models are designed to employ relays to further enhance the coverage and spectral efficiency of wireless networks [4]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In [4], a Multiple-Input Multiple-Output (MIMO) two-way relay system is introduced in which two transceivers exchange their information through a relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The authors in [5] designed the relay and source precoders by minimizing the bit error rate at the destination for a full-duplex MIMO relay system and SWIPT-enabled users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' A similar system with a half-duplex two-way relay is designed in [6] by minimizing the mean square error at the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Another impactful technology that is currently emerging is to use of the Intelligent Reflecting Surface (IRS) in wireless communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' This promising solution not only is capable of improving energy delivery but also can enhance the spectral efficiency of future wireless communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' An IRS is an array of large number of Reflecting Elements (REs) designed to have controllable electromagnetic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Each RE introduces a phase shift on the impinging signal, allowing beamforming/manipulation of the reflection waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Precisely, the IRS matrix allows controlling the reflected signal (amplify, attenuated, steer in the desired direction, and so on) toward optimal desired directions by purposefully designing the phase shift matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The IRS is exploited recently in SWIPT systems in [8]–[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The weighted sum harvested power maximization problem was studied in [8] for an IRS-aided SWIPT model in which a multiple-antennas access point serves multiple single-antenna users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In [9], the model in [8] is extended for a more practical multi-objective optimization problem by taking into account the trade-off between sum rate and sum harvested power maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In [10], the total transmission power is minimized for a Multiple-Input Single-Output (MISO) SWIPT system January 3, 2023 DRAFT 3 employing multiple IRSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In [11], the MISO SWIPT in [8]–[10] is extended to the MIMO case, and the weighted sum rate is maximized in an IRS-assisted system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The design of the energy waveform remarkably affects the performance of WPT-based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Indeed, an efficient waveform leads to significant improvement in the efficacy of power delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Experiments reveal that signals with high Peak to Average Power Ratio (PAPR) such as multi- sine signals provide more DC power at Energy Harvesting (EH) circuits than constant envelope signals with the same average RF power [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Based on this interesting observation, a multi-sine waveform design for WPT has been examined in several works [13]–[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Waveform design with a non-linear EH model was considered in [13] and [14] for MISO and Single-Input Single- Output (SISO) systems, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The authors of [15] proposed a low-complexity method for a waveform design in a SISO WPT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In [16], the transmit waveform was designed based on limited Channel State Information (CSI) feedback WPT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Then, the authors in [17] studied waveform design for an IRS-aided SWIPT MISO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The aforementioned methods for design of single-user systems were extended to the multi-user case in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Also, a waveform design was performed in [19] for wireless powered backscatter communication networks, and this work was extended to multi-user backscatter systems in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The authors of [21] investigated the waveform and transceiver design problem in a MISO SWIPT system and determined the multi-sine waveforms for Information Decoding (ID) and EH phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In this paper, we optimize a multi-user wireless powered relay/IRS system using a multi-sine waveform with the following main contributions: Relay model: To the best of our knowledge, a multi-sine waveform design for multiple user pairs in a wireless powered relay system has not been addressed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In this paper, we consider a multi-carrier Wireless Powered Communication (WPC) system for multi-user relay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Precisely, in our proposed model, an amplify-and-forward (AF) relay provides energy/information transmission from K transmitters to their receivers by adopting the TS scheme in all nodes1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Herein, the aim is to design the multi-carrier unmodulated energy waveforms and the allocated power for information waveforms at the transmitters, the amplification matrices in a relay, and the time durations for the EH and ID modes in order to maximize the minimum rate of the user pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In addition, we consider 1Note that the system in [22], where a TS scheme is only applied for a receiver, is considered as a special case of the proposed joint TS scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' January 3, 2023 DRAFT 4 the effect of the non-linearity of EH circuits in the design problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' IRS model: In the case of IRS-assisted communication, multi-pair WPC has not been considered in the literature, and therefore, herein, we consider this type of IRS-assisted systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Precisely, in this case, an active IRS2 replaced with the AF relay in the proposed system model mentioned in the above paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Also, some comparisons are made between relay and IRS models in terms of architecture and performance (see Remark 1-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Unified consideration of relay and IRS: Both proposed AF relay and active IRS-aided systems are modeled under a unified formulation, and we handle the resulting optimization problems under a unified mathematical umbrella.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We show that the problem is non-convex and consequently, is hard to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' To deal with this problem, we devise a method based on the Minorization-Maximization (MM) technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Interestingly, the proposed algorithm can deal with relay and IRS systems by switching between Kronecker and Hadamard products for parameters used in the algorithm (see Lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Sub-optimal methods: Some sub-optimal methods with lower signaling overhead and com- putational complexity are proposed and then, their performance are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Numerical result: Simulation results are reported to illustrate the effectiveness of the pro- posed method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' particularly, the impact of the relay/IRS matrix and energy waveform design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Also, numerical examples show that the minimum rate of users increases linearly/super- linearly with the number of antennas/REs in relay/IRS systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The rest of this paper is organized as follows: The signal and system models are explained in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In Section III, the minimum rate maximization problem is formulated, and a unified optimization framework is proposed for both relay and IRS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Section IV presents numerical examples to illustrate the effectiveness of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Finally, conclusions are drawn in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Notation: Bold lowercase (uppercase) letters are used for vectors (matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The notations arg(·), E[·], ℜ{·}, ∥·∥2, (·)T, (·)H, (·)∗, tr{·}, λmax(·), vec(·), Diag(·), ∇xf(·) and ∇2 xf(·) indicate the phase of a complex number, statistical expectation, real-part, l2-norm of a vector, transpose, Hermitian, complex conjugate, trace of a matrix, the principal eigenvalue of a matrix, stacking of the column of a matrix, a diagonal matrix formed by the entries, the gradient of a function with respect to (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=') x and the Hessian of a function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' x, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The symbols ⊗ and 2Note that in an active IRS, REs can amplify the reflected signals using their reflection-type amplifiers [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' January 3, 2023 DRAFT 5 T1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' τ T − τ Information Transmitter Energy Transmitter Tk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' TK AF Relay or Active IRS R1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Rk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' τ T − τ Information Decoder Energy Harvester RK Blockage Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Multi-user wireless powered relay/IRS system based on TS scheme with blocked direct path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' ⊙ stand for the Kronecker and Hadamard products of two matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We denote CN (ω, Σ) as a circularly symmetric complex Gaussian (CSCG) distribution with mean ω and covariance Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The set R+ represents non-negative real numbers and CN×N and DN×N are the set of N × N complex and complex diagonal matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The set of N × N positive (semi-)definite and identity matrices are denoted by SN ++ ⊂ CN×N (SN + ⊂ CN×N) and IN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The notation A ≻ B (A ⪰ B) means that A − B is positive (semi-)definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' SYSTEM MODEL We consider a multi-carrier wireless powered relay/IRS system with K user pairs {(Tk, Rk)}K k=1 as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 1, where the direct links between the transmitters and receivers are likely blocked (see [24] and [25], [26] for similar models of multiple user pairs with blocked direct path for relay and IRS systems, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The single-antenna transmitter Tk communicates with its receiver Rk through either an AF relay with MR antennas or an active IRS with MIRS REs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We assume that Rk harvests a part of its required power, whereas Tk and the relay/IRS have no energy concern [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In each time duration T, the relay/IRS helps Rk not only harvest energy from the signal of all {Tk}K k=1, but also decode the information from its corresponding transmitter Tk by using a joint TS scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Precisely, Tk, relay/IRS, and Rk switch simultaneously at time t = τ from their energy delivery modes to their communication modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We assume that all nodes are perfectly synchronized as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 2 for this switching [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We consider a multi- carrier system with a total bandwidth of Bt equally divided into N orthogonal subbands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We also model all channels to have a frequency-selective block fading, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=', the channel coefficients January 3, 2023 DRAFT 6 Relay: IRS: {Tk}K k=1 → IRS → {Rk}K k=1 {Tk}K k=1 → relay relay → {Rk}K k=1 {Tk}K k=1 → IRS → {Rk}K k=1 {Tk}K k=1 → relay relay → {Rk}K k=1 τ 2 τ (energy waveform) τ 2 T −τ 2 T − τ (information waveform) T −τ 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The transmission, amplification/reflection, and reception timeline for the proposed relay/IRS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' remain constant for at least T seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Let the complex random matrices HR n ∈ CMR×K and GR n ∈ CK×MR denote the channels from transmitters to the relay and the channels from the relay to the receivers for nth subband, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The elements of HR n and GR n are zero mean CSCG random variables in the case of Rayleigh fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Similarly, we denote the channels from transmitters to the IRS and the channels from the IRS to the receivers by HIRS n ∈ CMIRS×K and GIRS n ∈ CK×MIRS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In the sequel, Hn and Gn refer to either HR n and GR n or HIRS n and GIRS n , depending on the case under discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In addition, we assume that we can control the relay and IRS by collecting and using the CSI of all links [25]–[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' For example, a relay itself can act as a controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The CSI may be estimated in various ways, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=', by using orthogonal pilot sequences ( see [28], [29] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The CSI estimation is out of the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Also, we propose two low-complexity implementation methods mentioned in Remark 3 to reduce the signaling overhead in the controller node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Each Tk transmits a multi-sine energy waveform xE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k(t) and a multi-carrier modulated infor- mation waveform xI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k(t) to the relay/IRS during the first-hop transmission at the EH and ID time slots,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' as follows xE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k(t) = N � n=1 aE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='ncos(2πfnt + φE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' = ℜ � N � n=1 sE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nej2πfnt � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (1) xI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k(t) = N � n=1 aI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n(τ)cos(2πfnt + φI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n) = ℜ � N � n=1 sI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nej2πfnt � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (2) where sE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n = aE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nejφE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n and sI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n = aI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nejφI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n are the baseband complex signal represen- tations for the energy and information waveforms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We assume that the baseband information signals are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' CSCG random variable variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=', sI,k,n ∼ CN (0, pI,k,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The transmitted energy by Tk is constrained by τ 2ρ|sE,k,n|2 + T − τ 2ρ pI,k,n ≤ Tprf k,n, ∀k, n, (3) January 3, 2023 DRAFT 7 where prf k,n is the maximum power budget at Tk for the nth subband and ρ addresses both ρR = 2 for relay and ρIRS = 1 for IRS system according to the proposed timeline in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 2 (see also Remark 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' By defining sE,n = [sE,1,n, · · · , sE,K,n]T and sI,n = [sI,1,n, · · · , sI,K,n]T, the received signal at the relay/IRS is expressed as r(t) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 N� n=1 {HnsE,n + zn}, t ∈ TEH, for EH, N� n=1 {HnsI,n + zn}, t ∈ TID, for ID, (4) where TEH = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 0 ≤ t ≤ τ 2, for relay, 0 ≤ t ≤ τ, for IRS, TID = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 τ ≤ t ≤ τ + T−τ 2 , for relay, τ ≤ t ≤ T, for IRS, (5) and r denotes either rR or rIRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Furthermore, the AWGN zn denotes either zR n ∼ CN (0, σ2 R,nIMR) or zIRS n ∼ CN (0, σ2 IRS,nIMIRS) for relaying or reflecting modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In contrast to the passive IRS, an active IRS adds non-negligible noise (which is introduced by the active elements [23], [30]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' however, the added noise of an active IRS has considerably less impact compared to the relay noise (which is introduced by RF chains), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=', σ2 IRS,n ≤ σ2 R,n [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In the second-hop transmission, the relay/IRS amplifies the energy and information signals of Tk by amplification/reflection matrices and then forwards them to Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' For AF relay system, the amplification matrices is introduced as UR E,n and UR I,n ∈ CMR×MR, ∀n for energy and information phases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In the case of IRS-aided system, the reflection matrices is defined as UIRS E = Diag(θE) and UIRS I = Diag(θI) for energy and information time slots, respectively, where θE = [ηE,1ejθE,1, ηE,2ejθE,2, · · · , ηE,MIRSejθE,MIRS]T and θI = [ηI,1ejθI,1, ηI,2ejθI,2, · · · , ηI,MIRSejθI,MIRS]T with ηE,m, ηI,m ≥ 1 and θE,m, θI,m ∈ [0, 2π] respectively denote the reflection amplitude and the phase shift at the mth RE3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' An active IRS amplifies the signal without any significant delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' However, in an AF relay, the signal reception, amplification, and transmission at the RF chain cause a long delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Therefore, in practice, the AF relay requires twice time compared to the active IRS for transmission one information symbol [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 3Note that passive and passive lossless IRS require ηE,m, ηI,m ∈ [0, 1] and ηE,m = ηI,m = 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' January 3, 2023 DRAFT 8 We define UE,n and UI,n to address both UR E,n, UIRS E and UR I,n, UIRS I , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The forwarded signal by the relay/IRS is given by �r(t)= \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 �N n=1 UE,n (HnsE,n + zn) , t ∈ �TEH, for EH, �N n=1 UI,n (HnsI,n + zn) , t ∈ �TID, for ID, where �TEH = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 τ 2 ≤ t ≤ τ, for relay, 0 ≤ t ≤ τ, for IRS, �TID = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 τ + T−τ 2 ≤ t ≤ T, for relay, τ ≤ t ≤ T, for IRS, (6) and �r denotes either �rR or �rIRS for relay or IRS system, with a slight abuse of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Then, the power of �r(t) from the relay/IRS is written as E � ∥�r(t)∥2 2 � = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 1 2 N� n=1 � sH E,nVE,nsE,n + σ2 ntr � UE,nUH E,n �� , for EH, 1 2 N� n=1 � tr {QI,nVI,n} + σ2 ntr � UI,nUH I,n �� , for ID, (7) where σ2 n addresses both σ2 R,n and σ2 IRS,n, QI,n = Diag(pI,1,n, pI,2,n, · · · , pI,K,n) and VE,n = HH n UH E,nUE,nHn, ∀n, VI,n = HH n UH I,nUI,nHn, ∀n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (8) Using (7), the total consumed energy is bounded at the relay/IRS in t ∈ [0, T] as τ 2ρ � sH E,nVE,nsE,n + σ2 ntr{UE,nUH E,n} � + T − τ 2ρ � tr{QI,nVI,n} + σ2 ntr{UI,nUH I,n} � ≤ Tprf n , ∀n, (9) where prf n denotes either the maximum power budget at the relay prf R,n or IRS prf IRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We can write received signal at Rk as yk(t) = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 N� n=1 � gT k,nUE,n (HnsE,n + zn) + zk,n � , t ∈ �TEH, ∀k, for EH, N� n=1 � gT k,nUI,n (HnsI,n + zn) + zk,n + �zk,n � , t ∈ �TID, ∀k, for ID, where gk,n is the kth column vector of GT n, and zk,n as well as �zk,n are the AWGN from the antenna and baseband processing noises at Rk, respectively, with zk,n ∼ CN(0, σ2 k,n) and �zk,n ∼ CN (0, δ2 k,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The information signals at Rk corresponding to the nth subband can be expanded as yk,n(t) =gT k,nUI,nhk,nsI,k,n + gT k,nUI,n K � j=1,j̸=k hj,nsI,j,n + gT k,nUI,nzn + zk,n + �zk,n, ∀k, n, (10) January 3, 2023 DRAFT 9 where hk,n and gk,n are the kth column vector of Hn and GT n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' By defining pI,n = [pI,1,n, pI,2,n, · · · , pI,K,n]T, the SINR at the ID part for the nth subband is given by γk,n(pI,n, UI,n) = pI,k,nψk,k,n �K j=1,j̸=k pI,j,nψk,j,n + σ2n �ψk,n + δ2 k,n + σ2 k,n , ∀k, n, (11) where ψk,j,n = gT k,nUI,nhj,nhH j,nUH I,ng∗ k,n and �ψk,n = gT k,nUI,nUH I,ng∗ k,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' From Remark 1, we obtain the achievable rate at the kth pair as follows Rk � {pI,n}N n=1, {UI,n}N n=1, τ � = T − τ ρT N � n=1 log2 � 1 + γk,n(pI,n, UI,n) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (12) For the EH stream, we assume the noise power is negligible compared to the received signal power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We take into account the rectifier non-linearity by employing the results from [32] where the harvested energy at Rk is approximated by Ek � {sE,n}N n=1, {UE,n}N n=1, τ � = τ ρexp � �alog2pE,k � p �b E,kexp�c, ∀k, (13) where �a, �b, and �c are the curve fitting constants and pE,k is the average input power to Rk’s harvester as pE,k � {sE,n}N n=1, {UE,n}N n=1 � = 1 2 N � n=1 sH E,nΞk,nsE,n, ∀k, (14) with Ξk,n = HH n UH E,ng∗ k,ngT k,nUE,nHn, ∀k, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (15) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Note that the reflection matrix cannot be designed separately for each subband in the IRS system, while, thanks to the RF chain circuits in a relay, the amplification matrix design is considered for each subband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We note that an active IRS is considerably less expensive than an AF relay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' This is because an AF relay requires massive integrated circuits (including analog-to- digital/digital-to-analog converter, self-interference cancellation circuits, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The delay caused by RF chain processing of an AF relay contributes to latency, leads to lower transmission time, and requires more power for energy and information signals (see (3) and (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Therefore, a relay-IRS trade-off exists in the system performance (see (12) and (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' An approach with lower implementation complexity is considered in which only one amplification/reflection matrix needs to be designed for both energy and information time slots, called the t-static approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Also, one can consider another approach with only one amplification matrix design in both time slots and all subbands, referred to as t-f-static in the relay system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' These design methodologies lead to a lower signaling overhead and system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' January 3, 2023 DRAFT 10 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' THE PROPOSED MINIMUM RATE MAXIMIZATION METHOD In this section, the aim is to maximize the minimum rate of the multi-user relay/IRS WPC system w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' multi-sine energy waveforms sE,n, allocated power pI,n, amplification/reflection matrices UE,n, UI,n, and the time allocation parameter τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The unified minimum rate maximization problem for both relay and IRS systems is cast as max τ,{sE,n,pI,n,UE,n,UI,n}N n=1 min 1≤k≤K Rk (16) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' � τ, {sE,n, pI,n, UE,n, UI,n}N n=1 � ∈ Ω, where Ω = Ω0 ∩ Ωind with Ω0 = � C1 : 0 ≤ τ ≤ T, C2 : (3), pI,k,n ≥ 0, ∀k, n, C3 : (9), C4 : Ek ≥ Emin,k, ∀k � , (17) Ωind = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 CR : UE,n, UI,n ∈ CMR×MR, ∀n, for relay, CIRS : UE,n, UI,n ∈ DMIRS×MIRS, ∀n, |θE,m| ≥ 1, |θI,m| ≥ 1, ∀m, for IRS, (18) and Emin,k in C4 is the minimum required harvested energy for the kth user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The problem in (16) is non-convex due to the coupled design variables in the objective function and the constraints C2 − C4 and CIRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' To deal with this non-convex problem, we first solve the problem w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' {UE,n, UI,n} for fixed {sE,n, pI,n, τ}, then optimize {sE,n, pI,n} for given {UE,n, UI,n, τ}, and finally, solve the problem w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' τ via a closed-form solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The procedure is repeated until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Maximization over {UE,n, UI,n} Here, we first consider the relay problem, and then the IRS problem is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 1) Relay System: The problem in (16) for fixed {sE,n, pI,n} reduces to the following opti- mization max {UE,UI}N n=1 min 1≤k≤K N � n=1 log2 (1 + γk,n(UI,n)) (19) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' C3, C4, January 3, 2023 DRAFT 11 which is still a non-convex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' To start solving the problem, first we need to reformulate the obtained expressions for the relay power constraint (7), SINR (11), and the input power of harvesters (14) from Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We can rewrite (7) as (see Appendix A for the derivation) E � ∥�r(t)∥2 2 � = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 1 2 N� n=1 uH E,n �AR E,nuE,n, 0 ≤ t ≤ τ, for EH, 1 2 N� n=1 uH I,n �AR I,nuI,n, τ ≤ t ≤ T, for ID, (20) where uE,n = vec(UE,n), uI,n = vec(UI,n), and �AR E,n = � HnsE,nsH E,nHH n �T ⊗ IMR + σ2 nIM2 R, �AR I,n = � HnQI,nHH n �T ⊗ IMR + σ2 nIM2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (21) Therefore, we rewrite the relay power constraint in (9) by using (20) as τ 2ρuH E,n �AR E,nuE,n + T − τ 2ρ uH I,n �AR I,nuI,n ≤ Tprf R,n, ∀n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (22) Next, we rewrite the SINR and the input power at Rk’s harvester in (11) and (14) as γk,n(uI,n) = uH I,nAR k,nuI,n uH I,n �AR k,nuI,n + δ2 k,n + σ2 k,n , ∀k, n, (23) pE,k � {uE,n}N n=1 � = 1 2 N � n=1 uH E,n ¯AR k,nuE,n, ∀k, (24) where AR k,n = pI,k,n � hk,nhH k,n �T ⊗ g∗ k,ngT k,n, (25) �AR k,n= K � j=1,j̸=k pI,j,n � hj,nhH j,n �T ⊗ g∗ k,ngT k,n + σ2 nIMR ⊗ g∗ k,ngT k,n, (26) ¯AR k,n = � HnsE,nsH E,nHH n �T ⊗ g∗ k,ngT k,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (27) By using (22), (23), and (24) with an auxiliary variable αa the optimization problem in (19) can be equivalently rewritten as max αa,{uE,uI}N n=1 αa (28) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' C3 : (22), C4 : Ek � {uE,n}N n=1 � ≥ Emin,k, ∀k, C5 : N � n=1 log2 � 1 + uH I,nAR k,nuI,n uH I,n �AR k,nuI,n + ζk,n,a � ≥ αa, ∀k, where ζk,n,a = σ2 k,n + δ2 k,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The constraint C5 can be equivalently rewritten as C5 : N � n=1 � log2 � uH I,nBk,nuI,n + ζk,n,a � − log2 � uH I,n �AR k,nuI,n + ζk,n,a � � ≥ αa, (29) January 3, 2023 DRAFT 12 where Bk,n = �AR k,n + AR k,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' It is observed that this constraint is non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Therefore, we employ the MM technique to tackle its non-convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Precisely, we minorize the denominator term − log2 � uH I,n �AR k,nuI,n +ζk,n,b � by the using the following inequality log2(x) ≤ log2(x0) + log2 e x0 (x − x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (30) By setting x = uH I,n �AR k,nuI,n + ζk,n,a and x0 = � u0 I,n �H �AR k,nu0 I,n + ζk,n,a in (30) we obtain − log2 � uH I,n �AR k,nuI,n + ζk,n,a � ≥ − log2 �� u0 I,n �H �AR k,nu0 I,n + ζk,n,a � (31) − log2 e � uH I,n �AR k,nuI,n − � u0 I,n �H �AR k,nu0 I,n � � u0 I,n �H �AR k,nu0 I,n + ζk,n,a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Applying the above minorizer, the constraint C5 in (29) is rewritten at the ith iteration of the MM technique as N � n=1 � log2 � uH I,nBk,nuI,n + ζk,n,a � − log2 �� u(i−1) I,n �H �AR k,nu(i−1) I,n + ζk,n,a � (32) − log2 e � u(i−1) I,n �H �AR k,nu(i−1) I,n + ζk,n,a � uH I,n �AR k,nuI,n − � u(i−1) I,n �H �AR k,nu(i−1) I,n � � ≥ αa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The following lemma lays the ground for dealing with the first non-concave logarithmic term in (32) in light of the MM technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Let s(x) = − log2 � xHTx + ν � and xHQx ≤ P for any positive-definite matrices T, Q ∈ SN ++ and P ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Then, s(x) is bounded for all x and x0 as follows s(x) ≤ s(x0) + ℜ � bH(x − x0) � + (x − x0)HD(x − x0), where b = −2 log2 e xH 0 Tx0+νTx0, D = 4P wH 1 Qw1IM2 R, and w1 is the principal eigenvector of T and ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Using Lemma 1 and noting that the term τ 2uH E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �AR E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nuE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n in (22) is positive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' we obtain the following minorizer for the term log2(uH I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nBk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nuI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n + ζk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='a) in (32) at any given u0 I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n log2(uH I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nBk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nuI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n + ζk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='a) ≥ log2 �� u0 I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �H Bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nu0 I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n + ζk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='a � − ℜ � bH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n(uI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n − u0 I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n) � (33) − � uI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n − u0 I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �H Dk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n(uI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n − u0 I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' where bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n = −2 log2 e � u0 I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �H Bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nu0 I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n + ζk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='a Bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nu0 I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Dk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n = 16T T−τ prf R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �wH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �AR I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �wk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n IM2 R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 2023 DRAFT 13 and �wk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n denotes the principal eigenvector of Bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Applying (33),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' the constraint in (32) is restated as − N � n=1 � log2 e uH I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �AR k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nuI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n � u(i−1) I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �H �AR k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nu(i−1) I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n + ζk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='a + uH I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nDk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nuI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n + ℜ �� bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n − 2Dk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nu(i−1) I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �H uI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n � + d(i) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n � ≥ αa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' ∀k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (34) where d(i) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n = log2 � u(i−1) I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �H �AR k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nu(i−1) I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n + ζk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='a � u(i−1) I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �H Bk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nu(i−1) I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n + ζk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='a − ℜ � bH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nu(i−1) I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n � + � u(i−1) I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �H Dk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nu(i−1) I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n (35) − log2 e � u(i−1) I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �H �AR k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nu(i−1) I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n � u(i−1) I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �H �AR k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nu(i−1) I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n + ζk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Then, we can simplify constraint in (34) as − N � n=1 � uH I,nF(i) k,nuI,n + ℜ � (f(i) k,n)HuI,n � + d(i) k,n � ≥ αa, ∀k, (36) where F(i) k,n = log2 e �AR k,n � u(i−1) I,n �H �AR k,nu(i−1) I,n + ζk,n,a + Dk,n, f(i) k,n = bk,n − 2Dk,nu(i−1) I,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (37) Finally, we focus on the constraint C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' From (13) and (24), we see that in the left-hand side (LHS) of C4, Ek is neither convex nor concave w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' uE,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' To apply the MM technique on LHS of C4, we first define a parameter4 βk,n,a such that ∇2 uE,nEk � {uE,n}N n=1 � +βk,n,aIM2 R ⪰ 0, ∀k, n, and write Ek as the sum of a convex and a concave function as Ek � {uE,n}N n=1 � =Ek � {uE,n}N n=1 � + 1 2 N � n=1 βk,n,auH E,nuE,n − 1 2 N � n=1 βk,n,auH E,nuE,n, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (38) We now apply the MM technique to C4 and obtain a convex constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' To do so, we keep the concave part and minorize the convex part of (38) and rewrite C4 as Ek � {u(i−1) E,n }N n=1 � + 1 2 N � n=1 βk,n,a � u(i−1) E,n �H u(i−1) E,n + N � n=1 ℜ � ϑ(i) k,n,a � uE,n − u(i−1) E,n �� (39) − 1 2 N � n=1 βk,n,auH E,nuE,n ≥ Emin,k, ∀k, 4See Appendix C for a selection of βk,n,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' January 3, 2023 DRAFT 14 where ϑ(i) k,n,a =βk,n,a � u(i−1) E,n �H + τexp�c 2 exp � �alog2ω(i) k,a � � ω(i) k,a ��b−1 � 2�a log ω(i) k,a + �b � � u(i−1) E,n �H ¯AR k,n, with ω(i) k,a = 1 2 �N n=1 � u(i−1) E,n �H ¯AR k,nu(i−1) E,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Therefore, the ith MM iteration for (19) is the solution of the following convex problem max αa,{uE,n,uI,n}N n=1 αa (40) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' C3 : (22), C4 : (39), C5 : (36), which can be solved efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 2) IRS System: By considering UE,n = Diag(θE), UI,n = Diag(θI) and adding the constraint CIRS in (18), the optimization problem in (19) is considered in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Since UE,n and UI,n are diagonal matrices, the expressions in (22)-(24) are modified as τ 2θH E �AIRS E,nθE + T − τ 2 θH I �AIRS I,n θI ≤ Tprf IRS, ∀n, (41) γk,n(θI) = θH I AIRS k,n θI θH I �AIRS k,n θI + δ2 k,n + σ2 k,n , ∀k, n, (42) pE,k (θE) = 1 2 N � n=1 θH E ¯AIRS k,n θE, ∀k, (43) where their parameters are defined in Lemma 2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The parameters �AIRS E,n, �AIRS I,n , AIRS k,n , �AIRS k,n , and ¯AIRS k,n are expressed as follows: �AIRS E,n = � HnsE,nsH E,nHH n �T ⊙ IMIRS + σ2 nIMIRS, (44) �AIRS I,n = � HnQI,nHH n �T ⊙ IMIRS + σ2 nIMIRS, (45) AIRS k,n = pI,k,n � hk,nhH k,n �T ⊙ g∗ k,ngT k,n, (46) �AIRS k,n = K � j=1,j̸=k pI,j,n � hj,nhH j,n �T⊙ g∗ k,ngT k,n+ σ2 nIMIRS⊙ g∗ k,ngT k,n, (47) ¯AIRS k,n = � HnsE,nsH E,nHH n �T ⊙ g∗ k,ngT k,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (48) It is worth pointing out that the only difference between the parameters above and their corre- sponding expressions in (21) and (25)-(27), is the symbol of multiplication, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=', ⊗ and ⊙, in a January 3, 2023 DRAFT 15 proper dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The proper dimension consideration means MR → MIRS for all of the above parameters and IM2 R → IMIRS for the second terms of �AIRS E,n and �AIRS I,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Next, we focus on constraint CIRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' First, let us introduce the following minorizer [33] |x| ≥ ℜ � x∗ x0 |x0| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (49) Then, considering the above minorizer, the constraint CIRS is expressed as the ith iteration of MM as ℜ � θ∗ E,m θ(i−1) E,m |θ(i−1) E,m | � ≥ 1, ℜ � θ∗ I,m θ(i−1) I,m |θ(i−1) I,m | � ≥ 1, ∀m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (50) Therefore, the optimization problem in (28) is modified as max αa,θE,θI αa (51) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' C3 : (41), C4 : Ek (θE) ≥ Emin,k, ∀k, CIRS : (50), C5 : N � n=1 log2 � 1 + θH I AIRS k,n θI θH I �AIRS k,n θI + ζk,n,a � ≥ αa, ∀k, where the steps for constraints C3-C5 in Subsection III-A1 are used exactly here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Maximization over {sE,n, pI,n} By introducing an auxiliary variable αb, the relay/IRS problem in (16) for fixed {UE,n, UI,n, τ} boils down to the following optimization: max αb,{sE,n,pI,n}N n=1 αb (52) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' C2 : (3), pI,k,n ≥ 0, ∀k, n, C3 : (9), C4 : Ek � {sE}N n=1 � ≥ Emin,k, ∀k, C5 : N � n=1 log2 (1 + γk,n(pI,n)) ≥ αb, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The constraints C4 and C5 of this sub-problem are non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We first rewrite the SINR associated with the kth pair in (11) as γk,n(pI,n) = aT k,npI,n bT k,npI,n + σ2n �ψk,n + δ2 k,n + σ2 k,n , (53) where ak,n = ψk,k,nek, bk,n = [ψk,1,n, ψk,2,n, · · · , ψk,k−1,n, 0 , ψk,k+1,n, · · · , ψk,K,n]T, and ek is the kth unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Therefore, the LHS of C5 in (52) is written as N � n=1 � log2 � qT k,npI,n + ζk,n,b � − log2 � bT k,npI,n + ζk,n,b � � , (54) January 3, 2023 DRAFT 16 where qk,n = ak,n +bk,n and ζk,n,b = σ2 n �ψk,n +σ2 k,n +δ2 k,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Then, similar to the procedure in Sub- section III-A for C5, we resort to the MM technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Precisely, considering the inequality in (30), the second term in (54) is minorized by setting x = bT k,npI,n + ζk,n,b and x0 = bT k,np0 I,n + ζk,n,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' By substituting the minorizer in (54), the constraint C5 at the ith iteration is obtained as C5 : N � n=1 � log2 � qT k,npI,n + ζk,n,b � − log2(bT k,np(i−1) I,n + ζk,n,b) (55) − log2 e bT k,np(i−1) I,n + ζk,n,b bT k,n � pI,n − p(i−1) I,n � � ≥ αb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Next, we consider the non-convex constraint C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' It is observed that the term Ek � {sE,n}N n=1 � in the LHS of the this constraint is neither convex nor concave w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' sE,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' similar to the procedure in Subsection III-A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' we apply the MM by selecting βk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='b (see Appendix C) and minorize C4 at the ith iteration by Ek �� s(i−1) E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �N n=1 � + 1 2 N � n=1 βk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='b � s(i−1) E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �H s(i−1) E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n + N � n=1 ℜ � ϑ(i) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='b � sE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n − s(i−1) E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �� (56) − 1 2 N � n=1 βk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='bsH E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='nsE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n ≥ Emin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' ∀k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' where we define ϑ(i) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='b =βk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='b � s(i−1) E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �H + τ ρexp�cexp � �alog2ω(i) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='b � � ω(i) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='b ��b−1 � 2�a log ω(i) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='b + �b � � s(i−1) E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �H Ξk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' with ω(i) k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='b = 1 2 �N n=1 � s(i−1) E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n �H Ξk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='ns(i−1) E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Consequently, the ith iteration of the MM update for (52) is obtained easily as the interior point solution of the following convex problem max αb,{sE,n,pI,n}N n=1 αb (57) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' C2, C3, C4 : (56), C5 : (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Maximization over τ The optimization problem in (16) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' τ becomes min τ τ (58) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' C1 : 0 ≤ τ ≤ T, C2 : τvk ≤ �vk, ∀k, C3 : τ�v1 ≤ �v2, C4 : τ ≥ ¯vk, ∀k, where vk = 1 2ρ N � n=1 � |sE,k,n|2 − pI,k,n � , ∀k, �vk = T N � n=1 � prf k,n − pI,k,n 2ρ � , ∀k, (59) January 3, 2023 DRAFT 17 Algorithm 1 The Proposed Method for Minimum Rate Maximization in Relay/IRS Systems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Relay: Initialize U(l) E,n, U(l) I,n ∈ CMR×MR, τ (l) ∈ R+, l ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' IRS: Initialize θ(l) E , θ(l) I ∈ CMIRS, τ (l) ∈ R+, l ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' repeat 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Relay: Initialize U(i) E,n and U(i) I,n and set i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' IRS: Initialize θ(i) E , θ(i) I and set i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' repeat 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Relay: Solve (40) to obtain {UE,n, UI,n, αa}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' IRS: Solve (51) to obtain {θE, θI, αa}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Update i ← i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' until convergence 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Relay/IRS: Initialize s(i) E,n, p(i) I,n and set i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' repeat 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Relay/IRS: Solve the convex problem in (57) to obtain {sE,n, pI,n, αb}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Update i ← i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' until convergence 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Relay/IRS: Compute τ (l) via the closed-form solution in (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Update l ← l + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' until convergence �v1 = 1 2ρ N � n=1 � sH E,nVE,nsE,n + σ2 ntr � UE,nUH E,n � − tr{QI,nVI,n} − σ2 ntr{UI,nUH I,n} � , (60) �v2 = T N � n=1 � prf n − 1 2ρ � tr{QI,nVI,n} + σ2 ntr � UI,nUH I,n ��� , (61) ¯vk = ρEmin,k exp � �alog2pE,k � p�b E,kexp (�c) , ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (62) Therefore, a closed-form solution (for a non-empty feasible set5) can be obtained as τopt = max{¯v1, ¯v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=', ¯vK}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (63) 5The following conditions lead to a non-empty feasible set for the problem: 1) ¯vk ≤ T, ∀k, 2) �vk ≥ 0, ∀k, 3) �v2 ≥ 0, 4) �vj vj |K j=1 ≥ ¯vk, ∀k (for vj ≥ 0, ∀j), 5) �v2 �v1 ≥ ¯vk, ∀k (for v1 ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' January 3, 2023 DRAFT 18 TABLE I THE COMPUTATIONAL COMPLEXITY ORDER (PER INNER ITERATIONS) FOR STEP 3 OF THE ALGORITHM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Relay O �� 2NM 2 R(1 + 2N)(1 + K) �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='5� Relay (t-static) O �� NM 2 R(1 + N)(1 + 2K) �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='5� Relay (t-f-static) O �� 2M 2 R(1 + 2K) �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='5� IRS O � (6MIRS(N + K + 1))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='5� IRS (t-static) O � (2MIRS(N + 2K + 1))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='5� Algorithm 1 summarizes the discussions in Section III and represents the steps of the proposed method for maximizing the minimum rate of all user pairs in relay/IRS WPC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Note that similar mathematical derivations are used to develop t-f-static algorithm for relay system as well as t-static algorithm for both relay and IRS systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Remark 4 (convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' It has been shown that under some mild conditions, the MM technique converges to the stationary points of the problem [34], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Complexity Analysis The main computational burdens in Algorithm 1 are associated with steps 3, 6, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' At each inner iteration in step 3, the convex problems in (40) and (51) are solved via interior point methods for relay and IRS system design, respectively, with a computational complexity of O � (2NM2 R(1 + 2N)(1 + K))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='5� and O � (6MIRS(N + K + 1))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='5� [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Table I compares the computational complexity of step 3 for other versions of relay/IRS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Similar to step 3, the complexity (per inner iterations) for step 6 which solves (57) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=', by using the interior point methods) is O � (KN(1 + 2N)(5 + 2K))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='5� for all versions of relay/IRS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In step 8, the closed-form expression in (63) must be calculated leading to the complexity of6 O(N3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' NUMERICAL EXAMPLES Here, we evaluate the proposed relay/IRS method in different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The channels from transmitters to the relay and the channels from the relay to the receivers are modeled as Hn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='1 � �d1 d0 � −�γ 2 �Hn and Gn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='1 � �d2 d0 � −�γ 2 �Gn, respectively, where d0 = 1 m is a reference distance, �d1 is the distance between Tk and the relay, �d2 is the distance between the relay and Rk, and 6This can be decreased to O(N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='3) via finding the best order of matrix multiplications (see [37] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' January 3, 2023 DRAFT 19 T1 T2 TK Relay or IRS rT d3 d1 d2 rR R1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' R2 RK Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Simulation setup for relay/IRS WPC systems with K user pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' TABLE II THE BASELINE BENCHMARK METHODS Baseline 1 Baseline 2 Information Power Allocation ✓ ✓ Energy Waveform Design ✗ ✓ Time Allocation ✓ ✓ Energy/Information Relay Beamforming ✓ ✗ �γ = 3 is the path-loss exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' It is assumed that the elements of �Hn and �Gn are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' CSCG random variables with zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 3, the transmitters and receivers are distributed uniformly within a circle with radius rT and rR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We set the distance parameters as d1 = d2 = d3 = 10 m and rT = rR = 5 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The maximum power budget for Tk, relay, and IRS are set to prf k,n = prf R,n = 28 dBm, prf IRS = 20 dBm, ∀k, n, and the noise power at the relay, IRS and receivers are supposed to be σ2 R,n = σ2 k,n = δ2 k,n = −80 dBm, σ2 IRS,n = −100 dBm, ∀k, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The total bandwidth is fixed to Bt = 1 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We further assume the total operation time T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The curve fitting parameters for non-linear EH circuits are equal to �a = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='11, �b = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='17, and �c = −12 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Also, we set K = 5, N = 8, MR = 6, and Emin,k = Emin = 10 µW, ∀k, unless otherwise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We solve the convex optimization problems using CVX [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Relay System Here, we compare the results of the proposed algorithms with partially optimized methods (referred to as baseline schemes in the sequel) listed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' For the first baseline method, the energy signals are not optimized, and in the second baseline method, there is no optimization January 3, 2023 DRAFT 20 1 2 3 4 5 0 1 2 3 4 5 6 Minimum Rate (bps/Hz) Number of Outer Iterations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='8 Different Initializations (a) 1 2 3 4 5 6 7 0 1 2 3 4 Minimum Rate (bps/Hz) Number of Inner Iterations (b) 1 2 3 4 5 6 7 8 9 10 11 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='8 5 Number of Inner Iterations Minimum Rate (bps/Hz) (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Convergence behavior of the proposed method in Algorithm 1: (a) outer iterations for three random initial points, (b) inner iterations associated with the sub-problem III-A1 in the first outer iteration, (c) inner iterations associated with the sub-problem III-B in the first outer iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' for the relay beamformer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' more precisely, the relay amplification matrices are assumed to be identity matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' UR E,n = �αE,nIMR, ∀n, and UR I,n = �αI,nIMR, ∀n, where the scalar parameters �αE,n and �αI,n are employed to satisfy the feasible set Ω in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The convergence of the proposed algorithm for inner and outer iterations (see Algorithm 1) are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' This figure shows that the proposed algorithm converges within a few outer iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Also, in this example, the three different initializations lead to almost the same final value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='a, we illustrate the rate-energy region of the proposed method in comparison with the first baseline method for different number of subbands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We can observe that the minimum rate increases as N grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The optimal time allocation parameter τopt w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' the EH target is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' It is seen that the increased energy threshold Emin leads to a larger τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' As τ increases, the duration of the ID phase decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Therefore, as we observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='a, the minimum rate reduces with increasing Emin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Also, the impact of the energy waveform design is evident in both figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='b, we compare the minimum rate of the proposed optimal and sub- optimal approaches with baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' As we can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='a, increasing the number of pairs results in lower minimum rate for all methods with MR = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='b shows that a larger MR increases the minimum rate with an almost linear trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The importance of the energy waveform and relay beamforming design is observed through both figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We can see that the method with no relay beamforming has the worst performance compared to other methods since without a relay amplification matrix design, inter-pair interference cannot be managed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' January 3, 2023 DRAFT 21 10 20 30 40 50 60 70 80 90 100110120130 0 1 2 3 4 5 6 7 EH Target, Emin (µW ) Minimum Rate (bps/Hz) Proposed - N = 8 Baseline 1 - N = 8 Proposed - N = 9 Baseline 1 - N = 9 (a) the rate-energy region 10 20 30 40 50 60 70 80 90 100110120130 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='8 1 EH Target, Emin (µW ) Time Allocation Parameter (s) Proposed - N = 8 Baseline 1 - N = 8 Proposed - N = 9 Baseline 1 - N = 9 (b) the time allocation parameter τopt Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Comparison of the proposed and baseline 1 methods for different number of subbands N = 8, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 3 4 5 6 7 8 3 4 5 6 7 8 9 10 11 12 Number of Pairs, K Minimum Rate (bps/Hz) Proposed Baseline 1 Proposed (t−static) Proposed (t−f−static) Baseline 2 (a) minimum rate versus number of pairs K 5 6 7 8 9 10 2 3 4 5 6 7 8 9 Number of Antennas, MR Minimum Rate (bps/Hz) Proposed Baseline 1 Proposed (t−static) Proposed (t−f−static) Baseline 2 (b) minimum rate versus number of antennas MR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Comparison of the proposed optimal and sub-optimal methods with baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' IRS System In this subsection, the performance of the proposed IRS-assisted WPC system is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Since most of the studied scenarios for relay (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 4, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 5, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='a) have similar trends for IRS, we only consider the scenario of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='b, for the sake of brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' As we can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='7, in the case of IRS, the minimum rate has a super-linear ascent property versus increasing MIRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' CONCLUSION In this paper, the max-min rate maximization in a multi-carrier relay/IRS WPC system with a joint TS scheme was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' A unified framework was proposed to maximize the minimum rate of the user pairs in both relay and IRS systems by jointly designing the energy waveforms, January 3, 2023 DRAFT 22 7 10 13 16 19 22 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content='5 Number of REs, MIRS Minimum Rate (bps/Hz) Proposed Proposed (t−static) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The effect of the number of REs MIRS for the proposed optimal and sub-optimal IRS methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' power of information waveforms, amplification matrices, and the time allocation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The non-linearity in EH circuits was also considered in the design problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The non-convex problem was handled via the MM technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Numerical results demonstrated the effectiveness of the proposed algorithm in terms of the minimum rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' As a extended future work in this area, it might be interesting to develop a distributed algorithm for design of multi-user relay/IRS WPC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' APPENDIX A THE DERIVATION OF THE EXPRESSIONS IN (21) AND (25)-(27) Using tr � XHY � = vec(X)Hvec(Y) and vec(XYZ) = (ZT ⊗ X)vec(Y), the power of the relay signal for ID mode in (7) can be obtained as E � ∥�r(t)∥2 2 � = 1 2 N � n=1 � uH I,nvec � UI,nHnQI,nHH n � + σ2 nuH I,nuI,n � = 1 2 N � n=1 � uH I,n �� HnQI,nHH n �T ⊗ IMR � uI,n + σ2 nuH I,nuI,n � = 1 2 N � n=1 uH I,n �AR I,nuI,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Similarly, we can derive the power of the relay signal for the EH mode in (20) and the expressions in (25)–(27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' January 3, 2023 DRAFT 23 APPENDIX B PROOF OF LEMMA 1 By defining a positive semi-definite matrix D such that ∇2 xs(x) ⪯ D, we can write the following majorizer for s(x) as [39] s(x) ≤ s(x0) + ℜ � (∇xs(x))H |x=x0(x − x0) � + (x − x0)HD(x − x0), (64) where the gradient and Hessian of s(x) are respectively expressed as ∇xs(x) = −2 log2 e xHTx + ν Tx, (65) ∇2 xs(x) = � −2T xHTx + ν + 4TxxHT (xHTx + ν)2 � log2 e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Since T ⪰ 0, the term −2T xHTx+ν is negative semi-definite, and thus we obtain ξ > 0 such that for any ν ⩾ 0 4TxxHT (xHTx + ν)2 log2 e ⩽ 4TxxHT (xHTx)2 ⩽ ξIM2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Also, as TxxHT is a rank-one matrix, we can choose ξ as ξ ⩾ 4φ, where φ is given as φ = max x xHT2x (xHTx)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (66) Then by choosing a = VHx, where V is a full-rank matrix such that T = VVH, the following optimization is equivalently obtained from (66) as φ = max a aHVHVa aHa 1 aHa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (67) Using xHQx ≤ P and applying a similar procedure in [39, Appendix B], we can write φ ≤ Pλmax(T) vH 1 V−1QV−Hv1 , where v1 is the principal eigenvector of VHV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Finally, from (64), (65), and ξ = 4φ, we obtain b = ∇s(x)|x=x0 = −2 log2 e xH 0 Tx0+νTx0 and D = 4P wH 1 Qw1IM2 R, where w1 is the principal eigenvector of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' January 3, 2023 DRAFT 24 APPENDIX C A SELECTION OF βk,n,a AND βk,n,b The value of βk,n,b should be selected such that ∇2 sE,nEk � {sE,n}N n=1 � +βk,n,bIK ⪰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' The term ∇2 sE,nEk � {sE,n}N n=1 � is straightforwardly calculated as ∇2 sE,nEk � {sE,n}N n=1 � =̺k N � n=1 Ξk,n + ηk N � n=1 N � n′=1 Ξk,nsE,nsH E,n′Ξk,n′, (68) where ̺k = τ ρexp�c exp � �alog2pE,k � p �b−1 E,k � 2�a log pE,k + �b � , ηk = τexp�c exp � �alog2pE,k � p �b−2 E,k ρ � 4�a2log2pE,k + � 4�a�b − 2�a � log pE,k + �b2 −�b + 2�a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' As �a < 0,�b < 0, Ξk,n ⪰ 0, and Ξk,nsE,nsH E,nΞk,n ⪰ 0, it suffices to choose βk,n,b such that βk,n,bIK ⪰ − �̺k N � n=1 Ξk,n − �ηk N � n=1 N � n′=1 Ξk,nsE,nsH E,n′Ξk,n′, (69) where �̺k = τ ρ �b exp�c exp � �alog2pE,k � p �b−1 E,k , �ηk =τ ρexp�c exp � �alog2pE,k � p �b−2 E,k � log pE,k � 4�a�b − 2�a � + 2�a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Thus from (3), we can write ∥sE,n∥2 2 ≤ 2ρT τ K � k=1 prf k,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (70) Finally, using (16), (69), (70) and knowing that sH E,nΞk,nsE,n ≤ ∥sE,n∥2 2λmax (Ξk,n), we can select βk,n,b > βt k,n,b where βt k,n,b = − τ ρexp�c exp � 2�alog2T N � n=1 λmax (Ξk,n) K � k=1 prf k,n � �f �b−2 k � �� 4�a�b − 2�a � log �fk + 2�a � × N � n=1 N � n′=1 λmax (Ξk,nΞk,n′) K � k=1 � prf k,nprf k,n′ + �b �fk N � n=1 λmax (Ξk,n) � , with �fk = exp � −�b− � �b2−4�a log ρEmin,k τexp�c 2�a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' We can take similar steps for selecting βt k,n,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' January 3, 2023 DRAFT 25 APPENDIX D PROOF OF LEMMA 2 The ID part of the relay power constraint in (20) is uH I,n �AR I,nuI,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Only (iMIRS + i+ 1)th, 0 ≤ i ≤ MIRS − 1 entries of uI,n = vec(Diag(θI)) are non-zero for the IRS system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Thus, we can rewrite uH I,n �AR I,nuI,n for IRS system as θH I �AIRS I,n θI, where �AIRS I,n contains only the (kMIRS + k + 1, lMIRS +l+1)th, 0 ≤ k, l ≤ MIRS −1 entries of �AI,n which is the same as �AR I,n with replacing MR by MIRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' Therefore, from (21) and by using some matrix manipulations, we obtain �AIRS I,n = � HnQI,nHH n �T ⊙ IMIRS + σ2 nIMIRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9AyT4oBgHgl3EQfWfd0/content/2301.00164v1.pdf'} +page_content=' (71) Other expressions in 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We study regularity properties of frequency measures arising from ran- +dom substitutions, which are a generalisation of (deterministic) substitutions where +the substituted image of each letter is chosen independently from a fixed finite set. +In particular, for a natural class of such measures, we derive a closed-form analytic +formula for the Lq-spectrum and prove that the multifractal formalism holds. This +provides an interesting new class of measures satisfying the multifractal formalism. +More generally, we establish results concerning the Lq-spectrum of a broad class of +frequency measures. We introduce a new notion called the inflation word Lq-spectrum +of a random substitution and show that this coincides with the Lq-spectrum of the +corresponding frequency measure for all q ≥ 0. As an application, we obtain closed- +form formulas under separation conditions and recover known results for topological +and measure theoretic entropy. +CONTENTS +1. +Introduction +2 +1.1. +Entropy and Lq-spectra +4 +1.2. +Random substitutions +5 +1.3. +Statement and discussion of main results +5 +1.4. +Discussion and further work +9 +2. +Preliminaries +9 +2.1. +Symbolic notation +10 +2.2. +Dynamics, entropy and dimension +10 +2.3. +Lq-spectra and smoothness +12 +2.4. +Multifractal spectrum and multifractal formalism +14 +2.5. +Random substitutions and frequency measures +16 +2.6. +Primitive random substitutions +18 +2.7. +Compatible random substitutions +18 +2.8. +Frequency measures +20 +2.9. +Separation conditions and recognisability +21 +3. +Lq-spectra of frequency measures +23 +3.1. +Inflation word Lq-spectra +23 +3.2. +Lq-spectra for non-negative q +26 +2020 Mathematics Subject Classification. 37B10, 37C45, 52C23. +Key words and phrases. random substitution, multifractal analysis, Lq-spectrum, multifractal +formalism. +1 +arXiv:2301.04958v1 [math.DS] 12 Jan 2023 + +2 +ANDREW MITCHELL AND ALEX RUTAR +3.3. +Lq-spectra for negative q: lower bounds +29 +3.4. +Proof of general bounds for the Lq spectrum +31 +3.5. +Lq-spectra for negative q under recognisability +32 +3.6. +Recovering entropy from the Lq-spectrum +33 +4. +Recognisability and the multifractal formalism +34 +4.1. +Non-typical local dimensions +34 +4.2. +Proof of the multifractal formalism +40 +5. +Examples, counterexamples and applications +41 +5.1. +Failure of bounds for q < 0 without recognisability +41 +5.2. +Examples with recognisability +43 +5.3. +Examples without recognisability +46 +Acknowledgements +47 +References +47 +1. INTRODUCTION +A substitution is a combinatorial object consisting of a finite alphabet A along with a +set of transformation rules. The theory of substitutions, along with statistical properties +of the system under repeated iteration, is a large and actively researched field at +the interface of combinatorics and symbolic dynamics. A thorough introduction +to the statistical properties and dynamics of substitutions can be found in [2, 25]. +Associated with a (deterministic) substitution is a frequency measure, which encodes +the frequency of subwords under repeated iteration. Notably, the corresponding +subshift supporting this measure has zero topological entropy, and the frequency +measure is the unique ergodic measure supported on this subshift. +Random substitutions are a generalisation of (deterministic) substitutions [13] +where we apply a transformation rule to each letter randomly and independently +chosen from a finite set of possibilities. Similarly to the deterministic case, subshifts +associated with random substitutions support ergodic frequency measures which +capture the expected rate of occurrence of subwords under random iteration. But in +contrast to the deterministic case, the corresponding subshift typically has positive +topological entropy and supports uncountably many ergodic measures. Random +substitutions include examples exhibiting deterministic behaviour, while also includ- +ing examples which are subshifts of finite type. Moreover, there is a large amount +of intermediate behaviour: subshifts of random substitutions can simultaneously +exhibit long range correlation [3] (an indication of order) and positive topological +entropy [14] (an indication of disorder). +In this paper, we study the fine scaling properties of frequency measures associ- +ated with random substitutions from the perspective of multifractal analysis. This +perspective is relevant in a wide variety of contexts, such as the geometry of fractal +sets and measures and in dynamical systems, with typical applications to geometric + +Multifractal Random Substitutions +3 +measure theory and number theory. In our setting, our primary objects of study +are the Lq-spectrum, which is a parametrised family of quantities which capture the +inhomogeneous scaling properties of a measure, and the local dimension, which cap- +ture the exponential growth rate of a measure around a point. The Lq-spectrum and +local dimensions are related through a heuristic relationship known as the multifractal +formalism [18]. It is an important and well-studied question to determine settings +in which the multifractal formalism holds, and to determine qualitative conditions +describing its failure. +Much of the work on multifractal analysis has been done in the setting of local +dimensions of self-similar measures (for some examples, see [1, 10, 11, 19, 29]) and +Birkhoff sums of potentials in dynamical systems with a finite type property (see, +for example, [8, 24] and the reference therein). As a notable recent example, in +P. Shmerkin’s recent proof of the Furstenberg ×2 ×3 conjecture [29], he computes +the Lq-spectrum of a large class of dynamically self-similar measures and relates +such results to the multifractal analysis of slices of sets. This information about Lq- +spectra also implies Lp-smoothness properties in the question of absolute continuity +of Bernoulli convolutions (see [30] for some background on this classic problem). +For more detail on the geometry of measures and multifractal analysis, we refer the +reader to the foundational work by L. Olsen [22] and the classic texts of K. Falconer +[7] and Ya. Pesin [23]. +Returning to our setting, substitution dynamical systems have characteristic fea- +tures of (dynamical) self-similarity, but in many cases are far from being ergodic +measures on shifts of finite type. More generally, frequency measures provide a +rich family of shift-invariant ergodic measures which exhibit interesting and unique +properties in symbolic dynamics in a natural way. For example, it was proved in +[15, Theorem 4.8] that for a certain class of random substitutions, the corresponding +subshift supports a frequency measure that is the unique ergodic measure of maximal +entropy. However, this measure is not a Gibbs measure with respect to the zero +potential, and therefore the system does not satisfy the common specification property, +which is a well-known strategy for proving intrinsic ergodicity of symbolic dynamical +systems (see [5] and the references therein). Moreover, there are examples of random +substitutions such that the corresponding subshift supports multiple ergodic mea- +sures of maximal entropy [17, Example 6.5]. More generally, many key properties of +frequency measures associated with random substitutions are poorly understood. +In this paper, we derive symbolic expressions for the Lq-spectrum of frequency +measures associated with random substitutions under certain weak assumptions. +Then under an additional assumption (recognisability), we prove a closed-form +analytic expression for the Lq-spectrum and a variational formula which together +imply the multifractal formalism. We emphasise that this class of examples has +novel properties not witnessed before: in general, the unique frequency measure of +maximal dimension is not a Gibbs measure with respect to the zero potential and the + +4 +ANDREW MITCHELL AND ALEX RUTAR +corresponding subshift is not sofic. The techniques and results provide important +new perspectives on the geometry and dynamics of the respective measures. +1.1. Entropy and Lq-spectra. For a Borel probability measure in a compact metric +space, the Lq-spectrum is a well-studied quantity which encodes the scaling properties +of the measure, in a weak sub-exponential sense. Specifically, the Lq-spectrum of µ is +given by +τµ(q) = lim inf +r→0 +log sup � +i µ +� +B(xi, r) +�q +log r +. +where the supremum is taken over 2r-separated subsets {xi}i of the support of µ. +The Lq-spectrum encodes information about the local scaling of the measure µ. We +define the local dimension of µ at x by +dimloc(µ, x) = lim +r→0 +log µ +� +B(x, r) +� +log r +when the limit exists. We then define the multifractal spectrum of µ by +fµ(α) = dimH {x ∈ X : dimloc(µ, x) = α} . +In general, the structure of the set of local dimensions can be very complex—for +example, the level sets are often dense uncountable subsets of the support of µ. +However, the “multifractal miracle” is the phenomenon that, even though the level +sets are very complex, the multifractal spectrum is often a concave analytic function +of α. +In fact, the multifractal spectrum and the Lq-spectrum are related through a heuris- +tic relationship called the multifractal formalism [18], which speculates that under +certain regularity assumptions, the multifractal spectrum is given by the concave +conjugate of the Lq-spectrum, that is the quantity +τ ∗ +µ(α) = inf +q∈R(qα − τµ(q)). +Generally speaking, τ ∗ +µ(α) ≥ fµ(α) [19]: in particular, the slopes of the asymptotes of +the Lq-spectrum bound the exponential scaling of measures of balls B(x, r) uniformly +for all x ∈ supp µ. +In our specific setting, where our metric space is the two-sided shift AZ and the +measure µ is ergodic, the local dimension is precisely the scaling rate of the information +function of µ. In fact, the Shannon–McMillan–Breiman theorem states that the local +dimension of the measure (with an appropriate choice of the metric) is almost surely +the entropy of the measure. Thus the Lq-spectrum provides uniform control over the +scaling rate of the information function. More detail about these notions are given in +Section 2. + +Multifractal Random Substitutions +5 +1.2. Random substitutions. A (deterministic) substitution is a rule which replaces +each symbol in a finite or infinite string over an alphabet A with a finite word over +the same alphabet. Random substitutions generalise this notion by substituting a +randomly chosen word (according to a fixed finite distribution) independently for +each letter. We can also think of a random substitution as a (deterministic) set-valued +substitution ϑ, together with a choice of probabilities. +For example, given p ∈ (0, 1), the random Fibonacci substitution ϑp is defined by +ϑp : +� +� +� +� +� +� +� +a �→ +� +ab +with probability p, +ba +with probability 1 − p, +b �→ a. +To a given (primitive) random substitution ϑP , one can canonically a subshift Xϑ of +the two-sided shift AZ along with an ergodic frequency measure µP , which quantifies +the relative occurrence of a given word under repeated application of the random +substitution. +As highlighted in the introduction, primitive random substitutions give rise to +subshifts and measures with a wide variety of properties. As a result, we will impose +additional conditions. +Our main assumption, which we call compatibility, asserts that for each a ∈ A, the +number of occurrences of each b ∈ A is identical in every possible substituted image +of a. For example, the random Fibonacci substitution is compatible since in all the +possible images of a, a occurs once and b occurs once. The key feature of compatibility +is that the one can define a deterministic substitution matrix, such that the Perron– +Frobenius eigenvalue is the asymptotic growth rate of lengths of words, and the +corresponding right eigenvector encodes the asymptotic frequency with which the +individual letters appear. Compatibility is a common assumption: for example, it is +assumed in the main results of [3, 14, 21, 27]. +Another standard assumption is that of recognisability, which heuristically states +that each element of the subshift is the unique image of another element of the subshift. +Recognisability precludes the existence of periodic points [27] and is one of the +assumptions required to to establish intrinsic ergodicity in [15]. It is also assumed in +the main results of [12, 21]. +1.3. Statement and discussion of main results. We now give concise statements of +the main results in this paper. We refer the reader to Section 2 for full statements of +the notation and definitions used in this section. + +6 +ANDREW MITCHELL AND ALEX RUTAR +Fix a random substitution ϑP and let λ and R denote the Perron–Frobenius eigen- +value and corresponding right eigenvector of the substitution matrix of ϑP , respec- +tively. Given q ∈ R and k ∈ N, define +ϕϑP ,k(q) = ϕk(q) = − +� +a∈A +Ra log +� +� � +s∈ϑk(a) +P[ϑk +P (a) = s]q +� +� . +We define the inflation word Lq-spectrum of ϑP by +Tϑ,P (q) = lim inf +k→∞ +1 +λk ϕk(q). +We similarly define the upper variant T ϑ,P by taking a limit superior in place of the +limit inferior. Throughout, µP will denote the frequency measure associated with ϑP . +Heuristically, the inflation word spectrum approximates the frequency measure µP +by the probability distribution on the iterated system. The normalisation factors follow +from the observation that the words in the kth iteration have length approximately λk +when normalised by the left Perron–Frobenius eigenvector. +Our main general result bounding the Lq-spectrum is the following, which states +for q ≥ 0 that Tϑ,P and τµP coincide, and moreover provides bounds on τµP in terms +of the functions ϕk for all q ∈ R. +Theorem A. Let ϑP = (ϑ, P ) be a primitive and compatible random substitution with +corresponding frequency measure µP . Then the limits defining τµP (q) and Tϑ,P (q) exist and +coincide for all q ≥ 0. Moreover, +(1) For all 0 ≤ q ≤ 1, +(1.1) +1 +λk − 1ϕk(q) ≤ τµP (q) ≤ 1 +λk ϕk(q) +and (λ−kϕk(q))∞ +k=1 converges monotonically to τµP (q) from above. +(2) For all q ≥ 1, +(1.2) +1 +λk ϕk(q) ≤ τµP (q) ≤ +1 +λk − 1ϕk(q) +and (λ−kϕk(q))∞ +k=1 converges monotonically to τµP (q) from below. +(3) For all q < 0, +(1.3) +1 +λk − 1ϕk(q) ≤ τµP (q). +The notion of compatibility is defined in Section 2.7, which is key in order to obtain +the uniform estimate given in Lemma 2.9. For q < 0, it is not true in general that +τµP (q) and Tϑ,P (q) coincide (a counterexample is given in Example 5.2): the problem +is essentially “non-uniqueness of cutting points”, as highlighted by the averaging +procedure in the construction of the measure (see Lemma 2.12). In other words, the +corresponding upper bound in (1.3) does not hold in general. Despite this, it still +follows from (1.3) that Tϑ,P (q) provides a lower bound for τµP (q). + +Multifractal Random Substitutions +7 +In Proposition 3.1, we prove that if ϑP also satisfies the disjoint set condition, or the +identical set condition with identical production probabilities (see Definition 2.13), then +a closed-form expression can be obtained for the Lq-spectrum of the corresponding +frequency measure. By combining this result with Theorem A, we obtain the following +corollary. +Corollary B. Let ϑP be a primitive and compatible random substitution with corresponding +frequency measure µP . Then for all q ≥ 0: +(1) If ϑP satisfies the disjoint set condition, then +τµP (q) = +1 +λ − 1ϕ1(q). +(2) If ϑP satisfies the identical set condition and has identical production probabilities, +then +τµP (q) = 1 +λϕ1(q). +In particular, under the disjoint set condition or identical set condition with identical +production probabilities, the Lq-spectrum is analytic on (0, ∞). +In [15], a result analogous to Theorem A is obtained for entropy in terms of the +inflation word entropy, without the compatibility assumption. In fact, to highlight the +generality of our results on Lq-spectra, as a consequence of Theorem A, we obtain +new proofs of this result (as well as a result on topological entropy). +(a) We obtain the main result of [14] on topological entropy, which states that for +subshifts of primitive and compatible random substitutions, the topological +entropy can be characterised in terms of the asymptotic growth rate of inflation +words. +(b) For frequency measures corresponding to primitive and compatible random +substitutions, we also obtain the characterisation of (measure theoretic) en- +tropy obtained in [15, Theorem 3.3]—under the additional hypothesis that the +substitution is compatible. +This is described in the following corollary. +Corollary C. Let ϑP = (ϑ, P ) be a primitive and compatible random substitution with +associated subshift Xϑ and frequency measure µP . +(1) The limit +lim +k→∞ +1 +λk +� +a∈A +Ra log(#ϑk(a)) +exists and is equal to htop(Xϑ). +(2) The Lq-spectrum of µP is differentiable at 1. Moreover, the limit +lim +k→∞ +1 +λk +� +a∈A +Ra +� +v∈ϑk(a) +−P[ϑk +P (a) = v] log(P[ϑk +P (a) = v]) +exists and is equal to hµP (Xϑ) = dimH µP = τ ′ +µP (1). + +8 +ANDREW MITCHELL AND ALEX RUTAR +We now turn our attention to the multifractal spectrum. Firstly, while τµP (q) and +Tϑ,P (q) do not coincide in general for q < 0, if the random substitution that gives +rise to the frequency measure µP is additionally assumed to be recognisable (see +Definition 2.14), then the limits defining τµP (q) and Tϑ,P (q) both exist and coincide for +all q ∈ R. Moreover, under recognisability, we prove that the multifractal spectrum is +the concave conjugate of the Lq-spectrum: in other words, the multifractal formalism +holds for any associated frequency measure µP . In particular, we conclude that fµP is +a concave analytic function. +In fact, in Proposition 4.5 we prove a stronger variational formula for the multi- +fractal spectrum. For each α ∈ R, we construct measures ν such that dimH ν ≥ τ ∗(α) +and dimloc(µP , x) = α for ν-a.e. x ∈ Xϑ. In particular, we can take the measures to be +frequency measures associated with permissible probabilities for the substitution ϑ. +Theorem D. Let ϑP be a primitive, compatible, and recognisable random substitution with +corresponding frequency measure µP . Then for all q ∈ R, +τµP (q) = Tϑ,P (q) = +1 +λ − 1ϕ1(q). +Moreover, fµP (α) = τ ∗ +µP (α) is an analytic and concave function. In fact, for each α ∈ R +such that fµP (α) ≥ 0, there are permissible probabilities Q such that fµP (α) = dimH µQ and +dimloc(µP , x) = α for µQ-a.e. x ∈ Xϑ. +To conclude this section, we observe that our results on Lq-spectra also give uniform +bounds on the exponential scaling rate of the frequency measures. The following +result is a direct application of Theorem A and Theorem D, combined with Proposi- +tion 4.4. +Corollary E. Let ϑP = (ϑ, P ) be a primitive, compatible, and recognisable random substitu- +tion. Then +αmin := lim +q→∞ +τµP (q) +q += − +� +a∈A +Ra log +� +max +s∈ϑ(a) P[ϑP (a) = s] +� +αmax := lim +q→−∞ +τµP (q) +q += − +� +a∈A +Ra log +� +min +s∈ϑ(a) P[ϑP (a) = s] +� +. +and for all x ∈ Xϑ, αmin ≤ dimloc(µP , x) ≤ dimloc(µP , x) ≤ αmax. Moreover, +{dimloc(µP , x) : x ∈ Xϑ} = [αmin, αmax]. +In particular, when the probabilities P are chosen so that for each a ∈ A, P[ϑP (a) = +s] = 1/#ϑ(a) for all s ∈ ϑ(a), then the Lq-spectrum is the line with slope htop(Xϑ) +passing through (1, 0). Thus the local dimension of µP exists at every x ∈ Xϑ and +is given by the constant value αmin = αmax. This can be rephrased in terms of a +weak Gibbs-type property, which says that for every ϵ > 0, all n sufficiently large +(depending on ϵ), and u ∈ Ln, +(1.4) +exp(−|u|(htop(Xϑ) + ϵ)) ≤ µP ([u]) ≤ exp(−|u|(htop(Xϑ) − ϵ)); + +Multifractal Random Substitutions +9 +see, for example, [29, Lemma 1.4] for the short argument. In general, the error +term ϵ cannot be dropped by the addition of a constant factor. +Under certain +assumptions, one can show that there are infinitely many words with µP ([u]) ≈ +|u|−1 exp(−|u|(htop(Xϑ)), as explained in [15, Lemma 4.12]. These assumptions are +satisfied, for example, in Example 5.6. +Of course, similar one-sided results hold for q ≥ 0 only under the assumption of +compatibility, by iterating the formula for ϕk and taking an appropriate maximum at +each level. In fact, since τµP (q) is differentiable at 1, with derivative giving the entropy, +and since htop(Xϑ) = τµP (0), it follows that µP is a measure of maximal entropy if and +only if τ ′ +µP (q) exists and is constant on the interval (0, 1). +1.4. Discussion and further work. We conclude the introduction with a list of com- +ments and potentially interesting questions. +(1) What is the Lq-spectrum for a compatible substitution when q < 0? We do +not known this even for the random substitution given in Example 5.1, which +satisfies the identical set condition with identical production probabilities. +Obtaining results for q < 0 is substantially more challenging, since the sum +in τµP (q) depends on the measure of cylinders with very small (but non-zero) +measure. For example, in the self-similar case, without the presence of strong +separation assumptions, little is known (in contrast to the q ≥ 0 case). +(2) What happens without compatibility? Do the formulas in Theorem A hold +in general? In [15], it suffices to use an almost sure version of Lemma 2.9. +However, since the Lq-spectrum is sensitive to scaling at individual points as q +tends to ±∞, such an almost sure result in our case is insufficient. +(3) Outside the disjoint set condition and the identical set condition, what can +be said about differentiability of the Lq-spectrum? For q ≥ 0, we give the +Lq-spectrum as a uniform limit of analytic functions: however, aside from the +exceptional point q = 1 where we can say more, this is not enough to give +information about differentiability. +(4) Can our results on Lq-spectra and multifractal spectra (which hold for recog- +nisable substitutions) be relaxed to a weaker condition such as the disjoint set +condition (see Definition 2.13)? +(5) Can the error term in (1.4) be determined precisely, up to a constant? The +approximate Gibbs-type bounds discussed following Corollary E are closely +related to the bounds used in the proof of intrinsic ergodicity given in [15, +Theorem 4.8]. It could be worth exploring the relationship between intrinsic +ergodicity and Gibbs-type properties given by the Lq-spectrum. +2. PRELIMINARIES +In this section we introduce the key notation and definitions that we will use +throughout the paper. After introducing some basic notation, in Section 2.2 we +introduce symbolic dynamics on the two-sided shift, as well as our notions of entropy + +10 +ANDREW MITCHELL AND ALEX RUTAR +and dimension. In Section 2.3 we present the key definitions and basic results from +multifractal analysis that we work with throughout, including the definitions of the +Lq-spectrum and local dimensions of a measure. Then, in the following sections +we provide an introduction to random substitutions and their associated dynamical +systems. In Section 2.5 we give the definition of a random substitution via its action +on words, and define the subshift associated to a random substitution. Then, in +Section 2.6 and Section 2.7, we define what it means for a random substitution to be +primitive and compatible and present the key properties of such random substitutions. +In Section 2.8, we give the definition of the frequency measure associated to a random +substitution and state a key result used in the proof of our main results which relates +the measures of cylinder sets via the action of the random substitution. Finally, +in Section 2.9, we define what it means for a substitution to satisfy the disjoint or +identical set condition, and introduce recognisable random substitutions. +2.1. Symbolic notation. Throughout, we use the following symbolic notation, which +is essentially the same as the notation used in [2, 20]. +For a set B, we let #B be the cardinality of B and let F(B) be the set of non-empty +finite subsets of B. +We fix an alphabet A = {a1, . . . , ad}, for some d ∈ N, which is a finite set of letters +ai, and equip it with the discrete topology. Then a word u with letters in A is a finite +concatenation of letters, namely u = ai1 · · · ain for some n ∈ N. We write |u| = n for +the length of the word u, and for m ∈ N, we let Am denote the set of all words of +length m with letters in A. +We set A+ = � +m∈N Am and let +AZ = {(ain)n∈Z : ain ∈ A for all n ∈ Z} +denote the set of all bi-infinite sequences with elements in A, and endow AZ with the +product topology. We also fix a metric on AZ as follows. Given points x = (xn)n∈Z +and y = (yn)n∈Z, let N(x, y) = sup{n ∈ Z : xj = yj for all |j| ≤ n} and let d(x, y) = +e−2N(x,y)−1. The space X is compact with topology generated by the metric. +We will frequently write sequences (xn)n∈Z ∈ AZ as · · · x−1x0x1 · · · , with the corre- +sponding notation for finite sequences. +If i and j ∈ Z with i ≤ j, and x = · · · x−1x0x1 · · · ∈ AZ, then we let x[i,j] = +xixi+1 · · · xj. We use the same notation if v ∈ A+ and 1 ≤ i ≤ j ≤ |v|. For u and +v ∈ A+ (or v ∈ AZ), we write u ◁ v if u is a subword of v, namely if there exist i and +j ∈ Z with i ≤ j so that u = v[i,j]. For u and v ∈ A+, we set |v|u to be the number +of (possibly overlapping) occurrences of u as a subword of v. If u = ai1 · · · ain and +v = aj1 · · · ajm ∈ A+, for some n and m ∈ N, we write uv for the concatenation of u +and v. The abelianisation of a word u ∈ A+ is the vector Φ(u) ∈ N#A +0 +, defined by +Φ(u)a = |u|a for all a ∈ A. +2.2. Dynamics, entropy and dimension. We equip the space AZ with invertible +dynamics from the left-shift map S : AZ → AZ. Throughout, we will work with a + +Multifractal Random Substitutions +11 +subshift X ⊂ AZ, which is compact and shift-invariant, that is S−1(X) = X. Then µ will +denote an ergodic and S-invariant Borel probability measure with support contained +in X. +The metric structure on AZ enables us to define the Hausdorff dimension of Borel +subsets of X. Using this, we define the Hausdorff dimension of µ to be the quantity +dimH µ = inf{dimH E : µ(E) > 0} +where the infimum is taken over Borel sets E. Here, we use Hausdorff dimension +as inherited from the underlying metric; though it would also be appropriate to use +Bowen’s generalisation of topological entropy to non-compact sets [4]. We also define +the lower local dimension of µ at x by +dimloc(µ, x) = lim inf +r→0 +log µ +� +B(x, r) +� +log r +We define the upper local dimension dimloc(µ, x) analogously using the limit superior +in place of the limit inferior, and when the limits coincide, we refer to the shared +quantity as the local dimension and denote it by dimloc(µ, x). +Local dimensions and Hausdorff dimension are closely related: it follows from, for +instance, [7, Proposition 10.1] that +(2.1) +dimH µ = sup{s : dimloc(µ, x) ≥ s for µ-a.e. x}. +Now fix a partition ξ so that with ξk = �k +i=−k S−i(ξ), {ξk}∞ +k=1 generates the Borel +σ-algebra on X. We recall that the entropy of µ with respect to S is given by +hµ(X) = lim +k→∞ +1 +2k + 1 +� +A∈ξk +−µ(A) log +� +µ(A)). +where, by the classical Kolmogorov–Sina˘ı theorem, the quantity does not depend on +the choice of partition. +Now given x ∈ X, let ξk(x) denote the unique element in the partition ξk containing +x. Then the Shannon–McMillan–Breiman theorem states that the entropy of µ is +almost surely the information rate of µ, that is for µ-a.e. x ∈ X, +lim +k→∞ +− log µ +� +ξk(x) +� +2k + 1 += hµ(X). +We refer the reader to [6] for greater detail. +Now suppose ξ = {Ea}a∈A is the partition of X where Ea = {(xn)n∈Z ∈ X : x0 = a}. +For the remainder of this paper, ξ will always denote this partition. Then given +x = (xn)n∈Z ∈ X, +ξk(x) = {y ∈ X : xj = yj for all |j| ≤ k} = B(x, e−(2k+1)) +and therefore +dimloc(µ, x) = lim +k→∞ +− log µ +� +ξk(x) +� +2k + 1 + +12 +ANDREW MITCHELL AND ALEX RUTAR +where both limits exist if either limit exists. Since the limit on the right is µ almost +surely hµ(X), it follows from (2.1) that dimH µ = hµ(X). +Finally, the topological entropy of X is given by +htop(X) = lim +k→∞ +− log #{E ∈ ξk : E ∩ X ̸= ∅} +2k + 1 +. +Of course, htop(X) = dimB X, the box counting dimension of X. +2.3. Lq-spectra and smoothness. Given q ∈ R, we define +Sµ,r(q) = +sup +{xi}i∈P(r) +� +i +µ +� +B(xi, r) +�q +where P(r) is the set of discrete 2r-separated subsets of X, that is P(r) = {{xi}i : xi ∈ +X, d(xi, xj) ≥ 2r for i ̸= j}. We then define the Lq-spectrum of µ to be the function +τµ(q) = lim inf +q→0 +log Sµ,r(q) +log r +. +For convenience, we also denote the upper variant τ µ(q) by taking a limit superior +in place of the limit inferior. It is a standard consequence of Hölder’s inequality that +τµ(q) is a concave increasing function of q (note that this need not hold for τ µ(q)). +Of course, the preceding definitions hold more generally in an arbitrary metric +space, but in our particular setting we can rephrase the Lq-spectrum in terms of more +familiar sums over cylinders. Recall that ξ denotes the partition of Xϑ into cylinders +at 0 corresponding to the letters in A. Then set +Sµ,n(q) = +� +E∈ξk +µ(E)q. +Since distinct elements in the partition ξk are e−(2k+1)-separated, +Sµ,n(q) = Sµ,e−(2n+1)(q). +It follows immediately that +τµ(q) = lim inf +n→∞ +− log Sµ,n(q) +2n + 1 +with the analogous result for τ µ(q). In particular, we observe that τµ(0) = htop(X) +assuming µ is fully supported on X. +Finally, by shift invariance, we can characterise the subshift X in terms of a language +on X. Given n ∈ N, we set +Ln(X) = {w ∈ An : w ◁ x for some x ∈ X}. +Given w ∈ Ln(X), we let [w] = {(xn)n∈Z ∈ X : xi = wi for all 1 ≤ i ≤ n}. Of course, +by shift invariance, there is a measure-preserving bijection between L2n+1(X) and Xn, + +Multifractal Random Substitutions +13 +so it follows again that +τµ(q) = lim inf +n→∞ −1 +n log +� +u∈Ln(X) +µ([u])q. +We will primarily use this characterisation throughout the paper. +We first list some basic properties of the Lq-spectrum of the measure µ. Here, (a) +is a direct consequence of Hölder’s inequality, (b) is standard (see, for example, [29, +Lemma 1.4]) and (c) was proven in [9, Theorem 1.4]. +Lemma 2.1. Let µ be a shift-invariant measure on X. +(a) The Lq-spectrum τµ(q) is continuous, increasing and concave on R. +(b) Let αmin = limq→∞ τµ(q)/q and αmax = limq→−∞ τµ(q)/q. Then for every s < +αmin ≤ αmax < t, all n sufficiently large and u ∈ Ln, e−tn ≤ µ([u]) ≤ e−sn. In +particular, the local dimensions satisfy +αmin ≤ inf +x∈X dimloc(µ, x) ≤ sup +x∈X +dimloc(µ, x) ≤ αmax. +(c) The left and right derivatives of τµ at q = 1 bound the Hausdorff dimension of µ, that +is τ + +µ (1) ≤ dimH µ ≤ τ − +µ (1). +In fact, (a) gives intuition for why the Lq-spectrum encodes smoothness: rather +than obtain almost sure information on local dimensions, the Lq-spectrum contains +uniform information about local dimensions. +Finally, we prove a simple result concerning the Lq-spectrum which will be useful +later in the paper. +Lemma 2.2. Let ζ > 1 be arbitrary. Then +(2.2) +τµ(q) = 1 +ζ lim inf +n→∞ −1 +n log +� +� +� +u∈L⌊ζn⌋(X) +µ([u])q +� +� +and +(2.3) +τ µ(q) = 1 +ζ lim sup +n→∞ −1 +n log +� +� +� +u∈L⌊ζn⌋(X) +µ([u])q +� +� . +Proof. Of course, it always holds that +τµ(q) ≤ 1 +ζ lim inf +n→∞ −1 +n log +� +� +� +u∈L⌊ζn⌋(X) +µ([u])q +� +� +τ µ(q) ≥ 1 +ζ lim sup +n→∞ −1 +n log +� +� +� +u∈L⌊ζn⌋(X) +µ([u])q +� +� . + +14 +ANDREW MITCHELL AND ALEX RUTAR +First, let q < 0 and let n ∈ N be arbitrary. Let kn be minimal so that ⌊ζkn⌋ ≥ n. Observe +that there is some M ∈ N (independent of n) so that ⌊ζkn⌋ ≤ n + M: it follows that +limn→∞ n/kn = ζ. Then if v ∈ L⌊ζkn⌋(X) is arbitrary, [v] ⊂ [u] for some u ∈ Ln(X) and +µ([v])q ≥ µ([u])q. Thus +S⌊ζkn⌋,µ(q) ≥ Sn,µ(q). +which gives (2.2) for q < 0 since lim n/kn = ζ. +Similarly, for q ≥ 0, since there are at most (#A)M words v ∈ L⌊ζkn⌋(X) with +[v] ⊂ [u], pigeonholing, for each u ∈ Ln(X) there is some v(u) ∈ L⌊ζkn⌋(X) such that +µ([v(u)])q ≥ (#A)−qMµ([u])q. Thus +S⌊ζkn⌋,µ(q) ≥ (#A)−qMSn,µ(q). +This gives (2.2) for q ≥ 0. +The arguments for (2.3) follow analogously by choosing kn maximal so that ⌊ζkn⌋ ≤ +n. +□ +2.4. Multifractal spectrum and multifractal formalism. The Lq-spectrum of a mea- +sure is related to the (fine) multifractal spectrum. Let µ be a shift-invariant measure on +a subshift X. We recall that the local dimension of µ at x ∈ X is given by +dimloc(µ, x) = lim +n→∞ − +1 +2n + 1 log µ([x[−n,n]]) +when the limit exists. Given α ∈ R, set +Fµ(α) = {x ∈ X : dimloc(µ, x) = α} . +We then define the multifractal spectrum of µ by +fµ(α) = dimH Fµ(α) +using the convention that dimH ∅ = −∞. +The multifractal spectrum is related to the Lq-spectrum by the following result. Let +g: R → R ∪{−∞} be a concave function. For x ∈ R, we let g+(x) (resp. g−(x)) denote +the right (resp. left) derivative of g at x. Such limits necessarily exist by concavity. We +denote the subdifferential of g at x by ∂g(x) = [g+(x), g−(x)]. We then recall that the +concave conjugate of g is given by +g∗(α) = inf +q∈R{qα − g(q)}. +Note that g∗ is always concave since it is the infimum of a family of affine functions. +For more detail concerning the theory of concave functions, we refer the reader to +[26]. +Now, we say that µ satisfies the multifractal formalism when fµ = τ ∗ +µ. In general, +the multifractal formalism need not hold, but it is well-known that the concave +conjugate of the Lq-spectrum is an upper bound for the multifractal spectrum. For +the convenience of the reader, we provide a short self-contained proof, which follows +the main ideas of [19, Theorem 4.1]. + +Multifractal Random Substitutions +15 +Proposition 2.3. Let µ be a shift-invariant measure on a subshift X. Then fµ(α) ≤ τ ∗(α) +for all α ∈ R. +Proof. Recall that ξn denotes the partition of X into cylinders corresponding to words +of length 2n + 1, each of which has diameter precisely e−2n−1. For α ∈ R, n ∈ N and +ϵ > 0, let +Mn,ϵ(α) = +� +I ∈ ξn : e−(2n+1)(α+ϵ) ≤ µ(I) ≤ e−(2n+1)(α−ϵ)� +. +In other words, Mn,ϵ(α) is an ϵ-approximation of Fµ(α) at level n. Our strategy is to +control the size of the sets Mn,ϵ(α) in terms of the Lq-spectrum of µ, and then use +these sets to build a good cover of Fµ(α). Let q ∈ ∂τ ∗(α): we prove this in the case +that q ≥ 0; the case q < 0 is analogous. +First, observe that +(2.4) +S2n+1,µ(q) = +� +I∈ξn +µ(I)q ≥ +� +u∈Mn,ϵ(α) +µ(I)q ≥ e−(2n+1)(α+ϵ)q#Mn,ϵ(α). +Since τµ(q) = lim infn→∞(log S2n+1,µ(q))/(−2n−1) by Lemma 2.2, there is some Nϵ ∈ N +so that for all n ≥ Nϵ, S2n+1,µ(q) ≤ e−(2n+1)(τµ(q)−ϵ). Combining this with (2.4), +(2.5) +#Mn,ϵ(α) ≤ e−(2n+1)(τ(q)−ϵ) · e(2n+1)(α+ϵ)q = e(2n+1)(τ ∗(α)+(q+1)ϵ) +for all n ≥ Nϵ where we have used the fact that q ∈ ∂τ ∗(α). +Now for each x ∈ Fµ(α), we can find some nx ∈ N so that for all n ≥ nx, µ(ξn(x)) ≥ +e−(2n+1)(α+ϵ). In particular, +Gϵ := +∞ +� +n=Nϵ +Mn,ϵ(α) +is a Vitali cover for Fµ(α). +Now suppose {Ij}∞ +j=1 is any disjoint subcollection of Gϵ: then with s = τ ∗(α) + +2ϵ(1 + q), +∞ +� +j=1 +(diam Ij)s ≤ +∞ +� +n=Nϵ +� +I∈Mn,ϵ(α) +(diam I)s ≤ +∞ +� +n=Nϵ +e−(2n+1)s#Mn,ϵ(α) +≤ +∞ +� +n=Nϵ +e−(2n+1)se(2n+1)(τ ∗(α)+(q+1)ϵ) += +∞ +� +n=Nϵ +(e−(1+q)ϵ)2n+1 < ∞ +by (2.5). Thus by the Vitali covering theorem for Hausdorff measure, there is a cover +{Ei}∞ +i=1 for Fµ(α) such that +Hs(Fµ(α)) ≤ +∞ +� +i=1 +(diam Ei)s < ∞ + +16 +ANDREW MITCHELL AND ALEX RUTAR +and thus dimH Fµ(α) ≤ τ ∗(α) + 2ϵ(1 + q). But ϵ > 0 was arbitrary, so the desired result +follows. +□ +2.5. Random substitutions and frequency measures. We now introduce our pri- +mary objects of interest: random substitutions, and their associated frequency measures. +In a similar manner to [14, 15], we define a random substitution by the data required +to determine its action on letters. We then extend this to a random map on words. +Definition 2.4. Let A = {a1, . . . , ad} be a finite alphabet. A random substitution ϑP = +(ϑ, P ) is a set-valued substitution ϑ: A → F(A+) together with a set of non-degenerate +probability vectors +P = +� +pi = (pi,1, . . . , pi,ri) : ri = #ϑ(ai); pi ∈ (0, 1]ri; +ri +� +j=1 +pi,j = 1 for all i = 1, . . . , d +� +, +such that +ϑP : ai �→ +� +� +� +� +� +s(i,1) +with probability pi,1, +... +... +s(i,ri) +with probability pi,ri, +for every 1 ≤ i ≤ d, where ϑ(ai) = {s(i,j)}1≤j≤ri. +We call each s(i,j) a realisation of ϑP (ai). If ri = 1 for all i ∈ {1, . . . , d}, then we call +ϑP deterministic. +Example 2.5 (Random Fibonacci). Let A = {a, b}, and let p ∈ (0, 1). The random +Fibonacci substitution ϑP = (ϑ, P ) is the random substitution given by +ϑP : +� +� +� +� +� +� +� +a �→ +� +ab +with probability p, +ba +with probability 1 − p, +b �→ a +with defining data ra = 2, rb = 1, s(a,1) = ab, s(a,2) = ba, s(b,1) = a, P = {pa = (p, 1 − +p), pb = (1)} and corresponding set-valued substitution ϑ: a �→ {ab, ba}, b �→ {a}. +In the following we describe how a random substitution ϑP determines a (countable +state) Markov matrix Q, indexed by A+ × A+. We interpret the entry Qu,v as the +probability of mapping a word u to a word v under the random substitution. Formally, +Qai,s(i,j) = pi,j for all j ∈ {1, . . . , ri} and Qai,v = 0 if v /∈ ϑ(ai). We extend the action +of ϑP to finite words by mapping each letter independently to one of its realisations. +More precisely, given n ∈ N, u = ai1 · · · ain ∈ An and v ∈ A+ with |v| ≥ n, we let +Dn(v) = {(v(1), . . . , v(n)) ∈ (A+)n : v(1) · · · v(n) = v} + +Multifractal Random Substitutions +17 +denote the set of all decompositions of v into n individual words and set +Qu,v = +� +(v(1),...,v(n))∈Dn(v) +n +� +j=1 +Qaij ,v(j). +In words, ϑP (u) = v with probability Qu,v. +For u ∈ A+, let (ϑn +P (u))n∈N be a stationary Markov chain on some probability space +(Ωu, Fu, Pu), with Markov matrix given by Q; that is, +Pu[ϑn+1 +P +(u) = w | ϑn +P (u) = v] = Pv[ϑP (v) = w] = Qv,w +for all v and w ∈ A+, and n ∈ N. In particular, +Pu[ϑn +P (u) = v] = (Qn)u,v +for all u and v ∈ A+, and n ∈ N. We often write P for Pu if the initial word is +understood. In this case, we also write E for the expectation with respect to P. As +before, we call v a realisation of ϑn +P (u) if (Qn)u,v > 0 and set +ϑn(u) = {v ∈ A+ : (Qn)u,v > 0} +to be the set of all realisations of ϑn +P (u). Conversely, we may regard ϑn +P (u) as the set +ϑn(u) endowed with the additional structure of a probability vector. If u = a ∈ A is a +letter, we call a word v ∈ ϑk(a) a level-k inflation word, or exact inflation word. +To a given random substitution ϑP = (ϑ, P ) one can associate a subshift. First, +we say that a word u ∈ A+ is (ϑ-)legal if there exists an ai ∈ A and k ∈ N such +that u appears as a subword of some word in ϑk(ai). We define the language of ϑ by +Lϑ = {u ∈ A+ : u is ϑ-legal} and, for w ∈ A+ ∪ AZ, we let L(w) = {u ∈ A+ : u ◁ w} +denote the language of w. +Definition 2.6. The random substitution subshift of a random substitution ϑP = (ϑ, P ) +is the system (Xϑ, S), where Xϑ = {w ∈ AZ : L(w) ⊆ Lϑ} and S denotes the (left) +shift map, defined by S(w)i = wi+1 for each w ∈ Xϑ. +Under very mild assumptions, the space Xϑ is non-empty [17]. This holds, for +example, if the generating random substitution is primitive (we give a definition +in Section 2.6). We endow Xϑ with the subspace topology inherited from AZ, and +since Xϑ is defined in terms of a language, it is a compact S-invariant subspace of AZ. +Hence, Xϑ is a subshift. For n ∈ N, we write Ln +ϑ = Lϑ ∩ An and Ln(w) = L(w) ∩ An to +denote the subsets of Lϑ and L(w), respectively, consisting of words of length n. The +set-valued function ϑ naturally extends to Xϑ, where for w = · · · w−1w0w1 · · · ∈ Xϑ +we let ϑ(w) denotes the (infinite) set of sequences of the form v = · · · v−2v−1.v0v1 · · · , +with vj ∈ ϑ(wj) for all j ∈ Z. It is easily verified that ϑ(Xϑ) ⊂ Xϑ. +The notation Xϑ reflects the fact that the random substitution subshift does not +depend on the choice of (non-degenerate) probabilities P . In fact, this is the case for +many structural properties of ϑP . In these cases, one sometimes refers to ϑ instead of +ϑP as a random substitution, see for instance [14, 16, 27, 28]. On the other hand, for + +18 +ANDREW MITCHELL AND ALEX RUTAR +some applications, one needs additional structure on the probability space. In fact, +there is an underlying branching process, similar to a Galton–Watson process, that +allows one to construct more refined random variables, see [17] for further details. +The measure theoretic properties we consider are typically dependent on the choice +of probabilities; however, some of the auxiliary results we use only depend on the set- +valued substitution ϑ. To avoid confusion, for results where there is no dependence +on the choice of probabilities we will give the statement in terms of the set-valued +substitution ϑ and omit the dependence on P in the notation. +2.6. Primitive random substitutions. A standard assumption in the study of sub- +stitutions (both deterministic and random) is that of primitivity. Given a random +substitution ϑP = (ϑ, P ) over an alphabet A = {a1, . . . , ad} with cardinality d ∈ N, +we define the substitution matrix M = MϑP ∈ Rd×d of ϑP by +Mi,j = E[|ϑP (aj)|ai] = +rj +� +k=1 +pj,k|s(j,k)|ai. +Since M has only non-negative entries, it has a real eigenvalue λ of maximal modulus. +Observe that λ ≥ 1, with λ = 1 precisely if M is column-stochastic, so that the random +substitution is non-expanding. To avoid this degenerate situation, we will assume +that λ > 1. If the matrix M is primitive (that is if there exists a k ∈ N such that +all the entries of M k are positive), the Perron–Frobenius theorem gives that λ is a +simple eigenvalue and that the corresponding (right) eigenvector R = (R1, . . . , Rd) +can be chosen to have strictly positive entries. We will normalise this eigenvector so +that ∥R∥1 = 1. We will refer to λ as the Perron–Frobenius eigenvalue of the random +substitution, ϑP , with corresponding Perron–Frobenius eigenvector R. +Definition 2.7. We say that ϑP is primitive if M = MϑP is primitive and its Perron– +Frobenius eigenvalue satisfies λ > 1. +We emphasise that for a random substitution ϑP , being primitive is independent of +the (non-degenerate) choice of probabilities P . In this sense, primitivity is a property +of ϑ rather than ϑP . +Since M k +ϑP = Mϑk +P , the Perron–Frobenius eigenvalue of ϑk +P is λk. +2.7. Compatible random substitutions. Another standard assumption in the study +of random substitutions is that of compatibility, which gives that exact inflation words +have a well-defined abelianisation. In particular, the matrix of a compatible random +substitution is independent of the choice of probabilities, so the letter frequencies +are uniform and do not depend on the realisation. As discussed in the introduction, +the existence of uniform letter frequencies is fundamental in the proofs of our main +results. +Definition 2.8. We say that a random substitution ϑP = (ϑ, P ) is compatible if for all +a ∈ A, and u, v ∈ ϑ(a), we have Φ(u) = Φ(v). + +Multifractal Random Substitutions +19 +Observe that compatibility is independent of the choice of probabilities, and that +a random substitution ϑP = (ϑ, P ) is compatible if and only if for all u ∈ A+, we +have that |s|a = |t|a for all s and t ∈ ϑ(u), and a ∈ A. We write |ϑ(u)|a to denote +this common value, and let |ϑ(u)| denote the common length of words in ϑ(u). For +convenience, we also set |ϑ| = maxa∈A|ϑ(a)|. For a random substitution that is both +primitive and compatible, the (uniform) letter frequencies are encoded by the right +Perron–Frobenius eigenvector of the substitution matrix, which by compatibility is +independent of the choice of probabilities. In particular, we have the following (see +[25] for a proof in the deterministic case, which also holds in the random case by +compatibility). +Lemma 2.9 (Letter frequency bounds). If ϑP is a primitive and compatible random substi- +tution, then for all ε > 0 there is an integer N such that every word v of length at least N +satisfies +|v|(Ra − ε) < |v|a < |v|(Ra + ε) +for all a ∈ A. +The random Fibonacci substitution defined in Example 2.5 is compatible, since +Φ(ab) = Φ(ba) = (1, 1). It is also primitive, since the square of its substitution matrix +is positive. For any choice of probabilities, the right Perron–Frobenius eigenvector is +given by (τ −1, τ −2), where τ denotes the golden ratio. In terms of letter frequencies, +this means that in all sufficiently long legal words, approximately τ −1 proportion of +the letters are a and τ −2 proportion are b. +The following consequence of Lemma 2.9 is useful in the proof of Theorem A. +Lemma 2.10. Let ϑP = (ϑ, P ) be a primitive and compatible random substitution and let +q ≥ 1. For all ε > 0, there is an M ∈ N such that for every m ≥ M and v ∈ Lm +ϑ , +� +a∈A +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� +m(Ra+ε) +≤ +� +w∈ϑ(v) +P[ϑP (v) = w]q +≤ +� +a∈A +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� +m(Ra−ε) +. +For q ≤ 1, the same result holds with reversed inequalities. +Proof. Since ϑP is compatible, the cutting points of inflation tiles are well-defined, so +breaking the sum into inflation tiles we obtain +� +w∈ϑ(v) +P[ϑP (v) = w]q = +� +w1∈ϑ(v1) +P[ϑP (v1) = w]q +� +w2∈ϑ(v2) +· · · +� +wm∈ϑ(vm) +P[ϑP (vm) = wm]q += +� +a∈A +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� +|v|a +. + +20 +ANDREW MITCHELL AND ALEX RUTAR +The result then follows by applying Lemma 2.9 to bound |v|a, noting that for all a ∈ A +we have � +s∈ϑ(a) P[ϑP (a) = s]q ≤ 1 if q ≥ 1 and � +s∈ϑ(a) P[ϑP (a) = s]q ≥ 1 if q ≤ 1. +□ +2.8. Frequency measures. The main object that we associate with a given primitive +random substitution ϑP is the frequency measure µP . This measure quantifies the +relative occurrence of a given word in a random substitution. We now define this +measure precisely. +First, we define the expected frequency of a word v ∈ Lϑ by +freq(v) = lim +k→∞ +E[|ϑk +P (a)|v] +E[|ϑk +P (a)|] , +where, by primitivity, this limit is independent of the choice of a ∈ A. In fact, we +have the stronger property that the word frequencies exist P-almost surely in the +limit of large inflation words and are given by freq(v) for all v ∈ Lϑ (see [17] for +further details). Recalling that ξ(Xϑ) is the algebra of cylinder sets on Xϑ that specify +the origin, we define µP : ξ(Xϑ) ∪ {∅} → [0, 1] by µP (∅) = 0, µP (Xϑ) = 1, and +µP ([v]m) = freq(v) for v ∈ Lϑ and m ∈ {1 − |v|, 2 − |v|, . . . , 0}. This set function +extends to a unique measure. +Proposition 2.11 ([17, Proposition 5.3 and Theorem 5.9]). The set function µP is a +content with mass one which extends uniquely to a shift-invariant ergodic Borel probability +measure on Xϑ. +We call the measure µP defined in Proposition 2.11 the frequency measure correspond- +ing to the random substitution ϑP . Observe that frequency measures are dependent +on the probabilities of the substitution. As such, for the subshift of a primitive ran- +dom substitution that is non-deterministic, there exist uncountably many frequency +measures supported on this subshift [17]. In contrast, the subshift of a primitive +deterministic substitution has precisely one frequency measure, which is the unique +ergodic measure [25]. +Frequency measures corresponding to primitive and compatible random substi- +tutions satisfy the following renormalisation lemma, which relates the measure of +a cylinder set of a legal word to measures of cylinder sets of shorter words via the +production probabilities of the random substitution. This result first appeared in [17] +and is central to the proof of the main result in [15]. +Lemma 2.12 (Renormalisation). Let ϑP be a primitive and compatible random substitution +with corresponding frequency measure µP . Let n ∈ N and let k be an integer such that every +v ∈ Lk +ϑ has |ϑ(v)| ≥ n + |ϑ(v1)|. Then for every u ∈ Ln +ϑ, +µP ([u]) = 1 +λ +� +v∈Lk +ϑ +µP ([v]) +|ϑ(v1)| +� +j=1 +P[ϑP (v)[j,j+m−1] = u]. +Lemma 2.12 is key to the proof of Theorem A, as it relates the sums � +u∈Ln +ϑ µP ([u]) +to sums over smaller words via the production probabilities. This in turn allows us + +Multifractal Random Substitutions +21 +to obtain relations between τµP and ϕk. Under additional assumptions, simplified +reformulations of Lemma 2.12 can be obtained (see, for example, Lemma 2.18, which +is used in Theorem D). +2.9. Separation conditions and recognisability. In this section, we introduce addi- +tional common assumptions which either (1) impose a certain separation on inflation +words, or (2) impose a certain uniformity of the inflation and the probabilities. Under +these conditions, we can obtain closed-form formulas for the Lq-spectrum. +Definition 2.13. A random substitution ϑP = (ϑ, P ) satisfies the disjoint set condition +if +u and v ∈ ϑ(a) with u ̸= v =⇒ ϑk(u) ∩ ϑk(v) = ∅ +for all a ∈ A and k ∈ N. It satisfies the identical set condition if +u and v ∈ ϑ(a) =⇒ ϑk(u) = ϑk(v) +for all a ∈ A and k ∈ N. Moreover, we say that ϑP has identical production probabilities +if for all a ∈ A, k ∈ N and v ∈ ϑk(a), +P[ϑk−1 +P +(u1) = v] = P[ϑk−1 +P +(u2) = v] +for all u1 and u2 ∈ ϑ(a). +A consequence of the disjoint set condition is that for every a ∈ A, k ∈ N and +w ∈ ϑk(a), there is a unique v ∈ ϑk−1(a) such that w ∈ ϑ(v). In other words, every +exact inflation word can be uniquely de-substituted. The following definition extends +this idea of unique de-substitution from inflation words to all elements in the subshift. +Definition 2.14. Let ϑP = (ϑ, P ) be a primitive and compatible random substitution. +We call ϑP recognisable if for every x ∈ Xϑ there exists a unique y = · · · y−1y0y1 · · · ∈ Xϑ +and a unique integer k ∈ {0, . . . , |ϑ(y0)| − 1} with S−k(x) ∈ ϑ(y). +The following follows routinely from the definition of recognisability (a proof is +given in [15, Lemma 4.5]). +Lemma 2.15. If ϑP is a primitive, compatible and recognisable random substitution, then +ϑP satisfies the disjoint set condition. +In contrast to the disjoint set condition, recognisability is stable under taking powers +(see [15, Lemma 4.6]). +Lemma 2.16. Let ϑP be a primitive and compatible random substitution and m ∈ N. If ϑP +is recognisable, then so is ϑm +P . +An alternative characterisation of recognisability is the following local version. +Intuitively, local recognisability means that applying a finite window to a sequence is +enough to determine the position and the type of the inflation word in the middle +of that window. The following result is given in [15, Lemma 4.4] (see also [12, +Proposition 5.7]). + +22 +ANDREW MITCHELL AND ALEX RUTAR +Lemma 2.17. Let ϑP = (ϑ, P ) be a primitive and compatible random substitution. If ϑP is +recognisable, then there exists a smallest natural number κ(ϑ), called the recognisability +radius of ϑP , with the following property: if x ∈ ϑ([a]) for some a ∈ A and x[−κ(ϑ),κ(ϑ)] = +y[−κ(ϑ),κ(ϑ)] for some y ∈ Xϑ, then y ∈ ϑ([a]). +As a consequence of this local characterisation of recognisability, for every legal +word u with length greater than twice the radius of recognisability there exists an +inflation word w, appearing as a subword of u, which has a unique decomposition +into exact inflation words. We call the largest such w the recognisable core of u. +Local recognisability allows us to obtain a stronger version of Lemma 2.12 for +recognisable random substitutions. This result is key to obtaining the coincidence of +the Lq-spectrum and its inflation word analogue under recognisability for q < 0, and +thus the conclusion of Theorem D. +Lemma 2.18. Let ϑP = (ϑ, P ) be a primitive and compatible random substitution, with +corresponding frequency measure µP and u ∈ Lϑ. If v ∈ Lϑ and w ∈ ϑ(v) contains u as a +subword, then +µP ([u]) ≥ 1 +λµP ([v])P[ϑP (v) = w]. +If, additionally, ϑP is recognisable, |u| > 2κ(ϑ) and w′ is the recognisable core of u with +v′ ∈ Lϑ the unique legal word such that w′ ∈ ϑ(v′), then +µP ([u]) ≤ κ(ϑ) +λ µP ([v′])P[ϑP (v′) = w′]. +Proof. If u is a subword of w ∈ ϑ(v), then µP ([u]) ≥ µP ([w]). Thus by Lemma 2.12 +applied to µP ([w]), +µP ([u]) ≥ 1 +λµP ([v])P[ϑP (v) = w]. +Now, assume that ϑP is recognisable, |u| > 2κ(ϑ) and w′ ∈ ϑ(v′) is the recognisable +core of u. Let k be an integer such that every t ∈ Lk +ϑ has |ϑ(t)| ≥ k +|ϑ(v1)|. Since there +are at most κ(ϑ) letters of u preceding the recognisable core, if t ∈ Lk +ϑ is a word for +which u ∈ ϑ(t)[j,j+|u|−1] for some j ∈ {1, . . . , |ϑ(t1)|}, then ti · · · ti+|v|−1 = v′ for some +i ∈ {1, . . . , κ(ϑ)}. Moreover, since there is a unique way to decompose w′ into exact +inflation words, for each t ∈ Lk +ϑ there can be at most one j ∈ {1, . . . , ϑ(t1)} such that + +Multifractal Random Substitutions +23 +u ∈ ϑ(t)[j,j+|u|−1]. Hence, it follows by Lemma 2.12 that +µP ([u]) = 1 +λ +� +t∈Lk +µP ([t]) +|ϑ(t1)| +� +j=1 +P[ϑP (t)[j,j+|u|−1] = u] +≤ 1 +λ +κ(ϑ) +� +i=1 +� +t∈Lk +ϑ +ti···ti+|v|−1=v′ +µP ([t])P[ϑP (v′) = w′] += κ(ϑ) +λ µP ([v′])P[ϑP (v′) = w′], +which completes the proof. +□ +3. Lq-SPECTRA OF FREQUENCY MEASURES +In this section, we prove our main results on Lq-spectra of frequency measures. +Here, we relate the Lq-spectrum to a certain “symbolic” Lq-spectrum, which we +call the inflation word Lq-spectrum. Heuristically, the inflation word Lq-spectrum is +the natural guess for the Lq-spectrum if you do not account for non-uniqueness in +the positions in which legal words can appear in inflation words. This notion is +introduced in Section 3.1, where we also state and prove some of its key properties. In +particular, in Proposition 3.1, we prove a simple closed-form formula for the inflation +word Lq-spectrum under the disjoint set condition or the identical set condition with +identical production probabilities. In Proposition 3.2, we establish basic monotonicity +results. +Then, in Section 3.2 and Section 3.3, we establish the general bounds for the Lq- +spectrum in terms of the inflation word Lq-spectrum, giving Theorem A (the proof is +given in Section 3.4). We also prove that this bound is sharp in Section 3.5, under the +recognisability assumption. This proves the first part of Theorem D. However this +bound need not hold in general: we discuss a counterexample in Example 5.2. Finally, +in Section 3.6, we prove differentiability of the Lq-spectrum at q = 1 and show how to +recover known results for measure theoretic and topological entropy from our results +concerning Lq-spectra. +3.1. Inflation word Lq-spectra. Given a primitive random substitution ϑP = (ϑ, P ), +we can define an analogue of the Lq-spectrum in terms of its production probabilities, +in a similar manner to the inflation word analogue of entropy introduced in [15]. In +many cases, this notion coincides with the Lq-spectrum of the frequency measure +associated to ϑP . For each k ∈ N and q ∈ R, define +ϕk(q) = − +� +a∈A +Ra log +� +� � +s∈ϑk(a) +P[ϑk +P (a) = s]q +� +� , + +24 +ANDREW MITCHELL AND ALEX RUTAR +where R = (Ra)a∈A is the right Perron–Frobenius eigenvector of the substitution +matrix of ϑP . We define the inflation word Lq-spectrum of ϑP by +Tϑ,P (q) = lim inf +k→∞ +ϕk(q) +λk +. +We similarly define the upper variant T ϑ,P by taking a limit supremum in place of +the limit infimum. +We first state some key properties of Tϑ,P (q) which follow easily from the definition. +Firstly, if the random substitution ϑP is compatible and satisfies either the disjoint set +condition or the identical set condition with identical production probabilities, then +the limit defining Tϑ,P (q) exists for all q ∈ R and is given by a closed form expression. +For q ≥ 0, these properties transfer to the Lq-spectrum by Theorem A. +Proposition 3.1. Let ϑP be a primitive and compatible random substitution and q ∈ R. If +ϑP satisfies the disjoint set condition, then the limit defining Tϑ,P (q) exists and +Tϑ,P (q) = +1 +λ − 1ϕ1(q). +If ϑP satisfies the identical set condition and has identical production probabilities, then the +limit defining Tϑ,P (q) exists and +Tϑ,P (q) = 1 +λϕ1(q). +Proof. Fix q ∈ R. By the Markov property of ϑP , for all a ∈ A, k ∈ N and v ∈ ϑk(a), +(3.1) +P[ϑk +P (a) = v] = +� +s∈ϑ(a) +P[ϑP (a) = s] P[ϑk−1 +P +(s) = v]. +If ϑP satisfies the disjoint set condition, then for every v ∈ ϑk(a) there is a unique +s(v) ∈ ϑ(a) such that v ∈ ϑk−1(s(v)). Thus, for all s ∈ ϑ(a) such that s ̸= s(v), we have +P[ϑk−1 +P +(s) = v] = 0, and so it follows by (3.1) that +� +v∈ϑk(a) +P[ϑk +P (a) = v]q = +� +v∈ϑk(a) +P[ϑP (a) = s(v)]q P[ϑk−1 +P +(s(v)) = v]q += +� +s∈ϑ(a) +P[ϑP (a) = s]q +� +u∈ϑk−1(s) +P[ϑk−1 +P +(s) = u]q += +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� · +� +b∈A +� +� +� +u∈ϑk−1(b) +P[ϑk−1 +P +(b) = u]q +� +� +|ϑ(a)|b +, + +Multifractal Random Substitutions +25 +where in the final equality we use compatibility to split the second sum into inflation +tiles. Thus +ϕk(q) = − +� +a∈A +Ra +� +b∈A +|ϑ(a)|b log +� +� +� +u∈ϑk−1(b) +P[ϑk−1 +P +(b) = u]q +� +� +− +� +a∈A +Ra log +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� += λϕk−1(q) + ϕ1(q), +noting that � +a∈A Ra|ϑ(a)|b = λRb. It follows inductively that +1 +λk ϕk(q) = +k +� +j=1 +1 +λj ϕ1(q) +k→∞ +−−−→ +1 +λ − 1ϕ1(q), +so the limit defining Tϑ,P (q) exists and is equal to (λ − 1)−1ϕ1(q). +On the other hand, if ϑP satisfies the identical set condition and has identical +production probabilities, then P[ϑk−1 +P +(s1) = u] = P[ϑk−1 +P +(s2) = u] for all s1, s2 ∈ ϑ(a). +Hence, it follows by (3.1) that +� +v∈ϑk(a) +P[ϑk +P (a) = v]q = +� +v∈ϑk(a) +P[ϑk−1 +P +(s) = v]q +for any choice of s ∈ ϑ(a). By compatibility and the independence of the action, +� +v∈ϑk(a) +P[ϑk +P (a) = v]q = +� +b∈A +� +� +� +u∈ϑk−1(b) +P[ϑk−1 +P +(b) = u]q +� +� +|ϑ(a)|b +, +and thus +ϕk(q) = +� +b∈A +� +a∈A +Ra|ϑ(a)|b log +� +� +� +v∈ϑk−1(b) +P[ϑk−1 +P +(b) = v]q +� +� = λϕk−1(q), +noting that � +a∈A Ra|ϑ(a)|b = Rb. It follows by induction that ϕk(q)/λk = ϕ1(q)/λ for +all k ∈ N, so we conclude that Tϑ,P (q) exists and equals λ−1ϕ1(q). +□ +Proposition 3.2. Let ϑP be a primitive and compatible random substitution. For all q > 1 +and q < 0, the sequence (λ−kϕk(q))k is non-decreasing; and for all 0 < q < 1, the sequence is +non-increasing. +Proof. This is largely a consequence of Jensen’s inequality. Note that on the interval +(0, 1], the function x �→ xq is convex if q > 1 or q < 0, and concave if 0 < q < 1. We + +26 +ANDREW MITCHELL AND ALEX RUTAR +first prove this for the case when q > 1 or q < 0. Observe that for all a ∈ A, k ∈ N +with k ≥ 2 and v ∈ ϑk(a), it follows by the Markov property of ϑP that +� +v∈ϑk +P[ϑk +P(a) = v]q = +� +v∈ϑk(a) +� +� +� +s∈ϑ(a): v∈ϑk−1(s) +P[ϑP(a) = s]P[ϑk−1 +P (s) = v] +� +� +q +≤ +� +v∈ϑk(a) +�� +s∈ϑ(a): v∈ϑk−1(s) P[ϑP(a) = s]P[ϑk−1 +P (s) = v]q +� +s∈ϑ(a): v∈ϑk−1(s) P[ϑP(a) = s] +� +≤ +� +b∈A +� +� +� +w∈ϑk−1(b) +P[ϑk−1 +P (b) = w]q +� +� +|ϑ(a)|b +. +In the second line, we apply Jensen’s inequality, and in the last line, we use compat- +ibility to decompose each probability P[ϑk−1 +P (s) = w] into inflation tiles. It follows +that +1 +λk ϕk(q) ≥ − 1 +λk +� +b∈A +Rb +� +a∈A +Ra|ϑ(a)|b log +� +� +� +w∈ϑk−1(b) +P[ϑk−1 +P +(b) = w]q +� +� = +1 +λk−1ϕk−1(q), +noting that � +a∈A Ra|ϑ(a)|b = λ. +The 0 < q < 1 case follows similarly, with Jensen’s inequality giving the opposite +inequality since x �→ xq is concave. +□ +An analogous monotonicity result does not hold in general for the (λk − 1)−1ϕk(q) +bounds, even when q ≥ 0. A counterexample is given by the random period doubling +substitution (Example 5.7) with non-uniform probabilities. +3.2. Lq-spectra for non-negative q. The majority of the work in proving Theorem A +lies in proving the bounds in (1.1), (1.2) and (1.3). Observe that it suffices to prove +the bound for the case k = 1, since we then obtain the bound for other k ∈ N by +considering higher powers of the random substitution. We first prove the upper +bound for the case q > 1. +Throughout this section, we assume that the random substitution is primitive and +compatible. +Proposition 3.3. For all q > 1, +τ µP (q) ≤ +1 +λ − 1ϕ1(q). +Proof. Fix q > 1. Let ε > 0 and, for each n ∈ N, let m(n) be the integer defined by +m(n) = +� +n +λ − ε +� +. + +Multifractal Random Substitutions +27 +Then the integers n and m(n) satisfy the conditions of Lemma 2.12, so it follows that +� +u∈Ln +ϑ +µP ([u])q = +� +u∈Ln +ϑ +� +� +�1 +λ +� +v∈Lm(n) +ϑ +µP ([v]) +|ϑ(v1)| +� +j=1 +P[ϑP (v)[j,j+n−1] = u] +� +� +� +q +. +Since q > 1, the function x �→ xq is superadditive on the interval [0, 1], so +� +u∈Ln +ϑ +µP ([u])q ≥ +� +u∈Ln +ϑ +� +v∈Lm(n) +ϑ +µP ([v])q +� +�1 +λ +|ϑ(v1)| +� +j=1 +P[ϑP (v)[j,j+n−1] = u] +� +� +q +≥ 1 +λq +� +v∈Lm(n) +ϑ +µP ([v])q +|ϑ(v1)| +� +j=1 +� +u∈Ln +ϑ +P[ϑP (v)[j,j+n−1] = u]q. +We now bound the probability on the right of this expression by the production +probability of an inflation word. Observe that if w(u) ∈ ϑ(v) contains u as a subword +in position j, then P[ϑP (v)[j,j+n−1] = u] ≥ P[ϑP (v) = w(u)]. Hence, +� +u∈Ln +ϑ +P[ϑP (v)[j,j+n−1] = u]q ≥ +� +w∈ϑ(v) +P[ϑP (v) = w]q +for all j ∈ {1, . . . , |ϑ(v1)|}. +Since ϑP is compatible, by Lemma 2.10 there exists an N ∈ N such that for all n ≥ N +and all v ∈ Lm(n) +ϑ +� +w∈ϑ(v) +P[ϑP (v) = w]q ≥ +� +a∈A +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� +m(n)(Ra+ε) +. +Hence, +� +u∈Ln +ϑ +µP ([u])q ≥ 1 +λq +� +a∈A +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� +m(n)(Ra+ε) +� +v∈Lm(n) +ϑ +µP ([v])q. +Taking logarithms, rearranging and dividing by n gives +−1 +n log +� +� � +u∈Ln +ϑ +µP ([u])q +� +� ≤ − 1 +n log +� +� +� +� +v∈Lm(n) +ϑ +µP ([v])q +� +� +� + 1 +n log λq +− m(n) +n +� +a∈A +(Ra + ε) log +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� . + +28 +ANDREW MITCHELL AND ALEX RUTAR +Noting that m(n)/n → (λ − ε)−1 as n → ∞, it follows by Lemma 2.2 that +τ µP (q) ≤ +1 +λ − ετ µP (q) + +1 +λ − ε +� +a∈A +log +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� + cε +where c := (#A) maxa∈A log(� +s∈ϑ(a) P[ϑP (a) = s]q). But ε > 0 was arbitrary; letting +ε → 0 and rearranging we obtain +τ µP (q) ≤ +1 +λ − 1ϕ1(q), +which completes the proof. +□ +We now prove the corresponding lower bound. +Proposition 3.4. For all q > 1, +τµP (q) ≥ 1 +λϕ1(q). +Proof. Let ε > 0 and, for each n ∈ N, let m(n) be the integer defined by +m(n) = +� +n +λ − ε +� +. +Since q > 1, the function x �→ xq is convex on the interval [0, 1]. Hence, it follows by +Lemma 2.12 and two applications of Jensen’s inequality that +� +u∈Ln +ϑ +µP ([u])q = +� +u∈Ln +ϑ +� +� +�1 +λ +� +v∈Lm(n) +ϑ +µP ([v]) +|ϑ(v1)| +� +j=1 +P[ϑP (v)[j,j+n−1] = u] +� +� +� +q +≤ +� +v∈Lm(n) +ϑ +µP ([v]) +� +u∈Ln +ϑ +� +�1 +λ +|ϑ(v1)| +� +j=1 +P[ϑP (v)[j,j+n−1] = u] +� +� +q +≤ |ϑ|q−1 +λq +� +v∈Lm(n) +ϑ +µP ([v]) +|ϑ(v1)| +� +j=1 +� +u∈Ln +ϑ +P[ϑP (v)[j,j+n−1] = u]q. +We bound above the probability on the right of this expression by the production +probability of a sufficiently large inflation word contained in u. By compatibility, +there is an integer k(n) such that j + n ≤ |ϑ(v[1,m(n)−k(n)])| for all n ∈ N and v ∈ Lm(n) +ϑ +, +where lim k(n)/n = 0. In particular, for every v ∈ Ln +ϑ, a realisation of ϑ(v[2,m(n)−k(n)]) is +contained in u as an inflation word, so +� +u∈Ln +ϑ +P[ϑP (v)[j,j+n−1] = u]q ≤ +� +w∈ϑ(v2···vm(n)−k(n)) +P[ϑ(v2 · · · vm(n)−k(n)) = w]q. + +Multifractal Random Substitutions +29 +We now bound this quantity uniformly for all v ∈ Lm(n) +ϑ +. By Lemma 2.10 and the +above, there is an N ∈ N such that for all n ≥ N +� +u∈Ln +ϑ +µP ([u])q ≤ |ϑ|q−1 +λq +� +a∈A +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� +(m(n)−k(n)−1)(Ra−ε) +. +Taking logarithms, rearranging and dividing by n gives +−1 +n log +� +� � +u∈Ln +ϑ +µP ([u])q +� +� ≥ m(n) − k(n) − 1 +n +� +a∈A +(Ra − ε) log +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� +− log(|ϑ|q−1/λq) +n +n→∞ +−−−→ +1 +λ − ε +� +a∈A +(Ra − ε) log +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� , +But ε > 0 was arbitrarily, so +τµP (q) ≥ 1 +λϕ1(q), +which completes the proof. +□ +We now state the bounds for the q ∈ (0, 1) case. We do not give a proof here since +the arguments mirror the proofs of Proposition 3.3 and Proposition 3.4, except with +reversed inequalities since x �→ xq is concave rather than convex and subadditive as +opposed to superadditive. +Proposition 3.5. If q ∈ (0, 1), then +1 +λ − 1ϕ1(q) ≤ τµP (q) ≤ τ µP (q) ≤ 1 +λϕ1(q). +3.3. Lq-spectra for negative q: lower bounds. For q < 0, there exist primitive and +compatible random substitutions for which τµP (q) and Tϑ,P (q) do not coincide (see, +for instance, Example 5.2). However, we still obtain that τµP (q) ≥ Tϑ,P (q) for all q < 0. +To prove this, it suffices to show the sequence of bounds in (1.3) holds. Again, we only +need to prove the bound for k = 1 since the remaining bounds follow by considering +powers of the random substitution. +Proposition 3.6. If ϑP is a primitive and compatible random substitution, then for all q < 0, +τµP (q) ≥ +1 +λ − 1ϕ1(q). +Proof. Let ε > 0 be sufficiently small and for n sufficiently large, let m(n) be the integer +defined by +m(n) = +� +n +λ − ε +� +. + +30 +ANDREW MITCHELL AND ALEX RUTAR +To avoid division by zero, we rewrite Lemma 2.12 in a form where we do not sum +over elements equal to zero. Here, we write u ◀ ϑ(v) to mean there is a realisation +w of ϑ(v) for which u appears as a subword of w. For each v ∈ Lm(n) +ϑ +and u ∈ Ln +ϑ, let +J (v, u) = {j ∈ {1, . . . , |ϑ(v1)|} : u ∈ ϑ(v)[j,j+n−1]}. Observe that if j /∈ J (u, v), then +P[ϑP (v)[j,j+n−1] = u] = 0, and if u does not appear as a subword of any realisations of +ϑ(v), then J (u, v) = ∅. Therefore, we can rewrite Lemma 2.12 as +µP ([u]) = 1 +λ +� +v∈Lm(n) +ϑ +u◀ϑ(v) +µP ([v]) +� +j∈J (v,u) +P[ϑP (v)[j,j+n−1] = u]. +Hence, by the subadditivity of the function x �→ xq on the domain (0, 1], +� +u∈Ln +ϑ +µP ([u])q = +� +u∈Ln +ϑ +� +� +� +� +� +1 +λ +� +v∈Lm(n) +ϑ +u◀ϑ(v) +µP ([v]) +� +j∈J (v,u) +P[ϑP (v)[j,j+n−1] = u] +� +� +� +� +� +q +≤ 1 +λq +� +u∈Ln +ϑ +� +v∈Lm(n) +ϑ +u◀ϑ(v) +µP ([v])q +� +j∈J (v,u) +P[ϑP (v)[j,j+n−1] = u]q += 1 +λq +� +v∈Lm(n) +ϑ +µP ([v])q � +u∈Ln +ϑ +u◀ϑ(v) +� +j∈J (v,u) +P[ϑP (v)[j,j+n−1] = u]q. +For each j ∈ J (v, u), let wj(u) ∈ ϑ(v) be a word such that wj(u)[j,j+n−1] = u. Note +that there are at most K := 2|ϑ|(#A)|ϑ| different u ∈ Ln +ϑ such that wj(u)[j,j+n−1] = u. +Hence, +� +u∈Ln +ϑ +u◀ϑ(v) +� +j∈J (v,u) +P[ϑP (v)[j,j+n−1] = u]q ≤ +� +u∈Ln +ϑ +u◀ϑ(v) +� +j∈J (v,u) +P[ϑP (v) = wj(u)]q +≤ K +� +w∈ϑ(v) +P[ϑP (v) = w]q +and it follows that +� +u∈Ln +ϑ +µP ([u])q ≤ λ−qK +� +v∈Lm(n) +ϑ +µP ([v])q � +w∈ϑ(v) +P[ϑP (v) = w]q. +Thus, by Lemma 2.10, for all ε > 0 there is an integer N such that for all n ≥ N +� +u∈Ln +ϑ +µP ([u])q ≤ λ−qK +� +a∈A +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� +m(n)(Ra+ε) � +� +� +� +v∈Lm(n) +ϑ +µP ([v])q +� +� +� . + +Multifractal Random Substitutions +31 +Taking logarithms, rearranging and dividing by n gives +−1 +n log +� +� � +u∈Ln +ϑ +µP ([u])q +� +� ≥ − 1 +n log +� +� +� +� +v∈Lm(n) +ϑ +µP ([v])q +� +� +� + 1 +n log(λ−qK) +− m(n) +n +� +a∈A +(Ra + ε) log +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� . +Noting that m(n)/n → (λ − ε)−1 as n → ∞, it follows by Lemma 2.2 that +τµP (q) ≥ +1 +λ − ετµP (q) + +1 +λ − ε +� +a∈A +log +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� + cε +where c := (#A) maxa∈A log(� +s∈ϑ(a) P[ϑP (a) = s]q). Letting ε → 0 and rearranging, +we obtain +τ µP (q) ≥ +1 +λ − 1ϕ1(q), +which completes the proof. +□ +3.4. Proof of general bounds for the Lq spectrum. Using the bounds proven in the +prior two sections, we can now complete the proof of Theorem A. +Proof of Theorem A. Since, for each k ∈ N, the random substitution ϑk +P gives rise to +the same frequency measure as ϑP , applying Proposition 3.3, Proposition 3.4 and +Proposition 3.5 to ϑk +P , +1 +λk ϕk(q) ≤ τµP (q) ≤ τ µP (q) ≤ +1 +λk − 1ϕk(q) +for all q > 1 and +1 +λk − 1ϕk(q) ≤ τµP (q) ≤ τ µP (q) ≤ 1 +λk ϕk(q) +for 0 < q < 1. Letting k → ∞ gives +τµP (q) = τ µP (q) = Tϑ,P (q) = T ϑ,P (q) +for all q ∈ (0, 1) ∪ (1, ∞), so the limits defining τµP (q) and Tϑ,P (q) both exist and +coincide. The same holds for q = 0 and q = 1 by continuity. The monotonicity of +the bounds λ−kϕk(q) follows by Proposition 3.2. Finally for q < 0, for each k ∈ N, +applying Proposition 3.6 to ϑk +P gives that +τµP (q) ≥ +1 +λk − 1ϕk(q). +Passing to the limit completes the proof. +□ + +32 +ANDREW MITCHELL AND ALEX RUTAR +3.5. Lq-spectra for negative q under recognisability. While the upper bound does +not hold in general for q < 0, for recognisable random substitutions we can ob- +tain this using Lemma 2.18, which we recall is a refinement of Lemma 2.12 using +recognisability. +Proposition 3.7. If ϑP is a primitive, compatible and recognisable random substitution, then +for all q < 0, +τ µP (q) ≤ +1 +λ − 1ϕ1(q). +Proof. Let ε > 0 be sufficiently small and, for each n ∈ N sufficiently large, let m(n) be +the integer defined by +m(n) = +� +n +λ − ε +� +. +For each u ∈ Ln+2κ(ϑ) +ϑ +, let w(u) denote the recognisable core of u. Further, let v(u) +denote the unique legal word such that w(u) ∈ ϑ(v(u)). Then, by Lemma 2.18, we +have +(3.2) +µP ([u]) ≤ κ(ϑ) +λ µP ([v(u)])P[ϑ(v(u)) = w(u)]. +Observe that for all u ∈ Ln+2κ(ϑ) +ϑ +, the recognisable core w(u) has length at least n +so, by compatibility, there is an integer N such that if n ≥ N, then |v(u)| ≥ m(n) +for all u ∈ Ln+2κ(ϑ) +ϑ +. In particular, for every u there exists a v ∈ Lm(n) +ϑ +such that +µP ([v(u)]) ≤ µP ([v]) and a w ∈ ϑ(v) such that P[ϑP (v(u)) = w(u)] ≤ P[ϑP (v) = w]. +Hence, it follows by (3.2) and Lemma 2.10 that +� +u∈Ln+2κ(ϑ) +ϑ +µP ([u])q ≥ 1 +λq +� +v∈Lm(n) +ϑ +µP ([v])q � +w∈ϑ(v) +P[ϑP (v) = w]q +≥ 1 +λq +� +a∈A +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� +m(Ra−ε) +� +v∈Lm(n) +ϑ +µP ([v])q, +noting that since q < 0, the function x �→ xq is decreasing on (0, 1]. Taking logarithms, +rearranging and dividing by n gives +−1 +n log +� +� � +u∈Ln +ϑ +µP ([u])q +� +� ≤ − 1 +n log +� +� +� +� +v∈Lm(n) +ϑ +µP ([v])q +� +� +� + 1 +n log λq +− m(n) +n +� +a∈A +(Ra − ε) log +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� . + +Multifractal Random Substitutions +33 +Noting that m(n)/n → (λ − ε)−1 as n → ∞, it follows by Lemma 2.2 that +τ µP (q) ≤ +1 +λ − ετ µP (q) + +1 +λ − ε +� +a∈A +log +� +� � +s∈ϑ(a) +P[ϑP (a) = s]q +� +� + cε +where c := (#A) maxa∈A log(� +s∈ϑ(a) P[ϑP (a) = s]q). Letting ε → 0 and rearranging, +we obtain +τ µP (q) ≤ +1 +λ − 1ϕ1(q), +which completes the proof. +□ +3.6. Recovering entropy from the Lq-spectrum. Since the Lq-spectrum encodes both +topological and measure theoretic entropy, Theorem A provides an alternative means +of proving the coincidence of these quantities with the inflation word analogues +introduced in [14, 15]. +For notational simplicity, set +ρk = − +� +a∈A +Ra +� +s∈ϑk(a) +P[ϑk +P (a) = s] log(P[ϑk +P (a) = s]). +Proof of Corollary C. We first establish the result for topological entropy. By Theo- +rem A, the limit defining Tϑ,P (0) exists; in particular, +lim +m→∞ +1 +λk +� +a∈A +Ra log(#ϑm(a)) +exists. Since htop(Xϑ) = −τµP (0) = −Tϑ,P (0), we conclude that +htop(Xϑ) = − lim +m→∞ +1 +λk +� +a∈A +Ra log(#ϑm(a)) +as claimed. +Now we consider measure theoretic entropy. We first make the following elemen- +tary observation: if f and g are concave functions with f(1) = g(1) and f(x) ≤ g(x) +for all x ≥ 1, then f +(1) ≤ g+(1). Indeed, for all ϵ > 0, +f(1 + ϵ) − f(1) +ϵ +≤ g(1 + ϵ) − g(1) +ϵ +, +and taking the limit as ϵ goes to 0 (which always exists by concavity) yields the desired +inequality. +Recall that τµP and λ−kϕk are concave functions with τµP (1) = ϕk(1) = 0 for +all k ∈ N. Moreover, ϕk is differentiable for all k ∈ N with ϕ′ +k(1) = ρk and by +Proposition 3.2 and Theorem A, +� +λ−kϕk +�∞ +j=1 converges monotonically to τµP from + +34 +ANDREW MITCHELL AND ALEX RUTAR +below. In particular, ρk/λk is a monotonically increasing sequence bounded above by +τ + +µP (1), so that the limit indeed exists. Thus +τ + +µP (1) = lim +k→∞ +ρk +λk +since ϕk(q)/(λk − 1) ≥ τµP (q) for all q ∈ (0, ∞), using the preceding observation. +The result for τ − +µP (1) follows by an identical argument, instead using monotonicity +and the corresponding bounds for q ∈ (0, 1). Thus τ ′ +µP (1) = limk→∞ ρk/λk, so the +desired result follows by Lemma 2.1(c). +□ +4. RECOGNISABILITY AND THE MULTIFRACTAL FORMALISM +In this section we establish the multifractal formalism as stated in Theorem D. To +do this, we prove a variational principle by considering typical local dimensions of +one frequency measure µP relative to another frequency measure µQ. Our strategy is +to prove the almost-sure existence of relative letter frequencies in Lemma 4.3: this result, +combined with recognisability, gives Proposition 4.5. The multifractal formalism then +follows from this dimensional result combined with the formula for the Lq-spectrum +proven in Proposition 3.7—the proof is given in Section 4.2. +4.1. Non-typical local dimensions. To prove the multifractal formalism for a given +frequency measure µP , we show that for every α ∈ [αmin, αmax], there exists another +frequency measure µQ such that dimH µQ ≥ τ ∗ +µP (α) and dimloc(µP , x) = α for µQ- +almost every x ∈ Xϑ. Given a primitive set-valued substitution ϑ, permissible +probabilities P and Q, m ∈ N and a ∈ A, define the quantity Hm,a +P ,Q(ϑ) by +Hm,a +P ,Q(ϑ) = +� +v∈ϑm(a) +−P[ϑm +Q(a) = v] log P[ϑm +P (a) = v]. +Further, let Hm +P ,Q(ϑ) denote the vector (Hm,a +P ,Q(ϑ))a∈A. We first prove some properties +of the quantity Hm +P ,Q(ϑ) which we will use in the proof of Proposition 4.5. +Lemma 4.1. If ϑ is a primitive and compatible set-valued substitution and P and Q are +permissible probabilities, then for all m ∈ N, a ∈ A and s ∈ ϑ(a), +� +v∈ϑm(s) +P[ϑm +Q(s) = v] log P[ϑm +P (s) = v] = +� +b∈A +|ϑ(a)|b Hm,b +P ,Q(ϑ). +Proof. Since ϑ is compatible, we can decompose each v ∈ ϑm(s) into inflation words +v = v1 · · · v|ϑ(a)|. By the Markov property of ϑP (respectively ϑQ), we have that +P[ϑm +P (s) = v] = P[ϑm +P (s1) = v1] · · · P[ϑm +P (s|ϑ(a)|) = v|ϑ(a)|]. + +Multifractal Random Substitutions +35 +Therefore +� +v∈ϑm(s) +P[ϑm +Q(s) = v] log P[ϑm +P (s) = v] = +� +b∈A +|ϑ(a)|b +� +w∈ϑm(b) +P[ϑm +Q(b) = w] log P[ϑm +P (b) = w] += +� +b∈A +|ϑ(a)|b Hm,b +P ,Q(ϑ), +which completes the proof. +□ +Lemma 4.2. If ϑ is a primitive and compatible set-valued substitution satisfying the disjoint +set condition, with right Perron–Frobenius eigenvector R, and P and Q are permissible +probabilities, then the random substitutions ϑP = (ϑ, P ) and ϑQ = (ϑ, Q) satisfy +1 +λmHm +P ,Q(ϑ) · R → +1 +λ − 1H1 +P ,Q(ϑ) · R +as m → ∞. +Proof. Since ϑ satisfies the disjoint set condition, for all m ∈ N and a ∈ A, +Hm+1 +P ,Q (ϑ) · R = +� +a∈A +Ra +� +v∈ϑm+1(a) +P[ϑm+1 +Q +(a) = v] log P[ϑm+1 +P +(a) = v] += +� +a∈A +Ra +� +s∈ϑ(a) +P[ϑQ(a) = s] log P[ϑP (a) = s] ++ +� +a∈A +Ra +� +s∈ϑ(a) +P[ϑQ(a) = s] +� +v∈ϑm(s) +P[ϑm +Q(s) = v] log P[ϑm +P (s) = v] += H1 +P ,Q(ϑ) · R + +� +b∈A +Hm,b +P ,Q(ϑ) +� +a∈A +|ϑ(a)|bRa += H1 +P ,Q(ϑ) · R + λ +� +b∈A +RbHm,b +P ,Q(ϑ) += H1 +P ,Q(ϑ) · R + λHm +P ,Q(ϑ) · R. +In the second equality we use the Markov property of ϑP and ϑQ, laws of logarithms, +and that � +v∈ϑm(s) P[ϑm +Q(s) = v] = 1 for all s ∈ ϑ(a); in the third we apply Lemma 4.1; +in the fourth we use that MϑR = λR. Applying the above inductively, +1 +λmHm +P ,Q(ϑ) · R = +m +� +j=1 +1 +λj H1 +P ,Q(ϑ) · R +m→∞ +−−−→ +1 +λ − 1H1 +P ,Q(ϑ) · R, +which completes the proof. +□ +Any bi-infinite sequence x in the subshift of a recognisable random substitution can +be written as a bi-infinite concatenation of exact inflation words (wn,an), where wn,an +is an inflation word generated from the letter an. Given a recognisable set-valued + +36 +ANDREW MITCHELL AND ALEX RUTAR +substitution ϑ, a ∈ A and w ∈ ϑ(a), we define the inflation word frequency of (a, w) +in x ∈ Xϑ by +fx(a, w) = lim +n→∞ f n +x (a, w) +f n +x (a, w) = +1 +2n + 1#{m: am = a, wm,am = w, wm,am in x[−n,n]}, +provided the limit exists. For a given frequency measure µP , the inflation word +frequency of a µP -typical word is determined by the production probabilities. More +specifically, we have the following. +Lemma 4.3. Let ϑP = (ϑ, P ) be a primitive, compatible and recognisable random substitu- +tion with corresponding frequency measure µP . For µP -almost every x ∈ Xϑ , the inflation +word frequency exists and is given by +fx(a, w) = 1 +λRaP[ϑP (a) = w], +for all a ∈ A and w ∈ ϑ(a). +Proof. Let Aa,w be the set of points x ∈ Xϑ such that the above does not hold. We show +that Aa,w is a null set. Taking the complement and then the intersection over all a, w +gives a full-measure set with the required property. Given ε > 0, let E(n, ε) be the set +of x ∈ Xϑ such that +|f n +x (a, w) − 1 +λRaP[ϑP (a) = w]| > ε. +By the Borel–Cantelli lemma, it suffices to show that +� +n∈N +µP (E(n, ε)) < ∞ +for all ε > 0 in order to conclude that Aa,w is a nullset. To this end, we show that +µP (E(n, ε)) decays exponentially with n. Given u with |u| = 2n + 1 > 2κ(ϑ), let uR +denote the recognisable core of u, which has length at least |u| − 2κ(ϑ). Lemma 2.18 +gives that +µP ([u]) ≤ κ(ϑ) +λ µP ([v])P[ϑP (v) = uR] = κ(ϑ) +λ µP ([v]) +|v| +� +i=1 +P[ϑP (vi) = wi,vi] +where each wi,vi is the inflated image of vi in uR. By compatibility, we can choose +an integer N such that every v of length at least N satisfies |v|(Ra − ε/3) ≤ |v|a ≤ +|v|(Ra + ε/3) for all a ∈ A. For each v and a ∈ A, let Aa(v) denote the set of u′ ∈ ϑ(v) +such that the frequency of indices i ∈ {j : aj = a} with wi,a = w deviates from +P[ϑ(a) = w] by more than ε/3. Since ϑP acts independently on letters, it follows by +Cramér’s theorem that the sum � +u′∈A(v) P[ϑP (v) = u′] decays exponentially with |v|a + +Multifractal Random Substitutions +37 +(and hence with |v|). In particular, there is a constant C > 0, independent of the choice +of v, such that +(4.1) +� +u′∈A(v) +P[ϑP (v) = u′] ≤ e−Cn. +Note that if u is a sufficiently long legal word and has [u] ∩ E(n, ε) = ∅, then we +require that uR ∈ A(v). Indeed, if u′ /∈ A(v) and |v| ≥ N, then the relative inflation +word frequency of w is bounded above by +{j : aj = a} +|v| +|v| +|u| +� +P[ϑP (a) = w] + ε +3 +� +≤ 1 +λ +� +Ra + ε +3 +� � +P[ϑP (a) = w] + ε +3 +� +≤ 1 +λRaP[ϑP (a) = w] + ε +and, similarly, bounded below by RaP[ϑP (a) = w]/λ−ε; hence, [uR]∩E(n, ε) = ∅. Let +Vn denote set of all words which appear as the (unique) preimage of the recognisable +core of a word of length n. It then follows by Lemma 2.18 that +µP (E(n, ε)) ⩽ +� +u∈Ln +ϑ +[u]∩E(n,ε)̸=∅ +µP ([u]) ≤ κ(ϑ) +λ +� +v∈Vn +µP ([v]) +� +u′∈A(v) +P[ϑP (v) = u′] ⩽ e−Cn, +where in the final inequality we have used (4.1) and that +� +v∈Vn +µP ([v]) ≤ +n +� +j=1 +� +v∈Lj +ϑ +µP ([v]) ≤ n, +absorbing this contribution and the κ(ϑ)/λ factor into the constant C. It follows that +∞ +� +n=1 +µP (E(n, ε)) ≤ +∞ +� +n=1 +e−Cn < ∞, +and the result then follows by the Borel–Cantelli lemma. +□ +Finally, we require the following bounds on the exponential scaling rate of measures +of cylinders, which is essentially a consequence of Theorem A. In particular, these +give bounds on the possible local dimensions of the measure. +Proposition 4.4. If ϑP is a primitive and compatible random substitution, then there are +values 0 < s1 < s2 < ∞ and c1, c2 > 0 such that for all n ∈ N and v ∈ Ln(Xϑ), +s1 · n + c1 ≤ log µP ([v]) ≤ s2 · n + c2 +Proof. By Theorem A, for all k ∈ N and q > 1, +τµP (q) ≤ +1 +λk − 1ϕk(q); + +38 +ANDREW MITCHELL AND ALEX RUTAR +and for q < 0, +1 +λk − 1ϕk(q) ≤ τµP (q), +Moreover, for each k, with +βk,min := lim +q→∞ +ϕk(q) +q(λk − 1) = − +1 +λk − 1 +� +a∈A +Ra log +� +min +v∈ϑk(a) P[ϑk +P (a) = v] +� +βk,max := lim +q→−∞ +ϕk(q) +q(λk − 1) = − +1 +λk − 1 +� +a∈A +Ra log +� +max +v∈ϑk(a) P[ϑk +P (a) = v] +� +, +it follows that [βk,min, βk,max] ⊂ (0, ∞) is a decreasing nested sequence of intervals, so +with βmin = limk→∞ βk,min and βmax = limk→∞ βk,max, +0 < βmin ≤ lim +q→∞ τµP (q) ≤ lim +q→−∞ τµP (q) ≤ βmax < ∞. +Applying Lemma 2.1(b) gives the result. +□ +Finally, we obtain our main conclusion concerning relative local dimensions. +Proposition 4.5. Let ϑ be a primitive, compatible and recognisable set-valued substitution, +let P and Q be permissible probabilities, and let µP and µQ denote the frequency measure +corresponding to ϑ endowed with the probabilities P and Q, respectively. Then, for µQ-almost +all x ∈ Xϑ, +(4.2) +dimloc(µP , x) = +1 +λ − 1 +� +a∈A +Ra +� +v∈ϑ(a) +−P[ϑm +Q(a) = v] log P[ϑm +P (a) = v]. +Proof. Fix m ∈ N. It follows by Lemma 2.16 that since ϑP is recognisable, so is ϑm +P . +For each x ∈ Xϑ and n ∈ N with n > κ(ϑm), let un +−(x) denote the recognisable core of +x[−n,n] and let un ++(x) denote an inflation word of minimal length that contains x[−n,n]. +By compatibility, |un +−(x)|/(2n + 1) → λ−m and |un ++(x)|/(2n + 1) → λ−m as n → ∞. +Further, let vn +−(x) be the legal word such that un +−(x) ∈ ϑm(vn +−(x)) and vn ++(x) be the +legal word such that un ++(x) ∈ ϑm(vn ++(x)). Then, it follows by Lemma 2.18 and the +definition of local dimension that +lim inf +n→∞ +� +− +1 +2n + 1 log µP ([un +−(x)]) − +1 +2n + 1 log P[ϑP (vn +−(x)) = un +−(x)] +� +≤ dimloc(µP , x) ≤ dimloc(µP , x) +≤ lim sup +n→∞ +� +− +1 +2n + 1 log µP ([un ++(x)]) − +1 +2n + 1 log P[ϑP (vn ++(x)) = un ++(x)] +� +. +By Proposition 4.4, there exists a constant C ≥ 0 such that for all x ∈ Xϑ, +0 ≤ lim inf +n→∞ − +1 +2n + 1 log µP ([un +−(x)]) ≤ lim sup +n→∞ − +1 +2n + 1 log µP ([un ++(x)]) ≤ C. + +Multifractal Random Substitutions +39 +Hence, it follows from the above that +(4.3) +lim inf +n→∞ − +1 +2n + 1 log P[ϑP (vn +−(x)) = un +−(x)] +≤ dimloc(µP , x) ≤ dimloc(µP , x) +≤ lim sup +n→∞ − +1 +2n + 1 log P[ϑP (vn ++(x)) = un ++(x)] + C +λm. +We now show that for µQ-almost all x ∈ Xϑ, +lim inf +n→∞ −1 +n log P[ϑP (vn +−(x)) = un +−(x)] = lim sup +n→∞ −1 +n log P[ϑP (vn ++(x)) = un ++(x)] += 1 +λmHm +P ,Q(ϑ) · R. +By compatibility, we can decompose the production probabilities into inflation tiles as +P[ϑm +P (vn +−(x)) = un +−(x)] = +� +a∈A +� +w∈ϑm(a) +P[ϑm +P (a) = w]Na,w(x,n), +where, for each a ∈ A and w ∈ ϑm(a), Na,w(x, n) denotes the number of a’s in vn +−(x) +which map to w. It follows by Lemma 4.3, applied to ϑm +Q, that for µQ-almost all x ∈ Xϑ, +we have +1 +2n + 1Na,w(x, n) → 1 +λmRaP[ϑm +Q(a) = w] +for all a ∈ A and w ∈ ϑm(a). Hence, it follows that +lim +n→∞ − +1 +2n + 1 log P[ϑm +P (vn +−(x)) = un +−(x)] += 1 +λm +� +a∈A +Ra +� +v∈ϑm(a) +P[ϑm +Q(a) = v] log P[ϑm +P (a) = v] += 1 +λmHm +P ,Q(ϑ) · R, +with the same convergence holding for un ++(x) by identical arguments. Thus, it follows +from (4.3) that +1 +λmHm +P ,Q(ϑ) · R ≤ dimloc(µP , x) ≤ dimloc(µP , x) ≤ 1 +λmHm +P ,Q(ϑ) · R + C +λm. +Since the above holds for all m ∈ N, by letting m → ∞ it follows by Lemma 4.2 that +dimloc(µP , x) exists and +dimloc(µP , x) = +1 +λ − 1H1 +P ,Q(ϑ) · R, +which completes the proof. +□ + +40 +ANDREW MITCHELL AND ALEX RUTAR +4.2. Proof of the multifractal formalism. In this section, we apply the results ob- +tained in the previous section, along with results on the Lq-spectrum under recognis- +ability, to prove Theorem D. +Proof of Theorem D. We first obtain the results for the Lq-spectrum. Since every recog- +nisable random substitution satisfies the disjoint set condition, Proposition 3.1 gives +that Tϑ,P (q) = (λ − 1)−1ϕ1(q) for all q ∈ R. If q < 0, then by Theorem A and Proposi- +tion 3.7, +1 +λ − 1ϕ1(q) = Tϑ,P (q) ≤ τµP (q) ≤ τ µP (q) ≤ +1 +λ − 1ϕ1(q), +so we conclude that τµP (q) exists and equals (λ − 1)−1ϕ1(q). For q ≥ 0, the result +follows already from Corollary B. +We now obtain the results on the multifractal spectrum. In light of Proposition 2.3, +it remains to show that fµP (α) ≥ τ ∗ +µP (α) for each α ∈ R. As proven above, for all +q ∈ R, +τµP (q) = +1 +λ − 1ϕ1(q) = +1 +λ − 1 +� +a∈A +RaTa(q) +where for each a ∈ A +Ta(q) = − log +� +s∈ϑ(a) +P[ϑP (a) = s]q. +First, fix α ∈ (αmin, αmax) and let q ∈ R be chosen so that τ ′ +µP (q) = α. Observe that +qα − τµP (q) = τ ∗ +µP (α). Then define Q by the rule +P[ϑQ(a) = s] = P[ϑP (a) = s]qeTa(q) +for all a ∈ A and s ∈ ϑ(a). Then by Corollary C, +dimH µQ = +1 +λ − 1 +� +a∈A +Ra +� +− +� +v∈ϑ(a) +P[ϑQ(a) = v] log P[ϑQ(a) = v] +� += q · +1 +λ − 1 +� +a∈A +Ra +� +− +� +v∈ϑ(a) +P[ϑQ(a) = v] log P[ϑP (a) = v] +� +− +1 +λ − 1 +� +a∈A +RaTa(q) +� +v∈ϑ(a) +P[ϑQ(a) = v] += qα − τµP (q) = τ ∗ +µP (α) +since +τ ′ +µP (q) = +1 +λ − 1 +� +a∈A +Ra +− � +v∈ϑ(a) P[ϑP (a) = v]q log P[ϑP (a) = v] +e−Ta(q) += +1 +λ − 1 +� +a∈A +Ra +� +− +� +v∈ϑ(a) +P[ϑQ(a) = v] log P[ϑP (a) = v] +� +. + +Multifractal Random Substitutions +41 +In fact, this shows that dimloc(µP , x) = α for µQ-almost all x ∈ Xϑ by Proposition 4.5. +Thus fµP (α) ≥ dimH µQ = τ ∗ +µP (α), as required. +The result for α = αmin (resp. α = αmax) follows similarly by taking a degenerate +probability vector Q assigning equal value to the realisations of ϑ(a) with maximal +(resp. minimal) probabilities given by P , and zero otherwise. The corresponding +non-degenerate sub-substitution is also compatible and recognisable, so the same +arguments yield the corresponding bounds. +□ +5. EXAMPLES, COUNTEREXAMPLES AND APPLICATIONS +5.1. Failure of bounds for q < 0 without recognisability. In the following two +examples, we show the results in Theorem A do not extend in general to give an +upper bound for the Lq-spectrum in terms of the inflation word Lq-spectrum, for +q < 0. In Example 5.1, we construct a class of frequency measures on the full-shift +on two letters for which the Lq-spectrum and inflation word analogue differ in the +q < 0 case. The random substitutions that give rise to these frequency measures are +not compatible, but in Example 5.2 we present a compatible analogue. +In contrast, in Example 5.3, we give an example showing that the results for q < 0 +can hold for all q ∈ R under the identical set condition with identical production +probabilities. +Example 5.1. Let p1 < p2 ∈ (0, 1) such that p1 + 3p2 = 1 and let ϑP be the random +substitution defined by +ϑP : a, b �→ +� +� +� +� +� +� +� +� +� +ab +with probability p1 +ba +with probability p2 +aa +with probability p2 +bb +with probability p2 +We will show for all sufficiently small q < 0 that τµP (q) > Tϑ,P (q). Observe that, for +each k ∈ N, the word vk = (ab)2k ∈ ϑk+1(a) ∩ ϑk+1(b) occurs with probability +P[ϑk+1 +P +(a) = vk] = P[ϑk+1 +P +(b) = vk] = p2k +1 . +Clearly, this is the minimal possible probability with which a level-k inflation word +can occur, so it follows that +lim +q→−∞ +Tϑ,P (q) +q += −1 +2 log p1. +Now, let u ∈ L2k+1 +ϑ +be arbitrary. We show that µP ([u]) ≥ p2k−1 +1 +p2k−1 +2 +/2. Since ϑ(a) = ϑ(b) +with identical production probabilities, it follows by Lemma 2.12 that for any choice +of w ∈ L2k+1 +ϑ +µP ([u]) = 1 +2 +� +P[ϑP (w)[1,2k+1] = u] + [ϑP (w)[2,2k+1+1] = u] +� +. + +42 +ANDREW MITCHELL AND ALEX RUTAR +If P[ϑP (w)[1,2k+1] = u] ≥ p2k−1 +1 +p2k−1 +2 +, then we are done, otherwise at least half of the +letters in v must be sent to ab. But then for u to appear from the second letter, at least +half of the letters in v must be sent to ba or bb, so P[ϑP (w)[2,2k+1+1] = u] ≥ p2k−1 +1 +p2k−1 +2 +. +Hence, µP ([u]) ≥ p2k−1 +1 +p2k−1 +2 +/2 so, in particular, +min +u∈L2k+1 +ϑ +µP ([u]) ≥ 1 +2p2k−1 +1 +p2k−1 +2 +. +It follows that +lim +q→−∞ +τµP (q) +q +≤ −1 +4(log p1 + log p2) < −1 +2 log p1 = lim +q→−∞ +Tϑ,P (q) +q +. +By a slight modification of this example, we can construct a compatible random +substitution for which the two notions do not coincide. +Example 5.2. Let p1 < p2 ∈ (0, 1) such that p1 + 3p2 = 1 and let ϑP be the random +substitution defined by +ϑP : a, b �→ +� +� +� +� +� +� +� +� +� +ab ba +with probability p1 +ba ab +with probability p2 +ab ab +with probability p2 +ba ba +with probability p2 +By similar arguments to the previous example, we obtain +lim +q→−∞ +τµP (q) +q +≤ −1 +8(log p1 + log p2) < −1 +4 log p1 = lim +q→−∞ +Tϑ,P (q) +q +. +The random substitution in Example 5.2 satisfies the identical set condition with +identical production probabilities. These conditions are also satisfied by the following +example. However, here the Lq-spectrum and inflation word analogue coincide for +all q ∈ R by a direct argument. +Example 5.3. We will show that for the random substitution +ϑP : a, b �→ +� +ab +with probability p +ba +with probability 1 − p +the limit defining τµP (q) exists for all q ∈ R, and +τµP (q) = Tϑ,P (q) = 1 +λϕ1(q) = −1 +2 log(pq + (1 − p)q). +Corollary B gives the result for all q > 0 and that τµP (q) ≥ Tϑ,P (q) = 2−1ϕ1(q) for all +q < 0, so it only remains to verify for all q < 0 that +τ µP (q) ≤ Tϑ,P (q). + +Multifractal Random Substitutions +43 +Since ϑ(v1) = ϑ(v2) for all v1, v2 ∈ Lϑ, it follows from Lemma 2.12 that for all +u ∈ L2m +ϑ +and any v ∈ Lm+1 +ϑ +, +µP ([u]) = 1 +2 +� +P[ϑ(v)[1,1+2m−1] = u] + P[ϑ(v)[2,2+2m−1] = u] +� +. +Let V2m = {(ab)m, (ba)m}. If u ∈ L2m +ϑ +\ V2m, then u must contain bb as a subword. This +uniquely determines the cutting points in any inflation word decomposition, so there +exists a unique v and j(u) ∈ {1, 2} such that u ∈ ϑ(v)[j(u),2m+j(u)−1]. It follows that +� +u∈L2m +ϑ +µP ([u])q ≥ +� +u∈L2m +ϑ \V2m +�1 +2P[ϑP (v)[j(u),j(u)+2m−1] = u] +�q +≥ 1 +2q +� +u∈L2m +ϑ \V2m +P[ϑP (v2 · · · vm) = u[3−j(u),2−j(u)+2m]]q. +Now, for every w ∈ ϑ(v2 · · · vm) there is a u such that w = u[3−j(u),2−j(u)+2m]. Hence, +� +u∈L2m +ϑ +µP ([u])q ≥ 1 +2q +� +w∈ϑ(v2···vm) +P[ϑP (v2 · · · vm) = w]q +and the conclusion follows by similar arguments to those used in the proofs of the +main theorems. +5.2. Examples with recognisability. We first provide examples of random substitu- +tions for which the multifractal formalism holds. +Example 5.4. Let p > 0 and let ϑp be the random substitution defined by +ϑp : +� +� +� +� +� +a �→ +� +abb +with probability p +bab +with probability 1 − p +b �→ aa +Certainly ϑp is compatible, with corresponding primitive substitution matrix +M = +� +1 +2 +2 +0 +� +, +Perron–Frobenius eigenvalue (1 + +√ +17)/2, and (normalised) right Perron–Frobenius +eigenvector +� +−3 + +√ +17 +2 +, 5 − +√ +17 +2 +� +. +One can verify that ϑ is recognisable since every occurrence of aa intersects an image +of b and the adjacent letters then determine the cutting points. Thus by Theorem D, +for all q ∈ R +τµp(q) = Tϑ,P (q) = +1 +λ − 1ϕ1(q) = −7 − +√ +17 +8 +log(pq + (1 − p)q) + +44 +ANDREW MITCHELL AND ALEX RUTAR +τ1/5 +τ2/5 +(A) Lq-spectra +τ ∗ +1/5 +τ ∗ +2/5 +(B) Multifractal spectra +FIGURE 1. Lq-spectra and multifractal spectra corresponding to a recog- +nisable substitution for p ∈ {1/5, 2/5}. +and measure µp satisfies the multifractal formalism. The asymptotes have slopes −(7− +√ +17) log(p)/8 and −(7 − +√ +17) log(1 − p)/8. A plot of the Lq-spectra and multifractal +spectra for two choices of p is given in Figure 1. +For p = 1/2, the Lq-spectrum of the measure µp is a straight line and the multifractal +spectrum is equal to htop(Xϑ) at htop(Xϑ), and −∞ otherwise. +In the following example, we highlight that the multifractal spectrum need not +have value 0 at the endpoints. +Example 5.5. Let ϑp be the random substitution defined by +ϑp : +� +� +� +� +� +� +� +� +� +a �→ +� +� +� +� +� +abb +with probability p +bab +with probability p +bba +with probability 1 − 2p +b �→ aaa +Similarly to Example 5.4, ϑp is primitive, compatible and recognisable. Hence, Theo- +rem D gives that +τµp(q) = − 3 +10 log(2pq + (1 − 2p)q). +The asymptotes have slopes −3 log(p)/10 and −3 log(1 − 2p)/10. For p = 1/5 and +p = 2/5, the Lq-spectrum and multifractal spectrum of µp are plotted in Figure 2. +Here, we highlight that the endpoints of the multifractal spectrum need not be equal +to zero. + +Multifractal Random Substitutions +45 +τ1/5 +τ2/5 +(A) Lq-spectra +τ ∗ +1/5 +τ ∗ +2/5 +(B) Multifractal spectra +FIGURE 2. Lq-spectra and multifractal spectra corresponding to a recog- +nisable substitution for p ∈ {1/5, 2/5}. +Example 5.6. Consider the random substitution on the three-letter alphabet A = +{a, b, c} defined by +ϑP : +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +a �→ +� +bbc +with probability p1 +cbb +with probability 1 − p1 +b �→ +� +cca +with probability p2 +acc +with probability 1 − p2 +c �→ +� +aab +with probability p3 +baa +with probability 1 − p3 +for p1, p2, and p3 in (0, 1). It is immediate that this substitution is compatible, and +by considering the occurrences of 2, 3, or 4 letter repetitions, we observe that this +substitution is also recognisable. Moreover, the hypotheses of [15, Theorem 4.8] are +satisfied since ϑ is constant length and #ϑ(a) = #ϑ(b) = #ϑ(c). In particular, the +corresponding subshift Xϑ is intrinsically ergodic with unique measure of maximal +entropy given by taking p1 = p2 = p3 = 1/2. +It follows from [15, Lemma 4.12] that the measure of maximal entropy is not a +Gibbs measure with respect to the zero potential, so the system does not satisfy the +usual specification property. For this choice of uniform probabilities, the Lq-spectrum +is a straight line passing through the point (1, 0) with slope htop(Xϑ) = log(2)/2. More +generally, the Lq-spectrum is given for all q ∈ R by the formula +τµP (q) = −1 +6 +� +log +� +(1 − p1)q + pq +1 +� ++ log +� +(1 − p2)q + pq +2 +� ++ log +� +(1 − p3)q + pq +3 +�� +and the multifractal formalism is satisfied. + +46 +ANDREW MITCHELL AND ALEX RUTAR +For an example on an alphabet of size two, one may consider the random substitu- +tion +ϑP : +� +� +� +� +� +� +� +� +� +� +� +a �→ +� +ababbb +with probability p1 +abbabb +with probability 1 − p1 +b �→ +� +baabaa +with probability p2 +babaaa +with probability 1 − p2 +for p1 and p2 in (0, 1). The analysis of this example proceeds identically as above. +5.3. Examples without recognisability. Finally, we consider the two most commonly +studied examples of random substitutions: random period doubling and random +Fibonacci. +Example 5.7. Given p ∈ (0, 1), let ϑp be the random period doubling substitution +defined by +ϑp : +� +� +� +� +� +a �→ +� +ab +with probability p +ba +with probability 1 − p +b �→ aa +and let µp denote the corresponding frequency measure. The substitution ϑp satisfies +the disjoint set condition, so for all q ∈ [0, ∞), +τµp(q) = −2 +3 log(pq + (1 − p)q). +The asymptote as q → ∞ has slope −2 log(max{p, 1 − p})/3, which gives a sharp +lower bound on the local dimensions of µp. +If p = 1/2, then the measure µp has linear Lq-spectrum for q ≥ 0 given by +τµ1/2(q) = 2 +3(q − 1) log 2. +Since the substitution satisfies the disjoint set condition but is not recognisable, our +results do not give the Lq-spectrum for q < 0. +Example 5.8. The random Fibonacci substitution ϑp defined by +ϑp : +� +� +� +� +� +a �→ +� +ab +with probability p +ba +with probability 1 − p +b �→ a +does not satisfy either the identical set condition nor the disjoint set condition. Hence, +we cannot apply Corollary B to obtain a closed-form formula for τµp(q). However, we +can still apply Theorem A to obtain a sequence of lower and upper bounds. The case +k = 1 gives the following bounds for all 0 < q < 1: +− 1 +φ2 log(pq + (1 − p)q) = 1 +φϕ1(q) ≤ τµp(q) ≤ +1 +φ − 1ϕ1(q) = − log(pq + (1 − p)q), + +Multifractal Random Substitutions +47 +−1/2 +1/2 +1 +3/2 +2 +5/2 +1 +2 +3 +4 +5 +6 +ϕk/(λk − 1) +ϕk/λk +FIGURE 3. Upper and lower bounds on the Lq-spectrum of the fre- +quency measure corresponding to the random Fibonacci substitution +with p = 1/2, for k = 3, 5, 7. +where φ denotes the golden ratio. Reversing the inequalities yields the corresponding +bounds for q > 1. Of course, by considering larger k we can obtain better bounds. +For p = 1/2, the bounds given by Theorem A for k = 3, 5, 7 are shown in Figure 3. +ACKNOWLEDGEMENTS +The authors are grateful to Philipp Gohlke for his detailed comments on a draft +version of this manuscript, which helped to remove some technical assumptions from +Theorem D. They also thank Dan Rust and Tony Samuel for valuable input. AM +thanks SFB1283 and the Universität Bielefeld for supporting a research visit during +the summer of 2022, where some of the work on this project was undertaken. AM +was supported by EPSRC DTP and the University of Birmingham. AR was supported +by EPSRC Grant EP/V520123/1 and the Natural Sciences and Engineering Research +Council of Canada. The authors thank the organisers of the Junior Ergodic Theory +Meeting hosted at the ICMS in Edinburgh in March 2022, where this project began. +REFERENCES +1. Matthias Arbeiter and Norbert Patzschke, Random Self-Similar Multifractals, Math. 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Varjú, Recent progress on Bernoulli convolutions, Proceedings of the 7th European Congress +of Mathematics (Berlin), January 2018. 3 +SCHOOL OF MATHEMATICS, UNIVERSITY OF BIRMINGHAM, EDGBASTON, B15 2TT +Email address: acm925@student.bham.ac.uk +MATHEMATICAL INSTITUTE, UNIVERSITY OF ST ANDREWS, ST ANDREWS, KY16 9SS +Email address: alex@rutar.org + diff --git a/X9E4T4oBgHgl3EQfNwwL/content/tmp_files/load_file.txt b/X9E4T4oBgHgl3EQfNwwL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1f54045e528f03375c6a6440928d118e587cb176 --- /dev/null +++ b/X9E4T4oBgHgl3EQfNwwL/content/tmp_files/load_file.txt @@ -0,0 +1,1277 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf,len=1276 +page_content='Multifractal analysis of measures arising from random substitutions ANDREW MITCHELL AND ALEX RUTAR ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We study regularity properties of frequency measures arising from ran- dom substitutions, which are a generalisation of (deterministic) substitutions where the substituted image of each letter is chosen independently from a fixed finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, for a natural class of such measures, we derive a closed-form analytic formula for the Lq-spectrum and prove that the multifractal formalism holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This provides an interesting new class of measures satisfying the multifractal formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' More generally, we establish results concerning the Lq-spectrum of a broad class of frequency measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We introduce a new notion called the inflation word Lq-spectrum of a random substitution and show that this coincides with the Lq-spectrum of the corresponding frequency measure for all q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' As an application, we obtain closed- form formulas under separation conditions and recover known results for topological and measure theoretic entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' CONTENTS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Entropy and Lq-spectra 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Random substitutions 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Statement and discussion of main results 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Discussion and further work 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Preliminaries 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Symbolic notation 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Dynamics, entropy and dimension 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lq-spectra and smoothness 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal spectrum and multifractal formalism 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Random substitutions and frequency measures 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Primitive random substitutions 18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Compatible random substitutions 18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Frequency measures 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Separation conditions and recognisability 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lq-spectra of frequency measures 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Inflation word Lq-spectra 23 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lq-spectra for non-negative q 26 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 37B10, 37C45, 52C23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' random substitution, multifractal analysis, Lq-spectrum, multifractal formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='04958v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='DS] 12 Jan 2023 2 ANDREW MITCHELL AND ALEX RUTAR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lq-spectra for negative q: lower bounds 29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof of general bounds for the Lq spectrum 31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lq-spectra for negative q under recognisability 32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Recovering entropy from the Lq-spectrum 33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Recognisability and the multifractal formalism 34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Non-typical local dimensions 34 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof of the multifractal formalism 40 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Examples, counterexamples and applications 41 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Failure of bounds for q < 0 without recognisability 41 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Examples with recognisability 43 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Examples without recognisability 46 Acknowledgements 47 References 47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' INTRODUCTION A substitution is a combinatorial object consisting of a finite alphabet A along with a set of transformation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The theory of substitutions, along with statistical properties of the system under repeated iteration, is a large and actively researched field at the interface of combinatorics and symbolic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' A thorough introduction to the statistical properties and dynamics of substitutions can be found in [2, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Associated with a (deterministic) substitution is a frequency measure, which encodes the frequency of subwords under repeated iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Notably, the corresponding subshift supporting this measure has zero topological entropy, and the frequency measure is the unique ergodic measure supported on this subshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Random substitutions are a generalisation of (deterministic) substitutions [13] where we apply a transformation rule to each letter randomly and independently chosen from a finite set of possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Similarly to the deterministic case, subshifts associated with random substitutions support ergodic frequency measures which capture the expected rate of occurrence of subwords under random iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' But in contrast to the deterministic case, the corresponding subshift typically has positive topological entropy and supports uncountably many ergodic measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Random substitutions include examples exhibiting deterministic behaviour, while also includ- ing examples which are subshifts of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Moreover, there is a large amount of intermediate behaviour: subshifts of random substitutions can simultaneously exhibit long range correlation [3] (an indication of order) and positive topological entropy [14] (an indication of disorder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In this paper, we study the fine scaling properties of frequency measures associ- ated with random substitutions from the perspective of multifractal analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This perspective is relevant in a wide variety of contexts, such as the geometry of fractal sets and measures and in dynamical systems, with typical applications to geometric Multifractal Random Substitutions 3 measure theory and number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In our setting, our primary objects of study are the Lq-spectrum, which is a parametrised family of quantities which capture the inhomogeneous scaling properties of a measure, and the local dimension, which cap- ture the exponential growth rate of a measure around a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The Lq-spectrum and local dimensions are related through a heuristic relationship known as the multifractal formalism [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It is an important and well-studied question to determine settings in which the multifractal formalism holds, and to determine qualitative conditions describing its failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Much of the work on multifractal analysis has been done in the setting of local dimensions of self-similar measures (for some examples, see [1, 10, 11, 19, 29]) and Birkhoff sums of potentials in dynamical systems with a finite type property (see, for example, [8, 24] and the reference therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' As a notable recent example, in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Shmerkin’s recent proof of the Furstenberg ×2 ×3 conjecture [29], he computes the Lq-spectrum of a large class of dynamically self-similar measures and relates such results to the multifractal analysis of slices of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This information about Lq- spectra also implies Lp-smoothness properties in the question of absolute continuity of Bernoulli convolutions (see [30] for some background on this classic problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For more detail on the geometry of measures and multifractal analysis, we refer the reader to the foundational work by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Olsen [22] and the classic texts of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Falconer [7] and Ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Pesin [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Returning to our setting, substitution dynamical systems have characteristic fea- tures of (dynamical) self-similarity, but in many cases are far from being ergodic measures on shifts of finite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' More generally, frequency measures provide a rich family of shift-invariant ergodic measures which exhibit interesting and unique properties in symbolic dynamics in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For example, it was proved in [15, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='8] that for a certain class of random substitutions, the corresponding subshift supports a frequency measure that is the unique ergodic measure of maximal entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' However, this measure is not a Gibbs measure with respect to the zero potential, and therefore the system does not satisfy the common specification property, which is a well-known strategy for proving intrinsic ergodicity of symbolic dynamical systems (see [5] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Moreover, there are examples of random substitutions such that the corresponding subshift supports multiple ergodic mea- sures of maximal entropy [17, Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' More generally, many key properties of frequency measures associated with random substitutions are poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In this paper, we derive symbolic expressions for the Lq-spectrum of frequency measures associated with random substitutions under certain weak assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then under an additional assumption (recognisability), we prove a closed-form analytic expression for the Lq-spectrum and a variational formula which together imply the multifractal formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We emphasise that this class of examples has novel properties not witnessed before: in general, the unique frequency measure of maximal dimension is not a Gibbs measure with respect to the zero potential and the 4 ANDREW MITCHELL AND ALEX RUTAR corresponding subshift is not sofic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The techniques and results provide important new perspectives on the geometry and dynamics of the respective measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Entropy and Lq-spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For a Borel probability measure in a compact metric space, the Lq-spectrum is a well-studied quantity which encodes the scaling properties of the measure, in a weak sub-exponential sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Specifically, the Lq-spectrum of µ is given by τµ(q) = lim inf r→0 log sup � i µ � B(xi, r) �q log r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' where the supremum is taken over 2r-separated subsets {xi}i of the support of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The Lq-spectrum encodes information about the local scaling of the measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We define the local dimension of µ at x by dimloc(µ, x) = lim r→0 log µ � B(x, r) � log r when the limit exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We then define the multifractal spectrum of µ by fµ(α) = dimH {x ∈ X : dimloc(µ, x) = α} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In general, the structure of the set of local dimensions can be very complex—for example, the level sets are often dense uncountable subsets of the support of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' However, the “multifractal miracle” is the phenomenon that, even though the level sets are very complex, the multifractal spectrum is often a concave analytic function of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In fact, the multifractal spectrum and the Lq-spectrum are related through a heuris- tic relationship called the multifractal formalism [18], which speculates that under certain regularity assumptions, the multifractal spectrum is given by the concave conjugate of the Lq-spectrum, that is the quantity τ ∗ µ(α) = inf q∈R(qα − τµ(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Generally speaking, τ ∗ µ(α) ≥ fµ(α) [19]: in particular, the slopes of the asymptotes of the Lq-spectrum bound the exponential scaling of measures of balls B(x, r) uniformly for all x ∈ supp µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In our specific setting, where our metric space is the two-sided shift AZ and the measure µ is ergodic, the local dimension is precisely the scaling rate of the information function of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In fact, the Shannon–McMillan–Breiman theorem states that the local dimension of the measure (with an appropriate choice of the metric) is almost surely the entropy of the measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Thus the Lq-spectrum provides uniform control over the scaling rate of the information function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' More detail about these notions are given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal Random Substitutions 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Random substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' A (deterministic) substitution is a rule which replaces each symbol in a finite or infinite string over an alphabet A with a finite word over the same alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Random substitutions generalise this notion by substituting a randomly chosen word (according to a fixed finite distribution) independently for each letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We can also think of a random substitution as a (deterministic) set-valued substitution ϑ, together with a choice of probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For example, given p ∈ (0, 1), the random Fibonacci substitution ϑp is defined by ϑp : � � � � � � � a �→ � ab with probability p, ba with probability 1 − p, b �→ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' To a given (primitive) random substitution ϑP , one can canonically a subshift Xϑ of the two-sided shift AZ along with an ergodic frequency measure µP , which quantifies the relative occurrence of a given word under repeated application of the random substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' As highlighted in the introduction, primitive random substitutions give rise to subshifts and measures with a wide variety of properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' As a result, we will impose additional conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Our main assumption, which we call compatibility, asserts that for each a ∈ A, the number of occurrences of each b ∈ A is identical in every possible substituted image of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For example, the random Fibonacci substitution is compatible since in all the possible images of a, a occurs once and b occurs once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The key feature of compatibility is that the one can define a deterministic substitution matrix, such that the Perron– Frobenius eigenvalue is the asymptotic growth rate of lengths of words, and the corresponding right eigenvector encodes the asymptotic frequency with which the individual letters appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Compatibility is a common assumption: for example, it is assumed in the main results of [3, 14, 21, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Another standard assumption is that of recognisability, which heuristically states that each element of the subshift is the unique image of another element of the subshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Recognisability precludes the existence of periodic points [27] and is one of the assumptions required to to establish intrinsic ergodicity in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It is also assumed in the main results of [12, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Statement and discussion of main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We now give concise statements of the main results in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We refer the reader to Section 2 for full statements of the notation and definitions used in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 6 ANDREW MITCHELL AND ALEX RUTAR Fix a random substitution ϑP and let λ and R denote the Perron–Frobenius eigen- value and corresponding right eigenvector of the substitution matrix of ϑP , respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Given q ∈ R and k ∈ N, define ϕϑP ,k(q) = ϕk(q) = − � a∈A Ra log � � � s∈ϑk(a) P[ϑk P (a) = s]q � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We define the inflation word Lq-spectrum of ϑP by Tϑ,P (q) = lim inf k→∞ 1 λk ϕk(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We similarly define the upper variant T ϑ,P by taking a limit superior in place of the limit inferior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Throughout, µP will denote the frequency measure associated with ϑP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Heuristically, the inflation word spectrum approximates the frequency measure µP by the probability distribution on the iterated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The normalisation factors follow from the observation that the words in the kth iteration have length approximately λk when normalised by the left Perron–Frobenius eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Our main general result bounding the Lq-spectrum is the following, which states for q ≥ 0 that Tϑ,P and τµP coincide, and moreover provides bounds on τµP in terms of the functions ϕk for all q ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑP = (ϑ, P ) be a primitive and compatible random substitution with corresponding frequency measure µP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then the limits defining τµP (q) and Tϑ,P (q) exist and coincide for all q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Moreover, (1) For all 0 ≤ q ≤ 1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1) 1 λk − 1ϕk(q) ≤ τµP (q) ≤ 1 λk ϕk(q) and (λ−kϕk(q))∞ k=1 converges monotonically to τµP (q) from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' (2) For all q ≥ 1, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2) 1 λk ϕk(q) ≤ τµP (q) ≤ 1 λk − 1ϕk(q) and (λ−kϕk(q))∞ k=1 converges monotonically to τµP (q) from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' (3) For all q < 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3) 1 λk − 1ϕk(q) ≤ τµP (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The notion of compatibility is defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='7, which is key in order to obtain the uniform estimate given in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For q < 0, it is not true in general that τµP (q) and Tϑ,P (q) coincide (a counterexample is given in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2): the problem is essentially “non-uniqueness of cutting points”, as highlighted by the averaging procedure in the construction of the measure (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In other words, the corresponding upper bound in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3) does not hold in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Despite this, it still follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3) that Tϑ,P (q) provides a lower bound for τµP (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal Random Substitutions 7 In Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1, we prove that if ϑP also satisfies the disjoint set condition, or the identical set condition with identical production probabilities (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='13), then a closed-form expression can be obtained for the Lq-spectrum of the corresponding frequency measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' By combining this result with Theorem A, we obtain the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑP be a primitive and compatible random substitution with corresponding frequency measure µP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then for all q ≥ 0: (1) If ϑP satisfies the disjoint set condition, then τµP (q) = 1 λ − 1ϕ1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' (2) If ϑP satisfies the identical set condition and has identical production probabilities, then τµP (q) = 1 λϕ1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, under the disjoint set condition or identical set condition with identical production probabilities, the Lq-spectrum is analytic on (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In [15], a result analogous to Theorem A is obtained for entropy in terms of the inflation word entropy, without the compatibility assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In fact, to highlight the generality of our results on Lq-spectra, as a consequence of Theorem A, we obtain new proofs of this result (as well as a result on topological entropy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' (a) We obtain the main result of [14] on topological entropy, which states that for subshifts of primitive and compatible random substitutions, the topological entropy can be characterised in terms of the asymptotic growth rate of inflation words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' (b) For frequency measures corresponding to primitive and compatible random substitutions, we also obtain the characterisation of (measure theoretic) en- tropy obtained in [15, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3]—under the additional hypothesis that the substitution is compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This is described in the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑP = (ϑ, P ) be a primitive and compatible random substitution with associated subshift Xϑ and frequency measure µP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' (1) The limit lim k→∞ 1 λk � a∈A Ra log(#ϑk(a)) exists and is equal to htop(Xϑ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' (2) The Lq-spectrum of µP is differentiable at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Moreover, the limit lim k→∞ 1 λk � a∈A Ra � v∈ϑk(a) −P[ϑk P (a) = v] log(P[ϑk P (a) = v]) exists and is equal to hµP (Xϑ) = dimH µP = τ ′ µP (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 8 ANDREW MITCHELL AND ALEX RUTAR We now turn our attention to the multifractal spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Firstly, while τµP (q) and Tϑ,P (q) do not coincide in general for q < 0, if the random substitution that gives rise to the frequency measure µP is additionally assumed to be recognisable (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='14), then the limits defining τµP (q) and Tϑ,P (q) both exist and coincide for all q ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Moreover, under recognisability, we prove that the multifractal spectrum is the concave conjugate of the Lq-spectrum: in other words, the multifractal formalism holds for any associated frequency measure µP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, we conclude that fµP is a concave analytic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In fact, in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5 we prove a stronger variational formula for the multi- fractal spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For each α ∈ R, we construct measures ν such that dimH ν ≥ τ ∗(α) and dimloc(µP , x) = α for ν-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' x ∈ Xϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, we can take the measures to be frequency measures associated with permissible probabilities for the substitution ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑP be a primitive, compatible, and recognisable random substitution with corresponding frequency measure µP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then for all q ∈ R, τµP (q) = Tϑ,P (q) = 1 λ − 1ϕ1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Moreover, fµP (α) = τ ∗ µP (α) is an analytic and concave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In fact, for each α ∈ R such that fµP (α) ≥ 0, there are permissible probabilities Q such that fµP (α) = dimH µQ and dimloc(µP , x) = α for µQ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' x ∈ Xϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' To conclude this section, we observe that our results on Lq-spectra also give uniform bounds on the exponential scaling rate of the frequency measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The following result is a direct application of Theorem A and Theorem D, combined with Proposi- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Corollary E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑP = (ϑ, P ) be a primitive, compatible, and recognisable random substitu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then αmin := lim q→∞ τµP (q) q = − � a∈A Ra log � max s∈ϑ(a) P[ϑP (a) = s] � αmax := lim q→−∞ τµP (q) q = − � a∈A Ra log � min s∈ϑ(a) P[ϑP (a) = s] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' and for all x ∈ Xϑ, αmin ≤ dimloc(µP , x) ≤ dimloc(µP , x) ≤ αmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Moreover, {dimloc(µP , x) : x ∈ Xϑ} = [αmin, αmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, when the probabilities P are chosen so that for each a ∈ A, P[ϑP (a) = s] = 1/#ϑ(a) for all s ∈ ϑ(a), then the Lq-spectrum is the line with slope htop(Xϑ) passing through (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Thus the local dimension of µP exists at every x ∈ Xϑ and is given by the constant value αmin = αmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This can be rephrased in terms of a weak Gibbs-type property, which says that for every ϵ > 0, all n sufficiently large (depending on ϵ), and u ∈ Ln, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4) exp(−|u|(htop(Xϑ) + ϵ)) ≤ µP ([u]) ≤ exp(−|u|(htop(Xϑ) − ϵ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal Random Substitutions 9 see, for example, [29, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4] for the short argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In general, the error term ϵ cannot be dropped by the addition of a constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Under certain assumptions, one can show that there are infinitely many words with µP ([u]) ≈ |u|−1 exp(−|u|(htop(Xϑ)), as explained in [15, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' These assumptions are satisfied, for example, in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Of course, similar one-sided results hold for q ≥ 0 only under the assumption of compatibility, by iterating the formula for ϕk and taking an appropriate maximum at each level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In fact, since τµP (q) is differentiable at 1, with derivative giving the entropy, and since htop(Xϑ) = τµP (0), it follows that µP is a measure of maximal entropy if and only if τ ′ µP (q) exists and is constant on the interval (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Discussion and further work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We conclude the introduction with a list of com- ments and potentially interesting questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' (1) What is the Lq-spectrum for a compatible substitution when q < 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We do not known this even for the random substitution given in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1, which satisfies the identical set condition with identical production probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Obtaining results for q < 0 is substantially more challenging, since the sum in τµP (q) depends on the measure of cylinders with very small (but non-zero) measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For example, in the self-similar case, without the presence of strong separation assumptions, little is known (in contrast to the q ≥ 0 case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' (2) What happens without compatibility?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Do the formulas in Theorem A hold in general?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In [15], it suffices to use an almost sure version of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' However, since the Lq-spectrum is sensitive to scaling at individual points as q tends to ±∞, such an almost sure result in our case is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' (3) Outside the disjoint set condition and the identical set condition, what can be said about differentiability of the Lq-spectrum?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For q ≥ 0, we give the Lq-spectrum as a uniform limit of analytic functions: however, aside from the exceptional point q = 1 where we can say more, this is not enough to give information about differentiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' (4) Can our results on Lq-spectra and multifractal spectra (which hold for recog- nisable substitutions) be relaxed to a weaker condition such as the disjoint set condition (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='13)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' (5) Can the error term in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4) be determined precisely, up to a constant?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The approximate Gibbs-type bounds discussed following Corollary E are closely related to the bounds used in the proof of intrinsic ergodicity given in [15, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It could be worth exploring the relationship between intrinsic ergodicity and Gibbs-type properties given by the Lq-spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' PRELIMINARIES In this section we introduce the key notation and definitions that we will use throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' After introducing some basic notation, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2 we introduce symbolic dynamics on the two-sided shift, as well as our notions of entropy 10 ANDREW MITCHELL AND ALEX RUTAR and dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3 we present the key definitions and basic results from multifractal analysis that we work with throughout, including the definitions of the Lq-spectrum and local dimensions of a measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then, in the following sections we provide an introduction to random substitutions and their associated dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5 we give the definition of a random substitution via its action on words, and define the subshift associated to a random substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='6 and Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='7, we define what it means for a random substitution to be primitive and compatible and present the key properties of such random substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='8, we give the definition of the frequency measure associated to a random substitution and state a key result used in the proof of our main results which relates the measures of cylinder sets via the action of the random substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Finally, in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='9, we define what it means for a substitution to satisfy the disjoint or identical set condition, and introduce recognisable random substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Symbolic notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Throughout, we use the following symbolic notation, which is essentially the same as the notation used in [2, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For a set B, we let #B be the cardinality of B and let F(B) be the set of non-empty finite subsets of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We fix an alphabet A = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , ad}, for some d ∈ N, which is a finite set of letters ai, and equip it with the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then a word u with letters in A is a finite concatenation of letters, namely u = ai1 · · · ain for some n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We write |u| = n for the length of the word u, and for m ∈ N, we let Am denote the set of all words of length m with letters in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We set A+ = � m∈N Am and let AZ = {(ain)n∈Z : ain ∈ A for all n ∈ Z} denote the set of all bi-infinite sequences with elements in A, and endow AZ with the product topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We also fix a metric on AZ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Given points x = (xn)n∈Z and y = (yn)n∈Z, let N(x, y) = sup{n ∈ Z : xj = yj for all |j| ≤ n} and let d(x, y) = e−2N(x,y)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The space X is compact with topology generated by the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We will frequently write sequences (xn)n∈Z ∈ AZ as · · · x−1x0x1 · · · , with the corre- sponding notation for finite sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If i and j ∈ Z with i ≤ j, and x = · · · x−1x0x1 · · · ∈ AZ, then we let x[i,j] = xixi+1 · · · xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We use the same notation if v ∈ A+ and 1 ≤ i ≤ j ≤ |v|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For u and v ∈ A+ (or v ∈ AZ), we write u ◁ v if u is a subword of v, namely if there exist i and j ∈ Z with i ≤ j so that u = v[i,j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For u and v ∈ A+, we set |v|u to be the number of (possibly overlapping) occurrences of u as a subword of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If u = ai1 · · · ain and v = aj1 · · · ajm ∈ A+, for some n and m ∈ N, we write uv for the concatenation of u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The abelianisation of a word u ∈ A+ is the vector Φ(u) ∈ N#A 0 , defined by Φ(u)a = |u|a for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Dynamics, entropy and dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We equip the space AZ with invertible dynamics from the left-shift map S : AZ → AZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Throughout, we will work with a Multifractal Random Substitutions 11 subshift X ⊂ AZ, which is compact and shift-invariant, that is S−1(X) = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then µ will denote an ergodic and S-invariant Borel probability measure with support contained in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The metric structure on AZ enables us to define the Hausdorff dimension of Borel subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Using this, we define the Hausdorff dimension of µ to be the quantity dimH µ = inf{dimH E : µ(E) > 0} where the infimum is taken over Borel sets E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Here, we use Hausdorff dimension as inherited from the underlying metric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' though it would also be appropriate to use Bowen’s generalisation of topological entropy to non-compact sets [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We also define the lower local dimension of µ at x by dimloc(µ, x) = lim inf r→0 log µ � B(x, r) � log r We define the upper local dimension dimloc(µ, x) analogously using the limit superior in place of the limit inferior, and when the limits coincide, we refer to the shared quantity as the local dimension and denote it by dimloc(µ, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Local dimensions and Hausdorff dimension are closely related: it follows from, for instance, [7, Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1] that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1) dimH µ = sup{s : dimloc(µ, x) ≥ s for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' x}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Now fix a partition ξ so that with ξk = �k i=−k S−i(ξ), {ξk}∞ k=1 generates the Borel σ-algebra on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We recall that the entropy of µ with respect to S is given by hµ(X) = lim k→∞ 1 2k + 1 � A∈ξk −µ(A) log � µ(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' where, by the classical Kolmogorov–Sina˘ı theorem, the quantity does not depend on the choice of partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Now given x ∈ X, let ξk(x) denote the unique element in the partition ξk containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then the Shannon–McMillan–Breiman theorem states that the entropy of µ is almost surely the information rate of µ, that is for µ-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' x ∈ X, lim k→∞ − log µ � ξk(x) � 2k + 1 = hµ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We refer the reader to [6] for greater detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Now suppose ξ = {Ea}a∈A is the partition of X where Ea = {(xn)n∈Z ∈ X : x0 = a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For the remainder of this paper, ξ will always denote this partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then given x = (xn)n∈Z ∈ X, ξk(x) = {y ∈ X : xj = yj for all |j| ≤ k} = B(x, e−(2k+1)) and therefore dimloc(µ, x) = lim k→∞ − log µ � ξk(x) � 2k + 1 12 ANDREW MITCHELL AND ALEX RUTAR where both limits exist if either limit exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since the limit on the right is µ almost surely hµ(X), it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1) that dimH µ = hµ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Finally, the topological entropy of X is given by htop(X) = lim k→∞ − log #{E ∈ ξk : E ∩ X ̸= ∅} 2k + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Of course, htop(X) = dimB X, the box counting dimension of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lq-spectra and smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Given q ∈ R, we define Sµ,r(q) = sup {xi}i∈P(r) � i µ � B(xi, r) �q where P(r) is the set of discrete 2r-separated subsets of X, that is P(r) = {{xi}i : xi ∈ X, d(xi, xj) ≥ 2r for i ̸= j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We then define the Lq-spectrum of µ to be the function τµ(q) = lim inf q→0 log Sµ,r(q) log r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For convenience, we also denote the upper variant τ µ(q) by taking a limit superior in place of the limit inferior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It is a standard consequence of Hölder’s inequality that τµ(q) is a concave increasing function of q (note that this need not hold for τ µ(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Of course, the preceding definitions hold more generally in an arbitrary metric space, but in our particular setting we can rephrase the Lq-spectrum in terms of more familiar sums over cylinders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Recall that ξ denotes the partition of Xϑ into cylinders at 0 corresponding to the letters in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then set Sµ,n(q) = � E∈ξk µ(E)q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since distinct elements in the partition ξk are e−(2k+1)-separated, Sµ,n(q) = Sµ,e−(2n+1)(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It follows immediately that τµ(q) = lim inf n→∞ − log Sµ,n(q) 2n + 1 with the analogous result for τ µ(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, we observe that τµ(0) = htop(X) assuming µ is fully supported on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Finally, by shift invariance, we can characterise the subshift X in terms of a language on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Given n ∈ N, we set Ln(X) = {w ∈ An : w ◁ x for some x ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Given w ∈ Ln(X), we let [w] = {(xn)n∈Z ∈ X : xi = wi for all 1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Of course, by shift invariance, there is a measure-preserving bijection between L2n+1(X) and Xn, Multifractal Random Substitutions 13 so it follows again that τµ(q) = lim inf n→∞ −1 n log � u∈Ln(X) µ([u])q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We will primarily use this characterisation throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We first list some basic properties of the Lq-spectrum of the measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Here, (a) is a direct consequence of Hölder’s inequality, (b) is standard (see, for example, [29, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4]) and (c) was proven in [9, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let µ be a shift-invariant measure on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' (a) The Lq-spectrum τµ(q) is continuous, increasing and concave on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' (b) Let αmin = limq→∞ τµ(q)/q and αmax = limq→−∞ τµ(q)/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then for every s < αmin ≤ αmax < t, all n sufficiently large and u ∈ Ln, e−tn ≤ µ([u]) ≤ e−sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, the local dimensions satisfy αmin ≤ inf x∈X dimloc(µ, x) ≤ sup x∈X dimloc(µ, x) ≤ αmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' (c) The left and right derivatives of τµ at q = 1 bound the Hausdorff dimension of µ, that is τ + µ (1) ≤ dimH µ ≤ τ − µ (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In fact, (a) gives intuition for why the Lq-spectrum encodes smoothness: rather than obtain almost sure information on local dimensions, the Lq-spectrum contains uniform information about local dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Finally, we prove a simple result concerning the Lq-spectrum which will be useful later in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ζ > 1 be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2) τµ(q) = 1 ζ lim inf n→∞ −1 n log � � � u∈L⌊ζn⌋(X) µ([u])q � � and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3) τ µ(q) = 1 ζ lim sup n→∞ −1 n log � � � u∈L⌊ζn⌋(X) µ([u])q � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Of course, it always holds that τµ(q) ≤ 1 ζ lim inf n→∞ −1 n log � � � u∈L⌊ζn⌋(X) µ([u])q � � τ µ(q) ≥ 1 ζ lim sup n→∞ −1 n log � � � u∈L⌊ζn⌋(X) µ([u])q � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 14 ANDREW MITCHELL AND ALEX RUTAR First, let q < 0 and let n ∈ N be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let kn be minimal so that ⌊ζkn⌋ ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Observe that there is some M ∈ N (independent of n) so that ⌊ζkn⌋ ≤ n + M: it follows that limn→∞ n/kn = ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then if v ∈ L⌊ζkn⌋(X) is arbitrary, [v] ⊂ [u] for some u ∈ Ln(X) and µ([v])q ≥ µ([u])q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Thus S⌊ζkn⌋,µ(q) ≥ Sn,µ(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' which gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2) for q < 0 since lim n/kn = ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Similarly, for q ≥ 0, since there are at most (#A)M words v ∈ L⌊ζkn⌋(X) with [v] ⊂ [u], pigeonholing, for each u ∈ Ln(X) there is some v(u) ∈ L⌊ζkn⌋(X) such that µ([v(u)])q ≥ (#A)−qMµ([u])q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Thus S⌊ζkn⌋,µ(q) ≥ (#A)−qMSn,µ(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2) for q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The arguments for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3) follow analogously by choosing kn maximal so that ⌊ζkn⌋ ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal spectrum and multifractal formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The Lq-spectrum of a mea- sure is related to the (fine) multifractal spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let µ be a shift-invariant measure on a subshift X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We recall that the local dimension of µ at x ∈ X is given by dimloc(µ, x) = lim n→∞ − 1 2n + 1 log µ([x[−n,n]]) when the limit exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Given α ∈ R, set Fµ(α) = {x ∈ X : dimloc(µ, x) = α} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We then define the multifractal spectrum of µ by fµ(α) = dimH Fµ(α) using the convention that dimH ∅ = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The multifractal spectrum is related to the Lq-spectrum by the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let g: R → R ∪{−∞} be a concave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For x ∈ R, we let g+(x) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' g−(x)) denote the right (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' left) derivative of g at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Such limits necessarily exist by concavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We denote the subdifferential of g at x by ∂g(x) = [g+(x), g−(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We then recall that the concave conjugate of g is given by g∗(α) = inf q∈R{qα − g(q)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Note that g∗ is always concave since it is the infimum of a family of affine functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For more detail concerning the theory of concave functions, we refer the reader to [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Now, we say that µ satisfies the multifractal formalism when fµ = τ ∗ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In general, the multifractal formalism need not hold, but it is well-known that the concave conjugate of the Lq-spectrum is an upper bound for the multifractal spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For the convenience of the reader, we provide a short self-contained proof, which follows the main ideas of [19, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal Random Substitutions 15 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let µ be a shift-invariant measure on a subshift X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then fµ(α) ≤ τ ∗(α) for all α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Recall that ξn denotes the partition of X into cylinders corresponding to words of length 2n + 1, each of which has diameter precisely e−2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For α ∈ R, n ∈ N and ϵ > 0, let Mn,ϵ(α) = � I ∈ ξn : e−(2n+1)(α+ϵ) ≤ µ(I) ≤ e−(2n+1)(α−ϵ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In other words, Mn,ϵ(α) is an ϵ-approximation of Fµ(α) at level n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Our strategy is to control the size of the sets Mn,ϵ(α) in terms of the Lq-spectrum of µ, and then use these sets to build a good cover of Fµ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let q ∈ ∂τ ∗(α): we prove this in the case that q ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' the case q < 0 is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' First, observe that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4) S2n+1,µ(q) = � I∈ξn µ(I)q ≥ � u∈Mn,ϵ(α) µ(I)q ≥ e−(2n+1)(α+ϵ)q#Mn,ϵ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since τµ(q) = lim infn→∞(log S2n+1,µ(q))/(−2n−1) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2, there is some Nϵ ∈ N so that for all n ≥ Nϵ, S2n+1,µ(q) ≤ e−(2n+1)(τµ(q)−ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Combining this with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5) #Mn,ϵ(α) ≤ e−(2n+1)(τ(q)−ϵ) · e(2n+1)(α+ϵ)q = e(2n+1)(τ ∗(α)+(q+1)ϵ) for all n ≥ Nϵ where we have used the fact that q ∈ ∂τ ∗(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Now for each x ∈ Fµ(α), we can find some nx ∈ N so that for all n ≥ nx, µ(ξn(x)) ≥ e−(2n+1)(α+ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, Gϵ := ∞ � n=Nϵ Mn,ϵ(α) is a Vitali cover for Fµ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Now suppose {Ij}∞ j=1 is any disjoint subcollection of Gϵ: then with s = τ ∗(α) + 2ϵ(1 + q), ∞ � j=1 (diam Ij)s ≤ ∞ � n=Nϵ � I∈Mn,ϵ(α) (diam I)s ≤ ∞ � n=Nϵ e−(2n+1)s#Mn,ϵ(α) ≤ ∞ � n=Nϵ e−(2n+1)se(2n+1)(τ ∗(α)+(q+1)ϵ) = ∞ � n=Nϵ (e−(1+q)ϵ)2n+1 < ∞ by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Thus by the Vitali covering theorem for Hausdorff measure, there is a cover {Ei}∞ i=1 for Fµ(α) such that Hs(Fµ(α)) ≤ ∞ � i=1 (diam Ei)s < ∞ 16 ANDREW MITCHELL AND ALEX RUTAR and thus dimH Fµ(α) ≤ τ ∗(α) + 2ϵ(1 + q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' But ϵ > 0 was arbitrary, so the desired result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Random substitutions and frequency measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We now introduce our pri- mary objects of interest: random substitutions, and their associated frequency measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In a similar manner to [14, 15], we define a random substitution by the data required to determine its action on letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We then extend this to a random map on words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let A = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , ad} be a finite alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' A random substitution ϑP = (ϑ, P ) is a set-valued substitution ϑ: A → F(A+) together with a set of non-degenerate probability vectors P = � pi = (pi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , pi,ri) : ri = #ϑ(ai);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' pi ∈ (0, 1]ri;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' ri � j=1 pi,j = 1 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , d � , such that ϑP : ai �→ � � � � � s(i,1) with probability pi,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' s(i,ri) with probability pi,ri, for every 1 ≤ i ≤ d, where ϑ(ai) = {s(i,j)}1≤j≤ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We call each s(i,j) a realisation of ϑP (ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If ri = 1 for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , d}, then we call ϑP deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5 (Random Fibonacci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let A = {a, b}, and let p ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The random Fibonacci substitution ϑP = (ϑ, P ) is the random substitution given by ϑP : � � � � � � � a �→ � ab with probability p, ba with probability 1 − p, b �→ a with defining data ra = 2, rb = 1, s(a,1) = ab, s(a,2) = ba, s(b,1) = a, P = {pa = (p, 1 − p), pb = (1)} and corresponding set-valued substitution ϑ: a �→ {ab, ba}, b �→ {a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In the following we describe how a random substitution ϑP determines a (countable state) Markov matrix Q, indexed by A+ × A+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We interpret the entry Qu,v as the probability of mapping a word u to a word v under the random substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Formally, Qai,s(i,j) = pi,j for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , ri} and Qai,v = 0 if v /∈ ϑ(ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We extend the action of ϑP to finite words by mapping each letter independently to one of its realisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' More precisely, given n ∈ N, u = ai1 · · · ain ∈ An and v ∈ A+ with |v| ≥ n, we let Dn(v) = {(v(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , v(n)) ∈ (A+)n : v(1) · · · v(n) = v} Multifractal Random Substitutions 17 denote the set of all decompositions of v into n individual words and set Qu,v = � (v(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=',v(n))∈Dn(v) n � j=1 Qaij ,v(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In words, ϑP (u) = v with probability Qu,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For u ∈ A+, let (ϑn P (u))n∈N be a stationary Markov chain on some probability space (Ωu, Fu, Pu), with Markov matrix given by Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' that is, Pu[ϑn+1 P (u) = w | ϑn P (u) = v] = Pv[ϑP (v) = w] = Qv,w for all v and w ∈ A+, and n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, Pu[ϑn P (u) = v] = (Qn)u,v for all u and v ∈ A+, and n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We often write P for Pu if the initial word is understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In this case, we also write E for the expectation with respect to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' As before, we call v a realisation of ϑn P (u) if (Qn)u,v > 0 and set ϑn(u) = {v ∈ A+ : (Qn)u,v > 0} to be the set of all realisations of ϑn P (u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Conversely, we may regard ϑn P (u) as the set ϑn(u) endowed with the additional structure of a probability vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If u = a ∈ A is a letter, we call a word v ∈ ϑk(a) a level-k inflation word, or exact inflation word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' To a given random substitution ϑP = (ϑ, P ) one can associate a subshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' First, we say that a word u ∈ A+ is (ϑ-)legal if there exists an ai ∈ A and k ∈ N such that u appears as a subword of some word in ϑk(ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We define the language of ϑ by Lϑ = {u ∈ A+ : u is ϑ-legal} and, for w ∈ A+ ∪ AZ, we let L(w) = {u ∈ A+ : u ◁ w} denote the language of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The random substitution subshift of a random substitution ϑP = (ϑ, P ) is the system (Xϑ, S), where Xϑ = {w ∈ AZ : L(w) ⊆ Lϑ} and S denotes the (left) shift map, defined by S(w)i = wi+1 for each w ∈ Xϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Under very mild assumptions, the space Xϑ is non-empty [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This holds, for example, if the generating random substitution is primitive (we give a definition in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We endow Xϑ with the subspace topology inherited from AZ, and since Xϑ is defined in terms of a language, it is a compact S-invariant subspace of AZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Hence, Xϑ is a subshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For n ∈ N, we write Ln ϑ = Lϑ ∩ An and Ln(w) = L(w) ∩ An to denote the subsets of Lϑ and L(w), respectively, consisting of words of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The set-valued function ϑ naturally extends to Xϑ, where for w = · · · w−1w0w1 · · · ∈ Xϑ we let ϑ(w) denotes the (infinite) set of sequences of the form v = · · · v−2v−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='v0v1 · · · , with vj ∈ ϑ(wj) for all j ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It is easily verified that ϑ(Xϑ) ⊂ Xϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The notation Xϑ reflects the fact that the random substitution subshift does not depend on the choice of (non-degenerate) probabilities P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In fact, this is the case for many structural properties of ϑP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In these cases, one sometimes refers to ϑ instead of ϑP as a random substitution, see for instance [14, 16, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' On the other hand, for 18 ANDREW MITCHELL AND ALEX RUTAR some applications, one needs additional structure on the probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In fact, there is an underlying branching process, similar to a Galton–Watson process, that allows one to construct more refined random variables, see [17] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The measure theoretic properties we consider are typically dependent on the choice of probabilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' however, some of the auxiliary results we use only depend on the set- valued substitution ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' To avoid confusion, for results where there is no dependence on the choice of probabilities we will give the statement in terms of the set-valued substitution ϑ and omit the dependence on P in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Primitive random substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' A standard assumption in the study of sub- stitutions (both deterministic and random) is that of primitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Given a random substitution ϑP = (ϑ, P ) over an alphabet A = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , ad} with cardinality d ∈ N, we define the substitution matrix M = MϑP ∈ Rd×d of ϑP by Mi,j = E[|ϑP (aj)|ai] = rj � k=1 pj,k|s(j,k)|ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since M has only non-negative entries, it has a real eigenvalue λ of maximal modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Observe that λ ≥ 1, with λ = 1 precisely if M is column-stochastic, so that the random substitution is non-expanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' To avoid this degenerate situation, we will assume that λ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If the matrix M is primitive (that is if there exists a k ∈ N such that all the entries of M k are positive), the Perron–Frobenius theorem gives that λ is a simple eigenvalue and that the corresponding (right) eigenvector R = (R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , Rd) can be chosen to have strictly positive entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We will normalise this eigenvector so that ∥R∥1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We will refer to λ as the Perron–Frobenius eigenvalue of the random substitution, ϑP , with corresponding Perron–Frobenius eigenvector R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We say that ϑP is primitive if M = MϑP is primitive and its Perron– Frobenius eigenvalue satisfies λ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We emphasise that for a random substitution ϑP , being primitive is independent of the (non-degenerate) choice of probabilities P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In this sense, primitivity is a property of ϑ rather than ϑP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since M k ϑP = Mϑk P , the Perron–Frobenius eigenvalue of ϑk P is λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Compatible random substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Another standard assumption in the study of random substitutions is that of compatibility, which gives that exact inflation words have a well-defined abelianisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, the matrix of a compatible random substitution is independent of the choice of probabilities, so the letter frequencies are uniform and do not depend on the realisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' As discussed in the introduction, the existence of uniform letter frequencies is fundamental in the proofs of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We say that a random substitution ϑP = (ϑ, P ) is compatible if for all a ∈ A, and u, v ∈ ϑ(a), we have Φ(u) = Φ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal Random Substitutions 19 Observe that compatibility is independent of the choice of probabilities, and that a random substitution ϑP = (ϑ, P ) is compatible if and only if for all u ∈ A+, we have that |s|a = |t|a for all s and t ∈ ϑ(u), and a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We write |ϑ(u)|a to denote this common value, and let |ϑ(u)| denote the common length of words in ϑ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For convenience, we also set |ϑ| = maxa∈A|ϑ(a)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For a random substitution that is both primitive and compatible, the (uniform) letter frequencies are encoded by the right Perron–Frobenius eigenvector of the substitution matrix, which by compatibility is independent of the choice of probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, we have the following (see [25] for a proof in the deterministic case, which also holds in the random case by compatibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='9 (Letter frequency bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If ϑP is a primitive and compatible random substi- tution, then for all ε > 0 there is an integer N such that every word v of length at least N satisfies |v|(Ra − ε) < |v|a < |v|(Ra + ε) for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The random Fibonacci substitution defined in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5 is compatible, since Φ(ab) = Φ(ba) = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It is also primitive, since the square of its substitution matrix is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For any choice of probabilities, the right Perron–Frobenius eigenvector is given by (τ −1, τ −2), where τ denotes the golden ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In terms of letter frequencies, this means that in all sufficiently long legal words, approximately τ −1 proportion of the letters are a and τ −2 proportion are b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The following consequence of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='9 is useful in the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑP = (ϑ, P ) be a primitive and compatible random substitution and let q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For all ε > 0, there is an M ∈ N such that for every m ≥ M and v ∈ Lm ϑ , � a∈A � � � s∈ϑ(a) P[ϑP (a) = s]q � � m(Ra+ε) ≤ � w∈ϑ(v) P[ϑP (v) = w]q ≤ � a∈A � � � s∈ϑ(a) P[ϑP (a) = s]q � � m(Ra−ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For q ≤ 1, the same result holds with reversed inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since ϑP is compatible, the cutting points of inflation tiles are well-defined, so breaking the sum into inflation tiles we obtain � w∈ϑ(v) P[ϑP (v) = w]q = � w1∈ϑ(v1) P[ϑP (v1) = w]q � w2∈ϑ(v2) · · � wm∈ϑ(vm) P[ϑP (vm) = wm]q = � a∈A � � � s∈ϑ(a) P[ϑP (a) = s]q � � |v|a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 20 ANDREW MITCHELL AND ALEX RUTAR The result then follows by applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='9 to bound |v|a, noting that for all a ∈ A we have � s∈ϑ(a) P[ϑP (a) = s]q ≤ 1 if q ≥ 1 and � s∈ϑ(a) P[ϑP (a) = s]q ≥ 1 if q ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Frequency measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The main object that we associate with a given primitive random substitution ϑP is the frequency measure µP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This measure quantifies the relative occurrence of a given word in a random substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We now define this measure precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' First, we define the expected frequency of a word v ∈ Lϑ by freq(v) = lim k→∞ E[|ϑk P (a)|v] E[|ϑk P (a)|] , where, by primitivity, this limit is independent of the choice of a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In fact, we have the stronger property that the word frequencies exist P-almost surely in the limit of large inflation words and are given by freq(v) for all v ∈ Lϑ (see [17] for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Recalling that ξ(Xϑ) is the algebra of cylinder sets on Xϑ that specify the origin, we define µP : ξ(Xϑ) ∪ {∅} → [0, 1] by µP (∅) = 0, µP (Xϑ) = 1, and µP ([v]m) = freq(v) for v ∈ Lϑ and m ∈ {1 − |v|, 2 − |v|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This set function extends to a unique measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='11 ([17, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The set function µP is a content with mass one which extends uniquely to a shift-invariant ergodic Borel probability measure on Xϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We call the measure µP defined in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='11 the frequency measure correspond- ing to the random substitution ϑP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Observe that frequency measures are dependent on the probabilities of the substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' As such, for the subshift of a primitive ran- dom substitution that is non-deterministic, there exist uncountably many frequency measures supported on this subshift [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In contrast, the subshift of a primitive deterministic substitution has precisely one frequency measure, which is the unique ergodic measure [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Frequency measures corresponding to primitive and compatible random substi- tutions satisfy the following renormalisation lemma, which relates the measure of a cylinder set of a legal word to measures of cylinder sets of shorter words via the production probabilities of the random substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This result first appeared in [17] and is central to the proof of the main result in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12 (Renormalisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑP be a primitive and compatible random substitution with corresponding frequency measure µP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let n ∈ N and let k be an integer such that every v ∈ Lk ϑ has |ϑ(v)| ≥ n + |ϑ(v1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then for every u ∈ Ln ϑ, µP ([u]) = 1 λ � v∈Lk ϑ µP ([v]) |ϑ(v1)| � j=1 P[ϑP (v)[j,j+m−1] = u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12 is key to the proof of Theorem A, as it relates the sums � u∈Ln ϑ µP ([u]) to sums over smaller words via the production probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This in turn allows us Multifractal Random Substitutions 21 to obtain relations between τµP and ϕk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Under additional assumptions, simplified reformulations of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12 can be obtained (see, for example, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='18, which is used in Theorem D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Separation conditions and recognisability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In this section, we introduce addi- tional common assumptions which either (1) impose a certain separation on inflation words, or (2) impose a certain uniformity of the inflation and the probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Under these conditions, we can obtain closed-form formulas for the Lq-spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' A random substitution ϑP = (ϑ, P ) satisfies the disjoint set condition if u and v ∈ ϑ(a) with u ̸= v =⇒ ϑk(u) ∩ ϑk(v) = ∅ for all a ∈ A and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It satisfies the identical set condition if u and v ∈ ϑ(a) =⇒ ϑk(u) = ϑk(v) for all a ∈ A and k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Moreover, we say that ϑP has identical production probabilities if for all a ∈ A, k ∈ N and v ∈ ϑk(a), P[ϑk−1 P (u1) = v] = P[ϑk−1 P (u2) = v] for all u1 and u2 ∈ ϑ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' A consequence of the disjoint set condition is that for every a ∈ A, k ∈ N and w ∈ ϑk(a), there is a unique v ∈ ϑk−1(a) such that w ∈ ϑ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In other words, every exact inflation word can be uniquely de-substituted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The following definition extends this idea of unique de-substitution from inflation words to all elements in the subshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑP = (ϑ, P ) be a primitive and compatible random substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We call ϑP recognisable if for every x ∈ Xϑ there exists a unique y = · · · y−1y0y1 · · · ∈ Xϑ and a unique integer k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , |ϑ(y0)| − 1} with S−k(x) ∈ ϑ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The following follows routinely from the definition of recognisability (a proof is given in [15, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If ϑP is a primitive, compatible and recognisable random substitution, then ϑP satisfies the disjoint set condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In contrast to the disjoint set condition, recognisability is stable under taking powers (see [15, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑP be a primitive and compatible random substitution and m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If ϑP is recognisable, then so is ϑm P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' An alternative characterisation of recognisability is the following local version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Intuitively, local recognisability means that applying a finite window to a sequence is enough to determine the position and the type of the inflation word in the middle of that window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The following result is given in [15, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4] (see also [12, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 22 ANDREW MITCHELL AND ALEX RUTAR Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑP = (ϑ, P ) be a primitive and compatible random substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If ϑP is recognisable, then there exists a smallest natural number κ(ϑ), called the recognisability radius of ϑP , with the following property: if x ∈ ϑ([a]) for some a ∈ A and x[−κ(ϑ),κ(ϑ)] = y[−κ(ϑ),κ(ϑ)] for some y ∈ Xϑ, then y ∈ ϑ([a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' As a consequence of this local characterisation of recognisability, for every legal word u with length greater than twice the radius of recognisability there exists an inflation word w, appearing as a subword of u, which has a unique decomposition into exact inflation words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We call the largest such w the recognisable core of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Local recognisability allows us to obtain a stronger version of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12 for recognisable random substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This result is key to obtaining the coincidence of the Lq-spectrum and its inflation word analogue under recognisability for q < 0, and thus the conclusion of Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑP = (ϑ, P ) be a primitive and compatible random substitution, with corresponding frequency measure µP and u ∈ Lϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If v ∈ Lϑ and w ∈ ϑ(v) contains u as a subword, then µP ([u]) ≥ 1 λµP ([v])P[ϑP (v) = w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If, additionally, ϑP is recognisable, |u| > 2κ(ϑ) and w′ is the recognisable core of u with v′ ∈ Lϑ the unique legal word such that w′ ∈ ϑ(v′), then µP ([u]) ≤ κ(ϑ) λ µP ([v′])P[ϑP (v′) = w′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If u is a subword of w ∈ ϑ(v), then µP ([u]) ≥ µP ([w]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Thus by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12 applied to µP ([w]), µP ([u]) ≥ 1 λµP ([v])P[ϑP (v) = w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Now, assume that ϑP is recognisable, |u| > 2κ(ϑ) and w′ ∈ ϑ(v′) is the recognisable core of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let k be an integer such that every t ∈ Lk ϑ has |ϑ(t)| ≥ k +|ϑ(v1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since there are at most κ(ϑ) letters of u preceding the recognisable core, if t ∈ Lk ϑ is a word for which u ∈ ϑ(t)[j,j+|u|−1] for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , |ϑ(t1)|}, then ti · · · ti+|v|−1 = v′ for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , κ(ϑ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Moreover, since there is a unique way to decompose w′ into exact inflation words, for each t ∈ Lk ϑ there can be at most one j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , ϑ(t1)} such that Multifractal Random Substitutions 23 u ∈ ϑ(t)[j,j+|u|−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Hence, it follows by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12 that µP ([u]) = 1 λ � t∈Lk µP ([t]) |ϑ(t1)| � j=1 P[ϑP (t)[j,j+|u|−1] = u] ≤ 1 λ κ(ϑ) � i=1 � t∈Lk ϑ ti···ti+|v|−1=v′ µP ([t])P[ϑP (v′) = w′] = κ(ϑ) λ µP ([v′])P[ϑP (v′) = w′], which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lq-SPECTRA OF FREQUENCY MEASURES In this section, we prove our main results on Lq-spectra of frequency measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Here, we relate the Lq-spectrum to a certain “symbolic” Lq-spectrum, which we call the inflation word Lq-spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Heuristically, the inflation word Lq-spectrum is the natural guess for the Lq-spectrum if you do not account for non-uniqueness in the positions in which legal words can appear in inflation words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This notion is introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1, where we also state and prove some of its key properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1, we prove a simple closed-form formula for the inflation word Lq-spectrum under the disjoint set condition or the identical set condition with identical production probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2, we establish basic monotonicity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3, we establish the general bounds for the Lq- spectrum in terms of the inflation word Lq-spectrum, giving Theorem A (the proof is given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We also prove that this bound is sharp in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5, under the recognisability assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This proves the first part of Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' However this bound need not hold in general: we discuss a counterexample in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Finally, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='6, we prove differentiability of the Lq-spectrum at q = 1 and show how to recover known results for measure theoretic and topological entropy from our results concerning Lq-spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Inflation word Lq-spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Given a primitive random substitution ϑP = (ϑ, P ), we can define an analogue of the Lq-spectrum in terms of its production probabilities, in a similar manner to the inflation word analogue of entropy introduced in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In many cases, this notion coincides with the Lq-spectrum of the frequency measure associated to ϑP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For each k ∈ N and q ∈ R, define ϕk(q) = − � a∈A Ra log � � � s∈ϑk(a) P[ϑk P (a) = s]q � � , 24 ANDREW MITCHELL AND ALEX RUTAR where R = (Ra)a∈A is the right Perron–Frobenius eigenvector of the substitution matrix of ϑP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We define the inflation word Lq-spectrum of ϑP by Tϑ,P (q) = lim inf k→∞ ϕk(q) λk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We similarly define the upper variant T ϑ,P by taking a limit supremum in place of the limit infimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We first state some key properties of Tϑ,P (q) which follow easily from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Firstly, if the random substitution ϑP is compatible and satisfies either the disjoint set condition or the identical set condition with identical production probabilities, then the limit defining Tϑ,P (q) exists for all q ∈ R and is given by a closed form expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For q ≥ 0, these properties transfer to the Lq-spectrum by Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑP be a primitive and compatible random substitution and q ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If ϑP satisfies the disjoint set condition, then the limit defining Tϑ,P (q) exists and Tϑ,P (q) = 1 λ − 1ϕ1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If ϑP satisfies the identical set condition and has identical production probabilities, then the limit defining Tϑ,P (q) exists and Tϑ,P (q) = 1 λϕ1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Fix q ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' By the Markov property of ϑP , for all a ∈ A, k ∈ N and v ∈ ϑk(a), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1) P[ϑk P (a) = v] = � s∈ϑ(a) P[ϑP (a) = s] P[ϑk−1 P (s) = v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If ϑP satisfies the disjoint set condition, then for every v ∈ ϑk(a) there is a unique s(v) ∈ ϑ(a) such that v ∈ ϑk−1(s(v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Thus, for all s ∈ ϑ(a) such that s ̸= s(v), we have P[ϑk−1 P (s) = v] = 0, and so it follows by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1) that � v∈ϑk(a) P[ϑk P (a) = v]q = � v∈ϑk(a) P[ϑP (a) = s(v)]q P[ϑk−1 P (s(v)) = v]q = � s∈ϑ(a) P[ϑP (a) = s]q � u∈ϑk−1(s) P[ϑk−1 P (s) = u]q = � � � s∈ϑ(a) P[ϑP (a) = s]q � � · � b∈A � � � u∈ϑk−1(b) P[ϑk−1 P (b) = u]q � � |ϑ(a)|b , Multifractal Random Substitutions 25 where in the final equality we use compatibility to split the second sum into inflation tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Thus ϕk(q) = − � a∈A Ra � b∈A |ϑ(a)|b log � � � u∈ϑk−1(b) P[ϑk−1 P (b) = u]q � � − � a∈A Ra log � � � s∈ϑ(a) P[ϑP (a) = s]q � � = λϕk−1(q) + ϕ1(q), noting that � a∈A Ra|ϑ(a)|b = λRb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It follows inductively that 1 λk ϕk(q) = k � j=1 1 λj ϕ1(q) k→∞ −−−→ 1 λ − 1ϕ1(q), so the limit defining Tϑ,P (q) exists and is equal to (λ − 1)−1ϕ1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' On the other hand, if ϑP satisfies the identical set condition and has identical production probabilities, then P[ϑk−1 P (s1) = u] = P[ϑk−1 P (s2) = u] for all s1, s2 ∈ ϑ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Hence, it follows by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1) that � v∈ϑk(a) P[ϑk P (a) = v]q = � v∈ϑk(a) P[ϑk−1 P (s) = v]q for any choice of s ∈ ϑ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' By compatibility and the independence of the action, � v∈ϑk(a) P[ϑk P (a) = v]q = � b∈A � � � u∈ϑk−1(b) P[ϑk−1 P (b) = u]q � � |ϑ(a)|b , and thus ϕk(q) = � b∈A � a∈A Ra|ϑ(a)|b log � � � v∈ϑk−1(b) P[ϑk−1 P (b) = v]q � � = λϕk−1(q), noting that � a∈A Ra|ϑ(a)|b = Rb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It follows by induction that ϕk(q)/λk = ϕ1(q)/λ for all k ∈ N, so we conclude that Tϑ,P (q) exists and equals λ−1ϕ1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑP be a primitive and compatible random substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For all q > 1 and q < 0, the sequence (λ−kϕk(q))k is non-decreasing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' and for all 0 < q < 1, the sequence is non-increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This is largely a consequence of Jensen’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Note that on the interval (0, 1], the function x �→ xq is convex if q > 1 or q < 0, and concave if 0 < q < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We 26 ANDREW MITCHELL AND ALEX RUTAR first prove this for the case when q > 1 or q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Observe that for all a ∈ A, k ∈ N with k ≥ 2 and v ∈ ϑk(a), it follows by the Markov property of ϑP that � v∈ϑk P[ϑk P(a) = v]q = � v∈ϑk(a) � � � s∈ϑ(a): v∈ϑk−1(s) P[ϑP(a) = s]P[ϑk−1 P (s) = v] � � q ≤ � v∈ϑk(a) �� s∈ϑ(a): v∈ϑk−1(s) P[ϑP(a) = s]P[ϑk−1 P (s) = v]q � s∈ϑ(a): v∈ϑk−1(s) P[ϑP(a) = s] � ≤ � b∈A � � � w∈ϑk−1(b) P[ϑk−1 P (b) = w]q � � |ϑ(a)|b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In the second line, we apply Jensen’s inequality, and in the last line, we use compat- ibility to decompose each probability P[ϑk−1 P (s) = w] into inflation tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It follows that 1 λk ϕk(q) ≥ − 1 λk � b∈A Rb � a∈A Ra|ϑ(a)|b log � � � w∈ϑk−1(b) P[ϑk−1 P (b) = w]q � � = 1 λk−1ϕk−1(q), noting that � a∈A Ra|ϑ(a)|b = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The 0 < q < 1 case follows similarly, with Jensen’s inequality giving the opposite inequality since x �→ xq is concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ An analogous monotonicity result does not hold in general for the (λk − 1)−1ϕk(q) bounds, even when q ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' A counterexample is given by the random period doubling substitution (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='7) with non-uniform probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lq-spectra for non-negative q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The majority of the work in proving Theorem A lies in proving the bounds in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Observe that it suffices to prove the bound for the case k = 1, since we then obtain the bound for other k ∈ N by considering higher powers of the random substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We first prove the upper bound for the case q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Throughout this section, we assume that the random substitution is primitive and compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For all q > 1, τ µP (q) ≤ 1 λ − 1ϕ1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Fix q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ε > 0 and, for each n ∈ N, let m(n) be the integer defined by m(n) = � n λ − ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal Random Substitutions 27 Then the integers n and m(n) satisfy the conditions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12, so it follows that � u∈Ln ϑ µP ([u])q = � u∈Ln ϑ � � �1 λ � v∈Lm(n) ϑ µP ([v]) |ϑ(v1)| � j=1 P[ϑP (v)[j,j+n−1] = u] � � � q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since q > 1, the function x �→ xq is superadditive on the interval [0, 1], so � u∈Ln ϑ µP ([u])q ≥ � u∈Ln ϑ � v∈Lm(n) ϑ µP ([v])q � �1 λ |ϑ(v1)| � j=1 P[ϑP (v)[j,j+n−1] = u] � � q ≥ 1 λq � v∈Lm(n) ϑ µP ([v])q |ϑ(v1)| � j=1 � u∈Ln ϑ P[ϑP (v)[j,j+n−1] = u]q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We now bound the probability on the right of this expression by the production probability of an inflation word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Observe that if w(u) ∈ ϑ(v) contains u as a subword in position j, then P[ϑP (v)[j,j+n−1] = u] ≥ P[ϑP (v) = w(u)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Hence, � u∈Ln ϑ P[ϑP (v)[j,j+n−1] = u]q ≥ � w∈ϑ(v) P[ϑP (v) = w]q for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , |ϑ(v1)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since ϑP is compatible, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='10 there exists an N ∈ N such that for all n ≥ N and all v ∈ Lm(n) ϑ � w∈ϑ(v) P[ϑP (v) = w]q ≥ � a∈A � � � s∈ϑ(a) P[ϑP (a) = s]q � � m(n)(Ra+ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Hence, � u∈Ln ϑ µP ([u])q ≥ 1 λq � a∈A � � � s∈ϑ(a) P[ϑP (a) = s]q � � m(n)(Ra+ε) � v∈Lm(n) ϑ µP ([v])q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Taking logarithms, rearranging and dividing by n gives −1 n log � � � u∈Ln ϑ µP ([u])q � � ≤ − 1 n log � � � � v∈Lm(n) ϑ µP ([v])q � � � + 1 n log λq − m(n) n � a∈A (Ra + ε) log � � � s∈ϑ(a) P[ϑP (a) = s]q � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 28 ANDREW MITCHELL AND ALEX RUTAR Noting that m(n)/n → (λ − ε)−1 as n → ∞, it follows by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2 that τ µP (q) ≤ 1 λ − ετ µP (q) + 1 λ − ε � a∈A log � � � s∈ϑ(a) P[ϑP (a) = s]q � � + cε where c := (#A) maxa∈A log(� s∈ϑ(a) P[ϑP (a) = s]q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' But ε > 0 was arbitrary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' letting ε → 0 and rearranging we obtain τ µP (q) ≤ 1 λ − 1ϕ1(q), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ We now prove the corresponding lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For all q > 1, τµP (q) ≥ 1 λϕ1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ε > 0 and, for each n ∈ N, let m(n) be the integer defined by m(n) = � n λ − ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since q > 1, the function x �→ xq is convex on the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Hence, it follows by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12 and two applications of Jensen’s inequality that � u∈Ln ϑ µP ([u])q = � u∈Ln ϑ � � �1 λ � v∈Lm(n) ϑ µP ([v]) |ϑ(v1)| � j=1 P[ϑP (v)[j,j+n−1] = u] � � � q ≤ � v∈Lm(n) ϑ µP ([v]) � u∈Ln ϑ � �1 λ |ϑ(v1)| � j=1 P[ϑP (v)[j,j+n−1] = u] � � q ≤ |ϑ|q−1 λq � v∈Lm(n) ϑ µP ([v]) |ϑ(v1)| � j=1 � u∈Ln ϑ P[ϑP (v)[j,j+n−1] = u]q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We bound above the probability on the right of this expression by the production probability of a sufficiently large inflation word contained in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' By compatibility, there is an integer k(n) such that j + n ≤ |ϑ(v[1,m(n)−k(n)])| for all n ∈ N and v ∈ Lm(n) ϑ , where lim k(n)/n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, for every v ∈ Ln ϑ, a realisation of ϑ(v[2,m(n)−k(n)]) is contained in u as an inflation word, so � u∈Ln ϑ P[ϑP (v)[j,j+n−1] = u]q ≤ � w∈ϑ(v2···vm(n)−k(n)) P[ϑ(v2 · · · vm(n)−k(n)) = w]q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal Random Substitutions 29 We now bound this quantity uniformly for all v ∈ Lm(n) ϑ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='10 and the above, there is an N ∈ N such that for all n ≥ N � u∈Ln ϑ µP ([u])q ≤ |ϑ|q−1 λq � a∈A � � � s∈ϑ(a) P[ϑP (a) = s]q � � (m(n)−k(n)−1)(Ra−ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Taking logarithms, rearranging and dividing by n gives −1 n log � � � u∈Ln ϑ µP ([u])q � � ≥ m(n) − k(n) − 1 n � a∈A (Ra − ε) log � � � s∈ϑ(a) P[ϑP (a) = s]q � � − log(|ϑ|q−1/λq) n n→∞ −−−→ 1 λ − ε � a∈A (Ra − ε) log � � � s∈ϑ(a) P[ϑP (a) = s]q � � , But ε > 0 was arbitrarily, so τµP (q) ≥ 1 λϕ1(q), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ We now state the bounds for the q ∈ (0, 1) case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We do not give a proof here since the arguments mirror the proofs of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4, except with reversed inequalities since x �→ xq is concave rather than convex and subadditive as opposed to superadditive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If q ∈ (0, 1), then 1 λ − 1ϕ1(q) ≤ τµP (q) ≤ τ µP (q) ≤ 1 λϕ1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lq-spectra for negative q: lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For q < 0, there exist primitive and compatible random substitutions for which τµP (q) and Tϑ,P (q) do not coincide (see, for instance, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' However, we still obtain that τµP (q) ≥ Tϑ,P (q) for all q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' To prove this, it suffices to show the sequence of bounds in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Again, we only need to prove the bound for k = 1 since the remaining bounds follow by considering powers of the random substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If ϑP is a primitive and compatible random substitution, then for all q < 0, τµP (q) ≥ 1 λ − 1ϕ1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ε > 0 be sufficiently small and for n sufficiently large, let m(n) be the integer defined by m(n) = � n λ − ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 30 ANDREW MITCHELL AND ALEX RUTAR To avoid division by zero, we rewrite Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12 in a form where we do not sum over elements equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Here, we write u ◀ ϑ(v) to mean there is a realisation w of ϑ(v) for which u appears as a subword of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For each v ∈ Lm(n) ϑ and u ∈ Ln ϑ, let J (v, u) = {j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' , |ϑ(v1)|} : u ∈ ϑ(v)[j,j+n−1]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Observe that if j /∈ J (u, v), then P[ϑP (v)[j,j+n−1] = u] = 0, and if u does not appear as a subword of any realisations of ϑ(v), then J (u, v) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Therefore, we can rewrite Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12 as µP ([u]) = 1 λ � v∈Lm(n) ϑ u◀ϑ(v) µP ([v]) � j∈J (v,u) P[ϑP (v)[j,j+n−1] = u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Hence, by the subadditivity of the function x �→ xq on the domain (0, 1], � u∈Ln ϑ µP ([u])q = � u∈Ln ϑ � � � � � 1 λ � v∈Lm(n) ϑ u◀ϑ(v) µP ([v]) � j∈J (v,u) P[ϑP (v)[j,j+n−1] = u] � � � � � q ≤ 1 λq � u∈Ln ϑ � v∈Lm(n) ϑ u◀ϑ(v) µP ([v])q � j∈J (v,u) P[ϑP (v)[j,j+n−1] = u]q = 1 λq � v∈Lm(n) ϑ µP ([v])q � u∈Ln ϑ u◀ϑ(v) � j∈J (v,u) P[ϑP (v)[j,j+n−1] = u]q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For each j ∈ J (v, u), let wj(u) ∈ ϑ(v) be a word such that wj(u)[j,j+n−1] = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Note that there are at most K := 2|ϑ|(#A)|ϑ| different u ∈ Ln ϑ such that wj(u)[j,j+n−1] = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Hence, � u∈Ln ϑ u◀ϑ(v) � j∈J (v,u) P[ϑP (v)[j,j+n−1] = u]q ≤ � u∈Ln ϑ u◀ϑ(v) � j∈J (v,u) P[ϑP (v) = wj(u)]q ≤ K � w∈ϑ(v) P[ϑP (v) = w]q and it follows that � u∈Ln ϑ µP ([u])q ≤ λ−qK � v∈Lm(n) ϑ µP ([v])q � w∈ϑ(v) P[ϑP (v) = w]q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Thus, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='10, for all ε > 0 there is an integer N such that for all n ≥ N � u∈Ln ϑ µP ([u])q ≤ λ−qK � a∈A � � � s∈ϑ(a) P[ϑP (a) = s]q � � m(n)(Ra+ε) � � � � v∈Lm(n) ϑ µP ([v])q � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal Random Substitutions 31 Taking logarithms, rearranging and dividing by n gives −1 n log � � � u∈Ln ϑ µP ([u])q � � ≥ − 1 n log � � � � v∈Lm(n) ϑ µP ([v])q � � � + 1 n log(λ−qK) − m(n) n � a∈A (Ra + ε) log � � � s∈ϑ(a) P[ϑP (a) = s]q � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Noting that m(n)/n → (λ − ε)−1 as n → ∞, it follows by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2 that τµP (q) ≥ 1 λ − ετµP (q) + 1 λ − ε � a∈A log � � � s∈ϑ(a) P[ϑP (a) = s]q � � + cε where c := (#A) maxa∈A log(� s∈ϑ(a) P[ϑP (a) = s]q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Letting ε → 0 and rearranging, we obtain τ µP (q) ≥ 1 λ − 1ϕ1(q), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof of general bounds for the Lq spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Using the bounds proven in the prior two sections, we can now complete the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since, for each k ∈ N, the random substitution ϑk P gives rise to the same frequency measure as ϑP , applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5 to ϑk P , 1 λk ϕk(q) ≤ τµP (q) ≤ τ µP (q) ≤ 1 λk − 1ϕk(q) for all q > 1 and 1 λk − 1ϕk(q) ≤ τµP (q) ≤ τ µP (q) ≤ 1 λk ϕk(q) for 0 < q < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Letting k → ∞ gives τµP (q) = τ µP (q) = Tϑ,P (q) = T ϑ,P (q) for all q ∈ (0, 1) ∪ (1, ∞), so the limits defining τµP (q) and Tϑ,P (q) both exist and coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The same holds for q = 0 and q = 1 by continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The monotonicity of the bounds λ−kϕk(q) follows by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Finally for q < 0, for each k ∈ N, applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='6 to ϑk P gives that τµP (q) ≥ 1 λk − 1ϕk(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Passing to the limit completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ 32 ANDREW MITCHELL AND ALEX RUTAR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lq-spectra for negative q under recognisability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' While the upper bound does not hold in general for q < 0, for recognisable random substitutions we can ob- tain this using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='18, which we recall is a refinement of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12 using recognisability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If ϑP is a primitive, compatible and recognisable random substitution, then for all q < 0, τ µP (q) ≤ 1 λ − 1ϕ1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ε > 0 be sufficiently small and, for each n ∈ N sufficiently large, let m(n) be the integer defined by m(n) = � n λ − ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For each u ∈ Ln+2κ(ϑ) ϑ , let w(u) denote the recognisable core of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Further, let v(u) denote the unique legal word such that w(u) ∈ ϑ(v(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='18, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2) µP ([u]) ≤ κ(ϑ) λ µP ([v(u)])P[ϑ(v(u)) = w(u)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Observe that for all u ∈ Ln+2κ(ϑ) ϑ , the recognisable core w(u) has length at least n so, by compatibility, there is an integer N such that if n ≥ N, then |v(u)| ≥ m(n) for all u ∈ Ln+2κ(ϑ) ϑ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, for every u there exists a v ∈ Lm(n) ϑ such that µP ([v(u)]) ≤ µP ([v]) and a w ∈ ϑ(v) such that P[ϑP (v(u)) = w(u)] ≤ P[ϑP (v) = w].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Hence, it follows by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='10 that � u∈Ln+2κ(ϑ) ϑ µP ([u])q ≥ 1 λq � v∈Lm(n) ϑ µP ([v])q � w∈ϑ(v) P[ϑP (v) = w]q ≥ 1 λq � a∈A � � � s∈ϑ(a) P[ϑP (a) = s]q � � m(Ra−ε) � v∈Lm(n) ϑ µP ([v])q, noting that since q < 0, the function x �→ xq is decreasing on (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Taking logarithms, rearranging and dividing by n gives −1 n log � � � u∈Ln ϑ µP ([u])q � � ≤ − 1 n log � � � � v∈Lm(n) ϑ µP ([v])q � � � + 1 n log λq − m(n) n � a∈A (Ra − ε) log � � � s∈ϑ(a) P[ϑP (a) = s]q � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal Random Substitutions 33 Noting that m(n)/n → (λ − ε)−1 as n → ∞, it follows by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2 that τ µP (q) ≤ 1 λ − ετ µP (q) + 1 λ − ε � a∈A log � � � s∈ϑ(a) P[ϑP (a) = s]q � � + cε where c := (#A) maxa∈A log(� s∈ϑ(a) P[ϑP (a) = s]q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Letting ε → 0 and rearranging, we obtain τ µP (q) ≤ 1 λ − 1ϕ1(q), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Recovering entropy from the Lq-spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since the Lq-spectrum encodes both topological and measure theoretic entropy, Theorem A provides an alternative means of proving the coincidence of these quantities with the inflation word analogues introduced in [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For notational simplicity, set ρk = − � a∈A Ra � s∈ϑk(a) P[ϑk P (a) = s] log(P[ϑk P (a) = s]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof of Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We first establish the result for topological entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' By Theo- rem A, the limit defining Tϑ,P (0) exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' in particular, lim m→∞ 1 λk � a∈A Ra log(#ϑm(a)) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since htop(Xϑ) = −τµP (0) = −Tϑ,P (0), we conclude that htop(Xϑ) = − lim m→∞ 1 λk � a∈A Ra log(#ϑm(a)) as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Now we consider measure theoretic entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We first make the following elemen- tary observation: if f and g are concave functions with f(1) = g(1) and f(x) ≤ g(x) for all x ≥ 1, then f +(1) ≤ g+(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Indeed, for all ϵ > 0, f(1 + ϵ) − f(1) ϵ ≤ g(1 + ϵ) − g(1) ϵ , and taking the limit as ϵ goes to 0 (which always exists by concavity) yields the desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Recall that τµP and λ−kϕk are concave functions with τµP (1) = ϕk(1) = 0 for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Moreover, ϕk is differentiable for all k ∈ N with ϕ′ k(1) = ρk and by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2 and Theorem A, � λ−kϕk �∞ j=1 converges monotonically to τµP from 34 ANDREW MITCHELL AND ALEX RUTAR below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, ρk/λk is a monotonically increasing sequence bounded above by τ + µP (1), so that the limit indeed exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Thus τ + µP (1) = lim k→∞ ρk λk since ϕk(q)/(λk − 1) ≥ τµP (q) for all q ∈ (0, ∞), using the preceding observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The result for τ − µP (1) follows by an identical argument, instead using monotonicity and the corresponding bounds for q ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Thus τ ′ µP (1) = limk→∞ ρk/λk, so the desired result follows by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' RECOGNISABILITY AND THE MULTIFRACTAL FORMALISM In this section we establish the multifractal formalism as stated in Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' To do this, we prove a variational principle by considering typical local dimensions of one frequency measure µP relative to another frequency measure µQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Our strategy is to prove the almost-sure existence of relative letter frequencies in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3: this result, combined with recognisability, gives Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The multifractal formalism then follows from this dimensional result combined with the formula for the Lq-spectrum proven in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='7—the proof is given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Non-typical local dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' To prove the multifractal formalism for a given frequency measure µP , we show that for every α ∈ [αmin, αmax], there exists another frequency measure µQ such that dimH µQ ≥ τ ∗ µP (α) and dimloc(µP , x) = α for µQ- almost every x ∈ Xϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Given a primitive set-valued substitution ϑ, permissible probabilities P and Q, m ∈ N and a ∈ A, define the quantity Hm,a P ,Q(ϑ) by Hm,a P ,Q(ϑ) = � v∈ϑm(a) −P[ϑm Q(a) = v] log P[ϑm P (a) = v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Further, let Hm P ,Q(ϑ) denote the vector (Hm,a P ,Q(ϑ))a∈A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We first prove some properties of the quantity Hm P ,Q(ϑ) which we will use in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If ϑ is a primitive and compatible set-valued substitution and P and Q are permissible probabilities, then for all m ∈ N, a ∈ A and s ∈ ϑ(a), � v∈ϑm(s) P[ϑm Q(s) = v] log P[ϑm P (s) = v] = � b∈A |ϑ(a)|b Hm,b P ,Q(ϑ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since ϑ is compatible, we can decompose each v ∈ ϑm(s) into inflation words v = v1 · · · v|ϑ(a)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' By the Markov property of ϑP (respectively ϑQ), we have that P[ϑm P (s) = v] = P[ϑm P (s1) = v1] · · · P[ϑm P (s|ϑ(a)|) = v|ϑ(a)|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal Random Substitutions 35 Therefore � v∈ϑm(s) P[ϑm Q(s) = v] log P[ϑm P (s) = v] = � b∈A |ϑ(a)|b � w∈ϑm(b) P[ϑm Q(b) = w] log P[ϑm P (b) = w] = � b∈A |ϑ(a)|b Hm,b P ,Q(ϑ), which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If ϑ is a primitive and compatible set-valued substitution satisfying the disjoint set condition, with right Perron–Frobenius eigenvector R, and P and Q are permissible probabilities, then the random substitutions ϑP = (ϑ, P ) and ϑQ = (ϑ, Q) satisfy 1 λmHm P ,Q(ϑ) · R → 1 λ − 1H1 P ,Q(ϑ) · R as m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since ϑ satisfies the disjoint set condition, for all m ∈ N and a ∈ A, Hm+1 P ,Q (ϑ) · R = � a∈A Ra � v∈ϑm+1(a) P[ϑm+1 Q (a) = v] log P[ϑm+1 P (a) = v] = � a∈A Ra � s∈ϑ(a) P[ϑQ(a) = s] log P[ϑP (a) = s] + � a∈A Ra � s∈ϑ(a) P[ϑQ(a) = s] � v∈ϑm(s) P[ϑm Q(s) = v] log P[ϑm P (s) = v] = H1 P ,Q(ϑ) · R + � b∈A Hm,b P ,Q(ϑ) � a∈A |ϑ(a)|bRa = H1 P ,Q(ϑ) · R + λ � b∈A RbHm,b P ,Q(ϑ) = H1 P ,Q(ϑ) · R + λHm P ,Q(ϑ) · R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In the second equality we use the Markov property of ϑP and ϑQ, laws of logarithms, and that � v∈ϑm(s) P[ϑm Q(s) = v] = 1 for all s ∈ ϑ(a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' in the third we apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' in the fourth we use that MϑR = λR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Applying the above inductively, 1 λmHm P ,Q(ϑ) · R = m � j=1 1 λj H1 P ,Q(ϑ) · R m→∞ −−−→ 1 λ − 1H1 P ,Q(ϑ) · R, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ Any bi-infinite sequence x in the subshift of a recognisable random substitution can be written as a bi-infinite concatenation of exact inflation words (wn,an), where wn,an is an inflation word generated from the letter an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Given a recognisable set-valued 36 ANDREW MITCHELL AND ALEX RUTAR substitution ϑ, a ∈ A and w ∈ ϑ(a), we define the inflation word frequency of (a, w) in x ∈ Xϑ by fx(a, w) = lim n→∞ f n x (a, w) f n x (a, w) = 1 2n + 1#{m: am = a, wm,am = w, wm,am in x[−n,n]}, provided the limit exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For a given frequency measure µP , the inflation word frequency of a µP -typical word is determined by the production probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' More specifically, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑP = (ϑ, P ) be a primitive, compatible and recognisable random substitu- tion with corresponding frequency measure µP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For µP -almost every x ∈ Xϑ , the inflation word frequency exists and is given by fx(a, w) = 1 λRaP[ϑP (a) = w], for all a ∈ A and w ∈ ϑ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let Aa,w be the set of points x ∈ Xϑ such that the above does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We show that Aa,w is a null set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Taking the complement and then the intersection over all a, w gives a full-measure set with the required property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Given ε > 0, let E(n, ε) be the set of x ∈ Xϑ such that |f n x (a, w) − 1 λRaP[ϑP (a) = w]| > ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' By the Borel–Cantelli lemma, it suffices to show that � n∈N µP (E(n, ε)) < ∞ for all ε > 0 in order to conclude that Aa,w is a nullset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' To this end, we show that µP (E(n, ε)) decays exponentially with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Given u with |u| = 2n + 1 > 2κ(ϑ), let uR denote the recognisable core of u, which has length at least |u| − 2κ(ϑ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='18 gives that µP ([u]) ≤ κ(ϑ) λ µP ([v])P[ϑP (v) = uR] = κ(ϑ) λ µP ([v]) |v| � i=1 P[ϑP (vi) = wi,vi] where each wi,vi is the inflated image of vi in uR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' By compatibility, we can choose an integer N such that every v of length at least N satisfies |v|(Ra − ε/3) ≤ |v|a ≤ |v|(Ra + ε/3) for all a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For each v and a ∈ A, let Aa(v) denote the set of u′ ∈ ϑ(v) such that the frequency of indices i ∈ {j : aj = a} with wi,a = w deviates from P[ϑ(a) = w] by more than ε/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since ϑP acts independently on letters, it follows by Cramér’s theorem that the sum � u′∈A(v) P[ϑP (v) = u′] decays exponentially with |v|a Multifractal Random Substitutions 37 (and hence with |v|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, there is a constant C > 0, independent of the choice of v, such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1) � u′∈A(v) P[ϑP (v) = u′] ≤ e−Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Note that if u is a sufficiently long legal word and has [u] ∩ E(n, ε) = ∅, then we require that uR ∈ A(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Indeed, if u′ /∈ A(v) and |v| ≥ N, then the relative inflation word frequency of w is bounded above by {j : aj = a} |v| |v| |u| � P[ϑP (a) = w] + ε 3 � ≤ 1 λ � Ra + ε 3 � � P[ϑP (a) = w] + ε 3 � ≤ 1 λRaP[ϑP (a) = w] + ε and, similarly, bounded below by RaP[ϑP (a) = w]/λ−ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' hence, [uR]∩E(n, ε) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let Vn denote set of all words which appear as the (unique) preimage of the recognisable core of a word of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It then follows by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='18 that µP (E(n, ε)) ⩽ � u∈Ln ϑ [u]∩E(n,ε)̸=∅ µP ([u]) ≤ κ(ϑ) λ � v∈Vn µP ([v]) � u′∈A(v) P[ϑP (v) = u′] ⩽ e−Cn, where in the final inequality we have used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1) and that � v∈Vn µP ([v]) ≤ n � j=1 � v∈Lj ϑ µP ([v]) ≤ n, absorbing this contribution and the κ(ϑ)/λ factor into the constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It follows that ∞ � n=1 µP (E(n, ε)) ≤ ∞ � n=1 e−Cn < ∞, and the result then follows by the Borel–Cantelli lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ Finally, we require the following bounds on the exponential scaling rate of measures of cylinders, which is essentially a consequence of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, these give bounds on the possible local dimensions of the measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If ϑP is a primitive and compatible random substitution, then there are values 0 < s1 < s2 < ∞ and c1, c2 > 0 such that for all n ∈ N and v ∈ Ln(Xϑ), s1 · n + c1 ≤ log µP ([v]) ≤ s2 · n + c2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' By Theorem A, for all k ∈ N and q > 1, τµP (q) ≤ 1 λk − 1ϕk(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 38 ANDREW MITCHELL AND ALEX RUTAR and for q < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 1 λk − 1ϕk(q) ≤ τµP (q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' for each k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' with βk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='min := lim q→∞ ϕk(q) q(λk − 1) = − 1 λk − 1 � a∈A Ra log � min v∈ϑk(a) P[ϑk P (a) = v] � βk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='max := lim q→−∞ ϕk(q) q(λk − 1) = − 1 λk − 1 � a∈A Ra log � max v∈ϑk(a) P[ϑk P (a) = v] � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' it follows that [βk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' βk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='max] ⊂ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' ∞) is a decreasing nested sequence of intervals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' so with βmin = limk→∞ βk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='min and βmax = limk→∞ βk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 0 < βmin ≤ lim q→∞ τµP (q) ≤ lim q→−∞ τµP (q) ≤ βmax < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1(b) gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ Finally, we obtain our main conclusion concerning relative local dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑ be a primitive, compatible and recognisable set-valued substitution, let P and Q be permissible probabilities, and let µP and µQ denote the frequency measure corresponding to ϑ endowed with the probabilities P and Q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then, for µQ-almost all x ∈ Xϑ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2) dimloc(µP , x) = 1 λ − 1 � a∈A Ra � v∈ϑ(a) −P[ϑm Q(a) = v] log P[ϑm P (a) = v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Fix m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It follows by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='16 that since ϑP is recognisable, so is ϑm P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For each x ∈ Xϑ and n ∈ N with n > κ(ϑm), let un −(x) denote the recognisable core of x[−n,n] and let un +(x) denote an inflation word of minimal length that contains x[−n,n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' By compatibility, |un −(x)|/(2n + 1) → λ−m and |un +(x)|/(2n + 1) → λ−m as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Further, let vn −(x) be the legal word such that un −(x) ∈ ϑm(vn −(x)) and vn +(x) be the legal word such that un +(x) ∈ ϑm(vn +(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then, it follows by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='18 and the definition of local dimension that lim inf n→∞ � − 1 2n + 1 log µP ([un −(x)]) − 1 2n + 1 log P[ϑP (vn −(x)) = un −(x)] � ≤ dimloc(µP , x) ≤ dimloc(µP , x) ≤ lim sup n→∞ � − 1 2n + 1 log µP ([un +(x)]) − 1 2n + 1 log P[ϑP (vn +(x)) = un +(x)] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4, there exists a constant C ≥ 0 such that for all x ∈ Xϑ, 0 ≤ lim inf n→∞ − 1 2n + 1 log µP ([un −(x)]) ≤ lim sup n→∞ − 1 2n + 1 log µP ([un +(x)]) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal Random Substitutions 39 Hence, it follows from the above that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3) lim inf n→∞ − 1 2n + 1 log P[ϑP (vn −(x)) = un −(x)] ≤ dimloc(µP , x) ≤ dimloc(µP , x) ≤ lim sup n→∞ − 1 2n + 1 log P[ϑP (vn +(x)) = un +(x)] + C λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We now show that for µQ-almost all x ∈ Xϑ, lim inf n→∞ −1 n log P[ϑP (vn −(x)) = un −(x)] = lim sup n→∞ −1 n log P[ϑP (vn +(x)) = un +(x)] = 1 λmHm P ,Q(ϑ) · R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' By compatibility, we can decompose the production probabilities into inflation tiles as P[ϑm P (vn −(x)) = un −(x)] = � a∈A � w∈ϑm(a) P[ϑm P (a) = w]Na,w(x,n), where, for each a ∈ A and w ∈ ϑm(a), Na,w(x, n) denotes the number of a’s in vn −(x) which map to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It follows by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3, applied to ϑm Q, that for µQ-almost all x ∈ Xϑ, we have 1 2n + 1Na,w(x, n) → 1 λmRaP[ϑm Q(a) = w] for all a ∈ A and w ∈ ϑm(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Hence, it follows that lim n→∞ − 1 2n + 1 log P[ϑm P (vn −(x)) = un −(x)] = 1 λm � a∈A Ra � v∈ϑm(a) P[ϑm Q(a) = v] log P[ϑm P (a) = v] = 1 λmHm P ,Q(ϑ) · R, with the same convergence holding for un +(x) by identical arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Thus, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3) that 1 λmHm P ,Q(ϑ) · R ≤ dimloc(µP , x) ≤ dimloc(µP , x) ≤ 1 λmHm P ,Q(ϑ) · R + C λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since the above holds for all m ∈ N, by letting m → ∞ it follows by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2 that dimloc(µP , x) exists and dimloc(µP , x) = 1 λ − 1H1 P ,Q(ϑ) · R, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ 40 ANDREW MITCHELL AND ALEX RUTAR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof of the multifractal formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In this section, we apply the results ob- tained in the previous section, along with results on the Lq-spectrum under recognis- ability, to prove Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Proof of Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We first obtain the results for the Lq-spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since every recog- nisable random substitution satisfies the disjoint set condition, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1 gives that Tϑ,P (q) = (λ − 1)−1ϕ1(q) for all q ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If q < 0, then by Theorem A and Proposi- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='7, 1 λ − 1ϕ1(q) = Tϑ,P (q) ≤ τµP (q) ≤ τ µP (q) ≤ 1 λ − 1ϕ1(q), so we conclude that τµP (q) exists and equals (λ − 1)−1ϕ1(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For q ≥ 0, the result follows already from Corollary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We now obtain the results on the multifractal spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In light of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3, it remains to show that fµP (α) ≥ τ ∗ µP (α) for each α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' As proven above, for all q ∈ R, τµP (q) = 1 λ − 1ϕ1(q) = 1 λ − 1 � a∈A RaTa(q) where for each a ∈ A Ta(q) = − log � s∈ϑ(a) P[ϑP (a) = s]q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' First, fix α ∈ (αmin, αmax) and let q ∈ R be chosen so that τ ′ µP (q) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Observe that qα − τµP (q) = τ ∗ µP (α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then define Q by the rule P[ϑQ(a) = s] = P[ϑP (a) = s]qeTa(q) for all a ∈ A and s ∈ ϑ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Then by Corollary C, dimH µQ = 1 λ − 1 � a∈A Ra � − � v∈ϑ(a) P[ϑQ(a) = v] log P[ϑQ(a) = v] � = q · 1 λ − 1 � a∈A Ra � − � v∈ϑ(a) P[ϑQ(a) = v] log P[ϑP (a) = v] � − 1 λ − 1 � a∈A RaTa(q) � v∈ϑ(a) P[ϑQ(a) = v] = qα − τµP (q) = τ ∗ µP (α) since τ ′ µP (q) = 1 λ − 1 � a∈A Ra − � v∈ϑ(a) P[ϑP (a) = v]q log P[ϑP (a) = v] e−Ta(q) = 1 λ − 1 � a∈A Ra � − � v∈ϑ(a) P[ϑQ(a) = v] log P[ϑP (a) = v] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal Random Substitutions 41 In fact, this shows that dimloc(µP , x) = α for µQ-almost all x ∈ Xϑ by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Thus fµP (α) ≥ dimH µQ = τ ∗ µP (α), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The result for α = αmin (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' α = αmax) follows similarly by taking a degenerate probability vector Q assigning equal value to the realisations of ϑ(a) with maximal (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' minimal) probabilities given by P , and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The corresponding non-degenerate sub-substitution is also compatible and recognisable, so the same arguments yield the corresponding bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' EXAMPLES, COUNTEREXAMPLES AND APPLICATIONS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Failure of bounds for q < 0 without recognisability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In the following two examples, we show the results in Theorem A do not extend in general to give an upper bound for the Lq-spectrum in terms of the inflation word Lq-spectrum, for q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1, we construct a class of frequency measures on the full-shift on two letters for which the Lq-spectrum and inflation word analogue differ in the q < 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The random substitutions that give rise to these frequency measures are not compatible, but in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2 we present a compatible analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In contrast, in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3, we give an example showing that the results for q < 0 can hold for all q ∈ R under the identical set condition with identical production probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let p1 < p2 ∈ (0, 1) such that p1 + 3p2 = 1 and let ϑP be the random substitution defined by ϑP : a, b �→ � � � � � � � � � ab with probability p1 ba with probability p2 aa with probability p2 bb with probability p2 We will show for all sufficiently small q < 0 that τµP (q) > Tϑ,P (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Observe that, for each k ∈ N, the word vk = (ab)2k ∈ ϑk+1(a) ∩ ϑk+1(b) occurs with probability P[ϑk+1 P (a) = vk] = P[ϑk+1 P (b) = vk] = p2k 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Clearly, this is the minimal possible probability with which a level-k inflation word can occur, so it follows that lim q→−∞ Tϑ,P (q) q = −1 2 log p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Now, let u ∈ L2k+1 ϑ be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We show that µP ([u]) ≥ p2k−1 1 p2k−1 2 /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since ϑ(a) = ϑ(b) with identical production probabilities, it follows by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12 that for any choice of w ∈ L2k+1 ϑ µP ([u]) = 1 2 � P[ϑP (w)[1,2k+1] = u] + [ϑP (w)[2,2k+1+1] = u] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 42 ANDREW MITCHELL AND ALEX RUTAR If P[ϑP (w)[1,2k+1] = u] ≥ p2k−1 1 p2k−1 2 , then we are done, otherwise at least half of the letters in v must be sent to ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' But then for u to appear from the second letter, at least half of the letters in v must be sent to ba or bb, so P[ϑP (w)[2,2k+1+1] = u] ≥ p2k−1 1 p2k−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Hence, µP ([u]) ≥ p2k−1 1 p2k−1 2 /2 so, in particular, min u∈L2k+1 ϑ µP ([u]) ≥ 1 2p2k−1 1 p2k−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It follows that lim q→−∞ τµP (q) q ≤ −1 4(log p1 + log p2) < −1 2 log p1 = lim q→−∞ Tϑ,P (q) q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' By a slight modification of this example, we can construct a compatible random substitution for which the two notions do not coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let p1 < p2 ∈ (0, 1) such that p1 + 3p2 = 1 and let ϑP be the random substitution defined by ϑP : a, b �→ � � � � � � � � � ab ba with probability p1 ba ab with probability p2 ab ab with probability p2 ba ba with probability p2 By similar arguments to the previous example, we obtain lim q→−∞ τµP (q) q ≤ −1 8(log p1 + log p2) < −1 4 log p1 = lim q→−∞ Tϑ,P (q) q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The random substitution in Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2 satisfies the identical set condition with identical production probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' These conditions are also satisfied by the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' However, here the Lq-spectrum and inflation word analogue coincide for all q ∈ R by a direct argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We will show that for the random substitution ϑP : a, b �→ � ab with probability p ba with probability 1 − p the limit defining τµP (q) exists for all q ∈ R, and τµP (q) = Tϑ,P (q) = 1 λϕ1(q) = −1 2 log(pq + (1 − p)q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Corollary B gives the result for all q > 0 and that τµP (q) ≥ Tϑ,P (q) = 2−1ϕ1(q) for all q < 0, so it only remains to verify for all q < 0 that τ µP (q) ≤ Tϑ,P (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal Random Substitutions 43 Since ϑ(v1) = ϑ(v2) for all v1, v2 ∈ Lϑ, it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12 that for all u ∈ L2m ϑ and any v ∈ Lm+1 ϑ , µP ([u]) = 1 2 � P[ϑ(v)[1,1+2m−1] = u] + P[ϑ(v)[2,2+2m−1] = u] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let V2m = {(ab)m, (ba)m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If u ∈ L2m ϑ \\ V2m, then u must contain bb as a subword.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' This uniquely determines the cutting points in any inflation word decomposition, so there exists a unique v and j(u) ∈ {1, 2} such that u ∈ ϑ(v)[j(u),2m+j(u)−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It follows that � u∈L2m ϑ µP ([u])q ≥ � u∈L2m ϑ \\V2m �1 2P[ϑP (v)[j(u),j(u)+2m−1] = u] �q ≥ 1 2q � u∈L2m ϑ \\V2m P[ϑP (v2 · · · vm) = u[3−j(u),2−j(u)+2m]]q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Now, for every w ∈ ϑ(v2 · · · vm) there is a u such that w = u[3−j(u),2−j(u)+2m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Hence, � u∈L2m ϑ µP ([u])q ≥ 1 2q � w∈ϑ(v2···vm) P[ϑP (v2 · · · vm) = w]q and the conclusion follows by similar arguments to those used in the proofs of the main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Examples with recognisability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' We first provide examples of random substitu- tions for which the multifractal formalism holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let p > 0 and let ϑp be the random substitution defined by ϑp : � � � � � a �→ � abb with probability p bab with probability 1 − p b �→ aa Certainly ϑp is compatible, with corresponding primitive substitution matrix M = � 1 2 2 0 � , Perron–Frobenius eigenvalue (1 + √ 17)/2, and (normalised) right Perron–Frobenius eigenvector � −3 + √ 17 2 , 5 − √ 17 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' One can verify that ϑ is recognisable since every occurrence of aa intersects an image of b and the adjacent letters then determine the cutting points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Thus by Theorem D, for all q ∈ R τµp(q) = Tϑ,P (q) = 1 λ − 1ϕ1(q) = −7 − √ 17 8 log(pq + (1 − p)q) 44 ANDREW MITCHELL AND ALEX RUTAR τ1/5 τ2/5 (A) Lq-spectra τ ∗ 1/5 τ ∗ 2/5 (B) Multifractal spectra FIGURE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lq-spectra and multifractal spectra corresponding to a recog- nisable substitution for p ∈ {1/5, 2/5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' and measure µp satisfies the multifractal formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The asymptotes have slopes −(7− √ 17) log(p)/8 and −(7 − √ 17) log(1 − p)/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' A plot of the Lq-spectra and multifractal spectra for two choices of p is given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For p = 1/2, the Lq-spectrum of the measure µp is a straight line and the multifractal spectrum is equal to htop(Xϑ) at htop(Xϑ), and −∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In the following example, we highlight that the multifractal spectrum need not have value 0 at the endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Let ϑp be the random substitution defined by ϑp : � � � � � � � � � a �→ � � � � � abb with probability p bab with probability p bba with probability 1 − 2p b �→ aaa Similarly to Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='4, ϑp is primitive, compatible and recognisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Hence, Theo- rem D gives that τµp(q) = − 3 10 log(2pq + (1 − 2p)q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The asymptotes have slopes −3 log(p)/10 and −3 log(1 − 2p)/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For p = 1/5 and p = 2/5, the Lq-spectrum and multifractal spectrum of µp are plotted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Here, we highlight that the endpoints of the multifractal spectrum need not be equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Multifractal Random Substitutions 45 τ1/5 τ2/5 (A) Lq-spectra τ ∗ 1/5 τ ∗ 2/5 (B) Multifractal spectra FIGURE 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Lq-spectra and multifractal spectra corresponding to a recog- nisable substitution for p ∈ {1/5, 2/5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Consider the random substitution on the three-letter alphabet A = {a, b, c} defined by ϑP : � � � � � � � � � � � � � � � � � � � � � a �→ � bbc with probability p1 cbb with probability 1 − p1 b �→ � cca with probability p2 acc with probability 1 − p2 c �→ � aab with probability p3 baa with probability 1 − p3 for p1, p2, and p3 in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It is immediate that this substitution is compatible, and by considering the occurrences of 2, 3, or 4 letter repetitions, we observe that this substitution is also recognisable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Moreover, the hypotheses of [15, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='8] are satisfied since ϑ is constant length and #ϑ(a) = #ϑ(b) = #ϑ(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' In particular, the corresponding subshift Xϑ is intrinsically ergodic with unique measure of maximal entropy given by taking p1 = p2 = p3 = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' It follows from [15, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='12] that the measure of maximal entropy is not a Gibbs measure with respect to the zero potential, so the system does not satisfy the usual specification property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For this choice of uniform probabilities, the Lq-spectrum is a straight line passing through the point (1, 0) with slope htop(Xϑ) = log(2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' More generally, the Lq-spectrum is given for all q ∈ R by the formula τµP (q) = −1 6 � log � (1 − p1)q + pq 1 � + log � (1 − p2)q + pq 2 � + log � (1 − p3)q + pq 3 �� and the multifractal formalism is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 46 ANDREW MITCHELL AND ALEX RUTAR For an example on an alphabet of size two, one may consider the random substitu- tion ϑP : � � � � � � � � � � � a �→ � ababbb with probability p1 abbabb with probability 1 − p1 b �→ � baabaa with probability p2 babaaa with probability 1 − p2 for p1 and p2 in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The analysis of this example proceeds identically as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Examples without recognisability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Finally, we consider the two most commonly studied examples of random substitutions: random period doubling and random Fibonacci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Given p ∈ (0, 1), let ϑp be the random period doubling substitution defined by ϑp : � � � � � a �→ � ab with probability p ba with probability 1 − p b �→ aa and let µp denote the corresponding frequency measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The substitution ϑp satisfies the disjoint set condition, so for all q ∈ [0, ∞), τµp(q) = −2 3 log(pq + (1 − p)q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The asymptote as q → ∞ has slope −2 log(max{p, 1 − p})/3, which gives a sharp lower bound on the local dimensions of µp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' If p = 1/2, then the measure µp has linear Lq-spectrum for q ≥ 0 given by τµ1/2(q) = 2 3(q − 1) log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Since the substitution satisfies the disjoint set condition but is not recognisable, our results do not give the Lq-spectrum for q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The random Fibonacci substitution ϑp defined by ϑp : � � � � � a �→ � ab with probability p ba with probability 1 − p b �→ a does not satisfy either the identical set condition nor the disjoint set condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Hence, we cannot apply Corollary B to obtain a closed-form formula for τµp(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' However, we can still apply Theorem A to obtain a sequence of lower and upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' The case k = 1 gives the following bounds for all 0 < q < 1: − 1 φ2 log(pq + (1 − p)q) = 1 φϕ1(q) ≤ τµp(q) ≤ 1 φ − 1ϕ1(q) = − log(pq + (1 − p)q), Multifractal Random Substitutions 47 −1/2 1/2 1 3/2 2 5/2 1 2 3 4 5 6 ϕk/(λk − 1) ϕk/λk FIGURE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Upper and lower bounds on the Lq-spectrum of the fre- quency measure corresponding to the random Fibonacci substitution with p = 1/2, for k = 3, 5, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' where φ denotes the golden ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Reversing the inequalities yields the corresponding bounds for q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Of course, by considering larger k we can obtain better bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' For p = 1/2, the bounds given by Theorem A for k = 3, 5, 7 are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors are grateful to Philipp Gohlke for his detailed comments on a draft version of this manuscript, which helped to remove some technical assumptions from Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' They also thank Dan Rust and Tony Samuel for valuable input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' AM thanks SFB1283 and the Universität Bielefeld for supporting a research visit during the summer of 2022, where some of the work on this project was undertaken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' AM was supported by EPSRC DTP and the University of Birmingham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' AR was supported by EPSRC Grant EP/V520123/1 and the Natural Sciences and 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Péter P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' Varjú, Recent progress on Bernoulli convolutions, Proceedings of the 7th European Congress of Mathematics (Berlin), January 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content=' 3 SCHOOL OF MATHEMATICS, UNIVERSITY OF BIRMINGHAM, EDGBASTON, B15 2TT Email address: acm925@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='bham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='uk MATHEMATICAL INSTITUTE, UNIVERSITY OF ST ANDREWS, ST ANDREWS, KY16 9SS Email address: alex@rutar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} +page_content='org' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E4T4oBgHgl3EQfNwwL/content/2301.04958v1.pdf'} diff --git a/YdE2T4oBgHgl3EQfEQZV/content/tmp_files/2301.03633v1.pdf.txt b/YdE2T4oBgHgl3EQfEQZV/content/tmp_files/2301.03633v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe207c8b3e69c3b6c41e7355edbcbb21c546c1d5 --- /dev/null +++ b/YdE2T4oBgHgl3EQfEQZV/content/tmp_files/2301.03633v1.pdf.txt @@ -0,0 +1,11003 @@ +arXiv:2301.03633v1 [math.AP] 9 Jan 2023 +Smoothing Properties of a Linearization of the Three Waves Collision +Operator in the bosonic Boltzmann–Nordheim Equation +Jogia Bandyopadhyay∗ and Jani Lukkarinen †1 +1University of Helsinki, Department of Mathematics and Statistics +January 11, 2023 +Abstract +We consider the kinetic theory of a three-dimensional fluid of weakly interacting bosons in a non- +equilibrium state which includes both normal fluid and a condensate. More precisely, we look at the +previously postulated nonlinear Boltzmann–Nordheim equations for such systems, in a spatially homo- +geneous state which has an isotropic momentum distribution, and we linearize the equation around an +equilibrium state which has a condensate. We study the most singular part of the linearized operator com- +ing from the three waves collision operator for supercritical initial data. The operator has two types of +singularities, one of which is similar to the marginally smoothing operator defined by the symbol ln(1+p2). +Our main result in this context is that for initial data in a certain Banach space of functions satisfying +a H¨older type condition, at least for some finite time, evolution determined by the linearized operator +improves the H¨older regularity. The main difficulty in this problem arises from the combination of a +point singularity and a line singularity present in the linear operator, and we have to use some fine-tuned +function spaces in order to carry out our analysis. +1 +Introduction +An experimentally observed and also widely theoretically studied phenomenon of cold quantum fluids is +Bose condensation: a macroscopic number of fluid particles form a condensate whose mechanical properties +are very different from the properties of the same fluid at higher temperatures. For instance, the resistance +of the condensate can vanish. Many books have been written on the topic, for example, there is a recent +review of results including physics of partial condensation by Griffin, Nikuni and Zaremba [1]. +Mathematically, to study Bose condensation is a challenge; even the definition of the corresponding +equilibrium states for ideal Bose gas requires some effort (see, e.g., chapter 5.2.5. in [2]). For interacting +Bose gases, rigorous results on condensation have only recently started to appear. They mainly concern the +case of total condensation, or zero temperature, where all of the available particles lie in the condensate [3, 4]. +In the mean-field limit, the system can then be well-described by a factored state (i.e., the particles are not +correlated) determined by a single wave-function whose dynamics follow the Gross-Pitaevskii equation [5]. +∗jogiab@gmail.com +†jani.lukkarinen@helsinki.fi +1 + +As far as we are aware, there are no rigorous results about the dynamics of non-equilibrium states in other +than the mean-field limit with total condensation. In condensed matter physics, one commonly used tool +in the study of the time-evolution of bosonic quantum fluids is the bosonic Boltzmann-Nordheim equation +(also called Uehling-Uhlenbeck equation). It describes the evolution of the “phase space” density of the +particles, f(r, v, t) ≥ 0, with r ∈ R3 denoting position, v ∈ R3 velocity, and t ≥ 0 time, such that +∂tf(r, v, t) + v · ∇rf(r, v, t) = C4[f(r, ·, t)](v) . +(1.1) +The collision operator is given by +C4[h](v0) = 4π +� +(R3)3dv1dv2dv3 δ(v0 + v1 − v2 − v3)δ(ω0 + ω1 − ω2 − ω3) +� +˜h0˜h1h2h3 − h0h1˜h2˜h3 +� +, +(1.2) +where δ(·) denote Dirac δ-distributions: the first one is simply a shorthand notation for a convolution +integral, and the second enforces conservation of kinetic energy in the “collisions”; we have also used the +shorthand notations +hj = h(vj), +˜hj = 1 + hj, +ωj = Ekin +j += 1 +2v2 +j . +(1.3) +As we discuss in Section 1.1 and Appendix A, the main contribution to the evolution of a suitably scaled +perturbation ψt of an equilibrium solution containing a condensate density ¯n > 0 is obtained by linearization +of a certain three wave collision operator. This results in an evolution equation +d +dtψt = −¯nL3ψt, +(1.4) +where the operator L3 has the following explicit form +L3ψ(x) = +� ∞ +0 +dy K3(x, y)(ψ(x) − ψ(y)), +(1.5) +K3(x, y) = 4¯h(x)2xye− min(x,y)fBE(|x − y|) ˜fBE (max(x, y)) ˜fBE(x + y) +� +1 + e− max(x,y)� +, +(1.6) +¯h(x) = +� +x5/2fBE(x) ˜fBE(x) +�− 1 +2 , +fBE(x) := +1 +ex − 1 . +(1.7) +The integral kernel K3 has two types of singularities: there is boundary “point singularity” when x → 0, and +a 1/|x−y| -type “line singularity” as x → y. The defining integral is absolutely convergent for C(1)-functions +with sufficiently fast decay at x → 0 and x → ∞, but it is not clear if such properties could be preserved by +solutions to the equation. +In this paper, we clarify two issues about the evolution equation (1.4). We first define the action of +the operator, L3ψ, by the integral (1.5) for H¨older continuous functions ψ, with a weight ensuring that the +integral is absolutely convergent. We then show that this operator is non-negative in a certain weighted +L2-space, and we prove that it has a unique Friedrichs extension into a self-adjoint, non-negative operator +on this space. +The kernel of the operator is one-dimensional, and we prove that it has a spectral gap. +Hence, the semigroup generated by the self-adjoint operator is contractive in the orthocomplement of the +zero subspace. +We then study pointwise solutions to the evolution equation which the difference functions ∆t(x, y) := +ψt(x) − ψt(y) would satisfy if they were sufficiently regular. For initial data, the sufficient regularity is +guaranteed by picking them from a certain weighted Banach space of functions satisfying a H¨older-type +condition. We obtain unique solutions for the difference equation to these initial data in this Banach space. +2 + +However, in order to prove the smoothing, we then have to show that the solutions obtained thus, are +in fact differences of the form ψt(x) − ψt(y). In order to do this, we study a regularized version of the +original evolution equation for similar H¨older initial data, obtain unique solutions that are also difference +functions, take a limit to remove the regularization, and prove that in this limit the difference function +becomes identical to the ∆t solution obtained earlier. Our result proved for the ∆t solutions then imply +that H¨older regularity is improved by the time evolution almost everywhere for some finite time. +Finally, we show that the pointwise solutions coincide with the solutions given by the L2-semigroup. +This implies that also the semigroup is smoothing, at least for sufficiently regular initial data. +The main technical difficulty in the study of the linearized operator mentioned above is that the evolution +equation has two competing singularities, where the singularity connected with smoothing is marginal. Thus +we have to exercise considerable care in defining function spaces that remain invariant under the evolution +at least for some finite time. +To illustrate the point, let us first consider the semigroup generated by −L0, where L0 denotes the +positive operator on L2(Rd) corresponding to multiplication with ln(1+p2) in the Fourier space. An explicit +computation shows that the operator then acts on Schwarz functions as +� +dy K(x, y)(ψ(x) − ψ(y)) where +the integral kernel K has the same line singularity, 1/|x − y|, as L3. Then for t ≥ 0 the semigroup operator +e−tL0 is given by multiplication with (1 + p2)−t in the Fourier space, and thus the semigroup provides slow +smoothing of solutions: it maps the Sobolev space Hs to Hs+2t for any s. This behavior is different from +the standard case of semigroup of the Laplacian which has the symbol e−tp2, and thus immediately produces +smooth functions. In fact, it will become apparent later that the linearized operator, after a change of +variables to make it act on a weighted L2(R) space, in the present case closely resembles V0L0, where V0 +denotes multiplication with e−u/2, u ∈ R. However, this space is very large for purposes of using it directly +to study the full nonlinear problem. For example, it contains unphysical solutions of infinite mass and some +regularity is needed to make sense of the nonlinear collision integrals. +Our original motivation for studying the problem was to complete a nonlinear perturbation argument, +and it was clear there that some smoothing property of the linearized semigroup would be needed to control +the evolution. Indeed, this programme has been taken up the Escobedo, et al., in a series of papers and +preprints. An explicitly treatable asymptotic version of the linearized operator was considered in [11] where +its solutions were generated via a Green’s function method, yielding an integral formula with an controllable +kernel acting on sufficiently regular initial data. The control of the kernel leads to estimates which are +consistent with the smoothing which we prove here. +The full linearized operator has recently been studied in the preprint [12] where solutions to the evolution +equation (1.4) are generated by a perturbation argument on the kernel function derived in [11]. Uniqueness +and possible semigroup properties are not considered. Therefore, the present works gives at least an partial +answer, in the sense of L2 spaces. Since the explicit form of the linearized operator is slightly different from +the one in (1.5), we have explained in Appendix A how they nevertheless can be connected by a change of +variables. +Let us now briefly describe how the rest of this paper is organized. Section 2 contains an analysis of the +problem in a weighted L2 space; we define Friedrichs extensions for L3 and a sequence of approximating +operators denoted by Lε +3, prove some properties for the corresponding semigroup solutions and then prove +the most important result of this section, namely Theorem 2.5, which shows that the semigroup solutions +corresponding to the approximating operators converge to the L2 solution for our original operator. +Section 3 contains all of our Banach space results. The H¨older regularity is obtained in Theorem 3.3 as +a property in a certain Banach space of functions of two variables. We then show that these functions can +be identified with differences of the L2 solutions obtained in Section 2, at least for some finite time. In order +to do this we have to first obtain results such as Theorem 3.4 and Proposition 3.5 for the approximating +3 + +operators. The main smoothing result Theorem 3.2 follows as a straightforward consequence of our Banach +space results and Theorem 2.5. +Although the most important theorems, namely Theorems 3.3 and 3.4, are proved in these Banach +spaces, we have to rely on solutions in the weighted L2 space in order to connect these theorems and arrive +at the main smoothing result of Theorem 3.2. Thus, all our results in the weighted L2 space are presented +in Section 2 while Section 3 is reserved for the Banach space theorems. +We devote the rest of this section to a discussion of the physical connection and derivation of the +linearized three waves collision operator. +The necessary computational details of this derivation can be +found in Appendix A. +1.1 +Physical motivation of the nonlinear problem and the proposed linearization +The Bolztmann–Nordheim equation (1.1)–(1.2) has not yet been rigorously derived from evolution of a +bosonic quantum system but the following conjecture can be generalized from the discussion in [6]. Consider +a translation invariant quasi-free initial state on the bosonic Fock space determined by a two-point correlation +function g0(r1 −r2) where g0 is a rapidly decreasing function. Suppose that the N-particle dynamics is given +by a Hamiltonian with weak pair interactions, HN = �N +i=1 +1 +2p2 +i + λ 1 +2 +� +i̸=j V (ri − rj), 0 < λ ≪ 1. If V is +well-localized and +� +R3dr V (r) = 1, then up to times O(λ−2) the state should be translation invariant and +quasi-free apart from small corrections, and the following limit of the Fourier transform of the time-evolved +two-point function should exist: �gt(v)|t=λ−2˜t → W(v, ˜t) as λ → 0+ for any, not too large, ˜t ≥ 0. In addition, +this limit should be well approximated by solutions to the equation ∂tW(v, t) = C4[W(·, t)](v), with initial +data W(v, 0) = �g0(v). (In [6], for technical reasons, only a discretized version of this problem is considered. +The above conjecture is a generalization of Conjecture 5.1 in the bosonic case “θ = 1” there.) +Although the conjecture remains unproven at the moment, it relies on a perturbative expansion which +has been successfully applied to a related problem of equilibrium time-correlations for a discrete nonlinear +Schr¨odinger equation [7] (the connection to the above conjecture is outlined in Sec. 2.3 there) and, more +recently, also for its continuum version [8]. The proof in [7] uses quite heavily a property analogous to +supv |�g0(v)| < ∞ which in the above setup would be a consequence of the assumed sufficently fast decay of +correlations for large spatial separation. For a critical ideal Bose fluid at equilibrium one has W(v) equal to +the critical Bose-Einstein distribution which blows up as |v|−2 near v = 0. Thus this condition is violated in +a critical system, and one can justifiably question the validity of the perturbative derivation for any system +with critical and supercritical densities. +If W is homogeneous and isotropic, it depends only on x = ω(v) = 1 +2v2, and the evolution equation for +f(x, t) := W( +√ +2xˆe1, t) can be rewritten for x, t ≥ 0 as ∂tf(x, t) = C4[f(·, t)](x) after (with a slight abuse of +notation and neglecting an overall numerical constant) we define +C4[f](x0) := +1 +√x0 +� +R2 ++ +dx2dx3 +1(x1 ≥ 0) +min +j=0,1,2,3 +√xj +� +˜f0 ˜f1f2f3 − f0f1 ˜f2 ˜f3 +� +, +(1.8) +where x1 = x2 + x3 − x0, and fi = f(xi). (To show the connection is not completely straightforward. It +has been done in Appendix A of [9] for W which are Schwartz functions.) The numerical solutions to this +equation were studied by Semikoz and Tkachev in [9], and they found a finite time blowup at x = 0 for +smooth, but supercritical, initial data. In light of the doubts about the perturbative derivation for singular +cases, it is not clear how the solutions should be continued beyond the formation of the singularity. +One possibility is that the perturbative kinetic argument simply becomes inapplicable, and one has to +go back to the original time-evolution in Fock space to resolve the issue. Another possibility is that nothing +very special happens, and the equation continues to hold in its original (pointwise) sense, only restricted +4 + +to x > 0. The equation ∂tf = C4[f], x > 0, with f(x, 0) ∼ x−7/6, was studied by Escobedo, Mischler, +and Vel´azquez in [10]. (The value 7 +6 is related to Kolmogorov theory of wave turbulence [14].) They show +the existence of solutions locally in time, preserving the x−7/6 singularity. However, these solutions do not +conserve total mass of particles (which is obviously conserved by the microscopic dynamics). If one thinks +that the extra mass is exchanged with the condensate, this way of “extending” the solution would correspond +to adding the condensate mass as an extra degree of freedom with no backreaction to the normal fluid. +A different extension was considered by Lu in [15, 16]. He considers weak solutions, positive measures +µt(dx) such that t �→ µt is weakly continuously differentiable and ∂tµt = C4[µt] in the sense of distributions. +The existence of such solutions is proven in [15] wherein the precise meaning of how the measures are solutions +is defined on page 1611. In [16] he also proves that the solutions can be chosen so that they conserve both +mass and energy and converge to the physically expected equilibrium distribution as t → ∞. This occurs +even in the supercritical case, and it is shown that then a portion of the total mass condenses to x = 0 +(at least asymptotically as t → ∞). All of these results are deduced from subsequences of approximating +solutions to a regularized problem, and as such leave open the uniqueness of these solutions, and are not +amenable for numerical treatment. +In [9], Semikoz and Tkachev proposed a different method of continuing the solution after a condensate +has formed. On physical grounds, they postulated that the solution would be a positive measure of the +form ftot(x, t)√xdx = freg(x, t)√xdx + n(t)δ(x)dx, which corresponds to putting a mass n(t) ≥ 0 into a +δ-distribution at the origin v = 0 ∈ R3 and allowing only for a regular distribution for |v| > 0. (The square +roots are explained by the identity v2d|v| = +√ +2xdx.) With this ansatz they arrived at coupled equations of +the form +∂tfreg(t) = C4[freg(t)] + n(t)C3[freg(t)] , +d +dtn(t) = −n(t)ρ[C3[freg(t)]] , +(1.9) +where C3 is a new collision operator and ρ[f] denotes the mass functional. Since these equations involve +n(t), their solutions do not coincide with the singular solutions studied in [10]. The equations were again +solved numerically, and convergence towards the expected equilibrium was found in [9]. But even this set of +equations is somewhat problematic from the physical point of view: it does not answer how a condensate +can be generated (if n(0) = 0, it remains zero for all times), and for regular functions f one can prove that +ρ[C3[f]] ≥ 0, so it seems that n(t) can only decrease. A more detailed analysis of the ansatz was made by +Spohn in [17], and among other things, he showed that if freg(x, t) ≃ a(t)x−1, as would be the case for a +critical equilibrium distribution, then the second equation is equal to +d +dtn(t) = −n(t)(2ρ[√xfreg(x, t)] − c0a(t)2) , +with c0 = 1 +3π2 . +(1.10) +It follows that the critical freg are stationary solutions, and there can be an exchange of mass with either +sign between the regular fluid and the condensate. This computation and those in [10] clearly illustrate that, +for singular data, the meaning of the Boltzmann equation has to be carefully specified. +In this paper we look at a slight modification of the above evolution equations in which, at least in +principle, condensate can be freely created and destroyed. We use the same equations as before for the +regular part, but do not try to form a differential equation for n(t). Instead, as motivated by Lu’s results, +we impose a strict conservation of mass for all times and use this as a definition of n(t). Our aim is to +study the linearization of the three waves collision operator C3 around a stationary solution. To this end, +we consider the full distribution function for a Bose fluid at time t, in the presence of a condensate, given +by the measure µtot +t (dx) = freg(x, t)√xdx + n(t)δ0(dx) on R+ := [0, ∞). Here δ0 denotes the unit measure +concentrated at x = 0, and freg denotes a function on R+ with finite mass and energy, respectively defined +5 + +by the functionals +ρ[f] := +� ∞ +0 +dx√xf(x), +e[f] := +� ∞ +0 +dx√xxf(x) . +Including the contribution from the condensate, the total mass and energy are then defined as +M(t) = ρ[freg(t)] + n(t) , +E(t) = e[freg(t)] . +To summarize the conventions made so far: to get the “density” in the original 3-dimensional Boltzmann- +Nordheim equation, we use “f(r, v, t)dv” = +√ +2µtot +t (d(v2/2))dΩ, where dΩ denotes the standard integration +over the angular variables of v. The presence of the scaling factor +√ +2 here guarantees that “ ˜f(r, v, t)dv” = +√ +2˜µtot(d(v2/2))dΩ with ˜µtot(dx) := (1 + freg(x, t))√xdx + n(t)δ0(dx). +Suppose that the initial data is determined by n(0) = n0 ≥ 0 and freg(x, 0) = f0(x), where f0 ≥ 0 is +suitably regular. In particular, we assume that +e0 := e[f0], +ρ0 := ρ[f0] +are finite. If e0 = 0, then f0 = 0 almost everywhere, and defining freg(x, t) = 0, n(t) = n0, will yield a +solution, corresponding to total condensation. This case is not of interest here, let us only point out that it +is in line with the previous results on total condensation: the homogeneous solutions to the Gross-Pitaevskii +equation are wave-functions with a constant magnitude, thus leading to particle densities which do not vary +in time. +Let us thus assume that e0 > 0 when also ρ0 > 0. For such initial data, and with weak interactions, +one would physically expect the system to relax to an equilibrium distribution which is very close to that of +free particles. As discussed in the introduction, the Boltzmann-Nordheim equation is believed to arise in a +scaling limit with the strength of the interaction going to zero, thus its stationary solutions should be given +by the ideal gas equilibrium distributions. For subcritical initial data these depend on two parameters: an +inverse temperature β > 0 and the chemical potential µ ≤ 0. A particular case is when µ = 0 and the +corresponding distribution is called the critical Bose-Einstein distribution, +fβ,0(x) = fBE(βx) , +where fBE(x) := +1 +ex − 1 , +(1.11) +and the subcritical distributions are given by fβ,µ(x) := fBE(β(x − µ)), µ < 0. Since e0 > 0, there obviously +is a unique β > 0 such that e0 = e[fβ,0], and this is determined by +β := +�e[fBE] +e0 +� 2 +5 +. +(1.12) +We assume now that the initial state is supercritical, M(0) > ρ[fβ,0], i.e., +n0 + ρ0 +e3/5 +0 +> ρ[fBE] +e[fBE] +3 +5 +. +If this is true, there are no Bose-Einstein distributions which would have the right energy and particle +densities. These results can be found in many references, for instance, see Section 2 in [17] and Section 6 in +[15]. +On physical grounds, it is expected that the normal fluid relaxes towards the corresponding critical +distribution and that the additional particles are forming the condensate: since the particles in the condensate +6 + +do not contribute to energy density, this allows having the same mass and energy in the initial and the +equilibrium state. Motivated by this physical discussion, we define the equilibrium condensate density by +¯n := M(0) − ρ[fβ,0] = n0 + ρ0 − ρ[fβ,0] . +(1.13) +With these definitions, the assumption of a supercritical initial state is equivalent to assuming ¯n > 0. +We postulate that the time-evolution of the system conserves total mass, i.e., that +n(t) = M(0) − ρ[freg(t)] = ¯n − ρ[freg(t) − fβ,0] . +(1.14) +so the evolution equation is +d +dtfreg(x, t) = C4[freg(·, t)](x) + n(t)C3[freg(·, t)](x), +(1.15) +which becomes a closed equation for freg, after we insert the mass conservation law in (1.14). Here the first +collision operator is given by +C4[f](x0) = +1 +√x0 +� +R2 ++ +dx2dx3 +1(x1 ≥ 0)I(x) +� +˜f0 ˜f1f2f3 − f0f1 ˜f2 ˜f3 +� +x1=x2+x3−x0 +(1.16) +where fi := f(xi), ˜fi = 1 + fi, and +I(x) := +min +j=0,1,2,3 +√xj . +We have also ignored an overall explicit constant which can be recovered by rescaling time. From this form, +a formal substitution of the ansatz yields for the interaction term with the condensate +C3[f](x) = +2 +√x +� x +0 +dy +� +˜f(x)f(x − y)f(y) − f(x) ˜f(x − y) ˜f(y) +� +− 4 +√x +� ∞ +x +dy +� +˜f(y)f(y − x)f(x) − f(y) ˜f(y − x) ˜f(x) +� +. +(1.17) +As shown in [17], for f0 = fβ,0 one has C4[f0] = 0 = C3[f0]. Thus f(x, t) = fβ,0(x) yields a stationary +solution of this equation. (Note that then by definition also n(t) = ¯n = n0 is constant.) Here we inspect +if “small” perturbations of such states lead to solutions which relax towards a state of the same type (the +parameters of the new state do not need to be the same as those of the original one). A convenient way to +define the perturbation is to consider +ψt(x) := f(x, t) − fβ,0(x) +Rβ(x) +, +where Rβ(x) = βxfβ,0(x)(1 + fβ,0(x)) = R1(βx) . +As here R1 ≃ x−1, for solutions of the type considered by Spohn, f(x, t) ≃ a(t)x−1, one would have +a(t) = (ψt(0)+1)/β. Thus if continuous solutions with such asymptotics exist, then ψt would be continuous +also at 0. +Now we can solve the dependence on β, by scaling x′ = β−1x. Then C4 gains a factor of β2 and C3 a +factor of β1/2. Thus if we have a solution f1(x, t; ¯n) for β = 1, a solution to the generic case is given by +f(x, t; ¯n, β) := f1(βx, β−2t; β−3/2¯n). Therefore, it suffices to consider the case β = 1, when fβ,0 = fBE, and +ψt(x) = f(x, t) − fBE(x) +R(x) +, +where R(x) = R1(x) = xfBE(x) ˜fBE(x) . +(1.18) +7 + +Here and in the following, we employ a shorthand notation ˜fBE := 1 + fBE. The time-evolution of ψt(x) is +thus determined by +d +dtψt(x) = +1 +R(x) +� +C4[fBE + Rψt](x) + (¯n − ρ[Rψt]) C3[fBE + Rψt](x) +� +, +Now let −Li denote the linearization of Ci around fBE, and Qi the corresponding remainder: Qi[h] := +Ci[fBE + h] + Lih. Since Ci[fBE] = 0, Qi[h] is quadratic in h. Thus the evolution equation for ψt can be +written in the form +d +dtψt = −Lψt + Q[ψt] , +where +L = L4 + ¯nL3 , +Li = R−1LiR , +Q[ψ] = Q4[ψ] + ¯nQ3[ψ] − ρ[Rψ]Q3[ψ] + ρ[Rψ]L3ψ , +Qi[ψ] := R−1Qi[Rψ] . +It turns out that the linearized three waves collision operator is most singular and can thus be thought +of as dominant. Following the computations presented in Appendix A, we arrive at the evolution equation +(1.4) with the form (1.5) for the operator L3. Although it is not shown in this paper, our main result can +be extended without difficulty to cover also the subdominant operator L4, leading to a smoothing result for +the full linearized operator. +Acknowledgements +We are deeply grateful to Antti Kupiainen for many illuminating discussions on this problem as well as +valuable suggestions and comments on this manuscript. During the many years over which we have worked +on this project, we have benefited also from discussions with several other colleagues. In particular, we +would like to thank Cl´ement Mouhot, Herbert Spohn, and Juan J. L. Vel´azquez for their comments. We are +also thankful to Miguel Escobedo for correspondence about their newest work on the topic. The research +has been supported by the Academy of Finland, via an Academy project (project No. 339228), the Finnish +Centre of Excellence in Randomness and Structures (project No. 346306) and ERC Advanced Investigator +Grants 741487 and 227772. J Bandyopadhyay’s work in this paper is intended as a small tribute to their +father Raghab Bandyopadhyay. +2 +Solutions in a Weighted L2-space +In this section we consider, in a certain weighted L2 space of functions, the linear operator appearing in the +evolution equation (1.4) as well as a certain sequence of related linear operators. We first change variables +x → u = ln(ex − 1), x being the energy variable. Thus, while the old variable x was in R+, our new variable +u ∈ R, and we look at (1.4) in a weighted L2 space of functions on R. Also, we will henceforth include the +prefactor ¯n in the definition of L3 (while continuing to use the same symbol for the operator), so that the +right hand side of (1.4) now reads (−L3ψt). +As discussed in more detail in Appendix A, the resulting operator then naturally acts on a weighted L2 +space. The weight function ν is +ν(dw) = ν(w)dw = e−w (ln(1 + ew)) +5 +2 dw. +8 + +In this space the linearized operator acts on any suitably regular function ψ (say, compactly supported and +smooth) in its domain D(L3) as follows: +(L3ψ)(u) = +� +R +ν(dv)K3(u, v) +� +ψ(u) − ψ(v) +� +, +(2.1) +K3(u, v) = 4¯n [(ln(1 + eu))(ln(1 + ev))]− 3 +2 +e−|u−v| +1 + e− min(u,v) + e− max(u,v) +2 + emax(u,v) +1 − e−|u−v| . +The associated sesquilinear form is given by +˜Q +� +φ, ψ +� += 1 +2 +� +R2(ν × ν)(du, dv)K3(u, v) +� +φ(u) − φ(v) +�∗� +ψ(u) − ψ(v) +� +. +˜Q is evidently symmetric and non-negative. We now extend the form domain D( ˜Q) to cover all ψ for which +˜Q +� +ψ, ψ +� +< ∞ in the sense the above integral is convergent (note that in this case the integrand is real and +non-negative). The Cauchy–Schwartz inequality then implies that ˜Q(φ, ψ) is defined for all φ, ψ ∈ D( ˜Q) as +an absolutely convergent integral. +We will also look at a sequence of regularized linear operators Lε +3, defined for all ε0 > 0 in the following +way: +(Lε +3ψ)(u) = +� +R +ν(dv)K +ε +3(u, v) +� +ψ(u) − ψ(v) +� +, ∀ψ ∈ D(Lε +3), where, +K +ε +3(u, v) = K3(u, v)1 − e− min(ε(u,v),|u−v|) +1 − e−ε(u,v) +, ε(u, v) = ε(max(u, v)) = ε0 exp +� +− µ′ +γ0 +max(a1, max(u, v)) +� +. +(2.2) +Here µ′, γ0, and a1 are positive-valued parameters. For results connected with only the L2-solutions, the +values of these parameters in R+ do not matter. However, for our main result, proved in a certain Banach +space, these parameters have to be restricted to certain intervals in R+, as described in (2.4) and in Subsection +3.2. The sesquilinear form ˜Qε corresponding to the regularized linear operator is again non-negative and +symmetric. +In the rest of this section we will construct Friedrichs extensions of the operators L3 and Lε +3 in L2(ν), +show that these generate contractive semigroups in L2(ν), and finally, we prove that the semigroup solutions +associated with the Friedrichs extension of Lε +3 approximate those for the extension of L3 as ε0 → 0. +2.1 +Friedrichs Extensions and Their Properties +We will describe in detail the construction of the Friedrichs extension for L3. The extensions for Lε +3, ∀ε > 0 +can be constructed similarly. +Starting from the sesquilinear form ˜Q, we first define the following inner product +Q(φ, ψ) = ˜Q(φ, ψ) + (φ, ψ) , +on the earlier defined domain +D(Q) = D( ˜Q) = {ψ ∈ L2(ν) : Q(ψ, ψ) < ∞}. +This domain is in fact already complete, i.e., suitable for the Friedrichs extension, as the next result shows. +Lemma 2.1. D(Q) is complete under the inner product Q. +9 + +Proof. In order to prove completeness of D(Q), consider an arbitrary sequence ψn ∈ D(Q), such that ψn is +Cauchy under Q, i.e., Q(ψn − ψm, ψn − ψm) → 0 as m, n → ∞. Since the assumptions imply that (ψn) is +also Cauchy in L2(ν), there is ψ ∈ L2(ν) such that ψn → ψ. Thus we only need to show that i) ψ ∈ D(Q), +and, ii) Q(ψn − ψ, ψn − ψ) → 0 as n → ∞. +Let us first note that, for any r > 0, we can define the bounded kernel Kr(u, v) = +1(K3(u, v) < +1/r)K3(u, v), so that, Kr ≤ K3 and Kr converges pointwise to K3 as r → 0+. Then the corresponding +sesquilinear forms satisfy +lim +r→0+ Qr(φ, ψ) = Q(φ, ψ), ∀φ, ψ ∈ D(Q), by dominated convergence. +Also, for any r > 0, there exist positive constants C, C1 < ∞, such that +|Qr(φ, ψ)| ≤ |(φ, ψ)| + +� +R2(ν × ν)(du, dv)Kr(u, v)| +� +φ(u) − φ(v) +� +|| +� +ψ(u) − ψ(v) +� +| ≤ +� +1 + C +r +� +∥ψ∥L2∥φ∥L2, +and, ∥ψ∥2 +L2 ≤ Qr(ψ, ψ) ≤ +� +1 + C1 +r +� +∥ψ∥2 +L2. +The above inequalities imply that D(Qr) = L2(ν) and that the norm defined by Qr is equivalent to the +L2-norm. This means ψn → ψ ∈ D(Qr) and Qr(ψn, ψn) → Qr(ψ, ψ) as n → ∞. +Since ψn ∈ D(Q) is Q−Cauchy, it is Q−bounded. Thus there exists some positive constant C′ < ∞, +such that Qr(ψn, ψn) ≤ Q(ψn, ψn) ≤ C′ for all n, and for all r > 0. This means Qr(ψ, ψ) ≤ C′, for all r > 0 +and Qr(ψ, ψ) ր Q(ψ, ψ) ≤ C′ by monotone convergence. Thus ψ ∈ D(Q), and condition i) is fulfilled. +We can now prove ii). Since ψn +L2 +−→ ψ ∈ D(Q), the sequence φn = ψn −ψ is such that φn ∈ D(Q) for all +n and φn → 0 ∈ L2(ν). Choose ε > 0. Then, since the sequence φn is Cauchy in D(Q), there exists n0 such +that Q(φn − φm, φn − φm) < ε2, for all m, n > n0. Now Qr(φn, φn) ր Q(φn, φn), and thus for ε > 0 and +n > n0, there exists r = r(n, ε) such that 0 ≤ Q(φn, φn) − Qr(φn, φn) < ε. Then we have, for all m, n > n0, +the following: +Qr(φn, φn) = Qr(φn − φm, φn) − Qr(φm, φm − φn) + Qr(φm, φm) +≤ |Qr(φn − φm, φn)| + |Qr(φm, φm − φn)| + Qr(φm, φm) +≤ 4 +√ +C′ +� +Q(φn − φm, φn − φm) +�1/2 ++ Qr(φm, φm) +≤ 4 +√ +C′ε + Qr(φm, φm). +We can now let m → ∞, so that Qr(φm, φm) → 0. Thus +Q(φn, φn) < Qr(φn, φn) + ε ≤ (4 +√ +C′ + 2)ε, +which means Q(φn, φn) → 0 as n → ∞, proving ii). +In order to construct the Friedrichs extension we start from a simpler version of the operator L3, defined +on the space C0,α +c +of compactly supported α-H¨older continuous functions on R. Clearly this is a subspace +of D(Q) and dense in L2(ν). Then for ψ ∈ C0,α +c +(R), we define +(LR +3 ψ)(u) = +� +R +ν(dv)K3(u, v) +� +ψ(u) − ψ(v) +� +, +where the integral is absolutely convergent for all u, and it yields a function in L2(ν). We can use Fubini’s +theorem to conclude that for all ψ ∈ C0,α +c +(R) and φ ∈ D(Q) +(φ, LR +3 ψ) = Q(φ, ψ) − (φ, ψ), +10 + +so the form domain of LR +3 is contained in D(Q). We can then conclude that the Friedrichs extension L3 of +LR +3 is given by +(φ, L3ψ) = Q(φ, ψ) − (φ, ψ), ∀ψ ∈ D(L3), φ ∈ D(Q), +D(L3) = {ψ ∈ D(Q) | ∃C < ∞ such that |Q(φ, ψ)| ≤ C∥φ∥L2, ∀φ ∈ D(Q)}. +We refer to chapter VIII, pages 329-334, [20] and chapter VI, pages 322-326, [19] for more details. +Our main results are proved in the Banach space X of continuous functions on R satisfying a H¨older-type +condition (see (2.3)). From the definition (2.4) of the weight Γ0 characterizing X, it is obvious that +∀ψ ∈ X, Q(ψ, ψ) < ∞, since K3(u, v) +� +Γ0(u, v) +�2is absolutely integrable under ν × ν. +Thus for all φ ∈ D(Q), for all ψ ∈ X, the integral defining Q(φ, ψ) is absolutely convergent. It is easy to +check that +G ∈ L2(ν), where G(u) = +� +R +ν(dv)K3(u, v)|ψ(u) − ψ(v)|. +Therefore, we may find a constant C > 0 such that +|Q(φ, ψ)| ≤ C∥φ∥L2∥G∥L2, so ψ ∈ D(L3). +Thus X ⊂ D(L3). +In Section 3, where our main results are written, we drop the bar and simply denote the extended +operator by L3. Let us point out that, by a similar argument as used above, if ψ is bounded, measurable +and satisfies supu +� +R ν(dv)K3(u, v)|ψ(u) − ψ(v)| < ∞, then ψ ∈ D(L3) and L3ψ is given by the absolutely +convergent integral. +The Friedrichs extension L +ε +3 of the operator Lε +3 is constructed in an exactly similar manner as above and +it is easily seen that Xε ⊆ D(L +ε +3), where Xε is the corresponding Banach space. +We now prove the following result about these extensions. +Theorem 2.2. The linear operators L3 and L +ε +3 are non-negative and their zero subspace is spanned by the +constant function. In addition, all have spectral gaps in L2(ν) and generate contraction semigroups in this +space. +Proof. We first prove the claim for L3. The proof for L +ε +3 is similar. +If ψ ∈ D(L3), we have the following lower bound for the corresponding quadratic form +(ψ, L3ψ) = 1 +2 +� +R2(ν × ν)(du, dv)K3(u, v)|ψ(u) − ψ(v)|2 ≥ (ψ, L′ψ), +where +(L′ψ)(u) = V ′(u)ψ(u) − +� +R +ν(dv)K′(u, v)ψ(v), V ′(u) = +� +R +ν(dv)K′(u, v), +and, K′(u, v) = 2K′ +0(u, v) + K′ +1(u, v) +� +1(v > u/2) + +1(v < 2u) +� ++ K′ +2(u, v) +� +1(v > 3u/2) + +1(v < 2u/3) +� +. +Here K′ +0(u, v) = ¯n 1(− ln 2 ≤ u ≤ ln 2)1(− ln 2 ≤ v ≤ ln 2) +� +ln(1 + eu) ln(1 + ev) +�−3/2 +, +K′ +1(u, v) = ¯n1(u < − ln 2)1(v < − ln 2) +� +ln(1 + eu) ln(1 + ev) +�−3/2 +emin(u,v)e−|u−v|, +11 + +and, K′ +2(u, v) = ¯n 1(u > ln 2)1(v > ln 2) +� +ln(1 + eu) ln(1 + ev) +�−3/2 +emax(u,v)e−|u−v|. +Then it is easy to see that there exists C′ > 0, such that the potential V ′(u) has the following lower +bound: +V ′(u) ≥ C′¯n +�� +1 − 2e +1 +2u� +1(u < − ln 4) + e +3 +2 u +1(u < − ln 2) + e− 3 +2u +1(− ln 2 ≤ u ≤ ln 2) ++ u− 1 +2 e− 1 +2u +1(u > ln 2) + u− 3 +2 +� �2 +3u +�2 +− (ln 2)2� +1(u > 3 +2 ln 2) +� +. +Therefore, V ′(u) is strictly positive and there exists a∗ > 0 such that σ(V ) ⊂ (a∗, ∞). It is also easily +checked that the integral operator associated with the kernel K′ is Hilbert-Schmidt on L2(ν). This means, +firstly, that L′ is seld-adjoint (see theorem 4.3, chapter IV, [19]) and secondly, that L′ has the same essential +spectrum as V ′ (see theorem 5.35, chapter IV, [19]). The first implication tells us that σ(L′) ⊂ [0, ∞) (the +associated quadratic form being non-negative) and the second implication means that σ(L′)∪[0, a∗) contains +only discrete semi-simple eigenvalues, so that a∗ is the only possible accumulation point of the spectrum. +Clearly then, L′ and hence L3 have spectral gaps in L2(ν), and thus generate contraction semigroups in +this space. If ψ is a constant function, it belongs to the domain of L3 and L3ψ is given by the convergent +integral which yields zero. +On the other hand, if ψ ∈ D(L3) is not constant almost everywhere, then +(ψ, L3ψ) > 0 and thus also L3ψ ̸= 0. Therefore, the zero subspace of L3 is given by constant functions. +From the definition (2.2) of the kernel function K +ε +3 it is clear that, for all ψ ∈ D(L +ε +3), we again have the +following lower bound: +(ψ, L +ε +3ψ) = 1 +2 +� +R2(ν × ν)(du, dv)K +ε +3(u, v)|ψ(u) − ψ(v)|2 ≥ (ψ, L′ψ). +Then the exact same argument as before leads us to the conclusion that L +ε +3 has a spectral gap in L2(ν), it +generates a contraction semigroup there, its zero subspace is given by constant functions. +2.2 +Approximating the Semigroup Solution generated in L2(ν) by L3 +For the results proved in this subsection we will use certain lemmas and theorems proved in the next section. +The proofs of these lemmas/theorems from Section 3 rely on Theorem 2.2 but are independent of the results +proved in this subsection, so there is no circularity of argument. In what follows, we will mention explicitly +whenever we use any result from the next section. +We start with initial data ψ0 ∈ X, where X is the Banach space of continuous functions ψ on R such +that +∥ψ∥X = sup +v∈R +|ψ(v)| +˜Γ(v) ++ +sup +(v,r)∈R×R+ +|ψ(v) − ψ(v − r)| +Γ0(v, r) +< ∞, +(2.3) +where +˜Γ(v) = f(v) exp[µ max(a, c0v)], where f(v) = max +� +(ln(1 + ev))−α, (ln 2)−α� +and, +Γ0(v, r) = (f(v) + f(v − r)) exp[µ max(a, c0v, v − r)]g0(v, r), with g0(v, r) = +� +1 − e−κr�γ0 , +where 0 < α < 1/6, µ < 1 +2 − 3 +8α, µc0 ∈ (α, 1/4), a ≥ 9, γ0 ∈ (0, 1/8], and κ ≥ 7. +(2.4) +The parameters α, µ etc. appearing above do not have much bearing on the results in this subsection. +However, choosing the correct admissible values for them is critical for our results in the Banach spaces in +12 + +Section 3. The intervals to which these parameters are restricted are largely determined by computational +convenience in the proof of Theorem 3.3. For our computations in this paper, we choose 1/9 < α < 1/6.5, +µ ∈ [1/3, 7/16] and c0 = 0.52. These choices obviously satisfy the conditions in the last line of (2.4). In +Appendix B we describe in some detail how the weight function Γ0 and the choices for the above parameters +are arrived at. Let us also mention here that the parameter µ′ appearing in (2.2) is such that µ′ > µ, while +a1 = a/c0. +We denote by Y , the Banach space of continuous functions on R × R+, bounded with respect to the +weight function Γ0. Now +∀(u, v) ∈ R2, ψ ∈ L2(ν), |ψ(u) − ψ(v)| = |ψ(max(u, v)) − ψ(max(u, v) − r)|, where r = |u − v|. +In order to keep the notation simple, we will use, whenever convenient, the following equivalent definition +for the weight Γ0 without changing the symbol (and follow a similar convention for the weights Γε defined +subsequently) : +Γ0(u, v) = (f(u) + f(v)) exp[µ max(a, c0 max(u, v), min(u, v))]g0(u, v), +with g0(u, v) = +� +1 − e−κ|u−v|�γ0 , ∀(u, v) ∈ R2. +Evidently X is contained in both D(L3) and D(L +ε +3), and the results in the previous subsection guarantee +the existence of solutions e−tL3ψ0 and e−tL +ε +3ψ0, unique in L2(ν), for the Cauchy problems associated with L3 +and L +ε +3 respectively. In this subsection we will prove that e−tL3ψ0 is actually the limit function that e−tL +ε +3ψ0 +converges to, as ε0 → 0 for times t ∈ [0, T ∗]. The time T ∗ > 0 comes from Theorem 3.3 and Proposition 3.5, +and it has no dependence on ε0. +Given the initial data described above, let us define for any ε0 > 0, ϕt = e−tL +ε +3ψ0. Then by Theorem +3.5, there exists T ∗ > 0 such that |Dϕt| ≤ A1Γε, for all t ∈ [0, T ∗] and some constant A1 < ∞, where +Dϕt(v, r) = ϕt(v)−ϕt(v−r), for all (v, r) ∈ R×R+, both A1 and T ∗ being independent of ε0. . The weight +function Γε is defined as: +Γε(v, r) = (f(v) + f(v − r)) exp[µ max(a, c0v, v − r)]g(v, r), g(v, r) = +� +1 − e−κ(r+ε(v))�γ0 , γ0 = γ0/2. +(2.5) +Let us now consider a sequence εn for the regularization parameter, such that εn → 0. For any εn > 0 +taking the place of ε0 appearing in (2.2), we write the following for simplicity’s sake, by a slight abuse of +notation: +K +εn +3 (u, v) = K3(u, v)1 − e− min(εn(u,v),|u−v|) +1 − e−εn(u,v) +, εn(u, v) = εn exp +� +− µ′ +γ0 +max(a1, max(u, v)) +� +. +We also write +Γεn(v, r) = (f(v) + f(v − r)) exp[µ max(a, c0v, v − r)]gn(v, r), gn(v, r) = +� +1 − e−κ(r+εn(v))�γ0 . +The corresponding L2-solution is e−tL +εn +3 ψ0 and we write ϕn +t = e−tL +εn +3 ψ0. Then our results in the next section +(see Theorem 3.12 and Theorem 3.4) imply that for all ψ0 ∈ X, we have a unique solution ¯ϕn +t of the +associated Duhamel-integrated Cauchy problem in the Banach space Xεn, such that +∥ ¯ϕn +t ∥Xεn = sup +v∈R +| ¯ϕn +t (v)| +˜Γ(v) ++ +sup +(v,r)∈R×R+ +| ¯ϕn +t (v) − ¯ϕn +t (v − r)| +Γεn(v, r) +< ∞, ∀εn > 0, +13 + +and ¯ϕn +t = ϕn +t , ν-almost everywhere. +We denote by Y εn the Banach space of continuous functions on R × R+ bounded with respect to the weight +Γεn. Note that X ⊂ Xεn ⊂ D(L +εn +3 ), for all εn > 0. +For the rest of this section we will use some results, a couple of which have already been mentioned and +all of which are easily obtained via short, straightforward computations. Since these are essential for the +main theorem of this section, we collect them in the following lemma for ready reference. +Lemma 2.3. Given initial data ¯ϕ0 ∈ X and εn > 0, the following are true: +i) There exists A1 < ∞, T ∗ > 0, both independent of εn, such that |D ¯ϕn +t | ≤ A1∥D ¯ϕ0∥Y Γεn, for all +t ∈ [0, T ∗]. +ii) L +εn +3 ¯ϕn +t ∈ L2(ν) for all t ∈ R+. +iii) There exists a positive, measurable function �Γ on R2 such that +� +R2(ν × ν)(du, dv)K3(u, v)(�Γ(u, v))2 < ∞. +iv) There exists a positive constant C < ∞, such that +K +εn +3 (u, v) +� +Γ +εn(u, v) +�2 ≤ CK3(u, v) +��Γ(u, v) +�2, ∀(u, v) ∈ R2. +Proof. i) This estimate follows from the following upper bound obtained in Proposition 3.5: +|D ¯ϕn +t − ∆t| ≤ Q0 +� +Mεn +�p ln +� +min(M, ε−1 +n ) +� +∥∆∥Y , ∀t ∈ [0, T ∗], +where the constants Q0 and p > 0 are independent of εn, and ∆0 = D ¯ϕ0. +ii) From Theorem 3.4 we know that +¯ϕn ∈ Xεn, ∀εn > 0, ∀t > 0, i.e., +sup +t>0 +(v,r)∈R×R+ +|D ¯ϕn +t (v, r)| +Γεn(v, r) +< ∞. +It is straightforward to verify that G ∈ L2(ν), where +G(u) = +� +R +ν(dv)K +εn +3 (u, v)Γεn(u, v). +It then follows naturally that L +εn +3 ¯ϕn +t ∈ L2(ν). +iii) Define +�Γ(u, v) = (f(u) + f(v)) exp[µ max(a, c0 max(u, v), min(u, v))]¯g0(u, v), with ¯g0(u, v) = +� +1 − e−κ|u−v|�γ0 . +Then it is easy to check that +� +R2(ν × ν)(du, dv)K3(u, v) +��Γ(u, v) +�2 < ∞. +iv) For all εn > 0, and r > 0, there exist C1 > 0, C2 > 0, depending on κ, such that the following is true: +� +1 − e−κ(r+εn)�2γ0 +1 − e− max(r,εn) +≤ C1 +� +1 − e−κ(r+εn)�2γ0−1 ≤ C1 +� +1 − e−κr�2γ0−1 ≤ C2 +� +1 − e−κr�2γ0 +1 − e−r +. +14 + +It then follows directly from the definitions of the kernel functions and the weight functions that +K +εn +3 (u, v) +� +Γ +εn(u, v) +�2 ≤ CK3(u, v) +��Γ(u, v) +�2, ∀(u, v) ∈ R2, for some constant C > 0. +Given initial data ψ0 ∈ X, let us consider again the sequence εn of regularization parameters, with +εn → 0. Recall that corresponding to every εn, the regularized evolution equation (see 3.2.1) has a solution +¯ϕn ∈ Xεn for all times by Theorem 3.12. Then we can prove the following lemma. +Lemma 2.4. Let {εn} be a sequence of regularization parameters such that εn → 0 and ψ0 ∈ X be the initial +datum. Then the corresponding sequence { ¯ϕn} is Cauchy in C +� +[0, T ∗], L2(ν) +� +. +Proof. Consider m, n ∈ +N. For the sake of simplicity, let us assume min(εm, εn) = εm. Then we know that +¯ϕm ∈ Xεm and ¯ϕn ∈ Xεn and from the definitions of Γεm and Γεn it is clear that Xεm ⊂ Xεn. +Let ¯ϕn,m +t += ¯ϕn +t − ¯ϕm +t , then ¯ϕn,m +t +∈ Xεn. The function ¯ϕn,m +t +satisfies +∂t ¯ϕn,m +t += −L +εn +3 +� +¯ϕn +t − ¯ϕm +t + ¯ϕm +t +� ++ L +εm +3 +¯ϕm +t += −L +εn +3 ¯ϕn,m +t ++ gn,m +t +, +(2.6) +where gn,m +t +(u) = L +εm +3 +¯ϕm +t (u) − L +εn +3 ¯ϕm +t (u) += +� +R +ν(dv) +� +K +εm +3 (u, v) − K +εn +3 (u, v) +�� +¯ϕm +t (u) − ¯ϕm +t (v) +� += +� +R +ν(dv)S +εm,εn(u, v) +� +¯ϕm +t (u) − ¯ϕm +t (v) +� +, with S +εm,εn(u, v) = K +εm +3 (u, v) − K +εn +3 (u, v). +Now by part i) of Lemma 2.3 we can use: +�� ¯ϕm +t (u) − ¯ϕm +t (v) +�� ≤ A1∥D ¯ϕ0∥Y Γεm ≤ A1∥ψ0∥XΓεm, +to obtain the following estimate: +|gn,m +t +(u)| = +��� +� +R +ν(dv)S +εm,εn(u, v) +� +¯ϕm +t (u) − ¯ϕm +t (v) +���� ≤ Fn,m(u), +where Fn,m(u) = C0∥ψ0∥X +� +εn − εm +�γ0eµ max(a,u)f(u) +� +ln(1 + eu) +�− 1 +2e− µ′ +γ0 γ0 max(a1,u), C0 being +some positive contant independent of εm and εn. +Evidenly then, ∥Fn,m∥L2 → 0, as m, n → ∞. +Looking back at (2.6) we can now write the following for all t ∈ [0, T ∗]: +∂t∥ ¯ϕn,m +t +∥2 +L2 = 2 Re +� +¯ϕn,m +t +, ∂t ¯ϕn,m +t +� += 2 Re +� +¯ϕn,m +t +, gn,m +t +� +− 2 Re +� +¯ϕn,m +t +, L +εn +3 ¯ϕn,m +t +� +< 2 Re +� +¯ϕn,m +t +, gn,m +t +� +by the positivity of L +εn +3 , +≤ 2∥ ¯ϕn,m +t +∥L2∥gn,m +t +∥L2 +≤ 4∥ψ0∥L2∥Fn,m∥L2, +15 + +where in the last line we have used the fact that e−tL +εn +3 ψ0 is the L2-representative of ¯ϕn +t (see Lemma 3.10). +This means +sup +t∈[0,T ∗] +∥ ¯ϕn,m +t +∥L2 ≤ +� +4T ∗∥ψ0∥L2∥Fn,m∥L2. +Since the right-hand side of the above equation goes to zero as m, n → ∞, we conclude that the sequence +{ ¯ϕn} is Cauchy in C +� +[0, T ∗], L2(ν) +� +. +Since { ¯ϕn} is Cauchy, there exists a limit function in C +� +[0, T ∗], L2(ν) +� +that this sequence converges to. +Let us call it ˜ψ. In what follows, we prove that this limiting function is independent of the choice of the +sequence (εn) above, and we indeed then have limε→0 supt ∥ ¯ϕε +t − ˜ψt∥ = 0. For notational convenience, we +denote already from the start ¯ϕε → ˜ψ as ε → 0, although strictly speaking these first refer to ¯ϕn → ˜ψ for +some fixed choice of sequence εn → 0. We use ε to parametrize the regularized linear operator, ¯ϕε denotes +the Banach space solution associated with the operator L +ε +3, the corresponding Banach space is denoted by +Xε and we are interested in the limit ε → 0. +We are now in a position to prove the main result of this subsection, which states that this limit function +˜ψt is in fact the L2-solution associated with the original linearized operator L3. +Theorem 2.5. Given initial datum ψ0 ∈ X, let ˜ψt = limε→0 ¯ϕε +t in L2(ν) for t ∈ [0, T ∗]. +Then ˜ψt = +e−tL3ψ0, ν-almost everywhere. +Proof. We will first fix t ∈ [0, T ∗] and show that ˜ψt ∈ D(L3). Let us begin by demonstrating that ˜ψt ∈ D(Q). +There is now a subsequence of the original sequence (εn) such that along the subsequence ¯ϕε +t → ˜ψt a.e. +Since limε→0 K +ε +3(u, v) = K3(u, v) for all (u, v) ∈ R2, we have (along the subsequence) +K +ε +3(u, v) +�� ¯ϕε +t(u) − ¯ϕε +t(v) +��2 −→ K3(u, v) +�� ˜ψt(u) − ˜ψt(v) +��2 as ε → 0, +except on a set that has measure zero under ν × ν. Then, by parts i) and iv) of Lemma 2.3, there exist +constants A and C < ∞ such that +K +ε +3(u, v) +�� ¯ϕε +t(u) − ¯ϕε +t(v) +��2 ≤ A(∥ ¯ϕ0∥X)2K +ε +3(u, v) +� +Γε(u, v) +�2 ≤ CK3(u, v) +��Γ(u, v) +�2, +which implies the following by part iii) of Lemma 2.3, and the dominated convergence theorem: +lim +ε→0 +� +R2(ν × ν)(du, dv)K +ε +3(u, v) +�� ¯ϕε +t(u) − ¯ϕε +t(v) +��2 = +� +R2(ν × ν)(du, dv)K3(u, v) +�� ˜ψt(u) − ˜ψt(v) +��2 < ∞. +Thus, ˜ψt ∈ D(Q). +We then return to the original sequence (εn). +Let us now consider test functions φ from the space +C0,α0 +c +(R) of compactly supported α0-H¨older continuous functions on R. This is a dense subspace of L2(ν) +and a subspace of D(L +ε +3) for all ε > 0, as well as of D(L3). Then, since K +ε +3(u, v) ≤ K3(u, v) for all (u, v) ∈ R2 +and +� +R ν(dv)K3(u, v) +��φ(u) − φ(v) +�� is integrable, we have, by dominated convergence, +lim +ε→0 +� +L +ε +3φ +� +(u) = +� +L3φ +� +(u), pointwise as well as in L2(ν). +Also, by part i) of Lemma 2.3, we have, for all t ∈ [0, T ∗], the following: +��� +L +ε +3 ¯ϕε +t +� +(u) +�� ≤ C1 +� +R +ν(dv)K +ε +3(u, v)Γ +ε(u, v) ≤ C1H(u), +(2.7) +16 + +where the constant C1 is independent of ε, +H(u) = sup +ε>0 +� +R +ν(dv)K +ε +3(u, v)Γ +ε(u, v), and H ∈ L2(ν). +Then the following limit holds: +lim +ε→0 +� +φ, L +ε +3 ¯ϕε +t +� += lim +ε→0 +� +L +ε +3φ, ¯ϕε +t +� += +� +L3φ, ˜ψt +� += ˜Q(φ, ˜ψt), +where we have used the self-adjointness of L +ε +3, the limits obtained earlier and the fact that ˜ψt ∈ D(Q). +However, by (2.7), we have the following: +��� +φ, L +ε +3 ¯ϕε +t +��� ≤ C2∥φ∥L2∥H∥L2, where C2 is a constant independent of ε. +This means that +�� ˜Q(φ, ˜ψt) +�� ≤ C2∥φ∥L2∥H∥L2, ∀φ ∈ C0,α0 +c +(R). +Thus the map φ �→ Q(φ, ˜ψt) has a unique, bounded extension from the dense subspace containing our test +functions to L2(ν) and we conclude that ˜ψt ∈ D(L3). Then by self-adjointness of L3, ˜Q(φ, ˜ψt) = +� +L3φ, ˜ψt +� += +� +φ, L3 ˜ψt +� +for any φ ∈ D(L3). The above bounds, uniform in ε, allow us to conclude that +lim +ε→0 +� +φ′, L +ε +3 ¯ϕε +t +� += +� +φ′, L3 ˜ψt +� +, ∀φ′ ∈ L2(ν), i.e., L +ε +3 ¯ϕε +t−→L3 ˜ψt weakly in L2(ν), ∀t ∈ [0, T ∗]. +Now for any �ψ ∈ D(L3) and 0 ≤ s ≤ t < T ∗, we have the following: +∂s +� +e−(t−s)L3 �ψ, ¯ϕε +s +� += +� +L3e−(t−s)L3 �ψ, ¯ϕε +s +� +− +� +e−(t−s)L3 �ψ, L +ε +3 ¯ϕε +s +� += +� +L3e−(t−s)L3 �ψ, ¯ϕε +s − ˜ψs +� ++ +� +e−(t−s)L3 �ψ, L3 ˜ψs − L +ε +3 ¯ϕε +s +� +−→ 0, as ε → 0. +Continuity in s and uniform boundedness in ε for the terms on the right hand side above then implies that +we can apply the dominated convergence theorem to obtain +� t +0 +ds ∂s +� +e−(t−s)L3 �ψ, ¯ϕε +s +� += +� �ψ, ¯ϕε +t +� +− +� +e−tL3 �ψ, ψ0 +� +−→ 0 as ε → 0. +Since we know that +� �ψ, ¯ϕε +t +� +−→ +� �ψ, ˜ψt +� +as ε → 0, this means +� �ψ, ˜ψt +� += +� �ψ, e−tL3ψ0 +� +for any �ψ ∈ D(L3) . +Therefore, ˜ψt = e−tL3ψ0, ν-almost everywhere. +3 +Smoothing Solutions in a weighted Banach Space of H¨older-continuous +functions +In this section we study the linearized three waves collision operator described in (2.1) in a certain Banach +space, and prove that the corresponding time evolution has a smoothing property in this space. The proof of +this smoothing theorem relies on a few other results, proved in slightly different but related Banach spaces. +This section is organized as follows: first we state and briefly explain the main smoothing result as well +as the theorems leading to it; this is followed by two subsections containing the details of the proofs of +these theorems. In the first subsection we prove the existence and uniqueness of solutions of an initial value +17 + +problem derived from our original evolution equation in a weighted Banach space characterized by a time- +dependent H¨older-type condition. In the second subsection we consider a regularized evolution equation +which enables us to show that this solution is in fact identical to the one obtained in a space of differences of +functions, with the same initial value, so that the result proved in the first subsection implies the smoothing +of solutions of the original time evolution equation. +We begin by looking at the original evolution equation: +∂tψt(v) = −(L3ψt)(v) = +� +R +dw K3(v, w) +� +ψt(v) − ψt(w) +� +, +where the operator L3 is now written in terms of the kernel function in the flat space without the weight +ν. We keep in mind that L3, which we will analyze in a Banach space introduced in the previous section, is +the Friedrichs-extended operator constructed earlier. Then the difference function ψt(v) − ψt(v − r) evolves +in time as +∂t [ψt(v) − ψt(v − r)] += − [(L3ψt)(v) − (L3ψt)(v − r)] += − +�� ∞ +−∞ +dw K3(v, w) (ψt(v) − ψt(w)) − +� ∞ +−∞ +dw K3(v − r, w) (ψt(v − r) − ψt(w)) +� +. +(3.1) +Let us now write down the formulae defining K3 explicitly. +i) When v > w, +K3(v, w) = 4¯n (ln(1 + ev))− 3 +2 ln(1 + ew) ew + 2e−(v−w) +1 + ew + e−(v−w) +1 +1 − e−(v−w) , +and, ii) when w > v, +K3(v, w) = 4¯n (ln(1 + ev))− 3 +2 ln(1 + ew) ev + 2e−(w−v) +1 + ev + e−(w−v) +e−(w−v) +1 − e−(w−v) . +For our subsequent computations we will split the kernel function K3 as described below. This splitting is +done in order to separate out some terms that do not contain the line-singularity and consequently have +different asymptotic behaviors. +When v > w, K3(v, w) = K1 +3(v, w) + K2 +3(v, w), where +K1 +3(v, w) = 4¯n (ln(1 + ev))− 3 +2 ln(1 + ew) ew + 2e−(v−w) +1 + ew + e−(v−w) +e−(v−w) +1 − e−(v−w) , and +K2 +3(v, w) = 4¯n (ln(1 + ev))− 3 +2 ln(1 + ew) ew + 2e−(v−w) +1 + ew + e−(v−w) . +For the region w > v we will use the following splitting of K3(v, w) in some parts of our computation: +K3(v, w) = K3 +1(v, w) + K3 +2(v, w), +where K3 +2(v, w) = K3(v, w)e− 1 +2(w−v), and K3 +1(v, w) = K3(v, w) +� +1 − e− 1 +2 (w−v)� +. +Note that the part K2 +3 does not contain any line-singularity. Also, the integral of K2 +3(v, w) with respect +to the variable w yields a square-root-function-like growth for large, positive values of v, while K1 +3(v, w) +exhibits no such behavior due to the extra exponential decay in it. Let us also note that in the absence of +the point-singularity (when v does not assume arbitrarily large negative values), K3 +1(v, w) leads to bounded +integrals. We will use this splitting of the kernel function for w > v, to cut out certain bounded parts later. +The factor 1/2 appearing above has been chosen arbitrarily, but once fixed, this determines the behavior of +the weight function Γ(v, r) for large, positive values of v. +We now write down a crucial but easily derived property of the kernel functions in the form of a lemma. +18 + +Lemma 3.1. The kernel functions satisfy the following inequalities: +when v − r > w, K1 +3(v − r, w) ≥ K1 +3(v, w) and K2 +3(v − r, w) ≥ K2 +3(v, w) +∀r ≥ 0, +and when w > v, K3(v, w) ≥ K3(v − r, w), as well as, K3 +2(v, w) ≥ K3 +2(v − r, w) ∀r ≥ 0. +Proof. When v−r > w, the inequalities in the first line are obvious from the formulae for the kernel functions +K1 +3 and K2 +3. +When w > v, we can write the following: +∂ +∂v K3(v, w) = K3(v, w)b(v, w), +where b(v, w) = −3 +2 +ev +(1 + ev) ln(1 + ev) + 1 + +1 +ew−v − 1 + +ew−v +1 + ew + ew−v ≥ 0. +Therefore K3(v, w) ≥ K3(v − r, w), +∀r ≥ 0. +Similarly, K3 +2(v, w) ≥ K3 +2(v − r, w), +∀r ≥ 0. +Lemma 3.1 allows us to write a part of the right hand side of equation (3.1) as the combination of a +multiplication operator and a positivity-preserving operator, as we will see shortly. For now, we just write +the following: +[(L3ψt)(v) − (L3ψt)(v − r)] += +�� v−r +−∞ +dw K1 +3(v, w) (ψt(v) − ψt(w)) + +� v +v−r +dw K1 +3(v, w) (ψt(v) − ψt(w)) +− +� ∞ +v +dw K3(v, w) (ψt(w) − ψt(v)) − +� v−r +−∞ +dw K1 +3(v − r, w) (ψt(v − r) − ψt(w)) ++ +� v +v−r +dw K3(v − r, w) (ψt(w) − ψt(v − r)) + +� ∞ +v +dw K3(v − r, w) (ψt(w) − ψt(v − r)) +� ++ +�� v−r +−∞ +dw K2 +3(v, w) (ψt(v) − ψt(w)) + +� v +v−r +dw K2 +3(v, w) (ψt(v) − ψt(w)) +− +� v−r +−∞ +dw K2 +3(v − r, w) (ψt(v − r) − ψt(w)) +� +, +(3.2) +where the last two lines do not contain any line singularity. +We need to analyze the above operator in order to prove the main result of this paper. Before proceeding +to set the stage for that analysis, let us define the time-dependent weight function Γt(v, r), in terms of which +our main theorem is stated: +Γt(v, r) = +�� +f(v − r) + f(v) +� +exp +� +µ max (a, c0v, v − r) +�� +gt(v, r), +(3.3) +with the following H¨older-type time-dependent part gt : +gt(v, r) = +� +1 − e−κr�γt(v−r) , γt(v, r) = γ0 + a(t) +1 +1 + eβ(v−r) , +a(t) = 1 +8 min(1, ¯n) +t +1 + t, +κ ≥ 7, 0 < γ0 ≤ 1/8, 1 ≤ β ≤ κ/4. +(3.4) +19 + +The choices for the parameters β, κ and γ0 relating to the time-dependent H¨older-type condition are ex- +plained in Appendix C. Note that at t = 0 this is just the weight Γ0 characterizing the Banach spaces Y +and X defined in (2.4), with exponent γ0. The main result of this paper is then the following theorem: +Theorem 3.2. Given initial datum ψ0 ∈ X, let us define ∆0(v, r) = ψ0(v)−ψ0(v−r) for all (v, r) ∈ R×R+, +and ψt = e−tL3ψ0. Then there exists T ∗ > 0 such that for all t ∈ [0, T ∗] the following bound holds: +��ψt(v) − ψt(v − r) +�� ≤ CΓt(v, r)∥∆0∥Y +ν-almost everywhere on R × R+, +(3.5) +where C is a constant depending on the parameters appearing in the weight function Γt. +Theorem 3.2 is the obvious consequence of Theorem 2.5 and three other results which are stated below. +Before writing down the statements of these three theorems we briefly describe the evolution equations +considered in them as well as the function spaces in which these results are obtained. We will just sketch +out the schemes here, reserving the details of the constructions for later. +The first theorem deals with an evolution equation derived from the equation (3.1). The main idea here +is to introduce suitable cut-off functions δ1(v, r) and δ2(v, r), as well as cut-off parameters m0 and b0; and +then split the linear operator into an unbounded part Lu which is the sum of a potential function and a +positivity-preserving operator, a bounded part Lb and a perturbation Lδ (see Subsection 3.1). For t ≥ 0 and +(v, r) ∈ R × R+, we define the variable +∆t(v, r) = ψt(v) − ψt(v − r). +Then equation (3.1) is recast as the following evolution equation for the ∆-variable: +∂t∆t = −L∆t = −Lu∆t − Lb∆t − Lδ∆t += −Vu∆t + Ku∆t − Lb∆t − Lδ∆t, +(3.6) +where (Lu∆t)(v, r) = Vu(v, r)∆t(v, r) − (Ku∆t)(v, r) and the multiplication operator Vu is defined as +Vu(v, r) = +� v−r−δ1 +−∞ +dw K1 +3(v − r, w) + +� ∞ +v+δ2 +dw K3(v, w) + +� v +v−r+δ1 +dw K3(v − r, w) ++ +� v−δ2 +v−r +dw K1 +3(v, w) + +� v +v−r +dw K2 +3(v, w) + +� v−r +−∞ +dw K2 +3(v − r, w). +(3.7) +We consider (3.6) in the following Duhamel-integrated form: +∆t = e−tVu∆0 + +� t +0 +ds e−(t−s)VuKu[∆s] − +� t +0 +ds e−(t−s)VuLb[∆s] − +� t +0 +ds e−(t−s)VuLδ[∆s], +(3.8) +in the Banach space Y of functions in C ([0, T] × (R × R+)), for some T > 0, bounded with respect to Γt as +follows: +∥∆∥Y := sup +t∈[0,T] +sup +(v,r)∈R×R+ +|∆t(v, r)| +Γt(v, r) < ∞. +We then prove the following existence-uniqueness result for solutions of equation (3.8). +20 + +Theorem 3.3. There exists T ∗ > 0 depending on the cut-off parameters used in equation (3.6), and the +parameters α, µ and c0 appearing in the weight Γt, such that the following is true: +Given initial value ∆0 ∈ Y , a solution ∆ of (3.8) exists for all t ∈ [0, T ∗], such that +∥∆∥Y = +sup +t∈[0,T ∗] +sup +(v,r)∈R×R+ +|∆t(v, r)| +Γt(v, r) < ∞. +This solution is unique in Y and given by +∆t = (1 − F)−1 e−tVu∆0, +where F is the linear operator defined as +F[∆t] = +� t +0 +ds e−(t−s)VuKu[∆s] − +� t +0 +ds e−(t−s)VuLb[∆s] − +� t +0 +ds e−(t−s)VuLδ[∆s]. +(3.9) +In the theorem above, the time T ∗ is chosen so that +�� � t +0 ds e−(t−s)VuLb[∆s] +�� is small enough to guarantee +the invertibility of the operator (1−F) in Y. Observe that, ψ0 ∈ X (like in Theorem 3.2) implies that ∆0 ∈ Y , +with ∆0(v, r) = ψ0(v)−ψ0(v−r), Y being the Banach space of functions in C (R × R+) bounded with respect +to the weight function Γ0 defined in (2.4). However, the above theorem does not guarantee that, given initial +datum ∆0(v, r) = ψ0(v) − ψ0(v − r) in Y , the unique solution ∆t of (3.8) obtained via Theorem 3.3 is still +a difference of the ψ-variables. That this is indeed the case, is proved by Theorem 3.4 and Proposition 3.5. +Before stating these theorems, let us briefly describe the evolution equations and their solutions consid- +ered in them. We start from the regularized evolution equation associated with the operator in (2.2), +∂tψε +t (v) = −(Lε +3ψ)(u) = +� +R +dv Kε +3(u, v) +� +ψε +t (u) − ψε +t (v) +� +, +(3.10) +where Lε +3 is now written in terms of the kernel function Kε +3 in the flat space without the weight ν. We tag +solutions of this equation with ε in order to avoid confusion later, when we compare the solutions associated +with different operators such as L3 and Lε +3. We consider (3.10) for initial data ψε +0 ∈ X, where +X = +� +h ∈ C(R) : ∥h∥X < ∞ +� +, ∥h∥X = sup +v∈R +|h(v)| +˜Γ(v) +, +(3.11) +where ˜Γ is defined in (2.4). +Then Theorem 3.12 guarantees, for all t ≥ 0, the existence of a unique solution ψε +t ∈ X, of the following +Duhamel-integrated form of (3.10): +ψε +t =e−tV εψε +0 + +� t +0 +ds e−(t−s)V εKε +u[ψε +s] + +� t +0 +ds e−(t−s)V εKε +b [ψε +t ], +where V ε is a multiplication operator, Kε +b is a bounded operator on L2(ν), and a splitting akin to (3.6) +recasts the operator Lε +3 in the following form: +Lε +3ψε +t = V εψε +t − Kε +uψε +t − Kε +b ψε +t . +The existence of such a solution ψε +t implies the existence of the difference variable solution Dψε +t , with +Dψε +t (v, r) = ψε +t (v) − ψε +t (v − r), of the following evolution equation of differences derived from (3.10): +∂tDψε +t (v, r) = −( ˜LεDψε +t )(v, r) = −( ˜Lε +uDψε +t )(v, r) − ( ˜Lε +sDψε +t )(v, r) − Kε +b[ψε +t ](v, r), +(3.12) +21 + +where again the linear operator ˜Lε has been split, `a la (3.6), into an unbounded part ˜Lε +u, a perturbation +˜Lε +s, and a part Kε +b which can be bounded in terms of the L2(ν)-norm and the X-norm of ψε +t . Like before, +the unbounded part ˜Lε +u can be written as the sum ˜Lε +u = ˜Vε − ˜Kε +u, of the multiplication operator ˜Vε and a +positivity-preserving integral operator ˜Kε +u, leading to the following Duhamel-integrated form of (3.12): +Dψε +t = e−t˜VεDψε +0 + +� t +0 +ds e−(t−s)˜Vε ˜Kε +uDψε +s − +� t +0 +ds e−(t−s)˜Vε ˜Lε +sDψε +s − +� t +0 +ds e−(t−s)˜Vε ˜Kε +b[ψε +s]. +(3.13) +Then the next theorem shows the existence and uniqueness of solutions of (3.13). +Theorem 3.4. Given ε > 0, consider initial datum ψε +0 ∈ X such that +∥Dψε +0∥Y ε = +sup +(v,r)∈R×R+ +|ψε +0(v) − ψε +0(v − r)| +Γε(v, r) +< ∞, +Γε being the weight function defined in (2.5). Then there exists for all t > 0, a unique solution Dψε +t of +(3.13) with initial value ψε +0, such that ∥Dψε +t ∥Y ε < ∞. +The next theorem connects the solutions Dψε +t with the solutions ∆t obtained in Theorem 3.3 fot t ∈ +[0, T ∗]. Before stating the theorem let us define the variable whose time-evolution is dealt with in this result. +This variable is the following difference function, defined for all t ∈ [0, T ∗], +Dε +t = Dψε +t − ∆t. +(3.14) +From (3.6) and (3.12) we see (see Subsection 3.2 for details) that for t ∈ [0, T ∗], the time evolution of Dt is +governed by the equation +∂tDε +t = − ˜LεDε +t − ˜Lε +0[∆t], +(3.15) +where the operator ˜Lε is defined exactly like ˜L, but with the kernel function Kε +3 replacing the original kernel +function K3; the operator ˜Lε +0 also has the same structure as ˜L, but with K3 − Kε +3 replacing K3. Explicit +expressions of these operators are given in Subsection 3.2. Then the next result is +Proposition 3.5. Consider initial datum ψ0 ∈ X. This means Dψ0 = ∆0 ∈ Y and D0 = 0. Then there +exists T ∗ > 0 and a unique Duhamel-integrated solution Dε +t of (3.15), such that Dε +t ∈ Y ε for all t ∈ [0, T ∗]. +Moreover, there exists some positive constant Q < ∞ depending on the parameters α, κ and γ0, such that +|Dt| is bounded above as follows: +|Dε +t | ≤ Q (Mε0)p ln +� +min(M, ε−1 +0 ) +� +∥∆∥Y Γε, +where p = p(γ0) > 0 and the positive constant M < ∞ appears as a cut-off parameter in the evolution +equation (3.6) for ∆t. +The smoothing result of Theorem 3.2 can now be obtained in a very straightforward manner, as shown +below. +Proof of Theorem 3.2. Given initial datum ψ0 ∈ X, Proposition 3.5 and Theorem 2.5 imply the following +∀t ∈ [0, T ∗]: +��� +e−tL3ψ0 +� +(v) − +� +e−tL3� +ψ0(v − r) +�� = lim +ε0→0 |Dψε +t (v, r)| +ν-almost everywhere +22 + +≤ |∆t(v, r)| +≤ C∥∆0∥Y Γt(v, r), +where we have used Theorem 3.3 in the last line, C = ∥(1 − F)−1∥Y < ∞ and it is easy to see that +∥e−tVu∆0∥Y ≤ ∥∆0∥Y . +Let us make an important remark here about the time T ∗ mentioned in Theorem 3.2. This upper bound +has its origin in Theorem 3.3, where the condition t ≤ T ∗ is used to make +�� � t +0 ds e−(t−s)VuLb[∆s] +�� < +A(b0, m0, µ, c0)T ∗∥∆∥Y small enough to guarantee the invertibility of (1 − F) (see the proof of Theorem 3.3 +in 3.1.1). Now this result can be extended in time as follows: we first prove the theorem for t ∈ [0, T ∗], then +we can choose an initial time 0 < t0 < T ∗, and prove the existence of a unique solution of the corresponding +Duhamel-integrated equation for t ∈ [t0, t0 + T ∗]. Each such extension in time results in an extra factor of +the form (1 − F1)−1 (here F1 represents the operator corresponding to F in the new Duhamel equation) in +the formula for ∆t obtained in Theorem 3.3, and this process can be continued as long as the norm of the +product is finite. This in turn means that Proposition 3.5, Theorem 2.5 and finally, our main smoothing +result Theorem 3.2, all of which inherit the dependence on T ∗ from Theorem 3.3, can also be extended in +time. +The subsections that follow are devoted to the derivations of the evolution equations for ∆t, ψε +t and Dε +t, +and the proofs of Theorems 3.3, 3.4 and Proposition 3.5. Of these, the proof of Theorem 3.3 is the most +delicate and the other proofs rely on estimates that are similar to those employed there, so we use the next +subsection to first explain in detail how (3.6) is arrived at and then prove Theorem 3.3. +3.1 +Existence and Uniqueness of Solutions in a Banach space with a Time-dependent +H¨older-type Condition +3.1.1 +Derivation of Equation (3.6) +As mentioned before, (3.6) is obtained via the introduction of suitable cut-off functions δ1(v, r) and δ2(v, r) +in the original evolution equation (3.1) for the differences. Before defining these cut-off functions, let us +explain the idea behind the recasting of (3.1). Recall that eventually we want to solve a Duhamel-integrated +form of the evolution equation, where a potential function is used to control the rest of the terms (cf. the +multiplication operator Vu in (3.8)). When we split our linear operator into a potential Vu, a positivity- +preserving part Ku, a perturbation Lδ and the bounded part Lb, we thus want to make Vu as large as +possible. We start from the original equation (3.1) for the evolution of the differences. The right hand side +is then rewritten (without the minus sign) as +(L3ψt)(v) − (L3ψt)(v − r) += +� v−r−δ1 +−∞ +dw K1 +3(v − r, w) (ψt(v) − ψt(v − r)) − +� v−r−δ1 +−∞ +dw +� +K1 +3(v − r, w) − K1 +3(v, w) +� +(ψt(v) − ψt(w)) ++ +� ∞ +v+δ1 +dw K3(v, w) (ψt(v) − ψt(v − r)) − +� ∞ +v+δ1 +dw (K3(v, w) − K3(v − r, w)) (ψt(w) − ψt(v − r)) ++ +� v+δ1 +v+δ2 +dw K3(v, w) (ψt(v) − ψt(v − r)) − +� v+δ1 +v+δ2 +dw K3(v, w) (ψt(w) − ψt(v − r)) ++ +� v−δ2 +v−δ1 +dw K1 +3(v, w) (ψt(v) − ψt(v − r)) − +� v−δ2 +v−δ1 +dw K1 +3(v, w) (ψt(w) − ψt(v − r)) ++ +� v−δ1 +v−r +dw K1 +3(v, w) (ψt(v) − ψt(v − r)) − +� v−δ1 +v−r +dw K1 +3(v, w) (ψt(w) − ψt(v − r)) +23 + ++ +� v +v−r+δ1 +dw K3(v − r, w) (ψt(v) − ψt(v − r)) − +� v +v−r+δ1 +dw K3(v − r, w) (ψt(v) − ψt(w)) ++ +�� v−r +−∞ +dw K2 +3(v − r, w) (ψt(v) − ψt(v − r)) − +� v−r +−∞ +dw +� +K2 +3(v − r, w) − K2 +3(v, w) +� +(ψt(v) − ψt(w)) ++ +� v +v−r +dw K2 +3(v, w) (ψt(v) − ψt(v − r)) − +� v +v−r +dw K2 +3(v, w) (ψt(w) − ψt(v − r)) +� ++ +�� v−r +v−r−δ1 +dw K1 +3(v, w) (ψt(v) − ψt(w)) − +� v−r +v−r−δ1 +dw K1 +3(v − r, w) (ψt(v − r) − ψt(w)) ++ +� v+δ1 +v +dw K3(v − r, w) (ψt(w) − ψt(v − r)) − +� v+δ2 +v +dw K3(v, w) (ψt(w) − ψt(v)) ++ +� v +v−δ2 +dw K1 +3(v, w) (ψt(v) − ψt(w)) + +� v−r+δ1 +v−r +dw K3(v − r, w) (ψt(w) − ψt(v − r)) +� +. +The last three lines within square brackets constitute the Lδ-part; it is only when these terms with the +line-singularity are separated that the rest of the operator can be written in a (V − K)-form, with K +being positivity-preserving. We now move over to the ∆-variable, with ∆t(v, r) = ψt(v) − ψt(v − r) for all +(v, r) ∈ R × R+, and write the above equation, now in terms of ∆t, as +∂t∆t = −L∆t, +L∆t = Lpp∆t + Lδ∆t, where +Lδ∆t(v, r) = +� v−r +v−r−δ1 +dw K1 +3(v, w)∆t(v, v − w) + +� v+δ1 +v +dw K3(v − r, w)∆t(w, w − v + r) ++ +� v−r+δ1 +v−r +dw K3(v − r, w)∆t(w, w − v + r) − +� v−r +v−r−δ1 +dw K1 +3(v − r, w)∆t(v − r, v − r − w) ++ +� v +v−δ2 +dw K1 +3(v, w)∆t(v, v − w) − +� v+δ2 +v +dw K3(v, w)∆t(w, w − v), +(3.16) +and, Lpp∆t(v, r) = Vu(v, r)∆t(v, r) − +� v−r−δ1 +−∞ +dw +� +K1 +3(v − r, w) − K1 +3(v, w) +� +∆t(v, v − w) +− +� ∞ +v+δ1 +dw (K3(v, w) − K3(v − r, w)) ∆t(w, w − v + r) +− +� v−r +−∞ +dw +� +K2 +3(v − r, w) − K2 +3(v, w) +� +∆t(v, v − w) − +� v−δ1 +v−r +dw K1 +3(v, w)∆t(w, w − v + r) +− +� v +v−r+δ1 +dw K3(v − r, w)∆t(v, v − w) − +� v +v−r +dw K2 +3(v, w)∆t(w, w − v + r) +− +� v+δ1 +v+δ2 +dw K3(v, w)∆t(w, w − v + r) − +� v−δ2 +v−δ1 +dw K1 +3(v, w)∆t(w, w − v + r). +(3.17) +The potential Vu has already been defined in (3.7). The next step is then to carve out a bounded piece Lb +from Lpp as follows: +Lpp∆t(v, r) = Vu(v, r)∆t(v, r) − Ku∆t(v, r) − Lb∆t(v, r), +where, +Lb[∆t](v, r) +(3.18) +24 + += − +1(v < −b0) +� ∞ +a1 +dw +� +K3 +1(v, w) − K3 +1(v − r, w) +� +∆t(w, w − v + r) +− +1(v ≥ b0) +� ∞ +v+δ1 +dw +� +K3 +1(v, w) − K3 +1(v − r, w) +� +∆t(w, w − v + r) +− +1(v − r < −b0)1(v > 0) +� v +0 +dw K3(v − r, w)∆t(v, v − w) +− +1(v − r ≥ −b0) +� v +v−r+δ1 +dw K3 +1(v − r, w)∆t(v, v − w) +− +1(v < −m0) +� v−min(r,−v−b0) +v−r +dw K1 +3(v, w)∆t(w, w − v + r) +− +1(−m0 < v < m0) +� v +v−r +dw K2 +3(v, w)∆t(w, w − v + r) +− +1(v − r ≤ −m0) +� v−r−˜r +−∞ +dw +� +K2 +3(v − r, w) − K2 +3(v, w) +� +∆t(v, v − w) +− +1(−m0 < v − r < m0) +� v−r +−∞ +dw +� +K2 +3(v − r, w) − K2 +3(v, w) +� +∆t(v, v − w) +− +1(v − r ≥ m0) +� +1(v ≤ 3r) +� 0 +−∞ +dw +� +K2 +3(v − r, w) − K2 +3(v, w) +� +∆t(v, v − w) ++ 1(v > 3r) +� v−r +c0v +dw +� +K2 +3(v − r, w) − K2 +3(v, w) +� +∆t(v, v − w) +� +− +1(v ≤ −m0)1(v + r > −b0) +� 2v+b0 +v−r +dw K2 +3(v, w)∆t(w, w − v + r) +− +1(v ≥ m0) +� +1(v ≤ r) +� v +v−r +dw K2 +3(v, w)∆t(w, w − v + r) ++ 1(0 < v − r < c0v) +� v +c−1 +0 (v−r) +dw K2 +3(v, w)∆t(w, w − v + r) +� +, +(3.19) +where a1 = a/c0, and ˜r appearing in line 7 of the formula for Lb, is defined as: +˜r = max(−v − b0, r), +∀v < −b0, += max(−b0 − v + r, 0), +∀v ≥ −b0. +(3.20) +For our subsequent computations we choose b0 ≥ 10 and m0 ≥ max(2a1, b0 + 2). +Note that whenever +v ≥ −b0, ˜r = r − b0 − v, since this cut-off is used only when v − r ≤ −m0. We also observe that Lemma 3.1 +guarantees that Ku, as well as Lb, is positivity-preserving. +The above formulae hold true for all positive functions δ1 and δ2. The choice of these functions is then +determined by the requirement that |Lδ∆s| be made “small” compared to ∥∆∥Y VuΓs (see the estimate in +Lemma 3.7). Let us write down the explicit forms of these functions. +δ1(v, r) = +(1 − e−αr) +4 +γ0 +M exp( µ′ +γ0 max(a1, (v − r))) +δ2(v, r) = +(1 − e−αr) +4 +γ0 +M exp( µ′ +γ0 max(a1, v)) +, +(3.21) +25 + +where µ′ > µ and M is used as a sort of “tuning parameter” to make these cut-off functions “small”. This +“smallness” is inherited by Lδ[∆t](v, r), which consists of singular integrals over intervals of length δ1 and +δ2. We do not care much about the actual value of the constant M. It is quite large and that such a choice +0 < M < ∞ can be made is enough for us. +Before we conclude this part about the derivation of Equation (3.6), some remarks about the necessity +for using two different cut-off functions are in order. For the moment let us use δ as a stand-in for the cut-off +functions δ1 and δ2. Let us take the term +� v+δ +v +dw K3(v, w)∆t(w, w − v) from Lδ. Observe that, for large, +positive values of the variable v the expression +� v+δ +v +dw K3(v, w)Γt(w, w − v) = +� δ +0 +dr′ K3(v, v + r′) +� +f(v) + f(v + r′) +� +eµ max(a,v,c0(v+r′))gt(v + r′, r′), +contains an extra factor of eµ(1−c0)v relative to (VuΓt)(v, r). Thus the only way this term can be controlled +by VuΓt is by putting in a countervailing v-dependence in the definition of the cut-off function δ used here. +This explains the v-dependence in the definition of δ2 in (3.21). Note that with this definition of the cut-off +function the extra factor of eµ(1−c0)v gets cancelled. +Let us now note that the potential Vu(v, r) defined in (3.7) behaves as +Vu(v, r) ≃ ¯n +� � +ln(1 + ev−r) +�−1/2 ln δ−1 +1 ++ (ln(1 + ev))−1/2 ln δ−1 +2 ++ +� +max(1, v) +� +For large, positive values of v, ln δ−1 +2 +has a part that behaves like ln v. In order for the main estimate (see +Lemma 3.6) used in the proof of Theorem 3.3 to be true, Lu[Γt](v, r) = Vu(v, r)Γt(v, r) − Ku[Γt](v, r) must +have the same asymptotic behavior as Vu(v, r)Γt(v, r). It is clear from the proof of Lemma 3.6 that the +argument appearing in the point-singularity must be the same as that in the ln δ−1 multiplying it, otherwise +we would end up with terms of the form e− 1 +2(v−r) ln v, which would cause our estimate to fail in the region +v ≫ 0, v − r ≪ 0. Thus δ1 and δ2 have to be two different functions, as defined in (3.21). +3.1.2 +Solutions of (3.6): The proof of Theorem 3.3 +This part of subsection 3.1 is planned as follows: we first state three lemmas on which the proof of Theorem +3.3 rests; then we write down the proof of this theorem; finally, we prove the aforementioned lemmas. These +three lemmas concern the control of the three operators Ku, Lδ and Lb appearing in (3.6). Without further +ado, we state the lemmas below. +Lemma 3.6. There exists q0 > 0 depending on the parameters α, c0, µ′, γ0, such that the following bound +holds: +LuΓs + ˙Γs = VuΓs − KuΓs + ˙Γs ≥ +q0 +ln M VuΓs. +Lemma 3.7. There exists q1 > 0 depending on the parameters γ0 and κ, such that the following bound +holds: +|Lδ∆s| ≤ q1(κ, γ0) +Mγ0 ln M ∥∆∥Y +� +VuΓs + ˙Γs +� +. +Lemma 3.8. The bounded linear operator Lb satisfies +|Lb∆s| ≤ A(b0, m0, µ, c0)∥∆∥Y Γs, +for some A(b0, m0, µ, c0) > 0. +26 + +Let us now write down the proof of Theorem 3.3 on the basis of the above lemmas. +Proof of Theorem 3.3. Given initial datum ∆0 ∈ X, let us recall the Duhamel-integrated form (3.8) for the +evolution equation for ∆t: +∆t =e−tVu∆0 + F[∆t] +=e−tVu∆0 + +� t +0 +ds e−(t−s)VuKu[∆s] − +� t +0 +ds e−(t−s)VuLb[∆s] − +� t +0 +ds e−(t−s)VuLδ[∆s], +We observe that Lemma 3.6 implies the following upper bound: +KuΓs ≤ +� +1 − +q0 +ln M +� � +VuΓs + ˙Γs +� +, +which means we can write +����� +� t +0 +ds e−(t−s)VuKu[∆s] +����� ≤ ∥∆∥Y +� t +0 +ds e−(t−s)VuKuΓs +≤ +� +1 − +q0 +ln M +� +∥∆∥Y +� t +0 +ds ∂s +� +e−(t−s)VuΓs +� +≤ +� +1 − +q0 +ln M +� +∥∆∥Y Γt, +(3.22) +where in the first line we have used the fact that Ku is positivity-preserving. Similarly, Lemma 3.7 yields +the following estimate: +����� +� t +0 +ds e−(t−s)VuLδ[∆s] +����� ≤ +q1 +Mγ0 ln M ∥∆∥Y Γt. +(3.23) +Finally let us observe that for all s′ > 0, VuΓs′ + ˙Γs′ > 0, which means +� t +s +ds′∂s′ +� +e−(t−s′)VuΓs′ +� +> 0, ∀s ∈ (0, t), implying Γt > e−(t−s)VuΓs. +Lemma 3.8 then implies +����� +� t +0 +ds e−(t−s)VuLb[∆s] +����� ≤ A(b0, m0, µ, c0)T∥∆∥Y , +∀t ∈ [0, T], T > 0. +(3.24) +Putting together (3.22), (3.23) and (3.24), it is easy to see that +|F∆t| ≤ ∥∆∥Y Γt +� +1 − +q0 +ln M + AT + +q1 +Mγ0 ln M +� +, +(3.25) +where q0 and q1 have no dependence on M and it is evident that M < ∞ can be chosen large enough such +that the following is true: +0 < T ∗ < +1 +A ln M +� +q0 − +q1 +Mγ0 +� +, for some T ∗ ∈ R+. +Now that M is chosen in the above manner, clearly we have, for 0 < t ≤ T ∗, |F∆t| < ∥∆∥Y Γt,, i.e., +∥F∥Y < 1. The corresponding Neumann series then converges and we obtain the following unique solution +of our Duhamel-integrated equation: +∆t = (1 − F)−1 e−tVu∆0, ∀0 < t ≤ T ∗. +27 + +Note that, as we have mentioned already before the beginning of Subsection 3.1, the above estimate can +be extended in time. We now move on to the proofs of the lemmas stated above. We will prove Lemmas 3.7 +and 3.8 first and save the proof of Lemma 3.6, which is much more involved, for last. +Proof of Lemma 3.7. The time-derivative of Γs is +˙Γs(v, r) = −1 +8 min(1, ¯n) +1 +1 + eβ(v−r) · +1 +(1 + s)2 ln(1 − e−κr)−1Γs(v, r), +s > 0. +(3.26) +From the definition (3.7) of Vu, it is easy to see that +Vu(v, r) ≥ 2¯n +� +ln(1 + ev−r) +�−1/2 � +ln M + ln(1 − e−r)−1� +, +so that, Vu(v, r)Γs(v, r) + ˙Γs(v, r) > ¯n +� +ln(1 + ev−r) +�−1/2 � +ln M + ln(1 − e−r)−1� +Γs(v, r). +(3.27) +By definition (3.16) we have, +Lδ∆s(v, r) = +� δ1 +0 +dr′ K1 +3(v, v − r − r′)∆s(v, r + r′) + +� δ1 +0 +dr′ K3(v − r, v + r′)∆s(v + r′, r + r′) ++ +� δ1 +0 +dr′ K3(v − r, v − r + r′)∆s(v − r + r′, r′) − +� δ1 +0 +dr′ K1 +3(v − r, v − r − r′)∆s(v − r, r′) ++ +� δ2 +0 +dr′ K1 +3(v, v − r′)∆s(v, r′) − +� δ2 +0 +dr′ K3(v, v + r′)∆s(v + r′, r′). +We will write estimates for two of these terms. The rest can be estimated in a similar manner. +i) +� δ1 +0 +dr′ |K1 +3(v, v − r − r′)∆s(v, r + r′)| ≤ ∥∆∥Y +� δ1 +0 +dr′ K1 +3(v, v − r − r′)Γs(v, r + r′) +≤ 4¯n∥∆∥Y eµ max(a,c0v,v−r) (f(v − r) + f(v)) κ (ln(1 + ev))− 1 +2 × +× +� δ1 +0 +dr′ e−r−r′ � +1 − e−r−r′�γs(v−r)−1 +≤ 6κ¯n +γ0 +∥∆∥Y Γs(v, r) (ln(1 + ev))− 1 +2 δγs(v,r) +1 +, +ii) +� δ2 +0 +dr′ |K3(v, v + r′)∆s(v + r′, r′)| ≤ ∥∆∥Y +� δ2 +0 +dr′ K3(v, v + r′)Γs(v + r′, r′) +≤ 4¯n∥∆∥Y (ln(1 + ev))− 1 +2 κ (f(v − r) + f(v)) × +× +� δ2 +0 +dr′ e−r′ � +1 − e−r′�γs(v)−1 +eµ max(a,c0(v+r′),v) +≤ 6κ¯n +γ0 +∥∆∥Y eµ max(a,c0v) (f(v − r) + f(v)) (ln(1 + ev))− 1 +2 δγ0 +2 . +Estimating the rest of the terms in a similar manner, we can write the following upper bound: +|Lδ[∆s](v, r)| ≤ ∥∆∥Y +p1κ +γ0Mγ0 ¯n +� +ln(1 + ev−r) +�− 1 +2 (1 − e−αr)3 Γs(v, r), +where p1 is a numerical constant. Obviously then, +|Lδ[∆s](v, r)| ≤ ∥∆∥Y +q1(κ, γ0) +Mγ0 ln M +� +VuΓs(v, r) + ˙Γs(v, r) +� +, +∀(v, r) ∈ R × R+, +with q1 = 2p1κ +γ0 . +28 + +Lemma 3.8 is quite obvious from the definition (3.18) of the bounded operator Lb. +Proof of Lemma 3.8. It is clear from definition (3.18) that Lb∆s(v, r) does not contain any point or line +singularity. In addition, the exponential decay in the integrands ensures the finiteness of the integrals. The +bound in the lemma is then obtained by straightforward computations. +We now come to the most crucial estimate, i.e. the one contained in Lemma 3.6. The proof of this +lemma relies quite heavily on certain properties of the H¨older-type condition used in the definition (3.3) of +Γt. These properties are listed (and proved) in Appendix C. Since the computations are quite involved, we +will try to give a general idea of the scheme of the proof before writing down the proof formally. +The main idea of Lemma 3.6 is to establish that (LuΓs)(v, r) has the same asymptotic behavior as +Vu(v, r)Γs(v, r), for all (v, r) ∈ R × R+. Let us first observe that the potential Vu(v, r) grows exponentially +at (−∞), has a line singularity at r = 0 and grows like the square-root function at +∞, somewhat like the +function +V ′(v, r) ≃ ¯n +� +e− 1 +2 (v−r) ln(2 + 1/r) + +� +max(1, v) +� +. +In order to show that LuΓs has the same behavior, we will first split it into different parts and then show +how each of these parts produces the correct asymptotic behavior in different regions of R × R+. We write +Lu[Γt](v, r) = I1[Γt](v, r) + I2[Γt](v, r) + I3[Γt](v, r) + I4[Γt](v, r). +(3.28) +As the explicit formulae for the Ii’s (i ∈ {1, 2, 3, 4}) written below show, these terms consist of integrals +grouped together on the basis of intervals of integration and the type of singularities they contain. For +example, the terms in I3 do not contain any line singularity, unlike terms in I1 and I2. On the other hand, +the integrals in I4 contain an extra “smallness” because they are integrated over intervals of length at most +δ1, while in I2 the integrals are taken over intervals of length at most r. As will become apparent shortly, +the three groups I1, I2 and I3 produce different kinds of asymptotic behavior in LuΓt: +• I1 yields a term with point singularity which behaves like (ln(1+ev−r))−1/2 ln(1−e−r)−1, i.e., dominates +when r ≪ 1. +• From I2 one gets a term with the point singularity, which becomes dominant for r ≫ 1, behaving like +(ln(1 + ev−r))−1/2(1 − e−r)3. +• Finally, the group I3 contributes to LuΓt the necessary √v growth, when v ≫ 1 . +The Ii[Γt]’s are given by the following formulae: +I1[Γt](v, r) = +� ∞ +r+δ1 +dr′ � +K1 +3(v − r, v − r − r′)Γt(v, r) + K1 +3(v, v − r′)Γt(v, r′) − K1 +3(v − r, v − r − r′)Γt(v, r + r′) +� ++ +1(v < −b0) +� � ∞ +r+δ1 +dr′ K3(v, v + r′)Γt(v, r) − +1(a1 − v ≤ r + δ1) +�� ∞ +r+δ1 +dr′ K3 +2(v, v + r′)Γt(v + r′, r + r′) +− +� r+a1−v +r+δ1 +dr′ K3(v − r, v − r + r′)Γt(v − r + r′, r′) − +� ∞ +r+a1−v +dr′ K3 +2(v − r, v − r + r′)Γt(v − r + r′, r′) +� ++ +1(a1 − v > r + δ1) +�� a1−v +r+δ1 +dr′ K3(v, v + r′)Γt(v + r′, r + r′) + +� ∞ +a1−v +dr′ K3 +2(v, v + r′)Γt(v + r′, r + r′) +− +� r+a1−v +r+δ1 +dr′ K3(v − r, v − r + r′)Γt(v − r + r′, r′) − +� ∞ +r+a−v +dr′ K3 +2(v − r, v − r + r′)Γt(v − r + r′, r′) +� +29 + ++ +1(v ≥ −b0) +� � ∞ +r+δ1 +dr′ K3(v, v + r′)Γt(v, r) − +� ∞ +r+δ1 +dr′ K3 +2(v, v + r′)Γt(v + r′, r + r′) ++ +� ∞ +r+δ1 +dr′ K3 +2(v − r, v − r + r′)Γt(v − r + r′, r′) +� +, +(3.29) +I2[Γt](v, r) += +� r +δ1 +dr′ K1 +3(v, v − r′) +� +Γt(v, r) − Γt(v − r′, r − r′) +� ++ +� r +δ1 +dr′ K1 +3(v − r, v − r − r′) +� +Γt(v, r) − Γt(v, r + r′) +� ++ +� r +δ1 +dr′ K3(v − r, v − r + r′)Γt(v, r) − +� +1(v − r < −b0) +� min(r,r−v) +δ1 +dr′ K3(v − r, v − r + r′)Γt(v, r − r′) ++ 1(v − r ≥ −b0) +� r +δ1 +dr′ K3 +2(v − r, v − r + r′)Γt(v, r − r′) +� ++ +� r +δ1 +dr′ K3(v, v + r′)Γt(v, r) − +1(v < −b0) +� +1(r + δ1 ≤ a1 − v) +� r +δ1 +dr′ K3(v, v + r′)Γt(v + r′, r + r′) ++ 1(a1 − v < r + δ1) +�� min(r,a1−v) +δ1 +dr′ K3(v, v + r′)Γt(v + r′, r + r′) ++ 1(a1 − v < r) +� r +a1−v +dr′ K3 +2(v, v + r′)Γt(v + r′, r + r′) +�� +− +1(v ≥ −b0) +� r +δ1 +dr′ K3 +2(v, v + r′)Γt(v + r′, r + r′), +(3.30) +I3[Γt](v, r) = +� ∞ +0 +dr′ K2 +3(v − r, v − r − r′)Γt(v, r) + +� r +0 +dr′ K2 +3(v, v − r′)Γt(v, r) +− +1(v − r ≤ −m0) +� ˜r +0 +� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +Γt(v, r + r′) +− +1(v − r ≥ m0) +� +1(v ≤ 3r) +� v−r +0 +dr′ � +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +Γt(v, r + r′) ++ 1(v > 3r) +� ∞ +v−r−c0v +dr′ {K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′)}Γt(v, r + r′) +� +− +1(v ≤ −m0) +� +1(v + r < −b0) +� r +0 +dr′ K2 +3(v, v − r′)Γt(v − r′, r − r′) ++ +1(v + r ≥ −˜b0) +� −v−b0 +0 +dr′ K2 +3(v, v − r′)Γt(v − r′, r − r′) +� +− +1(v ≥ m0) +� +1(0 < v − r < c0v) +� r +v−c−1 +0 (v−r) +dr′ K2 +3(v, v − r′)Γt(v − r′, r − r′) ++ 1(v − r ≥ c0v) +� r +0 +dr′ K2 +3(v, v − r′)Γt(v − r′, r − r′) +� += I(1) +3 [Γt](v, r) + I(2) +3 [Γt](v, r), +(3.31) +30 + +where I(1) +3 +includes all the terms with K2 +3(v − r, v − r − r′) in the integrand, while I(2) +3 +includes those with +K2 +3(v, v − r′), and finally, +I4[Γt](v, r) = +� r+δ1 +r +dr′ K1 +3(v − r, v − r − r′) +� +Γt(v, r) − Γt(v, r + r′) +� ++ +� r+δ1 +r +dr′ K3(v, v + r′)Γt(v, r) +− +1(v < −b0) +� +1(r ≤ a1 − v < r + δ1) +�� a1−v +r +dr′ K3(v, v + r′)Γt(v + r′, r + r′) ++ +� r+δ1 +a1−v +dr′ K3 +2(v, v + r′)Γt(v + r′, r + r′) +� ++ + 1(a1 − v < r) +� r+δ1 +r +dr′ K3 +2(v, v + r′)Γt(v + r′, r + r′) ++ 1(a1 − v ≥ r + δ1) +� r+δ1 +r +dr′ K3(v, v + r′)Γt(v + r′, r + r′) +� +− +1(v ≥ −b0) +� r+δ1 +r +dr′ K3 +2(v, v + r′)Γt(v + r′, r + r′) ++ +� δ1 +δ2 +dr′ K3(v, v + r′) +� +Γt(v, r) − Γt(v + r′, r + r′) +� ++ +� δ1 +δ2 +dr′ K1 +3(v, v − r′) +� +Γt(v, r) − Γt(v, r′) +� +. (3.32) +Let us now proceed to the proof of Lemma 3.6, keeping in mind that the computational details referred to +in this proof are to be found in Appendix D. +Proof of 3.6. The proof of this lemma hinges on obtaining a suitable lower bound for LuΓs. We will outline +the general scheme of the estimates leading to this lower bound, writing down explicitly only those terms +that lead to the correct asymptotic behavior in different regions and refer to Appendix D for all other details. +We write +Γt = Γ1 +t + Γ2 +t, where Γ1 +t (v, r) = f(v − r) exp (µ max(a, c0v, v − r)) gt(v, r), +and Γ2 +t (v, r) = f(v) exp (µ max(a, c0v, v − r)) gt(v, r). +We will now look at each of the Ii[Γt]’s. +i) I1[Γt](v, r) : It is quite easy to obtain the following lower bound for I1[Γt](v, r): +I1[Γt](v, r) = I1[Γ1 +t ](v, r) + I1[Γ2 +t ](v, r) ≥ J0[Γ1 +t + Γ2 +t ](v, r) + J1[Γ1 +t ](v, r) + J2[Γ2 +t](v, r) + I[Γ1 +t + Γ2 +t](v, r), +where J0 denotes the dominant term close to the diagonal and is defined as +J0[Γ1 +t + Γ2 +t ](v, r) += f(v − r) +� +eµ max(a,c0v,v−r) +� ∞ +r+δ1 +dr′ K1 +3(v − r, v − r − r′) +� +gt(v, r) + gt(v, r′) − gt(v, r + r′) +� ++ +� ∞ +r+δ1 +dr′ K3(v, v + r′)e− 1 +2 r′eµ max(a,c0(v+r′),v−r) � +gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′) +� � ++ f(v)eµ max(a,c0v,v−r) +� ∞ +r+δ1 +dr′ K1 +3(v − r, v − r − r′) +� +gt(v, r) + gt(v, r′) − gt(v, r + r′) +� ++ +� ∞ +r+δ1 +dr′ K3(v, v + r′)e− 1 +2r′f(v + r′)eµ max(a,c0(v+r′),v−r) � +gt(v, r) + gt(v − r + r′, r′) +−gt(v + r′, r + r′) +� +. +(3.33) +31 + +J1, J2 and I contain only such integrals which do not contain the factor +� +1 − e−r′�−1 +in the integrand +and are thus sub-dominant to J0 close to the diagonal. Some of these integrals are negative and have to +controlled by the other parts. The explicit formulae for these tems as well as the computations showing how +they are controlled are in Appendix D. Then Lemma C.3 from Appendix C tells us: +� ∞ +r+δ1 +dr′ K1 +3(v − r, v − r − r′) +� +gt(v, r) + gt(v, r′) − gt(v, r + r′) +� +≥ 0.4gt(v, r) +� ∞ +r+δ1 +dr′K1 +3(v − r, v − r − r′), +which yields a (ln(1 + ev−r))−1/2 ln(1 − e−r)−1 term typifying the correct asymptotic behavior for r ≪ 1. +ii) I2[Γt](v, r) : We write I2[Γt](v, r) = I2[Γ1 +t ](v, r)+I2[Γ2 +t](v, r). It is the lower bound for I2[Γ1 +t ](v, r) that +produces the correct asymptotic behavior in the region away from the diagonal when the point singularity +dominates. This bound is: +I2[Γ1 +t](v, r) ≥ +1(v − r < −b0)eµ max{a,c0v} +� r +δ1 +dr′ � +K3(v − r, v − r + r′) +� +f(v − r) − (ln(1 + ev−r+r′))−α� +− K1 +3(v − r, v − r − r′) +� +f(v − r − r′) − f(v − r) +� � +gt(v, r) ++ +1(v < −b0)1(r > a1 − v)f(v − r) +� r +a1−v +dr′ K3(v, v + r′) +� +eµa − eµc0ve−( 1 +2−µc0)r′� +gt(v, r) ++ +1(v ≥ −b0)1(v − r < −b0) +� +f(v − r) +� r +δ1 +dr′ K3(v, v + r′) +� +eµ max{a,c0v} − e− 1 +2r′eµ max{a,c0(v+r′)}� +gt(v, r) ++ +� r +min(r,r−v) +dr′ K3(v − r, v − r + r′)(ln(1 + ev−r+r′))−α � +2gt(v, r) − gt(v, r + r′) +� � ++ I− +2 [Γ1 +t ](v, r), +(3.34) +where I− +2 [Γ1 +t](v, r) includes all the negative parts coming from I2[Γ1 +t ]; the integrals in I− +2 [Γ1 +t ] contain differ- +ences of the form (gt(., r + r′) − gt(., r)), which generate an extra exponential decay of e−κr, as explained in +Appendix D, and hence are “small” in the region away from the diagonal. +The relevant dominating behavior with point singularity for r ≫ 1, comes from the first term of the +lower bound (3.34) for I2[Γt](v, r). We call this term C3. Then +C3[Γ1 +t ](v, r) += 1(v − r < −b0)eµ max{a,c0v}gt(v, r) +� r +δ1 +dr′ � +K3(v − r, v − r + r′) +� +f(v − r) − (ln(1 + ev−r+r′))−α� +− K1 +3(v − r, v − r − r′) +� +f(v − r − r′) − f(v − r) +� � +(3.35) +≥ 1(v − r < −b0)4¯nΓ1 +t(v, r) +� +ln(1 + ev−r) +�− 1 +2 +� r +δ1 +dr′ +ev−r + 2e−r′ +1 + ev−r + e−r′ e−r′ +� +ln(1 + ev−r−r′) +ln(1 + ev−r) +�1−α +× +× +� +ln(1 + ev−r+r′) +ln(1 + ev−r−r′) +�1−2α � +1 − e− 248 +125 (1−2α)r′ +1 − e−r′ +� � +1 − +� +ln(1 + ev−r−r′) +ln(1 + ev−r) +�α� +iii) I3[Γt](v, r) : The square-root-like growth for large, positive values of v comes from I3[Γt](v, r) as follows: +1(v − r ≥ −m0)I1 +3[Γt](v, r) + +1(v ≥ m0)I2 +3[Γt](v, r) +32 + +≥ +1(v ≥ m0)¯nΓt(v, r) +� +1(v − r ≤ 0)(ln(1 + ev)) +1 +2 +� +1 − +� +ln 2 +ln(1 + ev) +�2 � ++ +1(0 < v − r < c0v)3 +2(ln(1 + ev)) +1 +2 +� +1 − +�v − r +c0v +�2 � ++ +1(v − r ≥ m0) +� +1(v − r ≤ 2 +3v) 4 +3 (ln(1 + ev)) +1 +2 +�ln(1 + ev−r) +ln(1 + ev) +�2 ++ +1(v − r > 2 +3v) 2 +� +ln(1 + ev−r) +� 1 +2 +� +1 − +� ln(1 + ec0v) +ln(1 + ev−r) +�2� �� ++ +1(−m0 < v − r < m0)eµ max(a,c0v,v−r)f(v − r)gt(v, r) +� ∞ +0 +dr′ K2 +3(v − r, v − r − r′) +The computations in Appendix D lead us to the following lower bound for LuΓs, for a suitable choice of the +constant M : +LuΓt(v, r) = I1[Γt](v, r) + I2[Γt](v, r) + I(1) +3 [Γt](v, r) + I(2) +3 [Γt](v, r) + I4[Γt](v, r) ≥ G[Γt](v, r), where +G[Γt](v, r) = ¯nΓt(v, r) +� +1(v − r ≤ −b0)¯b1(α) +� +ln(1 + ev−r) +�− 1 +2 � +1 − e−3αr�3 ++ 1 +2 +� +ln(1 + ev−r) +�− 1 +2 ln +� +1 − e− 7 +2 (r+δ1)�−1 ++ +1(−m0 < v − r < m0)1 +4 +� +ln(1 + ev−r) +�− 1 +2 ++ +1(0 < v − r < m0)1 +8 +� +ln(1 + ev−r) +� 1 +2 + +1(0 < v < m0)1 +2 (ln(1 + ev)) +1 +2 +� +1 − +�ln(1 + ev−r) +ln(1 + ev) +�2� ++ +1(v ≥ m0)b3(c0) (ln(1 + ev)) +1 +2 +� +, +where ¯b1(α) and b3(c0) are positive numbers bounded away from zero. Then using formula (3.26) for ˙Γs(v, r) +we can conclude that +LuΓs(v, r) + ˙Γs(v, r) ≥ G[Γs](v, r) + ˙Γs(v, r) ≥ G[Γs](v, r), +where +G[Γs](v, r) = ¯nΓs(v, r) +� � +ln(1 + ev−r) +�−1/2 +�3 +8 ln +� +1 − e− 7 +2(r+δ1)�−1 ++ +1(−m0 < v − r < 0)1 +4 ++ 1(v − r ≤ −b0)b1(α) +� +1 − e−3αr�3� ++ +1(0 ≤ v − r < m0)1 +8 +� � +ln(1 + ev−r) +�−1/2 + +� +ln(1 + ev−r) +�1/2 +� ++ +1(v > 0) (ln(1 + ev)) +1 +2 +� +1(0 < v < m0)1 +2 +� +1 − +�ln(1 + ev−r) +ln(1 + ev) +�2� ++ +1(v ≥ m0)b3(c0) +�� +. +From the definition (3.7) of the potential Vu, it is easy to see that there exist positive numbers C1 = +C1(µ′, γ0, a1) and C1 = C1(µ′, γ0, a1), such that the following is true for all (v, r) ∈ R × R+: +¯nC1 +� +(ln(1 + ev)) +1 +2 + +� +ln(1 + ev−r) +�− 1 +2 ln +� +2M(1 − e−αr)−1�� +33 + +≤ Vu(v, r) ≤ ¯nC1 +� +(ln(1 + ev)) +1 +2 + +� +ln(1 + ev−r) +�− 1 +2 ln +� +2M(1 − e−αr)−1�� +. +Then it is clear that there exists q0 > 0, depending on α, c0, µ′, γ0, b0, such that: +G[Γs](v, r) ≥ +q0 +ln M Vu(v, r)Γs(v, r), +∀(v, r) ∈ R × R+, +which implies the lower bound claimed in this lemma. +3.2 +Solutions of the Regularized Evolution Equation: +Let us recall the regularized evolution equation +∂tψt(v) = −(Lε +3ψt)(v) = − +� +R +dw Kε +3(v, w) +� +ψt(v) − ψt(w) +� +, where +Kε +3(v, w) = Kε(max(v,w)) +3 +(v, w) = K3(v, w)1 − e− min(ε(v,w),|v−w|) +1 − e−ε(v,w) +, +ε(v, w) = ε(max(v, w)) = ε0 exp +� +− µ′ +γ0 +max(a1, max(v, w)) +� +. +Recall that µ′ > µ and the admissible values for the parameters µ, a1 = a/c0 and γ0 have already been +defined in (2.4). In this subsection our goal is to prove Theorem 3.4 and Theorem 3.5. For our subsequent +computations it is useful to split the kernel function just like in the beginning of Section 2 (cf. Lemma 3.1). +When v > w, Kε +3(v, w) = K1,ε(v) +3 +(v, w) + K2,ε(v) +3 +(v, w), where +K1,ε(v) +3 +(v, w) = K1 +3(v, w)1 − e− min(ε(v),v−w) +1 − e−ε(v) +, K2,ε(v) +3 +(v, w) = K2 +3(v, w)1 − e− min(ε(v),v−w) +1 − e−ε(v) +. +Similarly in the region w > v, +Kε +3(v, w) = K3 +1,ε(w)(v, w) + K3 +2,ε(w)(v, w), where +K3 +2,ε(w)(v, w) = K3 +2(v, w)1 − e− min(ε(w),w−v) +1 − e−ε(w) +, and K3 +1,ε(w)(v, w) = K3 +1(v, w)1 − e− min(ε(w),w−v) +1 − e−ε(w) +. +Just like Lemma 3.1, we have the following result for the regularized case: +Lemma 3.9. The kernel functions satisfy the following inequalities for all r > 0: +i) For all w < v − r, K1,ε(v−r) +3 +(v − r, w) > K1,ε(v) +3 +(v, w), whenever either v ≤ a1 or v − w ≥ ε(v − r), +and K2,ε(v−r) +3 +(v − r, w) > K2,ε(v) +3 +(v, w), whenever v − r − w ≥ ε(v − r). +ii) For all w > v, Kε(w) +3 +(v, w) > Kε(w) +3 +(v − r, w), K3 +2,ε(w)(v, w) > K3 +2,ε(w)(v − r, w). +Proof. i) When w < v − r let us define r′ = v − r − w. Then: +K1,ε(v−r) +3 +(v − r, v − r − r′) − K1,ε(v) +3 +(v, v − r − r′) +≥ 4¯n(ln(1 + ev−r))−3/2 ln(1 + ev−r−r′) ev−r−r′ + 2e−r′ +1 + ev−r−r” + e−r′ +� +e−r′ +1 − e− max(r′,ε(v−r)) − +e−r−r′ +1 − e− max(r+r′,ε(v)) +� +Note that each of the two conditions, v ≤ a1 (which means ε(v) = ε(v − r)) and r + r′ ≥ ε(v − r), implies +that max(r +r′, ε(v)) ≥ max(r′, ε(v −r)), and consequently K1,ε(v−r) +3 +(v −r, v −r −r′) > K1,ε(v) +3 +(v, v −r −r′) +whenever either of these conditions is met. +34 + +Whenever v − r − w ≥ ε(v − r), K2,ε(v−r) +3 +(v − r, w) = K2 +3(v − r, w) and K2,ε(v) +3 +(v, w) = K2 +3(v, w), so the +corresponding inequality is proved by Lemma 3.1. +ii) For w > v we have: +Kε(w) +3 +(v, w) − Kε(w) +3 +(v − r, w) ≥ +1 +1 − emax(ε(w),w−v) +� +f(v, w) − f(v − r, w) +� +ln(1 + ew), +where f(v, w) = (ln(1 + ev))−3/2 ev + 2e−(w−v) +1 + ev + e−(w−v) e−(w−v). +Then ∂ +∂v f(v, w) ≥ f(v, w) +� +−3 +2 +ev +(1 + ev) ln(1 + ev) + 3 + ev +2 + ev +� +> 0. +The last inequality then is an obvious consequence of this. +Let us remember that Theorem 3.4 deals with the difference variable Dψ. In order to show the existence +of this difference variable, we will first prove the existence of a unique solution of the ψ-equation written +above. As mentioned before, we will tag solutions of the regularized equation with ε. +In what follows, we will first show, given a suitable initial value ψε +0, the existence of a unique Duhamel- +integrated solution of (3.10). Then we will go on to prove a similar existence-uniqueness result, namely +Theorem 3.4, for the difference variable. The final result of this subsection will be Proposition 3.5. The +proofs in this subsection follow closely the proofs of Lemmas 3.6-3.8 and Theorem 3.3 in the previous +subsection. +3.2.1 +Regularized Solution ψε +t +Let us consider the time-evolution, according to (3.10), of initial datum ψε +0 in the Banach space X, defined +in (3.11). Note that, since X ⊂ D(L +ε +3) for all ε > 0, Theorem 2.2 guarantees the existence of a unique (in +L2(ν)) solution, ϕt = e−tL +ε +3ψε +0. The evolution equation is first re-written as follows: +∂tψε +t (v) = − +� � ∞ +0 +dr′ Kε(v) +3 +(v, v − r′) + +� ∞ +0 +dr′ Kε(v+r′) +3 +(v, v + r′) +� +ψε +t (v) ++ +� ∞ +0 +dr′ Kε(v) +3 +(v, v − r′)ψε +t (v − r′) + +� ∞ +0 +dr′ Kε(v+r′) +3 +(v, v + r′)ψε +t (v + r′) += −V ε(v)ψε +t (v) + (Kε +uψε +t )(v) + (Kε +b ψε +t )(v), +where V ε(v) = +� ∞ +0 +dr′ Kε(v) +3 +(v, v − r′) + +� ∞ +0 +dr′ Kε(v+r′) +3 +(v, v + r′), +Kε +u consists of unbounded parts of the kernel function and Kε +b is L2-bounded. The explicit formulae for +them are given in Appendix E. +We will prove that there exists a unique Duhamel-integrated solution of the above evolution equation in +X, for all t > 0, given by +ψε +t = e−tV εψε +0 + +� t +0 +ds e−(t−s)V εKε +u[ψε +s] + +� t +0 +ds e−(t−s)V εKε +b[ψε +s]. +(3.36) +Before proving (3.36), we will consider the following equation, which differs from (3.36) only in that the +function ψε +s in the L2-bounded part Kb is replaced by the L2-solution ϕs : +ψε +t = e−tV εψε +0 + +� t +0 +ds e−(t−s)V εKε +u[ψε +s] + +� t +0 +ds e−(t−s)V εKε +b [e−sLε +3ψε +0]. +(3.37) +35 + +Equation (3.37) is just the Duhamel-integrated form of the evolution equation: +∂tψε +t (v) = −V ε(v)ψε +t (v) + (Kε +uψε +t )(v) + Kε +b [e−tLε +3ψε +0](v). +Lemma 3.10. Given initial value ψε +0 ∈ X, suppose ψε +t ∈ X solves equation (3.37) for all t > 0. Then +ψε +t = e−tLε +3ψε +0, ν-almost evrywhere. +Proof. ψε +t ∈ X ⊂ D(Lε +3), for all t > 0. This means, given any t > 0, we have, for all s ∈ (0, t) +∂s +� +e−(t−s)Lε +3ψε +s +� += e−(t−s)Lε +3 [Lε +3ψε +s + ∂sψε +s] += e−(t−s)Lε +3 [(V εψε +s − Kε +uψε +s − Kε +bψε +s) − (V εψε +s − Kε +uψε +s − Kε +b ϕs)] += e−(t−s)Lε +3Kε +b [ϕs − ψε +s], +where ϕs = e−sLε +3ψε +0. Integrating the above, we have +ψε +t − ϕt = +� t +0 +ds e−(t−s)Lε +3Kε +b [ϕs − ψε +s]. +Therefore by Minkowski’s integral inequality, ∥ψε +t − ϕt∥L2 ≤ +� t +0 +ds ∥Kε +b ∥L2∥ψε +s − ϕs∥L2. +Then by Gr¨onwall’s inequality ∥ψε +t − ϕt∥L2 = 0, i.e., ψε +t = ϕt ν-a.e. ∀t > 0. +Lemma 3.11. There exists a number s0 > 0, depending on the parameters α, µ′, γ0, such that: +V ε˜Γ − Kε +u˜Γ ≥ +s0 +ln ε−1 +0 +V ε˜Γ. +Proof. From the definition of the potential V ε, it is easy to see that there exist positive numbers C2 = +C2(µ′, γ0) > 1 and C2 = C2(µ′, γ0), such that the following is true: +¯n +�1 +4 (ln(1 + ev))− 1 +2 ln ε−1 +0 ++ C2 (ln(1 + ev)) +1 +2 +� +≤ V ε(v) +≤ C2¯n +� +(ln(1 + ev))− 1 +2 ln ε−1 +0 ++ (ln(1 + ev)) +1 +2 +� +, +∀v ∈ R. +From E.1 in Appendix E we get the following lower bound: +V ε(v)˜Γ(v) − (Kε +u˜Γ)(v) ≥ ˜Γ(v) +� +1(v ≤ 0)p2(α) (ln(1 + ev))− 1 +2 + +1(v > 0)p3 (ln(1 + ev)) +1 +2 +� +, +where p2 and p3 are positive constants bounded away from zero. +It is then obvious that for some s0 = s0(α, µ′, γ0) > 0 we can write +V ε(v)˜Γ(v) − (Kε +u˜Γ)(v) ≥ +s0 +ln ε−1 +0 +V ε(v)˜Γ(v), +∀v ∈ R. +36 + +Theorem 3.12. Given initial datum ψε +0 ∈ X, there exists, for all t > 0, a unique solution ψε +t of (3.36), such +that: +∥ψε∥′ = sup +t∈R+ +∥ψε +t ∥X = sup +t∈R+ +sup +v∈R +|ψε +t (v)| +˜Γ(v) +< ∞. +Proof. In order to prove the statement of this theorem, we will first show that given initial datum ψε +0 ∈ X, +there exists a unique solution ψε +t of (3.37), such that supt∈R+ ∥ψε +t ∥X < ∞. +Given such a solution, the +statement of this theorem is implied by Lemma 3.10. +Let F1[ψt] = +� t +0 +ds e−(t−s)V εKε +u[ψε +s], +and, Bt[ψε +0] = +� t +0 +ds e−(t−s)V εKε +b[e−sLε +3ψε +0]. +Then we can write (3.37) as +ψε +t =e−tV εψε +0 + F1[ψε +t ] + Bt[ψε +0]. +By Lemma 3.11 and the fact that Kε +u is positivity-preserving, we have: +|F1[ψε +t ](v)| ≤ ∥ψε∥′ +� +1 − +s0 +ln ε−1 +0 +� � t +0 +ds ∂s +� +e−(t−s)V ε ˜Γ +� +(v) +≤ ∥ψε∥′ +� +1 − +s0 +ln ε−1 +0 +� +˜Γ(v), +∀v ∈ R. +Thus ∥F1∥′ < 1, and the corresponding Neumann series converges. +The invertibility of F1 means we can write +ψε +t = (1 − F1)−1 � +e−tV εψε +0 + Bt[ψε +0] +� +, +which is clearly the unique solution of (3.37) corresponding to the initial value ψε +0. +Then, by Lemma 3.10, ψε +t = e−tLε +3ψε +0, for all t ≥ 0, ν-a.e., so that we can replace e−tLε +3ψε +0 by ψε +t in the +L2-bounded integrals and write +� t +0 +ds e−(t−s)V εKε +b[e−sLε +3ψε +0] = +� t +0 +ds e−(t−s)V εKε +b[ψε +s]. +Thus ψε +t solves (3.36). The uniqueness of this solution is straightforward since any such solution is also a +solution of (3.37). +3.2.2 +Regularized Evolution Equation for the Difference Variable Dψε +t and the Existence of +a Unique Solution +Our results above guarantee the existence of the difference variable Dψε, where Dψε +t (v, r) = ψε +t (v)−ψε +t (v−r), +in the Banach space Z of continuous functions defined on R+ × (R × R+), such that +∥Dψε∥Z = +sup +t>0 +(v,r)∈R×R+ +|Dψε +t (v, r)| +Γ(v, r) +< ∞, +where +Γ(v, r) = (f(v) + f(v − r)) eµ max(a,c0v,v−r). +37 + +Given ψε +0 ∈ X, this difference variable Dψε +t provides us with a solution of the following Cauchy problem +(derived from the corresponding problem for ψε +t ): +∂tDψε +t (v, r) = −( ˜LεDψε +t )(v, r) += − +� +wv +dw Kε +3(v, w)Dψε +t (w, w − v) ++ +� +wv−r +dw Kε +3(v − r, w)Dψε +t (w, w − v + r). +(3.38) +Given ψε +0, ψε +t is uniquely determined in X by Theorem 3.12, and we have the following trivial bound for +Dψε +t : +|Dψε +t (v, r)| = +����(1 − F1)−1 � +e−tV εψ0 + Bt[ψ0] +� +(v) − (1 − F1)−1 � +e−tV εψε +0 + Bt[ψε +0] +� +(v − r) +���� +≤ A0(α, µ′, γ0, b0, m0) +� +ln ε−1 +0 +� +(∥ψε +0∥X + ∥ψε +0∥L2) Γ(v, r). +(3.39) +Observe that although our results in 3.2.1 guarantee the existence of Dψε +t evolving according to (3.38), this +solution may not be unique in Z. +We will now show that equation (3.38) has a unique Duhamel-integrated solution in a certain subspace +Yε of Z, containing functions h bounded in the following norm: +∥h∥Yε = +sup +t>0 +(v,r)∈R×R+ +|ht(v, r)| +Γε(v, r) < ∞, +where +Γε(v, r) = (f(v) + f(v − r)) eµ max(a,c0v,v−r)g(v, r), +g(v, r) = +� +1 − e−κ(r+ε(v))�γ0 , +γ0 = γ0/2. +If we can prove that the Duhamel-integrated form of (3.38) has a unique solution in Z and also a +solution in Yε ⊂ Z, then we will have proved that this unique solution corresponding to the given initial +datum, satisfies a certain H¨older-like condition (see the definition of g). This result will be crucial in the +final step of the proof of the main smoothing result. In our computations to prove the existence-uniqueness +result in both Z and Yε, we will use the following generic weight function which covers both cases: +Γ +′ +ε(v, r) = (f(v) + f(v − r)) eµ max(a,c0v,v−r)˜g(v, r), +˜g(v, r) = +� +1 − e−κ(r+ε(v))�˜γ0 , +where ˜γ0 ∈ {0, γ0}. +The reason for using this weight is that almost all of our computations work for both Γ and Γε. So +our results are proved for the generic Γ +′ +ε. Whenever there is a difference, we point it out in the relevant +computations. We denote by ˜Yε the Banach space corresponding to the weight Γ +′ +ε, which means ˜Yε is either +Z or Yε. +The analysis of the evolution equation for Dψε +t is done along the same lines as the analysis of the ∆t- +equation in Subsection 3.1.1. Thus, here too the linear operator governing the time evolution is split into a +potential, a positivity-preserving part, a bounded part and a perturbation. Let us recall (3.12), where this +splitting has already been effected and the equation has been cast in a form amenable to our subsequent +computations: +∂tDψε +t (v, r) = −( ˜LεDψε +t )(v, r) = −( ˜Lε +uDψε +t )(v, r) − ( ˜Lε +sDψε +t )(v, r) − ˜Kε +b[ψε +t ](v, r). +38 + +Here +( ˜Lε +sDψε +t )(v, r) += − +1(v ≥ a1) +� v−r +−∞ +dw +� +K1,ε(v−r) +3 +(v − r, w) − K1,ε(v) +3 +(v, w) +� +Dψε +t (v, v − w)1(v − w ≤ ε(v − r)) +− +� v−r +v−r−ε(v−r) +dw +� +K2,ε(v−r) +3 +(v − r, w) − K2,ε(v) +3 +(v, w) +� +Dψε +t (v, v − w), +(3.40) +where ˜r has already been defined in (3.20). Also, as before, the unbounded operator ˜Lε +u can be written as +( ˜Lε +uDψε +t )(v, r) = ˜Vε(v, r)Dψε +t (v, r) − ( ˜Kε +uDψε +t )(v, r), with +˜Vε(v, r) = +� v−r +−∞ +dw Kε(v−r) +3 +(v − r, w) + +� v +v−r +dw Kε(v) +3 +(v, w) + +� v +v−r +dw Kε(w) +3 +(v − r, w) + +� ∞ +v +Kε(w) +3 +(v, w). +A useful bound for the Kε +b-part (see Appendix E.2) is obtained in a simple, straightforward way and is hence +stated without proof in the following lemma. +Lemma 3.13. Given initial datum ψε +0 ∈ X the following bound holds: +�� ˜Kε +bψε +t +�� ≤ M0(ln ε−1 +0 ) (∥ψε +0∥X + ∥ψε +0∥L2) Γ +′ +ε, +where M0 depends on the parameters α, µ, µ′, c0, a1, b0, γ0 and m0. +Lemma 3.14. There exists q2 > 0, depending on the parameters γ0, µ, µ′ and κ, such that the following +bound holds: +| ˜Lε +sDψε +t | ≤ q2(κ, µ, µ′, γ0)ε0 +ln ε−1 +0 +∥Dψε∥ ˜ +Yε ˜VεΓ +′ +ε. +Proof. There are two possible cases: +case 1: When we have ˜γ0 = 0 and Γ +′ +ε = Γ. It is quite easy to see there exist C3 = C3(µ′, γ0) > 0 and +C4 = C4(µ′, γ0) > 0 such that: +| ˜Lε +sDψε(v, r)| +≤ ∥Dψε∥ZΓ(v, r) +� +C3ε(v − r) +� +ln(1 + ev−r) +�− 1 +2 + C4 +1(v ≥ a1)1(r ≤ ε(v − r))(ε(v − r))2 (ln(1 + ev))− 1 +2 +� +. +case 2: When ˜Y = Y ε, we have ˜γ0 = γ0 and Γ +′ +ε = Γε. It is quite easy to see there exist C3 = C3(µ′, κ, µ, γ0) > +0 and C4 = C4(µ′, κ, µ, γ0) > 0 such that: +| ˜Lε +sDψε(v, r)| +≤ ∥Dψε∥YεΓε(v, r) +� +C3ε(v − r) +� +ln(1 + ev−r) +�− 1 +2 + C4 +1(v ≥ a1)1(r ≤ ε(v − r))(ε(v − r))2 (ln(1 + ev))− 1 +2 +� +. +Putting these together and using the lower bound of ˜Vε ( see Lemma 3.15), we obtain the required bound. +Lemma 3.15. There exists a number σ > 0, depending on the parameters α, c0, µ′, γ0, such that: +˜Lε +uΓ +′ +ε = ˜VεΓ +′ +ε − ˜Kε +uΓ +′ +ε ≥ +σ +ln ε−1 +0 +˜VεΓ +′ +ε. +39 + +Proof. It is easy to see that there exist positive numbers C3 = C3(µ′, γ0, a1) > 1 and C3 = C3(µ′, γ0, a1) +such that the following bound holds: +¯nC3(µ′, γ0, a1) +� +� +ln(1 + ev−r) +�− 1 +2 ln ε−1 +0 ++ (ln(1 + ev)) +1 +2 +� +≤ ˜Vε(v, r) +≤ ¯nC3(µ′, γ0, a1) +� +� +ln(1 + ev−r) +�− 1 +2 ln ε−1 +0 ++ (ln(1 + ev)) +1 +2 +� +, +∀(v, r) ∈ R × R+. +By computations outlined in Appendix E we have +˜Lε +uΓ +′ +ε(v, r) +≥ ¯nΓ +′ +ε(v, r) +� +ln(1 + ev−r) +�− 1 +2 +� +1(v − r < −b0)¯b1(α) +� +1 − e−3αr�2 � +1 − e−3α max(0,r−ε(v−r))� ++ 1 +2 ln +� +1 − e− 7 +2 max(r,ε(v−r))�−1 � ++ +1(−m0 < v − r < m0)¯nΓ +′ +ε(v, r) +� +1 +6 +� +ln(1 + ev−r) +�− 1 +2 + +1(v − r > ln 2)1 +9 +� +ln(1 + ev−r) +� 1 +2 +� ++ +1(ln 2 < v < m0) ¯n +2Γ +′ +ε(v, r) (ln(1 + ev)) +1 +2 +� +1 − +� +ln(1 + ev−r) +ln(1 + ev−min(r,ε(v))) +�2 � ++ +1(v ≥ m0)b4(c0)¯nΓ +′ +ε(v, r) (ln(1 + ev)) +1 +2 += G0[Γ +′ +ε](v, r). +From the upper bound on ˜Vε +u it is clearly seen that there exists σ > 0, depending on the parameters +α, µ′, γ0, a1 and c0, such that +G0[Γ +′ +ε](v, r) ≥ +σ +ln ε−1 +0 +˜Vε(v, r)Γ +′ +ε(v, r), +∀(v, r) ∈ R × R+, +as claimed. +We are now in a position to prove the main result about the solution Dψε +t , namely, Theorem 3.4. +Proof of Theorem 3.4. Let us begin by recalling that, for initial datum ψε +0 ∈ X, the Duhamel-integrated +form of the difference variable Dψε +t is given by equation (3.13). We can write +Dψε +t = e−t˜VεDψε +0 + +� t +0 +ds e−(t−s)˜Vε ˜Kε +uDψε +s − +� t +0 +ds e−(t−s)˜Vε ˜Lε +sDψε +s − +� t +0 +ds e−(t−s)˜Vε ˜Kε +b[ψε +s] += e−t˜VεDψε +0 + F[Dψε +t ] − Bt[ψε +0], +where +F[Dψε +t ] = +� t +0 +ds e−(t−s)˜Vε ˜Kε +uDψε +s − +� t +0 +ds e−(t−s)˜Vε ˜Lε +sDψε +s, +and, Bt[ψε +0] = +� t +0 +ds e−(t−s)˜Vε ˜Kε +b[ψε +s]. +40 + +Then by Lemmas 3.15 and 3.14, we arrive at the following estimate: +|F[Dψε +t ]| ≤ ∥Dψε∥ ˜YεΓ +′ +ε +� +1 − +σ +ln ε−1 +0 ++ q2ε0 +ln ε−1 +0 +� +. +Since σ and q2 have no dependence on ε0, we can choose the regularization parameter ε0 > 0 small enough +so that the following is true: +σ +ln ε−1 +0 +− q2ε0 +ln ε−1 +0 +> 0, +which implies ∥F∥ ˜Yε < 1. The corresponding Neumann series then converges and we have the following +unique solution for the Duhamel-integrated equation: +Dψε +t = (1 − F)−1 � +e−t˜VεDψε +0 − Bt[ψε +0] +� +. +Finally, it is easily seen that there exists A1 = A1(µ, µ′, c0, a1, b0, m0) > 0 such that +|Bt[ψε +0](v, r)| ≤ A1(ln ε−1 +0 )Γ +′ +ε(v, r) +� +∥ψε +0∥X0 + ∥ψε +0∥L2 +� +, +∀(v, r) ∈ R × R+, +which completes the proof of our assertion. +Thus, given ψε +0 ∈ X, we obtain a unique solution Dψε ∈ Yε of (3.13), ∀t > 0, satisfying +|Dψε +t (v, r)| ≤ C (f(v) + f(v − r)) eµ max(a,c0v,v−r) � +1 − e−κ(r+ε(v))�γ0 , ∀(v, r) ∈ R × R+ and some C < ∞. +3.2.3 +Connecting the Variables ∆t and Dψε +t : Evolution Equation for Dt = Dψε +t − ∆t +Recalling the original evolution equation (3.6) for ∆t, it is easily seen that we can write +∂t∆t = −Lε∆t − Lε +0∆t, +(3.41) +where the part Lε contains Kε and the part Lε +0 has exactly the same structure as Lε, but with K − Kε. The +explicit expressions are given in Appendix F. Let us now look back at (3.12). It is easily checked that +˜LεDψε +t (v, r) = LεDψε +t (v, r), +∀(v, r) ∈ R × R+. +Let us define the following difference function, for all t ∈ [0, T ∗]: +Dt = Dψε +t − ∆t. +Given ψε +0 ∈ X and ∆0 ∈ Y , such that Dψε +0 = ∆0, we now look at the Cauchy problem for the variable Dt. +By (3.41) and (3.12), we can write +∂tDt = −LεDt − Lε +0[∆t], +D0 = 0. +(3.42) +Let us define the Banach space Y ∗ +ε , analogous to Yε, but defined only for t ∈ [0, T ∗], via the following norm: +∥h∥Y ∗ +ε = +sup +t∈[0,T ∗] +(v,r)∈R×R+ +|ht(v, r)| +Γε(v, r) < ∞, +41 + +Since Y ⊆ Y ∗ +ε , and Dψε ∈ Y ∗ +ε , it follows that D ∈ Y ∗ +ε . +Our goal here is to show that Dt can be made arbitrarily small by choosing ε0 small enough. This will +be done by proving that a unique solution of the Duhamel-integrated form of (3.42) exists and can be made +arbitrarily small as claimed. +The analysis of (3.42) is done in the same manner as already seen for the evolution equations of the +∆t-variable and the Dψε +t -variable. Thus, we split the operator into three parts, namely i) Lε +u, which can +be written in the form of (Vε +u − Kε +u), Kε +u being positivity-preserving, ii) Lε +δ, which is to be controlled by +the potential as a “perturbation”, and, iii) a bounded part Lε +b. These operators and the expressions that +appear subsequently are only slightly different from the ones we have seen already. So we will dispense with +writing out most of the terms. We will mention only those terms for which the differences from analogous +expressions encountered in 3.1.1 and 3.2.2 become evident in the final estimates. We write +∂tDt = −Lε +uDt − Lε +δDt − Lε +bDt − Lε +0[∆t] = −Vε +uDt + Kε +uDt − Lε +δDt − Lε +bDt − Lε +0[∆t]. +The Duhamel-integrated form that we will be analyzing is: +Dt =e−tVε +uD0 + +� t +0 +ds e−(t−s)Vε +uKε +u[Ds] − +� t +0 +ds e−(t−s)Vε +uLε +b[Ds] − +� t +0 +ds e−(t−s)Vε +uLε +δ[Ds] +− +� t +0 +ds e−(t−s)Vε +uLε +0[∆s]. +(3.43) +The operator Lε +δ is somewhat different from Lδ, in the sense that it contains extra terms coming from the +perturbation ˜Lε +s, so we write it down below. +Lε +δDt(v, r) = +� � v−r +v−r−δ1 +dw K1,ε(v) +3 +(v, w)Dt(v, v − w) − +� v−r +v−r−δ1 +dw K1,ε(v−r) +3 +(v, w)Dt(v − r, v − r − w) ++ +� v+δ1 +v +dw Kε(w) +3 +(v − r, w)Dt(w, w − v + r) − +� v+δ2 +v +dw Kε(w) +3 +(v, w)Dt(w, w − v) ++ +� v +v−δ2 +dw K1,ε(v) +3 +(v, w)Dt(v, v − w) + +� v−r+δ1 +v−r +dw Kε(w) +3 +(v − r, w)Dt(w, w − v + r) +� +− +1(v ≥ a1) +� v−r +−∞ +dw +� +K1,ε(v−r) +3 +(v − r, w) − K1,ε(v) +3 +(v, w) +� +Dt(v, v − w)1(v − w ≤ ε(v − r)) +− +� v−r +v−r−ε(v−r) +dw +� +K2,ε(v−r) +3 +(v − r, w) − K2,ε(v) +3 +(v, w) +� +Dt(v, v − w) +(3.44) +The following lemmas are easy to obtain via simple and straightforward computations, so we state them +without proof. +Lemma 3.16. There exists q1 > 0, depending on the parameters a1, γ0 and κ, such that the following bound +holds: +|Lε +δDs| ≤ +q1(κ, γ0) +Mγ0 ln +� +min +� +M, ε−1 +0 +��∥D∥Y ∗ +ε Vε +uΓε. +and, +Lemma 3.17. The bounded linear operator Lb satisfies: +|Lε +bDs| ≤ A(b0, m0, µ, c0)∥D∥Y ∗ +ε Γε, +for some A(b0, m0, µ, c0) > 0. +42 + +We now state and prove the lemma which is the key to proving the existence of a unique solution Dt: +Lemma 3.18. There exists a number σ > 0, depending on the parameters α, c0, µ′, γ0, such that: +Lε +uΓε = Vε +uΓε − Kε +uΓε ≥ +σ +ln +� +min +� +M, ε−1 +0 +��Vε +uΓε. +Proof. As before, it is readily seen that there exist C4 = C4(µ′, γ0, a1) > 0 and C4 = C4(µ′, γ0, a1) > 0 such +that the following bound holds: +¯nC4(µ′, γ0, a1) +� +� +ln(1 + ev−r) +�− 1 +2 +� +ln +� +1 − e−αr�− 4 +γ0 + ln min +� +M, ε−1 +0 +� � ++ (ln(1 + ev)) +1 +2 +� +≤ ˜Vε +u(v, r) +≤ ¯nC4(µ′, γ0, a1) +� +� +ln(1 + ev−r) +�− 1 +2 +� +ln +� +1 − e−αr�− 4 +γ0 + ln min +� +M, ε−1 +0 +� � ++ (ln(1 + ev)) +1 +2 +� +. +Computations, with the like of which we have become quite familiar by now, also reveal the following: +Lε +uΓε(v, r) +≥ ¯nΓε(v, r) +� +ln(1 + ev−r) +�− 1 +2 +� +1(v − r < −b0)b2(α) +� +1 − e−3αr�2 � +1 − e−3α max(0,r−ε(v−r))� ++ 1 +2 ln +� +1 − e− 7 +2 max(r+δ1,ε(v−r))�−1 � ++ +1(−m0 < v − r < m0)¯nΓε(v, r) +� +1 +6 (ln(1 + ev))− 1 +2 + +1(v − r > ln 2)1 +9 +� +ln(1 + ev−r) +� 1 +2 +� ++ +1(ln 2 < v < m0) ¯n +2 Γε(v, r) (ln(1 + ev)) +1 +2 +� +1 − +� +ln(1 + ev−r) +ln(1 + ev−min(r,ε(v))) +�2 � ++ +1(v ≥ m0)b4(c0)¯nΓε(v, r) (ln(1 + ev)) +1 +2 += G0[Γε](v, r). +This makes it clear that there exists some σ > 0, depending on α, µ′, c0, γ0, such that +G0[Γε](v, r) ≥ +σ +ln +� +min +� +M, ε−1 +0 +�� ˜Vε +u(v, r)Γε(v, r), +∀(v, r) ∈ R × R+. +Before the next step let us note that henceforth we will assume Mε0 < 1 since M will be fixed while ε0 +will eventually be taken to zero. We note the following estimate for Lε +0[∆t]. +Lemma 3.19. There exist some positive constant C′ < ∞, depending κ, γ0 and α, and, p = p(γ0) > 0 such +that the following bound is true: +���Lε +0[∆t](v, r) +��� ≤ C′(κ, γ0, α)(Mε0)p∥∆∥Y Γε(v, r)Vε +u(v, r), +∀(v, r) ∈ R × R+. +43 + +Proof. We note that in many of the terms in Lε +0[∆], an extra “smallness” comes from the fact that the +integrand contains a term like ∆(., r′), where r′ is the variable of integration. However, there are some terms +for which this is not true. We will take one such term below and show what makes such a term still “small” +(compared to the potential). We keep in mind that now we can assume Mε0 < 1. +Consider the term +1(δ1 < ε(v − r)) +� ε(v−r) +δ1 +dr′ K1 +3(v − r, v − r − r′)e−r′ − e−ε(v−r) +1 − eε(v−r) +∆t(v, r). +δ1 < ε(v − r) =⇒ +1 +M +� +1 − e−αr�4/γ0 < ε0 =⇒ r < 1 +α ln +� +1 − (Mε0)γ0/4�−1 +=⇒ +� +1 − e−κr� +< 1 − +� +1 − (Mε0)γ0/4�κ/α +=⇒ +� +1 − e−κr�γ0−γ0 < +�κ +α +�γ0−γ0 (Mε0) +γ0 +4 (γ0−γ0). +Then we can write the following: +����� +1(δ1 < ε(v − r)) +� ε(v−r) +δ1 +dr′ K1 +3(v − r, v − r − r′)e−r′ − e−ε(v−r) +1 − eε(v−r) +∆t(v, r) +����� +≤ +1(δ1 < ε(v − r))∥∆∥Y +� +ln(1 + ev−r) +�− 1 +2 eµ max(a,c0v,v−r) (f(v − r) + f(v)) (1 − eκr)γ0 +� ε(v−r) +δ1 +dr′ +e−r′ +1 − e−r′ +≤ +1(δ1 < ε(v − r))∥∆∥Y Γε(v, r) +� +ln(1 + ev−r) +�− 1 +2 (1 − eκr)γ0−γ0 ln +� +ε0 +1 +M (1 − e−αr)4/γ0 +� +≤ +1(δ1 < ε(v − r))∥∆∥Y Γε(v, r) +� +ln(1 + ev−r) +�− 1 +2 (1 − eκr)γ0−γ0 4 +γ0 +ln +� +1 − e−αr�−1 +≤ +1(δ1 < ε(v − r))∥∆∥Y Γε(v, r) +� +ln(1 + ev−r) +�− 1 +2 4 +γ0 +�κ +α +� 1 +4γ0 (Mε0) +1 +16 γ0 � +(1 − eκr) +1 +4 γ0 ln +� +1 − e−αr�−1 � +, +where we have used the fact that γ0 = γ0/2. Following similar computations it is easy to see that there exist +p > 0 depending on γ0 and C′ = C′(κ, γ0, α) > 0, such that +���Lε +0[∆t](v, r) +��� ≤ C′(κ, γ0, α)(Mε0)p∥∆∥Y Γε(v, r)Vε +u(v, r), +(v, r) ∈ R × R+. +Finally, we are in a position to prove Proposition 3.5. +Proof of Proposition 3.5. We first write the Duhamel-integrated equation (3.43) as +Dt = e−tVε +uD0 + ˜F[Dt] − +� t +0 +ds e−(t−s)Vε +uLε +0[∆s], +where +˜F[Dt] = +� t +0 +ds e−(t−s)Vε +uKε +u[Ds] − +� t +0 +ds e−(t−s)Vε +uLε +b[Ds] − +� t +0 +ds e−(t−s)Vε +uLε +δ[Ds]. +Then, as before, Lemmas 3.18, 3.16 and 3.17 mean that, for all t ≤ T ∗: +| ˜FDt| ≤ ∥D∥Y ∗ +ε Γε +� +1 − +σ +ln +� +min +� +M, ε−1 +0 +�� + AT ∗ + +q1 +Mγ0 ln +� +min +� +M, ε−1 +0 +�� +� +. +(3.45) +44 + +Obviously then, we can choose M < ∞ large enough and ε−1 +0 +> M, so that we have, for some T ∗, +0 < T ∗ < +1 +A ln M +� +σ − +q1 +Mγ0 +� +. +This guarantees that ∥ ˜F∥Y ∗ +ε < 1. The convergence of the relevant Neumann series then gives us the following +unique solution of (3.43): +Dt = − +� +1 − ˜F +�−1 � t +0 +ds e−(t−s)Vε +uLε +0[∆s], ∀0 < t ≤ T ∗, +since the initial conditions are such that D0 = 0. +Finally, from lemma 3.19 it is clear that for some Q > 0, p > 0 depending on the parameters of the +weight functions, we have +|Dt| ≤ Q ln +� +min(M, ε−1 +0 ) +� +(Mε0)p ∥∆∥Y Γε, +∀t ∈ [0, T ∗]. +Note that the time T ∗ above may be different from the time that appears in Theorem 3.3, although the +same symbol has been used. This does not cause any problem because it is enough to choose the minimum +of these two times, name it our new T ∗, and understand that this minimum is the T ∗ that appears in our +main result Theorem 3.2 as well as Theorem 2.5. +A +Linearization of the Three-wave Collision Operator C3 +In what follows, we describe briefly how the expressions for the kernel functions appearing in (2.1) and +(3.1) are obtained from the linearization of the three waves collision operator C3. The linearized operator is +obtained first in terms of energy variables. We reserve the letters x and y (x, y are in R+) for these variables, +so that the kernel function is written, by a slight abuse of notation, as K3(x, y) in the flat metric (in the +weighted L2 space we use the notation K3(x, y)). Then we change variables to u and v, which take values in +R, and obtain the functions K3(u, v) and K3(u, v) appearing in (2.1) and (3.1) respectively. We also show +that our linearized operator is identical, upto a numerical factor, to the operator considered in [11] and [12]. +As mentioned in the introduction, the operator C3 is linearized around the equilibrium distribution fBE, +by considering perturbations fper of the form fper(x, t) = xfBE(x) ˜fBE(x)ψt(x). Considering the action of C3 +on ftot = fBE + fper and keeping in mind that C3[fBE] = 0, the following evolution equation for the variable +ψ in the linearized model is easily obtained: +∂tψt(x) = −L3ψt(x) += − +2¯n +√xgBE(x)˜gBE(x) +�� x +0 +dy x ˜fBE(x)fBE(y)fBE(x − y) +� +xψt(x) − yψt(y) − (x − y)ψt(x − y) +� ++2 +� ∞ +x +dy x ˜fBE(x)fBE(y) ˜fBE(y − x) +� +xψt(x) − yψt(y) + (y − x)ψt(y − x) +�� += −4¯n(¯h(x))2 +�� ∞ +0 +dy H(min(x, y), |x − y|)(ψt(x) − ψt(y)) +|x − y| +− +� ∞ +0 +dy H(x, y)(ψt(x) − ψt(y)) +x + y +� +, +(A.1) +where +H(x, y) = xfBE(x)yfBE(y)(x + y) ˜fBE(x + y), and ¯h(x) = +�√xgBE(x)˜gBE(x) +�− 1 +2 . +45 + +It is also easy to see that one can then write +L3ψt(x) = +� ∞ +0 +dy ˜H(x, y) (ψt(x) − ψt(y)) , +(A.2) +where +˜H(x, y) = 4¯n¯h(x)2xye− min(x,y)fBE(|x − y|) ˜fBE (max(x, y)) ˜fBE(x + y) +� +1 + e− max(x,y)� +, +and we have used the following identity: +1 − emin(x,y)fBE(x + y) +fBE(|x − y|) += +fBE(x + y) +fBE(min(x, y)) +� +1 + emax(x,y)� +. +From formula (A.2) it is evident that the operator L3 has a non-negative, symmetric sesquilinear form +on a dense domain of a weighted L2-space characterized by the weight ν(dx) = ¯h(x)−2dx, and that L3 is +conveniently defined as an unbounded operator in L2(ν) as follows: +L3ψt(x) = +� ∞ +0 +ν(dy)K3(x, y) (ψt(x) − ψt(y)) , +(A.3) +where K3(x, y) = 4¯n¯h(x)2¯h(y)2xye− min(x,y)fBE(|x − y|) ˜fBE (max(x, y)) ˜fBE(x + y) +� +1 + e− max(x,y)� +. +This yields (1.5). +We now change variables from x, y to u, v, where u = ln(ex − 1) and v = ln(ey − 1). For the convenience +of writing, we indulge in a slight abuse of notation and continue to use the same symbols ν and ψ as before +for the L2-weight and in the perturbed density respectively. Then it is obvious that in terms of the new +variable the weight in our L2-space can be written as ν(dv) = e−v (ln(1 + ev)) +5 +2 dv and that (2.1) is just +(A.3) written in terms of the new variables in R. +In [11] and [12] the same linearized model has been considered. Note that in these papers the model is +described in terms of the variable |p|/ +√ +2, p being the momentum, while we have used the energy variable +here. To establish the equivalence, let us agree to use letters x, y with primes for these new variables, so +that x and y in Equations (1.9) and (1.12) in [12] are now replaced by x′ and y′. Then Equations (1.9) and +(1.12) in [12] read: +∂u +∂t = pc(t) +� ∞ +0 +dy′ � +u(t, y′) − u(t, x′) +� +M(x′, y′), +and M(x′, y′) = +� +1 +sinh |x′2 − y′2| − +1 +sinh(x′2 + y′2) +� y′3 sinh x′2 +x′3 sinh y′2 , +respectively. The variable x′ is related to our energy variable x via x = 2x′2. Changing variables x′ → x, +retaining the same symbol for the perturbation u, we get +∂u +∂t = 1 +4pc(t) +� ∞ +0 +dy (u(t, y) − u(t, x)) M(x, y), where M(x, y) = M(x′, y′) +y′ +����� +x=2x′2,y=2y′2 +. +Using the fact that sinh x = 1 +2 +exp(−x) +fBE(2x) , we easily see that +M(x, y) = 2 +√ +2 +√x +y +x +fBE(y) +fBE(x)e− x−y +2 +� +e +1 +2 |x−y|fBE(|x − y|) − e +1 +2(x+y)fBE(x + y) +� += 2 +√ +2¯h(x)2xyfBE(x)fBE(y)ex+ye− min(x,y) � +fBE(|x − y|) − emin(x,y)fBE(x + y) +� += +√ +2 +2¯n +˜H(x, y). +46 + +B +Parameters α, c0 and µ appearing in the Weight Functions for the +Banach Spaces +As is evident from the arguments used in the proof of Theorem 3.3, the choice of the weight function Γ +is motivated by the requirement that Lu[Γt](v, r) = Vu(v, r)Γt(v, r) − Ku[Γt](v, r) mimics the asymptotic +behavior of Vu(v, r)Γt(v, r), so that Lemma 3.6 is true. Thus we want to cut out as much as possible of the +bounded part of Lpp (see (3.16)), and try to ensure the asymptotic dominance of KuΓ by VuΓ. +We know that the potential Vu(v, r) grows exponentially at (−∞), has a line singularity at r = 0 and +grows like the square-root function at +∞, somewhat like +V (v, r) = ¯n +� +e− 1 +2(v−r) ln(2 + 1/r) + +� +max(1, v) +� +. +The line singularity appearing in the kernel functions contributes the factor gt(v, r) = (1−e−κr)γt(v,r) to the +norm Γt(v, r). The nature of the dependence of the smoothing exponent γ on (v − r) is determined by the +way ˙Γt has to behave. The other factors in the norm are determined by the “desired” behavior of Lu[Γt](v, r) +for asymptotically large positive and negative values of the argument v. These considerations are simple to +understand when we look at the behavior of K3(v, v − r′) and K3(v, v + r′) in the regions v ≪ 0 and v ≫ 0 +respectively, as we explain below. +a) Behavior when v → −∞: +When v ≪ 0, we need to consider the case when the point singularity becomes dominant. To see how +it affects the behavior of our linearized operator (note that, since we are not looking at the line singularity +now, it makes sense to think about this in the space of ψ-variables instead of the differences), it is enough +to consider the following toy model: +T(v, v − r′) = e− 1 +2 ve−2r′ +T(v, v + r′) = e− 1 +2 ve−r′. +The important thing to note here is the relative tilt (extra factor of e−r′) on the left side of the diagonal +with respect to the right side. A quick back-of-the-envelope calculation then tells us the following: the +relative tilt in the kernel function means that the weight function for the ψ-variable should behave as e−αv +asymptotically for some α > 0, which in turn implies that, the asymptotic behavior of our Γt(v, r) should +be like e−α(v−r). A cheap upper bound is already imposed on α by the conservation of energy in the BEC +problem, so 0 < α < 1/2. +Coming back to our original linear operator Lu in the space of differences now, we observe that the choice +of a negative α generates some extra negative terms coming from the left side of the diagonal and these need +to be controlled. An easy way out is to add a sub-dominant factor e−αv to Γt(v, r). This generates some +extra positive terms. Finally, these singular factors are needed only when we are dealing with large, negative +values of the argument, so we include a cut-off in the relevant terms, e.g. we use max(e−α(v−r), (ln 2)α) +instead of the bare e−α(v−r). The inclusion of this cut-off makes our computations simpler for large positive +values of the arguments. +b) Behavior when v → +∞: +At +∞ the behavior is dominated by the part K2 +3 of the kernel function (this is in fact the motivation +behind the splitting of K3(v, v − r′) into K1 +3(v, v − r′) and K2 +3(v, v − r′)). Note that in this case there is no +relative tilt unlike for the region of large, negative values of the arguments. We have already mentioned that +47 + +for asymptotically large values of v, the potential grows like √v. Actually, the relevant behavior in this case +comes from the last two terms appearing in equation (3.7) defining Vu, i.e., the terms +� v +v−r dw K2 +3(v, w) and +� v−r +−∞ dw K2 +3(v − r, w). The latter term becomes important when v − r ≫ 0, especially when v and v − r are +comparable. +Let us then begin by comparing the contributions from K2 +3(v, v − r′) and K2 +3(v − r, v − r − r′) to the +potential term for v − r ≫ 0. +V1(v, r) = +� r +0 +dr′ K2 +3(v, v − r′) ∼ 4¯n√v +� +1 − +�v − r +v +�2� +, +V2(v, r) = +� ∞ +0 +dr′ K2 +3(v − r, v − r − r′) ∼ 4¯n +√ +v − r. +Now define x = v−r +v , and consider the quantity V2−V1. Let g(x) = x2+√x−1, so that V2−V1 = √vg(x). +Note that 0 < x < 1, for all v − r > 0. Evidently, g′(x) > 0. Thus, for all x ∈ (0, 1), g(x) has one and only +one zero. Let us call it c0. Then g(c0) = 0, and it is easy to ascertain that c0 ≈ 0.525. Thus x < c0 implies +V1 > V2, and x > c0 implies V2 > V1. +A similar growth for Lu can only come from the part I3[Γt](v, r), defined in (3.31). We obtain this +behavior is by making sure Lu[Γt](v, r) inherits as much as possible of the potential term. This motivates us +to look for a norm which renders the following terms bounded in certain regions for large values of (v − r) +and v: +� ∞ +0 +dr′ K2 +3(v − r, v − r − r′)Γt(v, r + r′) +and +� r +0 +dr′ K2 +3(v, v − r′)Γt(v − r′, r − r′). +Note that +∀ v − r − r′ ≫ 0, K2 +3(v − r, v − r − r′) ∼ 4¯n(v − r)− 3 +2 (v − r − r′), +and, ∀ v − r′ ≫ 0, K2 +3(v, v − r′) ∼ 4¯nv− 3 +2(v − r′). +The corresponding integrals can be rendered bounded if we choose the weight function Γt in such a way that +Γt(v, r +r′) grows exponentially as v −r −r′ → (+∞), and Γt(v −r′, r −r′) has a similar exponential growth +as v − r′ → (+∞). Clearly, we must include a factor h(v, r) in the weight function that behaves as follows: +h(v, r) = exp [µ max(v − r, c0v)] , for some suitably chosen c0 ∈ (0, 1). +Then our little computation above, comparing V1 and V2, gives us an idea about how the relative weight c0 +can be chosen. We choose c0 = 0.52. +We obtain upper bounds on permissible values of α and µ in the course of our computations described +in Appendix D. A brief description of how these bounds are arrived at is given below. +• Look at the combination of terms denoted by Comb.3 in Appendix D. The positivity of this combination +rests on the fact that α has a certain upper bound less than 1 +6. Essentially we do the following: +The term +� +−(K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′))(f(v − r − r′) − f(v − r)) +� +is controlled by +the term K +1 +3(v, v + r′)f(v + r′). +We are in the region v − r < −m0, v < −b0. Thus it makes sense to look at the toy model without +48 + +the line singularity. Then the lower bound +K +1 +3(v, v + r′)f(v + r′) > (K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′))(f(v − r − r′) − f(v − r)) +holds true for all − v − b0 > r′ > r, for all r > 0, +if e−( 1 +2+α)ve−(1+α)r > e−( 1 +2 +α)(v−r)e−(2−α)r, for all r > 0, or equivalently, if α < 1 +6. +• Our computations for the lower bound for I2[Γt](v, r) leads to upper bounds on µ and µc0. We choose +µc0 ≤ 1 +4, so that the following is true: +eµ max(˜a,v−r,c0(v−r′)) − e− 1 +2r′eµ max(˜a,v−r,c0(v+r′)) > 0. +The other upper bound chosen for computational convenience while controlling I2[Γt](v, r) is the fol- +lowing: +µ < 1 +2 − 3 +8α. +Finally, our computation for an estimate on I4[Γt](v, r) relies on the choice µc0 > α. +C +Properties of the H¨older-type Conditions +C.1 +The Time-dependent H¨older-type Condition +Let us recall the definition (3.4) of the time-dependent H¨older-type condition gt(v, r): +gt(v, r) = +� +1 − e−κr�γt(v,r) , γt(v, r) = γt(v − r) = γ0 + a(t) +1 +1 + eβ(v−r) , +a(t) = 1 +8 min(1, ¯n) +t +1 + t, +κ ≥ 7, 0 < γ0 ≤ 1/8, 1 ≤ β ≤ κ/4. +We will now prove a few bounds for certain combinations of gt, which are used in the computations detailed +in Appendix D and which will also elucidate the above bounds on β, γ0 and κ. +Lemma C.1. For all 0 < r′ ≤ r, the following are true: +i) gt(v, r) − gt(v, r − r′) ≥ 0, +gt(v, r + r′) − gt(v, r) ≥ 0, +ii) gt(v, r) − gt(v − r′, r − r′) ≥ 0, +gt(v + r′, r + r′) − gt(v, r) ≥ 0. +Proof. For part i) it is enough to notice that, for all r′′ ≥ 0, +∂gt(v, r′′) +∂r′′ +≥ (κ − β)gt(v, r′′) +a(t) +1 + eβ(v−r′′) +e−κr′′ +1 − e−κr′′ +> 0, +∀κ > β, +so that gt(v, r′′) is an increasing function in the second variable r′′, which implies the inequalities in part i). +The inequalities in part ii) are obvious from the definition of the H¨older condition gt(v, r), since in this +case the H¨older exponent is γt(v, r) for all the terms involved. +Lemma C.2. For all r′ > r, for all (v, r) ∈ R × R+, the following inequality holds: +gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′) > +� +2 − (1 + e−κr)γt(v,r)� +gt(v, r). +49 + +Proof. Let us define a function b0 as follows: b0(v, r, r′) = gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′). +Then +∂ +∂r′ b0 = κγt(v, r) +� +e−κr′ � +1 − e−κr′�γt(v,r)−1 +− e−κ(r+r′) � +1 − e−κ(r+r′)�γt(v,r)−1 � +> 0. +Then we can write the following: +b0(v, r, r′) > b0(v, r, r) = 2gt(v, r) − gt(v + r, 2r) += 2 +� +1 − e−κr�γt(v,r) − +� +1 − e−2κr�γt(v,r) += +� +2 − +� +1 + e−κr�γt(v,r) +� +gt(v, r). +Lemma C.3. For all r′ > r, for all (v, r) ∈ R × R+, the following inequality holds: +gt(v, r) + gt(v, r′) − gt(v, r + r′) ≥ +� +1 − +e−κr′ +(1 + e−κr)1−γt(v,r) +� +gt(v, r). +Proof. Note that: +gt(v, r + r′) − gt(v, r′) = +� +1 − e−κ(r+r′)�γt(v,r+r′) +− +� +1 − e−κr′�γt(v,r′) +≤ +� +1 − e−κ(r+r′)�γt(v,r′) +− +� +1 − e−κr′�γt(v,r′) +≤ +� +1 − e−κ(r+r′)�γt(v,r′) +� +1 − +� +1 − e−κr′ +1 − e−κr +1 − e−κ(r+r′) +�γt(v,r′)� +≤ +� +1 − e−κ(r+r′)�γt(v,r)−1 +e−κr′(1 − e−κr) +≤ +� +1 − e−2κr�γt(v,r)−1 e−κr′(1 − e−κr) +≤ +� +1 − e−κr�γt(v,r) +e−κr′ +(1 + e−κr)1−γt(v,r) , +which leads to the bound stated in the lemma. +As an aside, let us observe that, for all r′ > r: +e−κr′ +(1 + e−κr)1−γt(v,r) ≤ f1(r) = +e−κr +(1 + e−κr)3/4 . +Then it is easy to check that f1 is a decreasing function, so that the following estimate holds: +e−κr′ +(1 + e−κr)1−γt(v,r) ≤ f1(r) < 0.6, +and consequently, gt(v, r) + gt(v, r′) − gt(v, r + r′) > 0.4gt(v, r). +Lemma C.4. For all 0 ≤ r′ < r, for all (v, r) ∈ R × R+, the following inequality holds: +2gt(v, r) − gt(v, r − r′) − gt(v, r + r′) ≥ 0. +50 + +Proof. Let b2(v, r, r′) = −gt(v, r − r′) − gt(v, r + r′). +Then ∂b2 +∂r′ = γ0κ +� +e−κ(r−r′) � +1 − e−κ(r−r′)�γt(v−r+r′)−1 +− e−κ(r+r′) � +1 − e−κ(r+r′)�γt(v−r−r′)−1� ++ a(t) +� +H(v, r − r′) − H(v, r + r′) +� +, +where H(v, r′) = +1 +1 + eβ(v−r′) +� +1 − e−κr′�γt(v−r′) +� +κe−κr′ +1 − e−κr′ + +βeβ(v−r′) +1 + eβ(v−r′) ln(1 − e−κr′) +� += +1 +1 + eβ(v−r′) +� +1 − e−κr′�γt(v−r′) +F2(v, r′), +with F2(v, r′) = +κe−κr′ +1 − e−κr′ + +βeβ(v−r′) +1 + eβ(v−r′) ln(1 − e−κr′). +Then it is not difficult to see that: +∂ +∂r′ H(v, r′) += +1 +1 + eβ(v−r′) +� +1 − e−κr′�γt(v−r′) � +− +� +κ2 +e−κr′ +(1 − e−κr′)2 − β +eβ(v−r′) +1 + eβ(v−r′) +� +κe−κr′ +1 − e−κr′ − β ln(1 − e−κr′) +1 + eβ(v−r′) +�� ++ +� +κγ0 +e−κr′ +1 − e−κr′ + β +eβ(v−r′) +1 + eβ(v−r′) +� +F2(v, r′) + +a(t) +1 + eβ(v−r′) (F2(v, r′))2 +� +≤ +gt(v, r′) +1 + eβ(v−r′) +� +− κ2 +e−κr′ +(1 − e−κr′)2 +� +1 − β +κ − +�β +κ +�2� ++ +a(t) +1 + eβ(v−r′) +κ2e−2κr′ +(1 − e−κr′)2 ++ κ +e−κr′ +1 − e−κr′ +� +β +eβ(v−r′) +1 + eβ(v−r′) + κγ0 +e−κr′ +1 − e−κr′ +� � +< 0. +The upper bound in the last line holds because our parameters have been chosen so as to guarantee the +following inequality: +1 − γ0 − a(t) > 2β +κ + +�β +κ +�2 +. +The above inequality means that H is a strictly decreasing function of r′ and so, H(v, r − r′) ≥ H(v, r + r′). +On the other hand, note that γt(v − r + r′) ≤ γt(v − r − r′), and it is easily seen that +e−κ(r−r′) � +1 − e−κ(r−r′)�γt(v−r+r′)−1 +≥ e−κ(r+r′) � +1 − e−κ(r+r′)�γt(v−r−r′)−1 +. +Thus, +∂ +∂r′ b2 ≥ 0. +This means we have the following inequality: +2gt(v, r) − gt(v, r − r′) − gt(v, r + r′) = 2gt(v, r) + b2(v, r, r′) ≥ 2gt(v, r) + b2(v, r, r′) +�� +r′=0 = 0. +51 + +Lemma C.5. For all 0 ≤ r′ < r, for all (v, r) ∈ R × R+, the following inequality holds: +2gt(v, r) − gt(v − r′, r − r′) − gt(v + r′, r + r′) ≥ 0. +Proof. Taking the partial derivative with respect to r′, we see that: +∂ +∂r′ +� +2gt(v, r) − gt(v − r′, r − r′) − gt(v + r′, r + r′) +� += κγt(v, r) +� +e−κ(r−r′) � +1 − e−κ(r−r′)�γt(v−r)−1 +− e−κ(r+r′) � +1 − e−κ(r+r′)�γt(v−r)−1� +≥ 0, +so that, +2gt(v, r) − gt(v − r′, r − r′) − gt(v + r′, r + r′) ≥ +� +2gt(v, r) − gt(v − r′, r − r′) − gt(v + r′, r + r′) +� ���� +r′=0 += 0. +Lemma C.6. For all r′ > r, for all (v, r) ∈ R × R+, the following inequality holds: +2 +� +gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′) +� +− gt(v, r) > +� +1 − e−r� 1 +2 gt(v, r). +Proof. From Lemma C.2, we have, for all r′ > r: +2 +� +gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′) +� +− gt(v, r) > +� +2 +� +2 − +� +1 + e−κr�γt(v,r)� +− 1 +� +gt(v, r). +For all r < (ln 2)/3, it is obvious that 2 +� +2 − (1 + e−κr)γt(v,r)� +− 1 > (1 − e−r) +1 +2, since γt(v, r) < 1/4. +For all r ≥ (ln 2)/3, let us define f2(r) = 2 +� +2 − (1 + e−κr)1/4� +− 1 − (1 − e−r)1/2 . Then it is easy to see +that, +d +drf2 < 0, +so that, f2(r) > 0, which implies +� +2 +� +2 − (1 + e−κr)γt(v,r)� +− 1 +� +gt(v, r) > (1 − e−r)1/2 gt(v, r). +Lemma C.7. For all r′ ≥ r, for all p > γt(v, r), and for all ν ≤ 1, the following is true: +� +1 − e−νr�p gt(v, r′) ≤ +� +1 − e−νr�p gt(v − r + r′, r′) ≤ +� +1 − e−νr′�p +gt(v, r). +Proof. Let h1(r) = (1 − e−νr)(1 − e−κr)−1. Then it is easily computed that +d +dr′ h1 ≥ 0, so that, h1(r′) ≥ h1(r), ∀r′ ≥ r. +This means +1 − e−νr +1 − e−νr′ ≤ 1 − e−κr +1 − e−κr′ < 1, +52 + +which implies, +� 1 − e−νr +1 − e−νr′ +�p +≤ +� 1 − e−κr +1 − e−κr′ +�p +≤ +� 1 − e−κr +1 − e−κr′ +�γt(v,r) +, ∀p > γt(v, r), +that is, +� +1 − e−νr�p � +1 − e−κr′�γt(v,r) +≤ +� +1 − e−νr′�p � +1 − e−κr�γt(v,r) . +Since γt(v, r) is, by definition, an increasing function of the radial variable r, (1 − e−κr′)γt(v,r′) ≤ (1 − +e−κr′)γt(v,r), and this, together with the inequality obtained above, proves the statement of this lemma. +C.2 +The Regularized, Time-independent H¨older-type Condition +In this case, the H¨older exponent, denoted here by ˜γ0, is independent of time, and the results below hold for +each of the two values ˜γ0 is allowed to take in our computations for the regularized linear operator, namely +0 and γ0/2. The H¨older-type condition ˜g(v, r) depends on the first variable v only through the inclusion of +the regularization parameter ε(v) in the argument as follows: +˜g(v, r) = +� +1 − e−κ(r+ε(v))�˜γ0 , +˜γ0 ∈ {0, γ0/2}, and, ε(v) = ε0e− µ +γ0 max(a1,v). +Then the function ˜g has properties similar to those proved in lemmas C.1 -C.7 above. Most of the proofs +are also very similar to those recorded above, therefore we will write down the statements of the lemmas +while omitting some of the proofs. +Lemma C.8. For all 0 < r′ ≤ r, the following are true: +i) ˜gt(v, r) − ˜gt(v, r − r′) ≥ 0, +˜gt(v, r + r′) − ˜gt(v, r) ≥ 0, +ii) ˜gt(v, r) − ˜gt(v − r′, r − r′) ≥ 0, +˜gt(v + r′, r + r′) − ˜gt(v, r) ≥ 0. +Proof. The inequalities contained in part i) are obvious. +For part ii), let us observe that 0 ≤ ε(v − r′) − ε(v) < r′, and 0 ≤ ε(v) − ε(v + r′) < r′. So: +˜g(v − r′, r − r′) = +� +1 − e−κ(r+(ε(v−r′)−r′))�˜γ0 ≤ +� +1 − e−κ(r+ε(v))�˜γ0 = ˜g(v, r), +˜g(v + r′, r + r′) = +� +1 − e−κ(r+(ε(v+r′)+r′))�˜γ0 ≥ +� +1 − e−κ(r+ε(v))�˜γ0 = ˜g(v, r). +Lemma C.9. For all r′ > r, for all (v, r) ∈ R × R+, the following inequality holds: +˜g(v, r) + ˜g(v, r′) − ˜g(v, r + r′) ≥ +� +2 − (1 + e−κ(r+ε(v)))˜γ0� +˜g(v, r). +Proof. A straightforward differentiation reveals +∂ +∂r′ +� +˜g(v, r′) − ˜g(v, r + r′) +� +> 0. +So, ˜g(v, r) + ˜g(v, r′) − ˜g(v, r + r′) > ˜g(v, r) + +� +˜g(v, r′) − ˜g(v, r + r′) +� ���� +r′=r +≥ +� +2 − (1 + e−κ(r+ε(v)))˜γ0 +� +˜g(v, r). +53 + +Lemma C.10. For all r′ > r, for all (v, r) ∈ R × R+, the following inequality holds: +˜g(v, r) + ˜g(v − r + r′, r′) − ˜g(v + r′, r + r′) ≥ +� +2 − (1 + e−κ(r+ε(v)))˜γ0� +˜g(v, r). +Proof. In the region v + r′ ≤ a1, the statement of this lemma is the same as the previous lemma. +When v + r′ > a1, +˜g(v, r) + ˜g(v − r + r′, r′) − ˜g(v + r′, r + r′) +> ˜g(v, r) + ˜g(v + r′, r′) − ˜g(v + r′, r + r′) +> ˜g(v, r) + +� +˜g(v + r′, r′) − ˜g(v + r′, r + r′) +� ���� +r′=r +, by a simple differentiation, +≥ +� +2 − (1 + e−κ(r+ε(v)))˜γ0� +˜g(v, r). +Lemma C.11. For all 0 ≤ r′ < r, for all (v, r) ∈ R × R+, the following inequality holds: +2˜g(v, r) − ˜g(v, r − r′) − ˜g(v, r + r′) ≥ 0. +This lemma is obtained by following the proof of Lemma C.4 exactly and keeping in mind that the +exponent ˜γ0 is time-independent (so a(t) = 0) in this case. +Lemma C.12. For all 0 ≤ r′ < r, for all (v, r) ∈ R × R+, the following inequality holds: +2˜g(v, r) − ˜g(v − r′, r − r′) − ˜g(v + r′, r + r′) ≥ − +� � +1 − e−κ(r−r′+ε(v−r′))�˜γ0 − +� +1 − e−κ(r−r′+ε(v))�˜γ0 � +. +Proof. The inequality of this lemma is obtained by the following simple observation: +2˜g(v, r) − ˜g(v − r′, r − r′) − ˜g(v + r′, r + r′) +≥ 2˜g(v, r) − ˜g(v, r − r′) − ˜g(v, r + r′) − +�� +1 − e−κ(r−r′+ε(v−r′))�˜γ0 − +� +1 − e−κ(r−r′+ε(v))�˜γ0� +≥ − +�� +1 − e−κ(r−r′+ε(v−r′))�˜γ0 − +� +1 − e−κ(r−r′+ε(v))�˜γ0� +, +by Lemma C.11. +Lemma C.13. For all r′ > r, for all (v, r) ∈ R × R+, the following inequality holds: +2 +� +˜g(v, r) + ˜g(v − r + r′, r′) − ˜g(v + r′, r + r′) +� +− ˜g(v, r) > +� +1 − e−r� 1 +2 ˜g(v, r). +The proof of Lemma C.13 follows the corresponding proof for C.6. +Lemma C.14. For all r′ ≥ r, for all p > ˜γ0, for all ν ≤ 1, the following is true: +� +1 − e−νr�p ˜g(v, r′) ≤ +� +1 − e−νr′�p +˜g(v, r). +The proof is the same as that of Lemma C.7. +54 + +D +Computations for Lemma 3.6 +D.1 +Some Useful Inequalities involving the Kernel Function +In the computations for Lemma 3.6 we use certain properties of the kernel function K3 and the weight +function Γt, collected in the lemmas below. +Lemma D.1. For all v − r < −b0, for all 0 < r′ ≤ r the following inequality holds: +K3(v − r, v − r + r′) +� +f(v − r) − (ln(1 + ev−r+r′))−α� +− K1 +3(v − r, v − r − r′) +� +f(v − r − r′) − f(v − r) +� +≥ K3(v − r, v − r + r′) +� +ln(1 + ev−r+r′) +�−α +� +ln(1 + ev−r−r′) +ln(1 + ev−r+r′) +�α  + +1 − +� +ln(1 + ev−r−r′) +ln(1 + ev−r+r′) +�1−2α + + × +× +� +1 − +� +ln(1 + ev−r−r′) +ln(1 + ev−r) +�α� +Proof. Note that: +K3(v − r, v − r + r′) +� +f(v − r) − (ln(1 + ev−r+r′))−α� +− K1 +3(v − r, v − r − r′) +� +f(v − r − r′) − f(v − r) +� +≥ 4¯n +� +ln(1 + ev−r) +�−3/2 +e−r′ +1 − e−r′ +ev−r + 2e−r′ +1 + ev−r + e−r′ +� +ln(1 + ev−r+r′)(ln(1 + ev−r))−α +� +1 − +� ln(1 + ev−r) +ln(1 + ev−r+r′) +�α� +−(ln(1 + ev−r−r′))1−α +� +1 − +� +ln(1 + ev−r−r′) +ln(1 + ev−r) +�α�� +. +Now, +ln(1 + ev−r+r′)(ln(1 + ev−r))−α +� +1 − +� ln(1 + ev−r) +ln(1 + ev−r+r′) +�α� +− (ln(1 + ev−r−r′))1−α +� +1 − +� +ln(1 + ev−r−r′) +ln(1 + ev−r) +�α� += (ln(1 + ev−r+r′))1−2α +� +(ln(1 + ev−r+r′))α +�� +ln(1 + ev−r+r′) +ln(1 + ev−r) +�α +− 1 +�� +− (ln(1 + ev−r−r′))1−2α +� +(ln(1 + ev−r−r′))α +� +1 − +� +ln(1 + ev−r−r′) +ln(1 + ev−r) +�α�� +. +(D.1) +Now for all r′ ≥ 0, let us define the function h as follows: +(ln(1 + ev−r))αh(v − r, r′) = (ln(1 + ev−r+r′))α +�� +ln(1 + ev−r+r′) +ln(1 + ev−r) +�α +− 1 +� +− (ln(1 + ev−r−r′))α +� +1 − +� +ln(1 + ev−r−r′) +ln(1 + ev−r) +�α� +. +Then +∂ +∂r′ h(v − r, r′) = α(ln(1 + ev−r+r′))α−1 +ev−r+r′ +1 + ev−r+r′ +� +2(ln(1 + ev−r+r′))α − (ln(1 + ev−r))α� +55 + +− α(ln(1 + ev−r−r′))α−1 +ev−r−r′ +1 + ev−r−r′ +� +2(ln(1 + ev−r−r′))α − (ln(1 + ev−r))α� +. +Observe that for all v − r < −b0, e−r′ < ln(1+ev−r−r′) +ln(1+ev−r) +< e− 24 +25 r′, since b0 ≥ 10. Then for all r′ > 0 such thtat +� +ln(1 + ev−r−r′) +ln(1 + ev−r) +�α +< 1 +2, +it is obvious from the definition that +∂ +∂r′ h(v − r, r′) > 0. +On the other hand, for all r′ > 0 such that +� +ln(1 + ev−r−r′) +ln(1 + ev−r) +�α +≥ 1 +2, +we have e− 24 +25 αr′ > 1 +2, which, as one can easily check, implies v − r + r′ < −3.5, for all v − r < −b0, and this +in turn means, +(ln(1 + ev−r+r′))α−1 +ev−r+r′ +1 + ev−r+r′ > (ln(1 + ev−r−r′))α−1 +ev−r−r′ +1 + ev−r−r′ , +because +ew +1+ew (ln(1 + ew))α−1 is a strictly increasing function for all w < −3. Thus we have, again, +∂ +∂r′ h(v − r, r′) > 0 in this region. +Since we have proved h(v − r, r′) to be a strictly increasing function of r′, the following inequality holds: +(ln(1 + ev−r+r′))α +�� +ln(1 + ev−r+r′) +ln(1 + ev−r) +�α +− 1 +� +> (ln(1 + ev−r−r′))α +� +1 − +� +ln(1 + ev−r−r′) +ln(1 + ev−r) +�α� +. +Using the above we can refer to equation (D.1) and write: +(ln(1 + ev−r+r′))1−2α +� +(ln(1 + ev−r+r′))α +�� +ln(1 + ev−r+r′) +ln(1 + ev−r) +�α +− 1 +�� +− (ln(1 + ev−r−r′))1−2α +� +(ln(1 + ev−r−r′))α +� +1 − +� +ln(1 + ev−r−r′) +ln(1 + ev−r) +�α�� +> +� +ln(1 + ev−r+r′) +�1−α +� +ln(1 + ev−r−r′) +ln(1 + ev−r+r′) +�α  + +1 − +� +ln(1 + ev−r−r′) +ln(1 + ev−r+r′) +�1−2α + + × +× +� +1 − +� +ln(1 + ev−r−r′) +ln(1 + ev−r) +�α� +, +which obviously implies the inequality stated in this lemma. +Lemma D.2. For all v < −b0, for all 0 < r′ ≤ r and v + r < 0 the following inequality holds: +� r +δ1 +dr′K3(v, v + r′) +� +f(v) − f(v + r′) +� +gt(v, r) ≥ +� r +δ1 +dr′K1 +3(v, v − r′) +� +f(v − r′) − f(v) +� +gt(v − r′, r − r′). +56 + +Proof. Let us define +h2(v, r′) = ln(1+ev+r′) +� +(ln(1 + ev))−α − (ln(1 + ev+r′))−α� +−ln(1+ev−r′) +� +(ln(1 + ev−r′))−α − (ln(1 + ev))−α� +. +Then +∂ +∂r′ h2 = +ev+r′ +1 + ev+r′ +� +(ln(1 + ev))−α − (1 − α)(ln(1 + ev+r′))−α� +− +ev−r′ +1 + ev−r′ +� +(ln(1 + ev))−α − (1 − α)(ln(1 + ev−r′))−α� +> 0, +since (ln(1 + ev−r′))−α > (ln(1 + ev+r′))−α. +Thus h2(v, r′) > h2(v, 0) = 0, for all r′ > 0. Then the inequality stated in the lemma is obvious from the +definition of the kernel functions and the fact that gt(v, r) > gt(v − r′, r − r′), for all r′ > 0. +Lemma D.3. For all v + r′ < 0, for all r′ > r the following is true: +K3(v, v + r′)f(v + r′) − K1 +3(v − r, v − r − r′)f(v − r − r′) > 0. +Proof. +K3(v, v + r′)f(v + r′) − K1 +3(v − r, v − r − r′)f(v − r − r′) +≥ 4¯n +e−r′ +1 − e−r′ +ev + 2e−r′ +1 + ev + e−r′ (ln(1 + ev))− 3 +2(ln(1 + ev+r′))1−α + +1 − +� ln(1 + ev) +ln(1 + ev−r) +� 3 +2 +� +ln(1 + ev−r−r′) +ln(1 + ev+r′) +�1−α + +≥ 4¯n +e−r′ +1 − e−r′ +ev + 2e−r′ +1 + ev + e−r′ (ln(1 + ev))− 3 +2(ln(1 + ev+r′))1−α + +1 − +� +ln(1 + ev−r−r′) +ln(1 + ev−r) +�1−α � ln(1 + ev) +ln(1 + ev−r) +� 1 +2+α + + +> 0, +where we have used the fact that 1 − α > 1 +2 + α, for all α < 1/4, and that +ln(1 + ev−r) +ln(1 + ev−r−r′) > +ln(1 + ev) +ln(1 + ev−r′). +D.2 +Lower Bounds for Ii[Γt](v, r), i ∈ {1, 2, 3, 4} +D.2.1 +Estimating I1[Γt](v, r): +We start with the following lower bound, which can be derived quite easily by using Lemma C.7 proved +above: +I1[Γt](v, r) = I1[Γ1 +t](v, r) + I1[Γ2 +t](v, r) +≥ J0[Γ1 +t + Γ2 +t](v, r) + J1[Γ1 +t](v, r) + J2[Γ2 +t](v, r) + I[Γ1 +t + Γ2 +t ](v, r). +(D.2) +J0 has already been defined in (3.33). We now write down explicitly the other terms. As mentioned before, +these terms are sub-dominant to J0 close to the diagonal. +J1[Γ1 +t ](v, r) +57 + += f(v − r) +� +1(v < −b0) +� +eµa +� r+a1−v +r+δ1 +dr′ � +(K3(v − r, v − r + r′) − K3(v, v + r′))gt(v − r + r′, r′) +− (K1 +3(v − r, v − r − r′) − K1 +3(v, v − r′))gt(v, r′) +� ++ +� ∞ +r+a1−v +dr′ � +(K3(v − r, v − r + r′) − K3(v, v + r′))e− 1 +2 r′eµc0(v−r+r′)gt(v − r + r′, r′) +− (K1 +3(v − r, v − r − r′) − K1 +3(v, v − r′))eµagt(v, r′) +�� ++ 1(v ≥ −b0) +� ∞ +r+δ1 +dr′ � � +K3(v − r, v − r + r′) − K3(v, v + r′) +� +e− 1 +2 r′eµ max(a,c0(v−r+r′),v−r)gt(v − r + r′, r′) +− eµ max(a,c0v,v−r) � +K1 +3(v − r, v − r − r′) − K1 +3(v, v − r′) +� +gt(v, r′) +�� +, +(D.3) +J2[Γ2 +t ](v, r) +=1(v < −b0) +� +1(r ≤ a1 − v ≤ r + δ1) +� � ∞ +r+δ1 +dr′ � � +K3(v − r, v − r + r′) − K3(v, v + r′) +� +e− 1 +2r′ × +× exp +� +µ max{a, c0(v − r + r′), v − r} +� +f(v − r + r′)gt(v − r + r′, r′) +− +� +K1 +3(v − r, v − r − r′) − K1 +3(v, v − r′) +� +exp +� +µ max{a, c0v, v − r′} +� +f(v)gt(v, r′) +�� ++ +1(r + δ1 < a1 − v) +� � a1−v +r+δ1 +dr′ � � +K3(v − r, v − r + r′) − K3(v, v + r′) +� +× +× exp +� +µ max{a, c0(v − r + r′), v − r} +� +f(v + r′)gt(v − r + r′, r′) +− +� +K1 +3(v − r, v − r − r′) − K1 +3(v, v − r′) +� +exp +� +µ max{a, c0v, v − r′} +� +f(v)gt(v, r′) +� ++ +� ∞ +a1−v +dr′ � � +K3(v − r, v − r + r′) − K3(v, v + r′) +� +e− 1 +2r′ × +× exp +� +µ max{a, c0(v − r + r′), v − r} +� +f(v + r′)gt(v − r + r′, r′) +− +� +K1 +3(v − r, v − r − r′) − K1 +3(v, v − r′) +� +exp +� +µ max{a, c0v, v − r′} +� +f(v)gt(v, r′) +�� ++ +1(r > a1 − v) +� � r+a1−v +r+δ1 +dr′ � � +K3(v − r, v − r + r′) − K3 +2(v, v + r′) +� +× +× exp +� +µ max{a, c0(v − r + r′), v − r} +� +f(v + r′)gt(v − r + r′, r′) +− +� +K1 +3(v − r, v − r − r′) − K1 +3(v, v − r′) +� +exp +� +µ max{a, c0v, v − r′} +� +f(v)gt(v, r′) +� ++ +� ∞ +r+a1−v +dr′ � � +K3(v − r, v − r + r′) − K3(v, v + r′) +� +e− 1 +2r′ × +× exp +� +µ max{a, c0(v − r + r′), v − r} +� +f(v + r′)gt(v − r + r′, r′) +− +� +K1 +3(v − r, v − r − r′) − K1 +3(v, v − r′) +� +exp +� +µ max{a, c0v, v − r′} +� +f(v)gt(v, r′) +��� +58 + ++ +1(v ≥ −b0) +� � ∞ +r+δ1 +dr′ � � +K3(v − r, v − r + r′) − K3(v, v + r′) +� +e− 1 +2 r′ × +× exp +� +µ max{a, c0(v − r + r′), v − r} +� +f(v + r′)gt(v − r + r′, r′) +− +� +K1 +3(v − r, v − r − r′) − K1 +3(v, v − r′) +� +exp +� +µ max{a, c0v, v − r′} +� +f(v)gt(v, r′) +�� +, +(D.4) +and finally, +I[Γ1 +t + Γ2 +t ](v, r) += − eµ max(a,c0v) +� ∞ +r+δ1 +dr′ K1 +3(v, v − r′) +� +f(v − r − r′) − f(v − r′) +� +gt(v, r′) +− eµ max(a,c0v) +� ∞ +r+δ1 +dr′ � +K1 +3(v − r, v − r − r′) − K1 +3(v, v − r′) +� � +f(v − r − r′) − f(v − r) +� +gt(v, r′) +− eµ max(a,c0v) +� ∞ +r+δ1 +dr′ K1 +3(v − r, v − r − r′) +� +f(v − r − r′) − f(v − r) +� � +gt(v, r + r′) − gt(v, r′) +� ++ I(1)[Γ1 +t ](v, r) + I(2)[Γ2 +t ](v, r) + I(3)[Γ1 +t + Γ2 +t](v, r), +(D.5) +with +I(1)[Γ1 +t ](v, r) += 1(v < −b0)f(v − r) +� � r+a1−v +r+δ1 +dr′ K3(v, v + r′) +� +eµa − e− 1 +2 r′eµ max{a,v−r,c0(v+r′)}� +× +× +� +gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′) +� ++ +1(a1 − v ≥ r + δ1) +� r+a1−v +a1−v +dr′ K3(v, v + r′) +� +eµa − eµc0ve−( 1 +2−µc0)r′� +gt(v + r′, r + r′) ++ +1(a1 − v < r + δ1) +� r+a1−v +r+δ1 +dr′ K3(v, v + r′) +� +eµa − eµc0ve−( 1 +2 −µc0)r′� +gt(v + r′, r + r′) +� +, +(D.6) +I(2)[Γ2 +t ](v, r) += +1(v < −b0) +� � ∞ +max(r+δ1,a1−v) +dr′ K3(v, v + r′)f(v + r′) (eµa − eµc0v) gt(v, r) ++ +� ∞ +r+δ1 +dr′ K3(v, v + r′) +� +f(v) − f(v + r′) +� +eµagt(v, r) ++ +1(r ≤ a1 − v ≤ r + δ1) +� � r+a1−v +r+δ1 +dr′ K3 +2(v, v + r′)f(v + r′) +� +eµa − eµc0(v−r+r′)� +gt(v − r + r′, r′) ++ +� ∞ +r+δ1 +dr′ K3 +2(v, v + r′) +� +f(v − r + r′) − f(v + r′) +� +eµ max{a,c0(v−r+r′)}gt(v − r + r′, r′) ++ +� ∞ +r+δ1 +dr′ K3(v, v + r′)f(v + r′)eµc0v � +1 − 2e−( 1 +2−µc0)r′ + e−2( 1 +2 −µc0)r′� +gt(v, r) ++ +� r+a1−v +r+δ1 +dr′ K3 +1(v − r, v − r + r′)f(v − r + r′)eµagt(v − r + r′, r′) +� +59 + ++ +1(a1 − v > r + δ1) +� � ∞ +r+δ1 +dr′ K3 +2(v − r, v − r + r′) +� +f(v − r + r′) − f(v + r′) +� +× +× eµ max{a,c0(v−r+r′)}gt(v − r + r′, r′) ++ +� r+a1−v +r+δ1 +dr′ � +K3 +1(v − r, v − r + r′) − K3 +1(v, v + r′) +� � +f(v − r + r′) − f(v + r′) +� +× +× eµ max{a,c0(v−r+r′)}gt(v − r + r′, r′) ++ +� r+a1−v +a1−v +dr′ � +K3 +1(v − r, v − r + r′) − K3 +1(v, v + r′) +� +f(v + r′)eµ max{a,c0(v−r+r′)}gt(v − r + r′, r′) ++ eµa +� a1−v +r+δ1 +dr′ K3 +1(v, v + r′)f(v + r′) +� +gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′) +� ++ +� r+a1−v +a1−v +dr′ K3(v, v + r′)f(v + r′) +� +eµc0v � +1 − e−( 1 +2−µc0)r′� +gt(v, r)+ ++ +� +eµa − eµc0ve−( 1 +2−µc0)r′� +gt(v − r + r′, r′) +� ++ eµa +� r+a1−v +r+δ1 +dr′ K3 +1(v, v + r′) +� +f(v − r + r′) − f(v + r′) +� +gt(v − r + r′, r′) +� ++ +1(a1 − v < r) +� � r+a1−v +r+δ1 +dr′ K3 +2(v, v + r′)f(v + r′) +� +eµa − eµc0(v−r+r′)� +gt(v − r + r′, r′) ++ eµa +� r+a1−v +r+δ1 +dr′ K3 +2(v, v + r′) +� +f(v − r + r′) − f(v + r′) +� +gt(v − r + r′, r′) ++ +� r+a1−v +r+δ1 +dr′ K3 +1(v − r, v − r + r′)eµ max{a,c0(v−r+r′)} � +f(v − r + r′) − f(v + r′) +� +gt(v − r + r′, r′) ++ +� ∞ +r+δ1 +dr′ � +K3 +2(v − r, v − r + r′) − K3 +2(v, v + r′) +� � +f(v − r + r′) − f(v + r′) +� +× +× eµ max{a,c0(v−r+r′)}gt(v − r + r′, r′) +�� +, +(D.7) +and finally, +I(3)[Γ2 +t](v, r) += 1(v ≥ −b0) +� ∞ +r+δ1 +dr′ K3 +2(v − r, v − r + r′) +� +f(v − r + r′) − f(v + r′) +� +eµ max(a,v−r,c0(v−r+r′))gt(v − r + r′, r′) +(D.8) +The parts J1 and J2 yield some negative terms that will need to be controlled, as we will see shortly. +D.2.2 +Estimating I2[Γt](v, r) +I2[Γt](v, r) = I2[Γ1 +t ](v, r) + I2[Γ2 +t ](v, r). +60 + +1. A Lower Bound for I2[Γ1 +t](v, r) +Recall from (3.30) +I2[Γ1 +t ](v, r) +≥ +1(v − r < −b0)eµ max{a,c0v} +� r +δ1 +dr′ � +K3(v − r, v − r + r′) +� +f(v − r) − (ln(1 + ev−r+r′))−α� +− K1 +3(v − r, v − r − r′) +� +f(v − r − r′) − f(v − r) +� � +gt(v, r) ++ +1(v < −b0)1(r > a − v)f(v − r) +� r +a1−v +dr′ K3(v, v + r′) +� +eµa − eµc0ve−( 1 +2 −µc0)r′� +gt(v, r) ++ +1(v ≥ −b0)1(v − r < −b0) +� +f(v − r) +� r +δ1 +dr′ K3(v, v + r′) +� +eµ max{a,c0v} − e− 1 +2r′eµ max{a,c0(v+r′)}� +gt(v, r) ++ +� r +min(r,r−v) +dr′ K3(v − r, v − r + r′)(ln(1 + ev−r+r′))−α � +2gt(v, r) − gt(v, r + r′) +� � ++ I− +2 [Γ1 +t ](v, r), +where +I− +2 [Γ1 +t](v, r) += − f(v − r)eµ max(a,v−r,c0v) +� r +δ1 +dr′ � +K3(v, v + r′) − K1 +3(v, v − r′) +� � +gt(v + r′, r + r′) − gt(v, r + r′) +� +− 1(v < −b0)eµa� +4¯n +� +ln(1 + ev−r) +�− 3 +2 +� r +δ1 +dr′ � +ln(1 + ev−r+r′) +�1−α +e−r′ +1 − e−r′ × +× +� +1 − +� +ln(1 + ev−r−r′) +ln(1 + ev−r+r′) +�α� +� +gt(v, r + r′) − gt(v, r) +� ++ 4¯nf(v − r) (ln(1 + ev))− 3 +2 +� r +δ1 +dr′ ln(1 + ev+r′) +ev−r′(1 − e−r′)(1 − e− 5 +4 r′) +(1 + ev−r + e−r′)(1 + ev + e−r′)× +× +� +gt(v, r + r′) − gt(v, r) +� � +− 1(v ≥ −b0) +� +1(v − r < −b0)eµ max(a,c0v)� +4¯n +� +ln(1 + ev−r) +�− 3 +2 +� r +δ1 +dr′ � +ln(1 + ev−r−r′) +�1−α +× +× +e−r′ +1 − e−r′ +� +1 − +� +ln(1 + ev−r−r′) +ln(1 + ev−r+r′) +�α� +� +gt(v, r + r′) − gt(v, r) +� ++ 4¯nf(v − r) (ln(1 + ev))− 3 +2 +� r +δ1 +dr′ ln(1 + ev+r′)e−r′(1 − e−r−2r′) +(1 + ev−r + e−r′) × +� +gt(v, r + r′) − gt(v, r) +� � ++ +1(v − r ≥ −b0)eµ max(a,c0v,v−r)� +4¯nf(v − r) (ln(1 + ev))− 3 +2 +� r +δ1 +dr′ ln(1 + ev+r′) e−r′(1 − e−2r′) +(1 + ev−r + e−r′) × +× +� +gt(v, r + r′) − gt(v, r) +� ++ 4¯nf(v − r) (ln(1 + ev))− 3 +2 +� r +δ1 +dr′ ln(1 + ev−r′) +ev−r′(1 − e−r′) +(1 + ev−r′ + e−r′)(1 + ev + e−r′)× +× +� +gt(v, r + r′) − gt(v, r) +� �� +(D.9) +61 + +2. A Lower Bound for I2[Γ2 +t](v, r): +I2[Γ2 +t](v, r) ≥ +1(v < −b0) +� +eµa +� r +δ1 +dr′ K3(v, v + r′) +� +f(v) − f(v + r′) +� +gt(v, r) ++ +1(r > a1 − v) +� r +δ1 +dr′ K3(v, v + r′)f(v + r′) +� +eµa − eµc0ve−( 1 +2−µc0)r′� +gt(v, r) +� ++ +1(v ≥ −b0) +� r +δ1 +dr′ K3(v, v + r′)f(v) +� +eµ max(a,v−r,c0v) − e− 1 +2 r′eµ max(a,v−r,c0(v+r′)� +gt(v, r) ++ I− +2 [Γ2 +t](v, r), +(D.10) +where +I− +2 [Γ2 +t](v, r) += − eµ max(a,v−r,c0v)f(v) +� r +δ1 +dr′ � +K3(v, v + r′) − K1 +3(v, v − r′) +� � +gt(v + r′, r + r′) − gt(v, r + r′) +� +− 4¯n eµ max(a,v−r,c0v)f(v) (ln(1 + ev))− 3 +2 +� r +δ1 +dr′ ln(1 + ev+r′) e−r′(1 − e−2r′) +1 + ev−r + e−r′ +� +gt(v, r + r′) − gt(v, r) +� +− 4¯n eµ max(a,v−r,c0v)f(v) (ln(1 + ev))− 3 +2 +� r +δ1 +dr′ ln(1 + ev−r′) +ev−r′(1 − e−r′) +(1 + ev−r′ + e−r′)(1 + ev + e−r′)× +× +� +gt(v, r + r′) − gt(v, r) +� +D.2.3 +Lower Bound for I3[Γt](v, r): +I3[Γt](v, r) = I(1) +3 [Γt](v, r) + I(2) +3 [Γt](v, r). +For I3 we will just write out the relevant definitions explicitly. +Expression for I(1) +3 [Γt](v, r) : +a) I(1) +3 [Γ1 +t ](v, r) += +1(v − r ≤ −m0)eµ max(a,c0v)f(v − r) +� � ∞ +˜r +dr′ K2 +3(v − r, v − r − r′)gt(v, r) ++ +� ˜r +0 +dr′ � +K2 +3(v − r, v − r − r′)gt(v, r) − (K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′))gt(v, r + r′) +�� ++ +1(−m0 < v − r < m0)eµ max(a,v−r,c0v)f(v − r) +� ∞ +0 +dr′ K2 +3(v − r, v − r − r′)gt(v, r) ++ +1(v − r ≥ m0)f(v − r) +� +1(v ≤ 3r) +� +eµ max(a,v−r,c0v) +� ∞ +v−r +dr′K2 +3(v − r, v − r − r′)gt(v, r) ++ +� v−r +0 +dr′ � +eµ max(a,v−r,c0v)K2 +3(v − r, v − r − r′)gt(v, r) +− eµ max(a,v−r−r′,c0v) � +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +gt(v, r + r′) +�� ++ +1(v > 3r) +� +eµ max(a,v−r,c0v) +� v−r−c0v +0 +dr′ K2 +3(v − r, v − r − r′)gt(v, r) ++ +� ∞ +v−r−c0v +dr′ � +eµ max(a,v−r,c0v)K2 +3(v − r, v − r − r′)gt(v, r) +62 + +− eµ max(a,v−r−r′,c0v) � +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +gt(v, r + r′) +��� ++ A− +1 [Γ1 +t](v, r) + A− +2 [Γ1 +t](v, r) + A− +3 [Γ1 +t](v, r) + A− +4 [Γ1 +t](v, r), +(D.11) +where +A− +1 [Γ1 +t ](v, r) += − +1(v − r ≤ −m0)1(v < −b0)1(˜r > r + δ1) +� ˜r +r+δ1 +dr′ � +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +× +× eµ max(a,c0v) � +f(v − r − r′) − f(v − r) +� +gt(v, r + r′) +(D.12) +A− +2 [Γ1 +t](v, r) = − 1(v − r ≤ −m0) +� min(r,˜r) +min(δ1,˜r) +dr′ � +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +× +× eµ max(a,c0v) � +f(v − r − r′) − f(v − r) +� +gt(v, r + r′) +(D.13) +A− +3 [Γ1 +t](v, r) = − 1(v − r ≥ m0)1(v > 3r) +� ∞ +v−r +dr′ � +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +× +× eµ max(a,c0v) � +f(v − r − r′) − f(v − r) +� +gt(v, r + r′) +(D.14) +A− +4 [Γ1 +t](v, r) = − 1(v − r ≤ −m0)eµ max(a,c0v)� � min(δ1,˜r) +0 +dr′ � +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +× +× +� +f(v − r − r′) − f(v − r) +� +gt(v, r + r′) ++ +1(v < −b0) +� min(˜r,r+δ1) +r +dr′ � +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +× +× +� +f(v − r − r′) − f(v − r) +� +gt(v, r + r′) +� +. +(D.15) +b) I(1) +3 [Γ2 +t](v, r) +=1(v − r ≤ −m0)f(v)eµ max(a,c0v) +� � ∞ +˜r +dr′ K2 +3(v − r, v − r − r′)gt(v, r) ++ +� ˜r +0 +dr′ � +K2 +3(v − r, v − r − r′)gt(v, r) − +� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +gt(v, r + r′) +�� ++1(−m0 < v − r < m0)f(v)eµ max(a,v−r,c0v) +� ∞ +0 +dr′K2 +3(v − r, v − r − r′)gt(v, r) ++1(v − r ≥ m0)f(v) +� +1(v ≤ 3r) +� � v−r +0 +dr′ � +eµ max(a,v−r,c0v) − eµ max(a,v−r−r′,c0v)� +× +× K2 +3(v − r, v − r − r′)gt(v, r) +63 + ++ +� v−r +0 +dr′ eµ max(a,v−r−r′,c0v)� +K2 +3(v − r, v − r − r′)gt(v, r) − +− +� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +gt(v, r + r′) +� ++ eµ max(a,c0v,v−r) +� ∞ +v−r +dr′ K2 +3(v − r, v − r − r′)gt(v, r) +� ++ +1(v > 3r) +� +eµ max(a,c0v,v−r) +� v−r−c0v +0 +dr′ K2 +3(v − r, v − r − r′)gt(v, r) ++ +� ∞ +v−r−c0v +dr′ K2 +3(v − r, v − r − r′) +� +eµ max(a,c0v,v−r) − eµ max(a,c0v,v−r−r′)� +gt(v, r) ++ +� ∞ +v−r−c0v +dr′ eµ max(a,v−r−r′,c0v)� +K2 +3(v − r, v − r − r′)gt(v, r) +− +� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +gt(v, r + r′) +��� +. +(D.16) +Expression for I(2) +3 [Γt](v, r) : +a) I(2) +3 [Γ1 +t ](v, r) += +1(v ≤ −m0)eµaf(v − r) +� +1(v + r < −b0) +� r +0 +dr′ K2 +3(v, v − r′) +� +gt(v, r) − gt(v − r′, r − r′) +� ++ +1(v + r ≥ −b0) +� −v−b0 +0 +dr′ K2 +3(v, v − r′) +� +gt(v, r) − gt(v − r′, r − r′) +� ++ +1(v + r ≥ −b0) +� r +−v−b0 +dr′ K2 +3(v, v − r′)gt(v, r) +� ++ +1(−m0 < v < m0)eµ max(a,v−r,c0v)f(v − r) +� r +0 +dr′ K2 +3(v, v − r′)gt(v, r) ++ +1(v ≥ m0)f(v − r) +� +1(v ≤ r)eµ max(a,v−r,c0v) +� r +0 +dr′ K2 +3(v, v − r′)gt(v, r) ++ +1(0 < v − r < c0v) +� +eµ max(a,v−r,c0v) +� r +0 +dr′ K2 +3(v, v − r′)gt(v, r) +− +� r +v−c−1 +0 (v−r) +dr′ K2 +3(v, v − r′)eµ max(a,v−r,c0(v−r′))gt(v − r′, r − r′) +� ++ +1(v − r ≥ c0v) +� r +0 +dr′ K2 +3(v, v − r′) +� +eµ max(a,v−r,c0v)gt(v, r) − eµ max(a,v−r,c0(v−r′))gt(v − r′, r − r′) +�� +(D.17) +b) I(2) +3 [Γ2 +t ](v, r) += +1(v ≤ −m0)eµa� +− +1(v + r < −b0) +� r +0 +dr′ K2 +3(v, v − r′) +� +f(v − r′) − f(v) +� +gt(v − r′, r − r′) +− +1(v + r ≥ −b0) +� −v−b0 +0 +dr′ K2 +3(v, v − r′) +� +f(v − r′) − f(v) +� +gt(v − r′, r − r′) +64 + ++ +1(v + r < −b0)f(v) +� r +0 +dr′ K2 +3(v, v − r′) +� +gt(v, r) − gt(v − r′, r − r′) +� ++ +1(v + r ≥ −b0)f(v) +� � r +−v−b0 +dr′ K2 +3(v, v − r′)gt(v, r) ++ +� −v−b0 +0 +dr′ K2 +3(v, v − r′) +� +gt(v, r) − gt(v − r′, r − r′) +� �� ++ +1(−m0 < v < m0)eµ max(a,v−r,c0v)f(v) +� r +0 +dr′ K2 +3(v, v − r′)gt(v, r) ++ +1(v ≥ m0) +� +1(v ≤ r)eµ max(a,v−r,c0v)f(v) +� r +0 +dr′ K2 +3(v, v − r′)gt(v, r) ++ +1(0 < v − r < c0v) +� +eµ max(a,v−r,c0v)f(v) +� v−c−1 +0 (v−r) +0 +dr′ K2 +3(v, v − r′)gt(v, r) ++ +� r +v−c−1 +0 (v−r) +dr′ K2 +3(v, v − r′)f(v) +� +eµ max(a,v−r,c0v)gt(v, r) − eµ max(a,v−r,c0(v−r′))gt(v − r′, r − r′) +�� ++ +1(v − r ≥ c0v)eµ max(a,v−r)f(v) +� r +0 +dr′ K2 +3(v, v − r′) +� +gt(v, r) − gt(v − r′, r − r′) +� � +. +(D.18) +Let us recall that the correct asymptotic behavior of LuΓt for large, positive values of v comes from +1(v−r ≥ +m0)I(1) +3 [Γt](v, r) (when v and (v − r) are comparable) and +1(v ≥ m0)I(2) +3 [Γt](v, r) (when v is much bigger +than (v − r)). +We will look at I4 later because we already know it inherits a degree of “smallness” from the length of +the interval of integration. +D.3 +Combining the Estimates to arrive at the Main Lower Bound: +In this section we combine the terms written out in the previous subsection, in a way which will enable us +to find a useful lower bound on LuΓt(v, r). As mentioned already in Lemma 3.6, J0[Γt](v, r) from the lower +bound for I1[Γt](v, r) and C3[Γ1 +t](v, r), defined in (3.35), from the lower bound for I2[Γ1 +t](v, r), contribute to +LuΓt(v, r) the correct asymptotic behavior with point singularity. +Our main objective now is to tackle the negative terms obtained in the estimates above by using some +suitable combinations of integrals. We enumerate these combinations as Comb.1, Comb.2 etc. These com- +binations yield useful lower bounds for sums of the Ii’s, which eventually lead to our main estimate. In +what follows these lower bounds are denoted by LB1, LB2 etc. In other words, for ease of referencing, we +will denote the most important estimates as LB1, LB2 etc., the particular combinations of terms leading to +these estimates by Comb.1, Comb.2 etc. and reserve the usual numbering by section for everything else. +Note that terms of the form (gt(., r + r′) − gt(., r)) generate an extra exponential decay of e−κr. This +additional “smallness” makes it easy for us to deal with such negative terms. So we will focus first on those +negative terms which do not contain differences of the form (gt(., r + r′) − gt(., r)). +We will first look at the first two negative terms in (D.5). We combine the first negative term with terms +from I(2)[Γ2 +t](v, r) and I1 +3[Γ2 +t ](v, r) as follows: +− eµ max(a,c0v) +� ∞ +r+δ1 +dr′ K1 +3(v, v − r′) +� +f(v − r − r′) − f(v − r′) +� +gt(v, r′) +(Comb.1) ++ +1(v − r ≤ −m0)eµ max(a,c0v)f(v) +� ∞ +˜r +dr′ K2 +3(v − r, v − r − r′)gt(v, r) +65 + ++ +1(−m0 < v − r < m0)eµ max(a,v−r,c0v)f(v) +� ∞ +r+δ1 +dr′ K2 +3(v − r, v − r − r′)gt(v, r) ++ +1(v < −b0)1(r + δ1 < a1 − v) +� ∞ +r+δ1 +dr′ K3 +2(v − r, v − r + r′) +� +f(v − r + r′) − f(v + r′) +� +× +× eµ max(a,c0(v−r+r′))gt(v − r + r′, r′) +≥ − +1(v ≥ 0)eµ max(a,c0v) +� 2r+δ1 +r+δ1 +dr′ K1 +3(v, v − r′) +� +f(v − r − r′) − f(v − r′) +� +gt(v, r′) +− +1(v − r ≥ m0)eµ max(a,c0v) +� ∞ +2r+δ1 +dr′ K1 +3(v, v − r′) +� +f(v − r − r′) − f(v − r′) +� +gt(v, r′) + B+ +1 [Γ2 +t](v, r), +where +B+ +1 [Γ2 +t](v, r) += +1(v < −b0)eµaf(v)gt(v, r) +� +1(v − r < −m0) +� ∞ +max(−v−b0,r) +dr′ K2 +3(v − r, v − r − r′) × +× +� +1 − e−( 122 +250 −2α)r′ + e−( 122 +250 −α)re−r′ 1 − e−(1−α)r′ +1 − e−r′ +� ++ +1(v − r ≥ −m0) +� ∞ +r+δ1 +dr′ K2 +3(v − r, v − r − r′)× +× +� +1 − e−( 122 +250 −2α)r′ + e−( 122 +250 −α)re−r′ 1 − e−(1−α)r′ +1 − e−r′ +�� ++ 1(v ≥ −b0) +� +eµaf(v)gt(v, r)1(−b0 ≤ v < 0) +� ∞ +r+δ1 +dr′ K2 +3(v − r, v − r − r′) +� +1 − e−2( 1 +3−α)r′e−( 1 +3+α)(r′−r) ++ e−(r′−r)e−( 2 +3 −α)r 1 − e−(1−α)r′ +1 − e−r′ +� ++ +1(v ≥ 0)1(v − r < m0)f(v)gt(v, r) +� +eµ max(a,c0v) +� ∞ +r+δ1 +dr′ K2 +3(v − r, v − r − r′) × +× +� +1 − +�ln(1 + ev−r) +ln(1 + ev) +� 3 +2 +e−(1−2α)re−(1−α)r′� ++ 4¯n eµ max(a,c0v) (ln(1 + ev))− 3 +2 +� ∞ +r+δ1 +dr′ ln(1 + ev−r−r′) ev−r−r′ + 2e−r′ +1 + ev−r−r′ + e−r′ +1 − e−(1−α)(r+r′) +1 − e−r−r′ +e−r′e−(1−α)r ++ +� ∞ +r+δ1 +dr′ K2 +3(v − r, v − r − r′) +� +eµ max(a,v−r,c0v) − eµ max(a,c0v)��� +(D.19) +Now we look at the second term in (D.5). We combine this with a term from I(2)[Γ2 +t](v, r) and B+ +1 [Γ2 +t](v, r) +as follows. This is the second combination Comb.2. +− eµ max(a,c0v) +� ∞ +r+δ1 +dr′ � +K1 +3(v − r, v − r − r′) − K1 +3(v, v − r′) +� � +f(v − r − r′) − f(v − r) +� +gt(v, r′) +(Comb.2) ++ eµa +1(v < −b0)1(a1 − v > r + δ1) +� a1−v +r+δ1 +dr′ K3 +1(v, v + r′)f(v + r′)× +66 + +× +� +gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′) +� ++ B+ +1 [Γ2 +t ](v, r) +≥ − +1(v − r ≥ m0)eµ max(a,c0v) +� ∞ +r+δ1 +dr′ � +K1 +3(v − r, v − r − r′) − K1 +3(v, v − r′) +� +× +× +� +f(v − r − r′) − f(v − r) +� +gt(v, r′) ++ E1[Γ2 +t](v, r) + E2[Γ2 +t ](v, r), +where +E1[Γ2 +t ](v, r) += 1(v < −b0)eµa +� +1(−v − b0 > r + δ1) +� � a1−v +−v−b0 +dr′ K3 +1(v, v + r′)f(v + r′) +� +2 − (1 + e−κr)γt(v,r)� +gt(v, r) ++ +� −v−b0 +r+δ1 +dr′ K3 +1(v, v + r′)f(v + r′) +� +1 − e− 124 +125 (1−α)r′� � +1 − e−κr�2γ1 ++ f(v) +� ∞ +−v−b0 +dr′ K2 +3(v − r, v − r − r′) +� � +1 − e−( 122 +250 −2α)r′� � +1 − e−(1−2α)r′� ++ e− 116 +750 r′e−(1−2α)r′(1 − e−αr′)1 − e− 634 +750 r′ +1 − e−r′ +� +gt(v, r) +� ++ +1(−v − b0 ≤ r + δ1)eµa +� +1(r + δ1 ≤ a1 − v) +� a1−v +r+δ1 +dr′ K3 +1(v, v + r′)f(v + r′) × +× +� +2 − (1 + e−κr)γt(v,r)� +gt(v, r) ++ f(v) +� ∞ +r+δ1 +dr′ K2 +3(v − r, v − r − r′) +� +1 − e−( 122 +250 −2α)r′� � +1 − e−(1−2α)r′� +gt(v, r′) +� ++ eµaf(v) +� ∞ +max(−v−b0,r+δ1) +dr′ K2 +3(v − r, v − r − r′)e−r′e−( 122 +250 −α)r 1 − e−(1−α)r′ +1 − e−r′ +gt(v, r) +� +, +(D.20) +and, +E2[Γ2 +t](v, r) += 1(v ≥ −b0) +� +1(−b0 ≤ v < 0)eµaf(v) +� ∞ +r+δ1 +dr′ K2 +3(v − r, v − r − r′) +� � +1 − e−2( 1 +3 −α)r′� � +1 − e−(1−2α)r′� ++ e−(1−α)r′ � +1 − e−( 2 +3−3α)r′� ++ e−(r′−r)e−( 2 +3−α)r 1 − e−(1−α)r′ +1 − e−r′ +� +gt(v, r) ++ +1(v ≥ 0)1(v − r < m0)f(v)gt(v, r) +� +eµ max(a,c0v) +�ln(1 + ev−r) +ln(1 + ev) +� 3 +2 � ∞ +r+δ1 +dr′ K2 +3(v − r, v − r − r′)× +× e−r′e−(1−α)r 1 − e−(1−α)(r+r′) +1 − e−(r+r′) ++ +1(v − r ≤ max(a, c0v)) +� ∞ +r+δ1 +dr′ K2 +3(v − r, v − r − r′) +� � +1 − e−(1−2α)r′� � +1 − e−(1−α)r′e−(1−2α)r� +67 + ++ e−(1−α)r′ � +1 − e−(1−2α)(r+r′)� � +eµ max(a,c0v) ++ +1(v − r > max(a, c0v)) +� +eµ(v−r) − eµ max(a,c0v)� � ∞ +r+δ1 +dr′ K2 +3(v − r, v − r − r′) +� +1 − e−(1−2α)re−(1−α)r′� ++ +1(v − 2r − δ1 < max(a, c0v) < v − r) +� ∞ +r+δ1 +dr′ K2 +3(v − r, v − r − r′)e−αr′ � +1 − e−αr ++e−αr(1 − e−(1−3α)(r+r′)) +� +eµ max(a,c0v) ++ +1(v − 2r − δ1 ≥ max(a, c0v)) +� ∞ +r+δ1 +dr′ K2 +3(v − r, v − r − r′) +� � +1 − e−(1−2α)r′� � +1 − e−(1−α)r′e−(1−2α)r� ++ e−(1−α)r′ � +1 − e−(1−2α)(r+r′)� � +eµ max(a,c0v) +�� +(D.21) +Clearly, what the combinations Comb.1 and Comb.2 do to the two negative terms of I, is push them away +from the point singularities into regions where v − r > 0 and v > 0. +We now combine a term from E1[Γ2 +t ](v, r) with A−1 +1 [Γ1 +t](v, r) as follows, in Comb.3: +A− +1 [Γ1 +t](v, r) +(Comb.3) ++ +1(v < −b0)1(−v − b0 > r + δ1)eµa +� −v−b0 +r+δ1 +dr′ K3 +1(v, v + r′)f(v + r′) +� +1 − e− 124 +125 (1−α)r′� � +1 − e−κr�2γ1 +≥ 1(r + δ1 < −v − b0) +� +1(v − r < −m0)4¯n (ln(1 + ev))− 3 +2 +� −v−b0 +r+δ1 +dr′ +ev−r′(ln(1 + ev+r′))1−α +(1 + ev + e−r′)(1 + ev−r−r′ + e−r′) × +× (1 − e− 1 +2r′)(1 − e−(r′+r)) +� +1 − e− 124 +125 (1−α)r′� +(1 − e−κr)2γ1 ++ +1(v − r ≥ −m0) +� −v−b0 +r+δ1 +dr′ K3 +1(v, v + r′)f(v + r′) +� +1 − e− 124 +125 (1−α)r′� +(1 − e−κr)2γ1 +� +eµa. +Let us define: +E′ +1[Γ2 +t](v, r) += E1[Γ2 +t](v, r) − +1(−v − b0 > r + δ1)eµa +� −v−b0 +r+δ1 +dr′ K3 +1(v, v + r′)f(v + r′) +� +1 − e− 124 +125 (1−α)r′� � +1 − e−κr�2γ1 ++ 1(−v − b0 > r + δ1) +� +1(v − r < −m0)4¯n (ln(1 + ev))− 3 +2 +� −v−b0 +r+δ1 +dr′ +ev−r′(ln(1 + ev+r′))1−α +(1 + ev + e−r′)(1 + ev−r−r′ + e−r′) × +× (1 − e− 1 +2r′)(1 − e−(r′+r)) +� +1 − e− 124 +125 (1−α)r′� +(1 − e−κr)2γ1 ++ +1(v − r ≥ −m0) +� −v−b0 +r+δ1 +dr′ K3 +1(v, v + r′)f(v + r′) +� +1 − e− 124 +125 (1−α)r′� +(1 − e−κr)2γ1 +� +eµa. +(D.22) +68 + +This means we can put together the combinations (Comb.1), (Comb.2) and (Comb.3), and obtain the +following lower bound: +I[Γ1 +t + Γ2 +t](v, r) + I(1) +3 [Γ2 +t ](v, r) + A− +1 [Γ1 +t ](v, r) +(LB1) +≥ B− +1 [Γ1 +t ](v, r) + B2[Γ2 +t ](v, r) + I(1)[Γ1 +t](v, r) + B3[Γ2 +t](v, r) + E′ +1[Γ2 +t ](v, r) + E2[Γ2 +t](v, r) + I(3)[Γ2 +t ](v, r), +where +B− +1 [Γ1 +t](v, r) += − +1(v − r ≥ m0)eµ max(a,c0v)� � ∞ +2r+δ1 +dr′ K1 +3(v, v − r′) +� +f(v − r − r′) − f(v − r′) +� +gt(v, r′) ++ +� ∞ +r+δ1 +dr′ � +K1 +3(v − r, v − r − r′) − K1 +3(v, v − r′) +� � +f(v − r − r′) − f(v − r) +� +gt(v, r′) +� +− +1(v ≥ 0)eµ max(a,c0v) +� 2r+δ1 +r+δ1 +dr′ K1 +3(v, v − r′) +� +f(v − r − r′) − f(v − r′) +� +gt(v, r′) +− eµ max(a,c0v) +� ∞ +r+δ1 +dr′ K1 +3(v − r, v − r − r′) +� +f(v − r − r′) − f(v − r) +� +(gt(v, r + r′) − gt(v, r′)), (D.23) +B2[Γ2 +t](v, r) += I(1) +3 [Γ2 +t](v, r) − +1(v − r ≤ −m0)eµ max(a,c0v)f(v) +� ∞ +˜r +dr′ K2 +3(v − r, v − r − r′)gt(v, r) +− +1(−m0 < v − r < m0)eµ max(a,v−r,c0v)f(v) +� ∞ +r+δ1 +dr′ K2 +3(v − r, v − r − r′)gt(v, r), +(D.24) +and, +B3[Γ2 +t ](v, r) += I(2)[Γ2 +t](v, r) − +1(v < −b0)1(r + δ1 < a1 − v) +� � ∞ +r+δ1 +dr′ K3 +2(v − r, v − r + r′) +� +f(v − r + r′) − f(v + r′) +� +× +× eµ max(a,c0(v−r+r′))gt(v − r + r′, r′) ++ eµa +� a1−v +r+δ1 +dr′ K3 +1(v, v + r′)f(v + r′) +� +gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′) +� � +(D.25) +We will now turn to J1[Γ1 +t](v, r) and J2[Γ2 +t](v, r), defined in (D.3) and (D.4). It is not difficult to obtain +the following lower bound on the sum of these terms: +J1[]Γ1 +t ](v, r) + J2[Γ2 +t ](v, r) +≥ +1(v ≤ −b0) 4¯n +� +− eµaf(v − r) +� r+a1−v +r+δ1 +dr′ (ln(1 + ev))− 3 +2 +ev−r′(1 − e−r) ln(1 + ev+r′) +(1 + ev + e−r′)(1 + ev−r + e−r′)gt(v − r + r′, r′) +− f(v − r) +� ∞ +r+a1−v +dr′ (ln(1 + ev))− 3 +2 eµc0(v−r+r′) +ev− 3 +2r′(1 − e−r) ln(1 + ev+r′) +(1 + ev + e−r′)(1 + ev−r + e−r′)gt(v − r + r′, r′) +− +1(r ≤ a1 − v ≤ r + δ1) +� ∞ +r+δ1 +dr′ (ln(1 + ev))− 3 +2 +ev− 3 +2 r′(1 − e−r) ln(1 + ev+r′) +(1 + ev + e−r′)(1 + ev−r + e−r′)× +69 + +× +� +f(v − r + r′) − f(v + r′) +� +eµ max(a,c0(v−r+r′))gt(v − r + r′, r′) +− +� ∞ +r+δ1 +dr′ (ln(1 + ev))− 3 +2 +ev− 3 +2 r′(1 − e−r) ln(1 + ev+r′) +(1 + ev + e−r′)(1 + ev−r + e−r′)eµ max(a,c0(v−r+r′))f(v + r′)gt(v − r + r′, r′) +− +1(r + δ1 ≤ a1 − v) +� a1−v +r+δ1 +dr′ (ln(1 + ev))− 3 +2 ev−r′(1 − e− 1 +2r′)(1 − e−r) ln(1 + ev+r′) +(1 + ev + e−r′)(1 + ev−r + e−r′) +× +× eµ max(a,c0(v−r+r′))f(v + r′)gt(v − r + r′, r′) ++ 2eµa +� r+a1−v +r+δ1 +dr′ � +ln(1 + ev−r) +�− 3 +2 +e−2r′ ln(1 + ev−r+r′) +(1 − e−r′)(1 + ev + e−r′) +� +1 − ln(1 + ev−r−r′) +ln(1 + ev−r+r′) +� +× +× +� +1 − +�ln(1 + ev−r) +ln(1 + ev) +� 3 +2 +ln(1 + ev+r′) +ln(1 + ev−r+r′) +� +f(v − r)gt(v − r + r′, r′) ++ 2 +� ∞ +r+a1−v +dr′ � +ln(1 + ev−r) +�− 3 +2 +e− 5 +2r′ ln(1 + ev−r+r′) +(1 − e−r′)(1 + ev + e−r′) +� +1 − e +1 +2 r′ ln(1 + ev−r−r′) +ln(1 + ev−r+r′) +� +× +× +� +1 − +�ln(1 + ev−r) +ln(1 + ev) +� 3 +2 +ln(1 + ev+r′) +ln(1 + ev−r+r′) +� +eµc0(v−r+r′)f(v − r)gt(v − r + r′, r′) ++ +1(r > a1 − v) +� r+a1−v +r+δ1 +dr′ K3 +1(v − r, v − r + r′)f(v + r′)eµ max(a,c0(v−r+r′))gt(v − r + r′, r′) +� +− +1(v > −b0) +� +4¯n (ln(1 + ev))− 3 +2 +� ∞ +r+δ1 +dr′ � +f(v − r) + f(v + r′) +� +ev− 3 +2r′(1 − e−r) ln(1 + ev+r′) +(1 + ev + e−r′)(1 + ev−r + e−r′)× +× eµ max(a,v−r,c0(v−r+r′))gt(v − r + r′, r′) ++ +1(v − r ≥ 0) eµ max(a,v−r,c0v) +� ∞ +r+δ1 +dr′ K1 +3(v − r, v − r − r′) (f(v − r) + f(v)) × +× (1 − e− 1 +2 r′)(1 − e− 1 +2r)gt(v − r + r′, r′) +� +We now combine J0[Γ1 +t + Γ2 +t](v, r), I(1)[Γ1 +t](v, r) and E′ +1[Γ2 +t](v, r) with the lower bound for J1[Γ1 +t ](v, r) + +J2[Γ2 +t ](v, r) written above, to arrive at the following estimate denoted by Comb.4. +J1[Γ1 +t](v, r) + J2[Γ2 +t](v, r) + J0[Γ1 +t + Γ2 +t](v, r) + I(1)[Γ1 +t ](v, r) + E′ +1[Γ2 +t](v, r) +(Comb.4) +≥ J 0[Γ1 +t + Γ2 +t](v, r) + E[Γ1 +t + Γ2 +t](v, r), +70 + +where +J 0[Γ1 +t + Γ2 +t ](v, r) += − +1(v < −b0)1(r ≤ a1 − v ≤ r + δ1)4¯n (ln(1 + ev))− 3 +2 +� r−v +r+δ1 +dr′ +ev− 3 +2r′(1 − e− 1 +2r′)(1 − e−r) +(1 + ev + e−r′)(1 + ev−r + e−r′)× +× eµa ln(1 + ev+r′) +� +f(v − r + r′) − f(v + r′) +� +gt(v − r + r′, r′) ++ +1(v < −b0)eµa (f(v − r) + f(v)) +� ∞ +r+δ1 +dr′ K1 +3(v − r, v − r − r′) +� +gt(v, r) + gt(v, r′) − gt(v, r + r′) +� ++ +1(v ≥ −b0)eµ max(a,c0v,v−r) (f(v − r) + f(v)) +� +1(v − r < 0) +� ∞ +r+δ1 +dr′ K1 +3(v − r, v − r − r′) × +× +� +gt(v, r) + gt(v, r′) − gt(v, r + r′) +� ++ +1(v − r ≥ 0) +� ∞ +r+δ1 +dr′ K1 +3(v − r, v − r − r′)e− 1 +2r′ � +gt(v, r) + gt(v, r′) − gt(v, r + r′) +� � ++ 4¯n (ln(1 + ev))− 3 +2 +� ∞ +r+δ1 +dr′ ln(1 + ev+r′) +ev + 2 +1 + ev + e−r′ +e− 5 +2 r′ +1 − e−r′ eµ max(a,v−r,c0(v+r′))× +× +� +f(v − r) + f(v + r′) +� � +gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′) +� ++ +1(v ≥ −b0)4¯n (ln(1 + ev))− 3 +2 +� ∞ +r+δ1 +dr′ ln(1 + ev+r′) +ev−r + e−r′ +1 + ev−r + e−r′ +ev− 3 +2r′ +1 + ev + e−r′ × +× +� +f(v − r) + f(v + r′) +� � +2 − (1 + e−κr)γt(v,r)� +eµ max(a,v−r,c0(v+r′))gt(v, r), +(D.26) +and, +E[Γ1 +t + Γ2 +t](v, r) += +1(v < −b0) +� +4¯nf(v − r) (ln(1 + ev))− 3 +2 +� r+a1−v +r+δ1 +dr′ +ev + 2 +1 + ev + e−r′ +e−2r′ +1 − e−r′ ln(1 + ev+r′) × +× +� +eµa − e− 1 +2r′eµ max(a,v−r,c0(v+r′))� � +gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′) +� ++ eµaf(v) +� ∞ +max(−v−b0,r+δ1) +dr′ K2 +3(v − r, v − r − r′)e−r′e−( 122 +250 −α)r +� +1 − e−(1−α)r′ +1 − e−r′ +� +gt(v, r) ++ +1(r + δ1 < −v − b0) +� � ∞ +r+δ1 +dr′ K3 +2(v − r, v − r + r′) +� +f(v − r + r′) − f(v + r′) +� +× +× +� +eµ max(a,c0(v−r+r′)) − eµa� +gt(v − r + r′, r′) ++ eµaf(v) +� ∞ +−v−b0 +dr′ K2 +3(v − r, v − r − r′) +� � +1 − e−( 122 +250 −2α)r′� � +1 − e−(1−2α)r′� ++ e− 116 +750 r′e−(1−2α)r′(1 − e−αr′) +� +1 − e− 634 +750 r′ +1 − e−r′ +� � +gt(v, r) +� ++ +1(r + δ1 > −v − b0)eµaf(v) +� ∞ +max(−v−b0,r) +dr′ K2 +3(v − r, v − r − r′) +� +1 − e−( 122 +250 −2α)r′� +× +× +� +1 − e−(1−2α)r′� +gt(v, r) +71 + ++ +1(r > a1 − v) +� r+a1−v +r+δ1 +dr′ K3 +1(v − r, v − r + r′)f(v + r′)eµ max(a,c0(v−r+r′))gt(v − r + r′, r′) ++ +1(r + δ1 > a1 − v)f(v − r) +� r+a1−v +r+δ1 +dr′ K3(v, v + r′) +� +eµa − eµc0ve−( 1 +2 −µc0)r′� +gt(v + r′, r + r′) ++ +1(a1 − v ≥ r + δ1)f(v − r) +� r+a1−v +a1−v +dr′ K3(v, v + r′) +� +eµa − eµc0ve−( 1 +2 −µc0)r′� +gt(v + r′, r + r′) +� +. +(D.27) +This leads us to consolidate (Comb.4) and (LB1) into the second lower bound (LB2) +I1[Γ1 +t + Γ2 +t ](v, r) + I(1) +3 [Γ2 +t ](v, r) + A− +1 [Γ1 +t ](v, r) +(LB2) +≥ B− +1 [Γ1 +t ](v, r) + B2[Γ2 +t ](v, r) + B3[Γ2 +t](v, r) + E2[Γ2 +t](v, r) + I(3)[Γ2 +t ](v, r) ++ J 0[Γ1 +t + Γ2 +t ](v, r) + E[Γ1 +t + Γ2 +t](v, r), +Recall now the lower bound on I2[Γ1 +t](v, r) given by (3.34) and the following definition (cf. (3.35)): +C3[Γ1 +t](v, r) += +1(v − r < −b0)eµ max{a,c0v}gt(v, r) +� r +δ1 +dr′ � +K3(v − r, v − r + r′) +� +f(v − r) − (ln(1 + ev−r+r′))−α� +− K1 +3(v − r, v − r − r′) +� +f(v − r − r′) − f(v − r) +� � +. +This term now has to be used to offset the negative term A− +2 [Γ1 +t](v, r). +By virtue of Lemma D.1, this +combination yields the following inequality: +C3[Γ1 +t](v, r) + A− +2 [Γ1 +t](v, r) +(Comb.5) +≥ +1(−m0 < v − r < −b0)C3[Γ1 +t ](v, r) ++ +1(v − r ≤ −m0)eµ max(a,c0v)4¯n +� +ln(1 + ev−r) +�− 3 +2 +� min(r,˜r) +δ1 +dr′ ev−r−r′ + 2e−2r′ +1 + ev−r + e−r′ +� +ln(1 + ev−r−r′) +�1−α +× +× +� +ln(1 + ev−r+r′) +ln(1 + ev−r−r′) +�1−2α �� +1 − e− 248 +125 (1−2α)r′ +1 − e−r′ +� +− er′e− 248 +125 (1−2α)r′ +� � +1 − +� +ln(1 + ev−r−r′) +ln(1 + ev−r) +�α� +gt(v, r) +≥ +1(−m0 < v − r < −b0)C3[Γ1 +t ](v, r) + +1(v − r ≤ −m0)¯n +� +ln(1 + ev−r) +�− 1 +2 Γ1 +t (v, r)b1(α) +� +1 − e−3αr�3 . +Let us define +˜B[Γ1 +t](v, r) = +1(v − r ≤ −m0)eµ max(a,c0v)f(v − r)b1(α)¯n +� +ln(1 + ev−r) +�− 1 +2 � +1 − e−3αr�3 gt(v, r), +(D.28) +where b1 is positive and bounded away from 0, for all α > 0. Note that ˜B has the requisite singular behavior +like (ln(1 + ev−r))−1/2 for r ≫ 1. +Then we combine the rest of I2[Γt](v, r) with E[Γ1 +t + Γ2 +t ](v, r) and J 0[Γ1 +t + Γ2 +t ](v, r), and obtain the +following lower bound: +I1[Γt](v, r) + I2[Γt](v, r) + I(1) +3 [Γ2 +t](v, r) + A− +1 [Γ1 +t ](v, r) + A− +2 [Γ1 +t ](v, r) +(LB3) +≥ B− +1 [Γ1 +t](v, r) + +1(−m0 < v − r < −b0)C3[Γ1 +t ](v, r) + ˜B[Γ1 +t ](v, r) + B2[Γ2 +t ](v, r) + Bf +3 [Γ2 +t](v, r) +72 + ++ E2[Γ2 +t ](v, r) + I(3)[Γ2 +t](v, r) + J +f +0[Γ1 +t + Γ2 +t](v, r) + E +f[Γ1 +t + Γ2 +t](v, r), +where +B2[Γ2 +t](v, r) += B2[Γ2 +t ](v, r) + +1(v < −b0)eµa +� r +δ1 +dr′ K3(v, v + r′)(f(v) − f(v + r′))gt(v, r) ++ +1(v ≥ −b0) +� r +δ1 +dr′ K3(v, v + r′)f(v) +� +eµ max(a,v−r,c0v) − e− 1 +2 r′eµ max(a,v−r,c0(v+r′))� +gt(v, r), +(D.29) +J +f +0[Γ1 +t + Γ2 +t ](v, r) += +1(v < −b0)eµa (f(v − r) + f(v)) +� ∞ +r+δ1 +dr′ K1 +3(v − r, v − r − r′) +� +gt(v, r) + gt(v, r′) − gt(v, r + r′) +� ++ +1(v ≥ −b0)eµ max(a,c0v,v−r) (f(v − r) + f(v)) +� +1(v − r < 0) +� ∞ +r+δ1 +dr′ K1 +3(v − r, v − r − r′) × +× +� +gt(v, r) + gt(v, r′) − gt(v, r + r′) +� ++ +1(v − r ≥ 0) +� ∞ +r+δ1 +dr′ K1 +3(v − r, v − r − r′)e− 1 +2r′ � +gt(v, r) + gt(v, r′) − gt(v, r + r′) +� � ++ 4¯n (ln(1 + ev))− 3 +2 +� ∞ +2r +dr′ ln(1 + ev+r′) e− 5 +2 r′ +1 − e−r′ × +× eµ max(a,v−r,c0v)f(v + r′) +� +2 − (1 + e−κr)γt(v,r)� +gt(v, r) ++ +1(v ≥ −b0)4¯n (ln(1 + ev))− 3 +2 +� ∞ +2r +dr′ ln(1 + ev+r′) e− 5 +2r′ +1 − e−r′ × +× eµ max(a,v−r,c0v)f(v − r) +� +2 − (1 + e−κr)γt(v,r)� +gt(v, r), +(D.30) +E +f[Γ1 +t + Γ2 +t](v, r) += +1(v < −b0) +� +eµaf(v) +� ∞ +max(−v−b0,r+δ1) +dr′ K2 +3(v − r, v − r − r′)e−r′e−( 122 +250 −α)r +� +1 − e−(1−α)r′ +1 − e−r′ +� +gt(v, r) ++ +1(r + δ1 < −v − b0) +� � ∞ +r+δ1 +dr′ K3 +2(v − r, v − r + r′) +� +f(v − r + r′) − f(v + r′) +� +× +× +� +eµ max(a,c0(v−r+r′)) − eµa� +gt(v − r + r′, r′) ++ eµaf(v) +� ∞ +−v−b0 +dr′ K2 +3(v − r, v − r − r′) +� � +1 − e−( 122 +250 −2α)r′� � +1 − e−(1−2α)r′� ++ e− 116 +750 r′e−(1−2α)r′(1 − e−αr′) +� +1 − e− 634 +750 r′ +1 − e−r′ +� � +gt(v, r′) +� ++ +1(r + δ1 ≥ −v − b0)eµaf(v) +� ∞ +max(−v−b0,r) +dr′ K2 +3(v − r, v − r − r′) +� +1 − e−( 122 +250 −2α)r′� +× +73 + +× +� +1 − e−(1−2α)r′� +gt(v, r′) ++ +1(r > a1 − v) +� r+a1−v +r+δ1 +dr′ K3 +1(v − r, v − r + r′)f(v + r′)eµ max(a,c0(v−r+r′))gt(v − r + r′, r′) +� +, (D.31) +and +Bf +3 [Γ2 +t ](v, r) += +1(v < −b0) +� � ∞ +max(r+δ1,a1−v) +dr′ K3(v, v + r′)f(v + r′) (eµa − eµc0v) gt(v, r) ++ +� ∞ +r+δ1 +dr′ K3(v, v + r′) +� +f(v) − f(v + r′) +� +eµagt(v, r) ++ +1(r ≤ a1 − v ≤ r + δ1) +� � r+a1−v +r+δ1 +dr′ K3 +2(v, v + r′)f(v + r′) +� +eµa − eµc0(v−r+r′)� +gt(v − r + r′, r′) ++ +� r−v +r+δ1 +dr′ K3 +2(v, v + r′)e− 1 +2r′ � +f(v − r + r′) − f(v + r′) +� +eµagt(v − r + r′, r′) ++ +� ∞ +r+δ1 +dr′ K3(v, v + r′)f(v + r′)eµc0v � +1 − 2e−( 1 +2−µc0)r′ + e−2( 1 +2−µc0)r′� +gt(v, r) ++ +� r+a−v +r+δ1 +dr′ K3 +1(v − r, v − r + r′)f(v − r + r′)eµagt(v − r + r′, r′) +� ++ +1(a1 − v > r + δ1) +� � a1−v +r+δ1 +dr′ K3 +1(v − r, v − r + r′) +� +f(v − r + r′) − f(v + r′) +� +eµagt(v − r + r′, r′) ++ +� r+a1−v +a1−v +dr′ K3 +2(v, v + r′)f(v + r′) +� +eµa − eµc0ve−( 1 +2−µc0)r′� +gt(v − r + r′, r′) ++ 8¯n eµc0v +� r+a1−v +a1−v +dr′ (ln(1 + ev))− 3 +2 +� +ln(1 + ev+r′) +�1−α e−2r′(1 − e− 1 +2 r′)(1 − e−( 1 +2−µc0)r′) +(1 − e−r′)(1 + ev + e−r′) +gt(v, r) ++ 8¯n eµa +� r+a1−v +a1−v +dr′ � +ln(1 + ev−r) +�− 3 +2 +� +ln(1 + ev−r+r′) +�1−α +e−2r′(1 − e− 1 +2 r′)2 +(1 − e−r′)(1 + ev + e−r′)gt(v − r + r′, r′) +� ++ +1(a1 − v < r) +� � r+a1−v +r+δ1 +dr′ K3 +2(v − r, v − r + r′) +� +f(v − r + r′) − f(v + r′) +� +eµagt(v − r + r′, r′) ++ +� r+a1−v +r+δ1 +dr′ K3 +2(v, v + r′) +� +eµa − eµc0(v−r+r′)� +f(v + r′)gt(v − r + r′, r′) ++ +� r+a1−v +r+δ1 +dr′ K3 +1(v − r, v − r + r′)eµa � +f(v − r + r′) − f(v + r′) +� +gt(v − r + r′, r′) +�� +. +Let us now turn to the negative terms in +1(v < −m0)I(2) +3 [Γ2 +t ](v, r) and observe that we can use a term from +B2[Γ2 +t ](v, r) to control these negative terms as follows: +1(v ≤ −m0)eµa� +− +1(v + r < b0) +� r +0 +dr′ K2 +3(v, v − r′) +� +f(v − r′) − f(v) +� +gt(v − r′, r − r′) +74 + +− +1(v + r ≥ b0) +� −v−b0 +0 +dr′ K2 +3(v, v − r′) +� +f(v − r′) − f(v) +� +gt(v − r′, r − r′) +� ++ +1(v < −b0)eµa +� r +δ1 +dr′ K3(v, v + r′)(f(v) − f(v + r′))gt(v, r) +≥ +1(−m0 < v < −b0)eµa +� r +δ1 +dr′ K3(v, v + r′)(f(v) − f(v + r′))gt(v, r), +so that we can write +B2[Γ2 +t ](v, r) + I(2) +3 [Γ2 +t](v, r) +(Comb.6) +≥ B ++ +2 [Γ2 +t ](v, r) + +1(v − r ≥ m0)I(1) +3 [Γ2 +t](v, r) + +1(v ≥ m0)I(2) +3 [Γ2 +t ](v, r), ++ +1(−m0 < v < m0)eµ max(a,c0v,v−r)f(v) +� r +0 +dr′ K2 +3(v, v − r′)gt(v, r), +where +B ++ +2 [Γ2 +t ](v, r) += +1(v − r ≤ −m0)eµ max(a,c0v)f(v) +� ˜r +0 +dr′ � +K2 +3(v − r, v − r − r′)gt(v, r) +− +� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +gt(v, r + r′) +� ++ +1(v ≤ −m0)eµaf(v) +� +1(v + r < b0) +� r +0 +dr′ K2 +3(v, v − r′) +� +gt(v, r) − gt(v − r′, r − r′) +� ++ +1(v + r ≥ −b0)f(v) +� � r +−v−b0 +dr′ K2 +3(v, v − r′)gt(v, r) ++ +� −v−b0 +0 +dr′ K2 +3(v, v − r′) +� +gt(v, r) − gt(v − r′, r − r′) +� +�� ++ +1(−m0 < v < −b0)eµa +� r +δ1 +dr′ K3(v, v + r′)(f(v) − f(v + r′))gt(v, r) ++ +1(v ≥ −b0) +� r +δ1 +dr′ K3(v, v + r′)f(v) +� +eµ max(a,v−r,c0v) − e− 1 +2 r′eµ max(a,v−r,c0(v+r′))� +gt(v, r). +(D.32) +We control the negative term A− +3 [Γ1 +t ] by a term from I(1) +3 [Γ2 +t ] as follows: +1(v − r ≥ m0)1(v > 3r)f(v) +� +eµ max(a,v−r,c0v) − eµ max(a,c0v)� +gt(v, r) +� ∞ +v−r−c0v +dr′ K2 +3(v − r, v − r − r′) ++ A− +3 [Γ1 +t](v, r) +> +1(v − r ≥ m0)1(v > 3r)4¯n +� +ln(1 + ev−r) +�− 3 +2 +� ∞ +v−r +dr′ +ev−r−r′ + 2e−r′ +1 + ev−r−r′ + e−r′ × +× +� +ln(1 + ev−r−r′) +�1−α +e− 2 +3r � +1 − e−(κ− 2 +3)r� +eµc0vgt(v, r + r′), +which allows us to arrive at the following estimate: +1(v − r ≥ m0) +� +I(1) +3 [Γ1 +t](v, r) + I(1) +3 [Γ2 +t ](v, r) +� +75 + +≥ B4[Γt](v, r) + +1(v − r ≥ m0)I +1,+ +3 +[Γt](v, r), +where +B4[Γt](v, r) += +1(v − r ≥ m0)1(v > 3r)4¯n +� +ln(1 + ev−r) +�− 3 +2 +� ∞ +v−r +dr′ +ev−r−r′ + 2e−r′ +1 + ev−r−r′ + e−r′ × +× +� +ln(1 + ev−r−r′) +�1−α +e− 2 +3 r � +1 − e−(κ− 2 +3)r� +eµc0vgt(v, r + r′) ++ +1(v − r ≥ m0) +� +1(v ≤ 3r) (f(v − r) + f(v)) +� +¯n(ln 2)2 � +ln(1 + ev−r) +�− 3 +2 eµ max(a,v−r,c0v)gt(v, r) ++ +� v−r +0 +dr′ � +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� � +eµ max(a,v−r,c0v) − eµ max(a,v−r−r′,c0v)� +gt(v, r) +� ++ +1(v > 3r) +� +f(v − r) +� ∞ +v−r−c0v +dr′ K2 +3(v − r, v − r − r′) +� +eµ max(a,v−r,c0v) − eµ max(a,c0v)� +gt(v, r) ++ (f(v − r) + f(v)) eµ max(a,c0v) +� ∞ +v−r−c0v +dr′ � +K2 +3(v − r, v − r − r′)gt(v, r) +− +� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +gt(v, r + r′) +��� +, +(D.33) +and +I +1,+ +3 +[Γt](v, r) += +1(v − r ≥ m0)Γt(v, r) +� +1(v ≤ 3r) 4 +3 ¯n (ln(1 + ev)) +1 +2 +�ln(1 + ev−r) +ln(1 + ev) +�2 ++ +1(v > 3r) 2¯n +� +ln(1 + ev−r) +� 1 +2 +� +1 − +� ln(1 + ec0v) +ln(1 + ev−r) +�2� � +(D.34) +I ++ +3 contributes the correct asymptotic behavior for large, positive values of v − r, when v and v − r are +comparable. +Putting the above estimates together, we can write the following lower bound: +I1[Γt](v, r) + I2[Γt](v, r) + I(1) +3 [Γt](v, r) + I(2) +3 [Γt](v, r) +(LB4) +≥ B− +1 [Γ1 +t](v, r) + A− +4 [Γ1 +t](v, r) + +1(−m0 < v − r < −b0)C3[Γ1 +t ](v, r) + ˜B[Γ1 +t](v, r) ++ B ++ +2 [Γ2 +t](v, r) + Bf +3 [Γ2 +t ](v, r) + E2[Γ2 +t](v, r) + J +f +0[Γt](v, r) + E +f[Γt](v, r) + B4[Γt](v, r) ++ +1(−m0 < v − r < m0)I(1) +3 [Γ1 +t](v, r) + +1(−m0 < v < m0)I(2) +3 [Γt](v, r) ++ I +1,+ +3 +[Γt](v, r) + +1(v ≥ m0)I2 +3[Γt](v, r) + I(3)[Γ2 +t ](v, r). +Let us reflect for a moment on the asymptotic behavior for large, positive values of v. Note that v ≤ 3r ⇐⇒ +v − r ≤ 2 +3v and v > 3r ⇐⇒ v − r > 2 +3v. The desired behavior mimicking that of the potential Vu(v, r) +76 + +comes from I +1,+ +3 +[Γt](v, r), +1(−m0 < v − r < m0)I(1) +3 [Γt](v, r) and +1(v ≥ m0)I2 +3[Γt](v, r), as is evident from +the following lower bound: +I +1,+ +3 +[Γt](v, r) + +1(−m0 < v − r < m0)I(1) +3 [Γt](v, r) + +1(v ≥ m0)I2 +3[Γt](v, r) +≥ +1(v ≥ m0) +� +1(v − r ≤ 0)¯n(ln(1 + ev)) +1 +2 +� +1 − +� +ln 2 +ln(1 + ev) +�2 � +Γt(v, r) ++ +1(−m0 < v − r < m0)eµ max(a,c0v,v−r)f(v − r)gt(v, r) +� ∞ +0 +dr′ K2 +3(v − r, v − r − r′) ++ +1(0 < v − r < c0v)3 +2 ¯n(ln(1 + ev)) +1 +2 +� +1 − +�v − r +c0v +�2 � +Γt(v, r) ++ +1(v − r ≥ m0)Γt(v, r) +� +1(v − r ≤ 2 +3v) 4 +3 ¯n (ln(1 + ev)) +1 +2 +�ln(1 + ev−r) +ln(1 + ev) +�2 ++ +1(v − r > 2 +3v) 2¯n +� +ln(1 + ev−r) +� 1 +2 +� +1 − +� ln(1 + ec0v) +ln(1 + ev−r) +�2� �� +(D.35) +In the lower bound (LB4) the first two terms, i.e., B− +1 [Γ1 +t ] and A− +4 [Γ1 +t ], are the only non-positive ones. +Between these two, A4 has a δ1-smallness (i.e. while the term may have a point singularity, it also contains +a factor of δ1, which we can choose to be arbitrarily small), as is evident from (D.15). Our next step is +to combine some suitable positive terms with B− +1 , so that we are left with a negative term which has a +δ1-smallness. Thus, at the end of the next step all the negative terms will have this kind of smallness. +We combine some terms from B ++ +2 [Γ2 +t ], +1(−m0 < v < m0)I(2) +3 [Γ2 +t ] and J +f +0[Γt] to control the negative term +B− +1 [Γ1 +t] as follows: +B− +1 [Γ1 +t](v, r) + +1(v ≥ −b0)f(v) +� r +δ1 +dr′ K3(v, v + r′) +� +eµ max(a,c0v,v−r) − e− 1 +2 r′eµ max(a,c0(v+r′),v−r)� +gt(v, r) +(Comb.7) ++ eµ max(a,c0v,v−r)f(v) +� ∞ +r+δ1 +dr′ K1 +3(v − r, v − r − r′)e− 1 +2r′ � +gt(v, r) + gt(v, r′) − gt(v, r + r′) +� ++ +1(0 ≤ v < m0)eµ max(a,c0v,v−r)f(v) +� r +0 +dr′ K2 +3(v, v − r′)gt(v, r) +≥ B− +1 [Γ1 +t ](v, r) + ˜B+ +1 [Γ2 +t ](v, r), +where +B− +1 [Γ1 +t ](v, r) = − 1(v ≥ 0) +� 2r+δ1 +2r +dr′ K1 +3(v, v − r′)eµ max(a,c0v) � +f(v − r − r′) − f(v − r′) +� +gt(v, r′), +and +˜B+ +1 [Γ2 +t](v, r) = +1(0 ≤ v < m0)eµ max(a,c0v,v−r)f(v)gt(v, r) +1 +1 + e− 1 +2r +� r +0 +dr′ K2 +3(v, v − r′) ++ 1 +2eµ max(a,c0v,v−r)f(v) +� ∞ +r+δ1 +dr′ K1 +3(v − r, v − r − r′)e− 1 +2 r′ � +gt(v, r) + gt(v, r′) − gt(v, r + r′) +� ++ 1 +2 +� +2 − 2γ0 − 1 +2 +� +eµ max(a,c0v,v−r)f(v)gt(v, r) +� ∞ +r+δ1 +dr′ K1 +3(v − r, v − r − r′)e− 1 +2 r′(1 − e−(κ−1)r′) +77 + ++ +1(v ≥ 0)eµ max(a,c0v,v−r)f(v)gt(v, r) +� r +δ1 +dr′ K3(v, v + r′) +� +1 − e−( 1 +2−µc0)r′� +. +At this stage, all the negative terms we are left with, are integrals over intervals of lengths δ1 and (δ1−δ2). +Let us denote the sum of these negative terms by S− +δ1,δ2[Γt](v, r), as follows: +S− +δ1,δ2[Γt](v, r) = B− +1 [Γ1 +t ](v, r) + I4[Γt](v, r) + A− +4 [Γ1 +t](v, r). +Since S− +δ1,δ2 has a δ1-smallness, the constant M in the definition of the cut-off functions (see (3.21)) may be +chosen large enough, so that the positive terms appearing in the estimate (LB4) can be used to offset this +negative term, leading to the following lower bound: +I1[Γt](v, r) + I2[Γt](v, r) + I(1) +3 [Γt](v, r) + I(2) +3 [Γt](v, r) + I4[Γt](v, r) ≥ G[Γt](v, r), +(LB5) +where +G[Γt](v, r) += +1(v − r ≤ −b0)¯b1(α)¯nΓt(v, r) +� +ln(1 + ev−r) +�− 1 +2 � +1 − e−3αr�3 ++ ¯n +2 +� +ln(1 + ev−r) +�− 1 +2 ln +� +1 − e− 7 +2(r+δ1)�−1 +Γt(v, r) + +1(−m0 < v − r < m0) ¯n +4 +� +ln(1 + ev−r) +�− 1 +2 Γt(v, r) ++ +1(0 < v < m0) ¯n +2 (ln(1 + ev)) +1 +2 +� +1 − +�ln(1 + ev−r) +ln(1 + ev) +�2� +Γt(v, r) ++ +1(0 < v − r < m0) ¯n +8 +� +ln(1 + ev−r) +� 1 +2 Γt(v, r) ++ +1(v ≥ m0)b3(c0)¯n (ln(1 + ev)) +1 +2 Γt(v, r), +(D.36) +where ¯b1(α) and b3(c0) are positive numbers bounded away from zero. +It is important to note here that, for controlling S− +δ1,δ2, we need to put a suitable upper bound on the +constant M. Let us describe briefly how that comes about. The bound comes from the fact that ˜B[Γ1 +t](v, r) +is used to control A− +4 [Γ1 +t ](v, r) as follows: +A− +4 [Γ1 +t](v, r) ≥ − eµ max(a,c0v)f(v − r)gt(v, r)¯n +� +ln(1 + ev−r) +�− 1 +2 +1(v − r ≤ −m0) +� +3 +4 (min(δ1, ˜r))2 ++ +1(v < −b0)1(r < −b0 − v)2e−rδ1 +� +, +and we choose M such that +1 +M2 + 1 +M ≤ b1(α) +10 , +so that ˜B[Γ1 +t ](v, r) (see (D.28)) dominates A− +4 [Γ1 +t](v, r). +E +Computations pertaining to the Regularized Problem +E.1 +Formulae and Computations Relating to Theorem 3.12 +The the unbounded operator Ku and the L2-bounded part Kb are defined as follows, via cut-off parameters +b′ +0 and m′ +0: +(Kε +uψε +t )(v) +78 + += 1(v < −b′ +0) +� � −v−b′ +0 +0 +dr′ K1,ε(v) +3 +(v, v − r′)ψε +t (v − r′) + +� ∞ +−v−b′ +0 +dr′ K1,ε(v) +3 +(v, v − r′)e−r′ψε +t (v − r′) ++ +� −v−b′ +0 +0 +dr′ Kε(v+r′) +3 +(v, v + r′)ψε +t (v + r′) + +� ∞ +−v−b′ +0 +dr′ Kε(v+r′) +3 +(v, v + r′)e−r′ψε +t (v + r′) +� ++ +1(v ≥ −b′ +0) +� +1(−b′ +0 ≤ v ≤ m′ +0) +� ∞ +0 +dr′ K1,ε(v) +3 +(v, v − r′)e−r′ψε +t (v − r′) ++ +1(v > m′ +0) +� v +0 +dr′ K1,ε(v) +3 +(v, v − r′)ψε +t (v − r′) + +1(v ≤ m′ +0) +� ∞ +0 +dr′ Kε(v+r′) +3 +(v, v + r′)e−r′ψε +t (v + r′) ++ +1(v > m′ +0) +� ∞ +0 +dr′ Kε(v+r′) +3 +(v, v + r′)ψε +t (v + r′) +� ++ +1(v < −m′ +0) +� r0 +0 +dr′ K2,ε(v) +3 +(v, v − r′)ψε +t (v − r′) + +1(v > m′ +0) +� v−a1 +0 +dr′ K2,ε(v) +3 +(v, v − r′)ψε +t (v − r′), +and +(Kε +b ψε +t )(v) +=1(v < −b′ +0) +� � ∞ +−v−b′ +0 +dr′ K1,ε(v) +3 +(v, v − r′)(1 − e−r′)ψε +t (v − r′) ++ +� ∞ +−v−b′ +0 +dr′ Kε(v+r′) +3 +(v, v + r′)(1 − e−r′)ψε +t (v + r′) +� ++ +1(v ≥ −b′ +0) +� +1(v ≤ m′ +0) +� ∞ +0 +dr′ K1,ε(v) +3 +(v, v − r′)(1 − e−r′)ψε +t (v − r′) ++ +1(v > m′ +0) +� ∞ +v +dr′ K1,ε(v) +3 +(v, v − r′)ψε +t (v − r′) ++ +1(−b′ +0 ≤ v ≤ m0) +� ∞ +0 +dr′ Kε(v+r′) +3 +(v, v + r′)(1 − e−r′)ψε +t (v + r′) +� ++ +1(v < −m′ +0) +� ∞ +r0 +dr′ K2,ε(v) +3 +(v, v − r′)ψε +t (v − r′) + +1(−m′ +0 ≤ v ≤ m′ +0) +� ∞ +0 +dr′ K2,ε(v) +3 +(v, v − r′)ψε +t (v − r′) ++ +1(v > m′ +0) +� ∞ +v−a1 +dr′ K2,ε(v) +3 +(v, v − r′)ψε +t (v − r′). +In the formulae above r0 = −v − b′ +0 plays the same role as ˜r in the previous section. m′ +0 > max(b′ +0, 2a1) has +to be chosen large enough. +The computations for Lemma 3.11 follow the same scheme as employed in Appendix D and are quite +straightforward, so we skip them and write the resulting estimate, for some positive constants p2 and p3 +bounded away from zero: +V ε(v)˜Γ(v) − (Kε +u˜Γ)(v) ≥ ˜Γ(v) +� +1(v ≤ 0)p2(α) (ln(1 + ev))− 1 +2 + +1(v > 0)p3 (ln(1 + ev)) +1 +2 +� +. +E.2 +Formulae and Computations Relating to Theorem 3.4 +The computations for Theorem 3.4 are almost the same as those described in Appendix D (which lead to a +similar result for the ∆-variable, namely Theorem 3.3), so there is nothing to be gained by repeating those +arguments and estimates here. We will only write down the explicit forms of the operators, since the limits +79 + +of the integrals are slightly different now (owing to the difference between the cut-off functions δ and ε in +these two cases). +Recall that the solution we are seeking proves the equation (3.13). The operator ˜Lε +s has already been +defined in (3.40). We now write down the expressions for the other operators. +( ˜Lε +uDψε +t )(v, r) += +1(v < a1) +� � v−r +−∞ +dw K1,ε(v−r) +3 +(v − r, w)Dψε +t (v, r) +− +� v−r +−∞ +dw +� +K1,ε(v−r) +3 +(v − r, w) − K1,ε(v−r) +3 +(v, w) +� +Dψε +t (v, v − w) +� ++ +1(v ≥ a1) +� � v−r +−∞ +dw K1,ε(v−r) +3 +(v − r, w)Dψε +t (v, r) +− +� v−r +−∞ +dw +� +K1,ε(v−r) +3 +(v − r, w) − K1,ε(v) +3 +(v, w) +� +Dψε +t (v, v − w)1(v − w > ε(v − r)) +� ++ +� ∞ +v +dw Kε(w) +3 +(v, w)Dψε +t (v, r) − +� +1(v < −b0) +� � a1 +v +dw +� +Kε(w) +3 +(v, w) − Kε(w) +3 +(v − r, w) +� +Dψε +t (w, w − v + r) ++ +� ∞ +a1 +dw +� +K3 +2,ε(w)(v, w) − K3 +2,ε(w)(v − r, w) +� +Dψε +t (w, w − v + r) +� ++ +1(v ≥ −b0) +� ∞ +v +dw +� +K3 +2,ε(w)(v, w) − K3 +2,ε(w)(v − r, w) +� +Dψε +t (w, w − v + r) +� ++ +� v +v−r +dw Kε(w) +3 +(v − r, w)Dψε +t (v, r) − +� +1(v − r < −b0) +� +1(v > 0) +� 0 +v−r +dw Kε(w) +3 +(v − r, w)Dψε +t (v, v − w) ++ 1(v ≤ 0) +� v +v−r +dw Kε(w) +3 +(v − r, w)Dψε +t (v, v − w) +� ++ +1(v − r ≥ −b0) +� v +v−r +dwK3 +2,ε(w)(v − r, w)Dψε +t (v, v − w) +� ++ +� v +v−r +dw K1,ε(v) +3 +(v, w)Dψε +t (v, r) +− 1(v < −m0) +� v +v−r +dw +1(v − w < min(r, −v − b0))K1,ε(v) +3 +(v, w)Dψε +t (w, w − v + r) +− 1(v ≥ −m0) +� v +v−r +dw K1,ε(v) +3 +(v, w)e−(v−w)Dψε +t (w, w − v + r) + +� v−r +−∞ +dw K2,ε(v−r) +3 +(v − r, w)Dψε +t (v, r) +− 1(v − r ≤ −m0) +� v−r−ε(v−r) +v−r−max(˜r,ε(v−r)) +dw +� +K2,ε(v−r) +3 +(v − r, w) − K2,ε(v−r) +3 +(v, w) +� +Dψε +t (v, v − w) +− 1(v − r ≥ m0) +� +1(v ≤ 3r) +� v−r−ε(v−r) +0 +dw +� +K2,ε(v−r) +3 +(v − r, w) − K2,ε(v−r) +3 +(v, w) +� +Dψε +t (v, v − w) ++ +1(v > 3r) +� c0v +−∞ +dw +� +K2,ε(v−r) +3 +(v − r, w) − K2,ε(v−r) +3 +(v, w) +� +Dψε +t (v, v − w) +� ++ +� v +v−r +dw K2,ε(v) +3 +(v, w)Dψε +t (v, r) − +1(v ≤ −m0) +� +1(v + r < −b0) +� v +v−r +dw K2,ε(v) +3 +(v, w)Dψε +t (w, w − v + r) +80 + ++ +1(v + r ≥ −b0) +� v +2v+b0 +dw K2,ε(v) +3 +(v, w)Dψε +t (w, w − v + r) +� +− +1(v ≥ m0) +� +1(0 < v − r < c0v) +� c−1 +0 (v−r) +v−r +dw K2,ε(v) +3 +(v, w)Dψε +t (w, w − v + r) ++ +1(v − r ≥ c0v) +� v +v−r +dw K2,ε(v) +3 +(v, w)Dψε +t (w, w − v + r) +� +, +which means we can write +( ˜Lε +uDψε +t )(v, r) = ˜Vε(v, r)Dψε +t (v, r) − ( ˜Kε +uDψε +t )(v, r), +the definitions of ˜Vε and ˜Kε +u being obvious from the formula for ˜Lε +u written above. Finally, +Kε +b[ψε +t ](v, r) += − +1(v < −b0) +� ∞ +a1 +dw +� +K3 +1,ε(w)(v, w) − K3 +1,ε(w)(v − r, w) +� +(ψε +t (w) − ψε +t (v − r)) +− +1(v ≥ −b0) +� ∞ +v +dw +� +K3 +1,ε(w)(v, w) − K3 +1,ε(w)(v − r, w) +� +(ψε +t (w) − ψε +t (v − r)) +− +1(v − r < −b0)1(v > 0) +� v +0 +dw Kε(w) +3 +(v − r, w) (ψε +t (v) − ψε +t (w)) +− +1(v − r ≥ −b0) +� v +v−r +dw K3 +1,ε(w)(v − r, w) (ψε +t (v) − ψε +t (w)) +− +1(v < −m0) +� v−min(r,−v−b0) +v−r +dw K1,ε(v) +3 +(v, w) (ψε +t (w) − ψε +t (v − r)) +− +1(v − r ≤ −m0) +� v−r−max(˜r,ε(v−r)) +−∞ +dw +� +K3 +2,ε(v−r)(v − r, w) − K3 +2,ε(v−r)(v, w) +� +(ψε +t (v) − ψε +t (w)) +− +1(−m0 < v − r < m0) +� v−r−ε(v−r) +−∞ +dw +� +K3 +2,ε(v−r)(v − r, w) − K3 +2,ε(v−r)(v, w) +� +(ψε +t (v) − ψε +t (w)) +− +1(v − r ≥ m0) +� +1(v ≤ 3r) +� 0 +−∞ +dw +� +K3 +2,ε(v−r)(v − r, w) − K3 +2,ε(v−r)(v, w) +� +(ψε +t (v) − ψε +t (w)) ++ +1(v > 3r) +� v−r−ε(v−r) +c0v +dw +� +K3 +2,ε(v−r)(v − r, w) − K3 +2,ε(v−r)(v, w) +� +(ψε +t (v) − ψε +t (w)) +� +− +1(v ≤ −m0)1(v + r ≥ −b0) +� 2v+b0 +v−r +dw K2,ε(v) +3 +(v, w) (ψε +t (w) − ψε +t (v − r)) +− +1(−m0 < v < m0) +� v +v−r +dw K2,ε(v) +3 +(v, w) (ψε +t (w) − ψε +t (v − r)) +− +1(v ≥ m0) +� +1(0 < v − r < c0v) +� v +c−1 +0 (v−r) +dw K2,ε(v) +3 +(v, w) (ψε +t (w) − ψε +t (v − r)) ++ +1(v − r ≤ 0) +� v +v−r +dw K2,ε(v) +3 +(v, w) (ψε +t (w) − ψε +t (v − r)) +� +. +The results for Dψε are proved via methods which are completely analogous to those employed in section +1 for the variable ∆. Thus it is critical to establish that ( ˜Lε +uΓ)(v, r) has the same asymptotic behavior as +81 + +˜Vε(v, r)Γ(v, r), ∀(v, r) ∈ R×R+. Like in 3.1.2 (cf. equation (3.28)), we will now split ˜Lε +uΓ into several terms +as shown below. We use the symbol Ii (recall Ii from 3.1.2) for the main integral terms. Then it is easy to +see: +( ˜Lε +uΓ +′ +ε)(v, r) +=I1[Γ](v, r) + I2[Γ +′ +ε](v, r) + I3[Γ +′ +ε](v, r) + +8 +� +i=1 +ei[Γ +′ +ε](v, r) + I +ε +4[Γ +′ +ε](v, r). +Let us first define I1, I2 and I3. +1) I1[Γ +′ +ε](v, r) += +∞ +� +max(r,ε(v−r)) +dr′� +K1 +3(v − r, v − r − r′)Γ +′ +ε(v, r) + K1 +3(v, v − r′)Γ +′ +ε(v, r′) − K1 +3(v − r, v − r − r′)Γ +′ +ε(v, r + r′) +� ++ +1(v < −b0) +� +∞ +� +max(r,ε(v−r)) +dr′ K3(v, v + r′)Γ +′ +ε(v, r) +− +1(r ≤ a1 − v) +� +a1−v +� +max(r,ε(v−r)) +dr′ K3(v, v + r′)Γ +′ +ε(v + r′, r + r′) + +� ∞ +a1−v +dr′ K3 +2(v, v + r′)Γ +′ +ε(v + r′, r + r′) +− +r+a1−v +� +max(r,ε(v−r)) +dr′ K3(v − r, v − r + r′)Γ +′ +ε(v − r + r′, r′) − +∞ +� +r+a1−v +dr′ K3 +2(v − r, v − r + r′)Γ +′ +ε(v − r + r′, r′) +� +− 1(r > a1 − v) +� +∞ +� +max(r,ε(v−r)) +dr′ K3 +2(v, v + r′)Γ +′ +ε(v + r′, r + r′) +− +r+a1−v +� +max(r,ε(v−r)) +dr′ K3(v − r, v − r + r′)Γ +′ +ε(v − r + r′, r′) − +∞ +� +r+a1−v +dr′ K3 +2(v − r, v − r + r′)Γ +′ +ε(v − r + r′, r′) +�� ++ +1(v ≥ −b0) +∞ +� +max(r,ε(v−r)) +dr′ � +K3(v, v + r′)Γ +′ +ε(v, r) + K3 +2(v − r, v − r + r′)Γ +′ +ε(v − r + r′, r′) +− K3 +2(v, v + r′)Γ +′ +ε(v + r′, r + r′) +� +, +2) I2[Γ +′ +ε](v, r) += +1(v − r < −b0) +� +min(r,r−v) +� +min(r,ε(v−r)) +dr′ K3(v − r, v − r + r′) +� +Γ +′ +ε(v, r) − Γ +′ +ε(v, r − r′) +� ++ +r +� +min(r−v,r) +dr′ K3(v − r, v − r + r′)Γ +′ +ε(v, r) − +r +� +min(r,ε(v−r)) +dr′ K1 +3(v − r, v − r − r′) +� +Γ +′ +ε(v, r + r′) − Γ +′ +ε(v, r) +� � +82 + ++ +1(v − r ≥ −b0) +� +r +� +min(r,ε(v−r)) +dr′ K3(v − r, v − r + r′) +� +Γ +′ +ε(v, r) − e− 1 +2r′Γ +′ +ε(v, r − r′) +� +− +r +� +min(r,ε(v−r)) +dr′ K1 +3(v − r, v − r − r′) +� +Γ +′ +ε(v, r + r′) − Γ +′ +ε(v, r) +� � ++ +1(v < −b0) +� +r +� +min(r,ε(v−r)) +dr′ K1 +3(v, v − r′) +� +Γ +′ +ε(v, r) − Γ +′ +ε(v − r′, r − r′) +� +− +min(r,a1−v) +� +min(r,ε(v−r)) +dr′K3(v, v + r′) +� +Γ +′ +ε(v + r′, r + r′) − Γ +′ +ε(v, r) +� ++ +r +� +min(r,a1−v) +dr′ K3(v, v + r′) +� +e− 1 +2 r′Γ +′ +ε(v + r′, r + r′) − Γ +′ +ε(v, r) +� � ++ +1(v ≥ −b0) +� +r +� +min(r,ε(v−r)) +dr′ K1 +3(v, v − r′) +� +Γ +′ +ε(v, r) − Γ +′ +ε(v − r′, r − r′) +� +− +r +� +min(r,ε(v−r)) +dr′ K3(v, v + r′) +� +e− 1 +2r′Γ +′ +ε(v + r′, r + r′) − Γ +′ +ε(v, r) +� � ++ +1(v < −m0) +r +� +min(r,−v−b0) +dr′K1 +3(v, v − r′)Γ +′ +ε(v − r′, r − r′) ++ +1(v ≥ −m0) +r +� +min(r,ε(v−r)) +dr′K1 +3(v, v − r′)(1 − e−r′)Γ +′ +ε(v − r′, r − r′), +and +3) I3[Γ +′ +ε](v, r) = I +(1) +3 [Γ +′ +ε](v, r) + I +(2) +3 [Γ +′ +ε](v, r), +where +a) I +(1) +3 [Γ +′ +ε](v, r) += 1(v − r ≤ −m0) +� +1(v < −b0) +� +1(˜r < max(r, ε(v − r))) +∞ +� +ε(v−r) +dr′ K2 +3(v − r, v − r − r′)Γ +′ +ε(v, r) ++ +1(˜r ≥ max(r, ε(v − r))) +� +1(r > ε(v − r)) +r +� +ε(v−r) +dr′eµa(f(v − r) + f(v)) +� +K2 +3(v − r, v − r − r′)˜g(v, r) +− +� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +˜g(v, r + r′) +� +83 + ++ +˜r +� +max(r,ε(v−r)) +dr′eµa(f(v − r) + f(v)) +� +K2 +3(v − r, v − r − r′)˜g(v, r) +− +� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +˜g(v, r + r′) +� ++ +� ∞ +˜r +dr′ K2 +3(v − r, v − r − r′)Γ +′ +ε(v, r) +�� ++ +1(v ≥ −b0) +� +∞ +� +max(˜r,ε(v−r)) +dr′ K2 +3(v − r, v − r − r′)Γ +′ +ε(v, r) ++ +˜r +� +min(˜r,ε(v−r)) +dr′ +� +K2 +3(v − r, v − r − r′)Γ +′ +ε(v, r) +− +� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +eµ max(a,c0v,v−r−r′)(f(v) + f(v − r))˜g(v, r + r′) +��� ++ +1(−m0 < v − r < m0) +� ∞ +ε(v−r) +dr′ K2 +3(v − r, v − r − r′)Γ +′ +ε(v, r) ++ +1(v − r ≥ m0) +� +1(v ≤ 3r) +� � ∞ +v−r +dr′ K2 +3(v − r, v − r − r′)Γ +′ +ε(v, r) ++ +� v−r +ε(v−r) +(f(v) + f(v − r)) +� +K2 +3(v − r, v − r − r′)eµ max(a,c0v,v−r)˜g(v, r) +− eµ max(a,c0v,v−r−r′) � +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +˜g(v, r + r′) +�� ++ +1(v > 3r) +� v−r−c0v +� +ε(v−r) +dr′ K2 +3(v − r, v − r − r′)Γ +′ +ε(v, r) ++ +∞ +� +v−r−c0v +(f(v) + f(v − r)) +� +K2 +3(v − r, v − r − r′)eµ max(a,c0v,v−r)˜g(v, r) +− eµ max(a,c0v,v−r−r′) � +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +˜g(v, r + r′) +�� ++ A +− +1 [Γ +′,1 +ε ](v, r) + A +− +2 [Γ +′,1 +ε ](v, r) + A +− +3 [Γ +′,1 +ε ](v, r), +with +A +− +1 [Γ +′,1 +ε ](v, r) = − 1(v − r ≤ −m0)1(v < −b0)1(˜r > max(r, ε(v − r)))eµa× +× +˜r +� +max(r,ε(v−r)) +dr′� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� � +f(v − r − r′) − f(v − r) +� +˜g(v, r + r′), +84 + +A +− +2 [Γ +′,1 +ε ](v, r) = − +1(v − r ≤ −m0) +� +1(v < −b0)1(r > ε(v − r))eµa × +× +r +� +ε(v−r) +dr′� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� � +f(v − r − r′) − f(v − r) +� +˜g(v, r + r′) ++ +1(v ≥ −b0)1(˜r > ε(v − r)) +˜r +� +ε(v−r) +dr′� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +× +× +� +f(v − r − r′) − f(v − r) +� +˜g(v, r + r′) +� +, +A +− +3 [Γ +′,1 +ε ](v, r) = − +1(v − r > m0)1(v > 3r) +∞ +� +v−r +dr′� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +× +× eµ max(a,c0v,v−r−r′) � +f(v − r − r′) − f(v − r) +� +˜g(v, r + r′), +and b) I +(2) +3 [Γ +′ +ε](v, r) += +r +� +min(r,ε(v−r)) +dr′ K2 +3(v, v − r′)Γ +′ +ε(v, r) +− +1(v ≤ −m0) +� +1(v + r < −b0) +r +� +min(r,ε(v−r)) +dr′ K2 +3(v, v − r′)Γ +′ +ε(v − r′, r − r′) ++ +1(v + r ≥ −b0) +−v−b0 +� +ε(v−r) +dr′ K2 +3(v, v − r′)Γ +′ +ε(v − r′, r − r′) +� +− +1(v ≥ m0) +� +1(0 < v − r < c0v) +r +� +max(v−c−1 +0 (v−r),ε(v−r)) +dr′ K2 +3(v, v − r′)Γ +′ +ε(v − r′, r − r′) ++ +1(v − r ≥ c0v) +r +� +min(r,ε(v−r)) +dr′ K2 +3(v, v − r′)Γ +′ +ε(v − r′, r − r′) +� +. +The other terms are “small”, as seen from the definitions below: +4) I +ε +4[Γ +′ +ε](v, r) = +1(v ≥ a1)1(r ≤ ε(v − r)) +� ε(v−r) +� +r +dr′ K1,ε(v−r) +3 +(v − r, v − r − r′)Γ +′ +ε(v, r) +− +ε(v−r) +� +max(r,ε(v−r)−r) +dr′ K1,ε(v−r) +3 +(v − r, v − r − r′)Γ +′ +ε(v, r + r′) +� +85 + ++ +1(v ≥ −b0) +� ∞ +0 +dr′ � +K3 +2,ε(v−r+r′)(v − r, v − r + r′) − K3 +2,ε(v+r′)(v − r, v − r + r′) +� +Γ +′ +ε(v − r + r′, r′), +5) e1[Γ +′ +ε](v, r) +=1(v < a1) +max(r,ε(v−r)) +� +r +dr′ � +K1,ε(v−r) +3 +(v − r, v − r − r′)Γ +′ +ε(v, r) − K1,ε(v−r) +3 +(v − r, v − r − r′)Γ +′ +ε(v, r + r′) +K1,ε(v−r) +3 +(v, v − r′)Γ +′ +ε(v, r′) +� +, +6) e2[Γ +′ +ε](v, r) =1(v < −b0) +max(r,ε(v−r)) +� +r +dr′ � +Kε(v+r′) +3 +(v, v + r′)Γ +′ +ε(v, r) − Kε(v+r′) +3 +(v, v + r′)Γ +′ +ε(v + r′, r + r′) ++Kε(v+r′) +3 +(v − r, v − r + r′)Γ +′ +ε(v − r + r′, r′) +� +, +7) e3[Γ +′ +ε](v, r) =1(v ≥ −b0) +max(r,ε(v−r)) +� +r +dr′ � +Kε(v+r′) +3 +(v, v + r′)Γ +′ +ε(v, r) − K3 +2,ε(v+r′)(v, v + r′)Γ +′ +ε(v + r′, r + r′) ++K3 +2,ε(v+r′)(v − r, v − r + r′)Γ +′ +ε(v − r + r′, r′) +� +, +8) e4[Γ +′ +ε](v, r) = +1(v − r < −b0) +min(r,ε(v−r)) +� +0 +dr′ Kε(v−r+r′) +3 +(v − r, v − r + r′) +� +Γ +′ +ε(v, r) − Γ +′ +ε(v, r − r′) +� ++ +1(v − r ≥ −b0) +min(r,ε(v−r)) +� +0 +dr′ Kε(v−r+r′) +3 +(v − r, v − r + r′) +� +Γ +′ +ε(v, r) − e− 1 +2r′Γ +′ +ε(v, r − r′) +� +− +1(v < a1) +min(r,ε(v−r)) +� +0 +dr′ K1,ε(v−r) +3 +(v − r, v − r − r′) +� +Γ +′ +ε(v, r + r′) − Γ +′ +ε(v, r) +� +− +1(v ≥ a1) +min(r,ε(v−r)) +� +0 +dr′ +1(r + r′ > ε(v − r))K1,ε(v−r) +3 +(v − r, v − r − r′) +� +Γ +′ +ε(v, r + r′) − Γ +′ +ε(v, r) +� ++ +1(v ≥ a1) +min(r,ε(v−r)) +� +0 +dr′ +1(r + r′ ≤ ε(v − r))K1,ε(v−r) +3 +(v − r, v − r − r′)Γ +′ +ε(v, r), +9) e5[Γ +′ +ε](v, r) = +min(r,ε(v−r)) +� +0 +dr′ K1,ε(v) +3 +(v, v − r′) +� +Γ +′ +ε(v, r) − Γ +′ +ε(v − r′, r − r′) +� +86 + +− +min(r,ε(v−r)) +� +0 +dr′ Kε(v+r′) +3 +(v, v + r′) +� +Γ +′ +ε(v + r′, r + r′) − Γ +′ +ε(v, r) +� ++ +1(v ≥ −b0) +min(r,ε(v−r)) +� +0 +dr′ Kε(v+r′) +3 +(v, v + r′)(1 − e− 1 +2 r′)Γ +′ +ε(v + r′, r + r′), +10) e6[Γ +′ +ε](v, r) = +1(v ≥ −m0) +min(r,ε(v−r)) +� +0 +dr′ K1,ε(v) +3 +(v, v − r′)(1 − e−r′)Γ +′ +ε(v + r′, r + r′), +11) e7[Γ +′ +ε](v, r) = +� ε(v−r) +0 +dr′ Kv−r,v−r−r′ +3 +Γ +′ +ε(v, r), +and +12) e8[Γ +′ +ε](v, r) = +min(r,ε(v−r)) +� +0 +dr′K2,ε(v) +3 +(v, v − r′)Γ +′ +ε(v, r) +− +1(v ≤ −m0) +� +1(v + r < −b0) +min(r,ε(v−r)) +� +0 +dr′K2,ε(v) +3 +(v, v − r′)Γ +′ +ε(v − r′, r − r′) ++ +1(v + r ≥ −b0) +� ε(v−r) +0 +dr′ K2,ε(v) +3 +(v, v − r′)Γ +′ +ε(v − r′, r − r′) +� +− +1(v ≥ m0) +� +1(0 < v − r < c0v) +max(ε(v−r,v−c−1 +0 (v−r))) +� +v−c−1 +0 (v−r) +dr′ K2,ε(v) +3 +(v, v − r′)Γ +′ +ε(v − r′, r − r′) ++ +1(v − r ≥ c0v) +min(r,ε(v−r)) +� +0 +dr′ K2,ε(v) +3 +(v, v − r′)Γ +′ +ε(v − r′, r − r′) +� +. +F +Computations Relating to the Evolution of Dt = Dψε +t − ∆t +In the evolution equation (3.41) the operator Lε is given by: +Lε∆t(v, r) += +� v−r−δ1 +−∞ +dw K1,ε(v−r) +3 +(v − r, w)∆t(v, r) − +� v−r−δ1 +−∞ +dw +� +K1,ε(v−r) +3 +(v − r, w) − K1,ε(v) +3 +(v, w) +� +∆t(v, v − w) ++ +� ∞ +v+δ1 +dw Kε(w) +3 +(v, w)∆t(v, r) − +� ∞ +v+δ1 +dw +� +Kε(w) +3 +(v, w) − Kε(w) +3 +(v − r, w) +� +∆t(w, w − v + r) ++ +� v−δ1 +v−r +dw K1,ε(v) +3 +(v, w) (∆t(v, r) − ∆t(w, w − v + r)) + +� v +v−r+δ1 +dw Kε(w) +3 +(v − r, w) (∆t(v, r) − ∆t(v, v − w)) ++ +� v+δ1 +v+δ2 +dw Kε(w) +3 +(v, w) (∆t(v, r) − ∆t(w, w − v + r)) + +� v−δ2 +v−δ1 +dw K1,ε(v) +3 +(v, w) (∆t(v, r) − ∆t(w, w − v + r)) +87 + ++ +� v +v−r +dw K2,ε(v) +3 +(v, w) (∆t(v, r) − ∆t(w, w − v + r)) ++ +� v−r +−∞ +dw K2,ε(v−r) +3 +(v − r, w)∆t(v, r) − +� v−r +−∞ +dw +� +K2,ε(v−r) +3 +(v − r, w) − K2,ε(v) +3 +(v, w) +� +∆t(v, w) ++ +� � v−r +v−r−δ1 +dw K1,ε(v) +3 +(v, w)∆t(v, v − w) − +� v−r +v−r−δ1 +dw K1,ε(v−r) +3 +(v, w)∆t(v − r, v − r − w) ++ +� v+δ1 +v +dw Kε(w) +3 +(v − r, w)∆t(w, w − v + r) − +� v+δ2 +v +dw Kε(w) +3 +(v, w)∆t(w, w − v) ++ +� v +v−δ2 +dw K1,ε(v) +3 +(v, w)∆t(v, v − w) + +� v−r+δ1 +v−r +dw Kε(w) +3 +(v − r, w)∆t(w, w − v + r) +� +. +Note that the last six terms in square brackets contribute towards our new “Lδ∆t(v, r)”. The other part Lε +0 +is defined exactly as above, but with Kε( , ) +3 +( , ) substituted by K3( , ) − Kε( , ) +3 +( , ). +Like before, Lε is separated into three parts as Lε = Lε +u + Lε +δ + Lε +b and then Lε +uΓε is written as: +Lε +uΓε(v, r) =˜I1[Γε](v, r) + ˜I2[Γε](v, r) + ˜I3[Γε](v, r) + +8 +� +i=1 +˜ei[Γε](v, r) + ˜Iε +4[Γε](v, r). +We will write the definitions for ˜I1, ˜I2 and ˜I3 below. These terms determine the asymptotic behavior of +Lε +uΓε. The lower limits of the relevant integrals are different from similar terms seen before, owing to the +new interplay between the two kinds of “smallness”-parameters, namely the δ-functions and the ε-functions, +so we write below their complete expressions. ˜I4 and all the ˜ei’s have “smallness” coming from either ε or δ, +and we have already seen how such terms are controlled, so we will skip writing down the explicit formulae +for them. +1) ˜I1[Γε](v, r) += +∞ +� +max(r+δ1,ε(v−r)) +dr′� +K1 +3(v − r, v − r − r′)Γε(v, r) + K1 +3(v, v − r′)Γε(v, r′) − K1 +3(v − r, v − r − r′)Γε(v, r + r′) +� ++ 1(v < −b0) +� +∞ +� +max(r+δ1,ε(v−r)) +dr′ K3(v, v + r′)Γε(v, r) +− 1(r + δ1 ≤ a1 − v) +� +a1−v +� +max(r+δ1,ε(v−r)) +dr′ K3(v, v + r′)Γε(v + r′, r + r′) ++ +� ∞ +a1−v +dr′ K3 +2(v, v + r′)Γε(v + r′, r + r′) +− +r+a1−v +� +max(r+δ1,ε(v−r)) +dr′ K3(v − r, v − r + r′)Γε(v − r + r′, r′) − +∞ +� +r+a1−v +dr′ K3 +2(v − r, v − r + r′)Γε(v − r + r′, r′) +� +− 1(r + δ1 > a1 − v) +� +∞ +� +max(r+δ1,ε(v−r)) +dr′ K3 +2(v, v + r′)Γε(v + r′, r + r′) +88 + +− +r+a1−v +� +max(r,ε(v−r)) +dr′ K3(v − r, v − r + r′)Γε(v − r + r′, r′) − +∞ +� +r+a1−v +dr′ K3 +2(v − r, v − r + r′)Γε(v − r + r′, r′) +�� ++ 1(v ≥ −b0) +∞ +� +max(r+δ1,ε(v−r)) +dr′ � +K3(v, v + r′)Γε(v, r) + K3 +2(v − r, v − r + r′)Γε(v − r + r′, r′) +− K3 +2(v, v + r′)Γε(v + r′, r + r′) +� +, +2) ˜I2[Γε](v, r) += +1(v − r < −b0) +� +min(r,r−v) +� +min(r,max(δ1,ε(v−r))) +dr′ K3(v − r, v − r + r′) +� +Γε(v, r) − Γε(v, r − r′) +� ++ +r +� +min(r−v,r) +dr′ K3(v − r, v − r + r′)Γε(v, r) +− +r +� +min(r,max(δ1,ε(v−r))) +dr′ K1 +3(v − r, v − r − r′) +� +Γε(v, r + r′) − Γε(v, r) +� +� ++1(v − r ≥ −b0) +� +r +� +min(r,max(δ1,ε(v−r))) +dr′ K3(v − r, v − r + r′) +� +Γε(v, r) − e− 1 +2r′Γε(v, r − r′) +� +− +r +� +min(r,max(δ1,ε(v−r))) +dr′ K1 +3(v − r, v − r − r′) +� +Γε(v, r + r′) − Γε(v, r) +� +� ++1(v < −b0) +� +r +� +min(r,max(δ1,ε(v−r))) +dr′ K1 +3(v, v − r′) +� +Γε(v, r) − Γε(v − r′, r − r′) +� +− +min(r,a1−v) +� +min(r,max(δ1,ε(v−r))) +dr′K3(v, v + r′) +� +Γε(v + r′, r + r′) − Γε(v, r) +� ++ +r +� +min(r,a1−v) +dr′ K3(v, v + r′) +� +e− 1 +2 r′Γε(v + r′, r + r′) − Γε(v, r) +� � ++1(v ≥ −b0) +� +r +� +min(r,max(δ1,ε(v−r))) +dr′ K1 +3(v, v − r′) +� +Γε(v, r) − Γε(v − r′, r − r′) +� +− +r +� +min(r,max(δ1,ε(v−r))) +dr′ K3(v, v + r′) +� +e− 1 +2r′Γε(v + r′, r + r′) − Γε(v, r) +� � +89 + ++ +1(v < −m0) +r +� +min(r,−v−b0) +dr′K1 +3(v, v − r′)Γε(v − r′, r − r′) ++ +1(v ≥ −m0) +r +� +min(r,max(δ1,ε(v−r))) +dr′K1 +3(v, v − r′)(1 − e−r′)Γε(v − r′, r − r′), +3) ˜I3[Γε](v, r) = ˜I(1) +3 [Γε](v, r) + ˜I(2) +3 [Γε](v, r), +where +a) ˜I(1) +3 [Γε](v, r) += 1(v − r ≤ −m0) +� +1(v < −b0) +� +∞ +� +max(˜r,ε(v−r)) +dr′ K2 +3(v − r, v − r − r′)Γε(v, r) ++ +1(r > ε(v − r)) +r +� +max(δ1,ε(v−r)) +dr′ eµa(f(v − r) + f(v)) +� +K2 +3(v − r, v − r − r′)g(v, r) +− (K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′))g(v, r + r′) +� ++ +1(˜r > max(r + δ1, ε(v − r))) +˜r +� +max(r+δ1,ε(v−r)) +dr′eµa(f(v − r) + f(v)) +� +K2 +3(v − r, v − r − r′)g(v, r) +− (K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′))g(v, r + r′) +�� ++ +1(v ≥ −b0) +� ∞ +� +˜r +dr′ K2 +3(v − r, v − r − r′)Γε(v, r) ++ +1(˜r > max(δ1, ε(v − r))) +˜r +� +max(δ1,ε(v−r)) +dr′ +� +K2 +3(v − r, v − r − r′)Γε(v, r) +− (K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′))eµ max(a,c0v,v−r−r′)(f(v) + f(v − r))g(v, r + r′) +��� ++ +1(−m0 < v − r < m0) +� ∞ +ε(v−r) +dr′ K2 +3(v − r, v − r − r′)Γε(v, r) ++ +1(v − r ≥ m0) +� +1(v ≤ 3r) +� � ∞ +v−r +dr′ K2 +3(v − r, v − r − r′)Γε(v, r) ++ +� v−r +ε(v−r) +(f(v) + f(v − r)) +� +K2 +3(v − r, v − r − r′)eµ max(a,c0v,v−r)g(v, r) +− eµ max(a,c0v,v−r−r′)(K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′))g(v, r + r′) +� � +90 + ++ +1(v > 3r) +� v−r−c0v +� +ε(v−r) +dr′ K2 +3(v − r, v − r − r′)Γε(v, r) ++ +∞ +� +v−r−c0v +(f(v) + f(v − r)) +� +K2 +3(v − r, v − r − r′)eµ max(a,c0v,v−r)g(v, r) +− eµ max(a,c0v,v−r−r′)(K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′))g(v, r + r′) +� � ++ ˜A− +1 [Γ +1 +ε](v, r) + ˜A− +2 [Γ +1 +ε](v, r) + ˜A− +3 [Γ +1 +ε](v, r), +with +˜A− +1 [Γ +1 +ε](v, r) = − 1(v − r ≤ −m0)1(v < −b0)1(˜r > max(r + δ1, ε(v − r)))eµa× +× +˜r +� +max(r+δ1,ε(v−r)) +dr′ � +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� � +f(v − r − r′) − f(v − r) +� +g(v, r + r′), +˜A− +2 [Γ +1 +ε](v, r) = − 1(v − r ≤ −m0) +� +1(v < −b0)1(r > ε(v − r))eµa × +× +r +� +max(δ1,ε(v−r)) +dr′� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� � +f(v − r − r′) − f(v − r) +� +g(v, r + r′) ++ +1(v ≥ −b0)1(˜r > max(δ1, ε(v − r))) +˜r +� +min(δ1,ε(v−r)) +dr′� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +× +× +� +f(v − r − r′) − f(v − r) +� +g(v, r + r′) +� +, +˜A− +3 [Γ +1 +ε](v, r) = − 1(v − r > m0)1(v > 3r) +∞ +� +v−r +dr′� +K2 +3(v − r, v − r − r′) − K2 +3(v, v − r − r′) +� +× +× eµ max(a,c0v,v−r−r′) � +f(v − r − r′) − f(v − r) +� +g(v, r + r′), +and +b) ˜I(2) +3 [Γε](v, r) = +1(v ≤ −m0) +� +min(r,−v−b0) +� +min(r,max(ε(v−r),δ1)) +dr′ K2 +3(v, v − r′) +� +Γε(v, r) − Γε(v − r′, r − r′) +� +− +r +� +min(r,−v−b0) +dr′ K2 +3(v, v − r′)Γε(v, r) +� ++ +1(−m0 < v < m0) +� r +min(ε(v),r) +dr′ K2 +3(v, v − r′)Γε(v, r) +91 + ++ +1(v ≥ m0) +� +1(0 < v − r < c0v) +� +r +� +max(v−c−1 +0 (v−r),ε(v−r)) +dr′ K2 +3(v, v − r′) +� +Γε(v, r) − Γε(v − r′, r − r′) +� ++ +v−c−1 +0 (v−r) +� +min(ε(v−r),v−c−1 +0 (v−r)) +dr′ K2 +3(v, v − r′)Γε(v, r) +� ++ +1(v − r ≥ c0v) +r +� +min(r,max(δ1,ε(v−r))) +dr′ K2 +3(v, v − r′) +� +Γε(v, r) − Γε(v − r′, r − r′) +�� +. +References +[1] A. Griffin, T. Nikuni, and E. Zaremba, Bose-condensed gases at finite temperatures. Cambridge Uni- +versity Press, Cambridge, 2009. +[2] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics II . Springer, +New York, 1981. +[3] E. H. Lieb, R. Seiringer, and J. Yngvason, Bosons in a trap: A rigorous derivation of the Gross- +Pitaevskii energy functional, Phys. Rev. A 61(4) (2000) 043602. +[4] E. H. Lieb and R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases, Com- +mun. Math. Phys. 264(2) (2006) 505–537. +[5] L. Erd˝os, B. Schlein, and H.-T. Yau, Derivation of the Gross-Pitaevskii equation for the dynamics of +Bose-Einstein condensate, Ann. Math. 172(1) (2010) 291–370. +[6] J. Lukkarinen and H. Spohn, Not to normal order—Notes on the kinetic limit for weakly interacting +quantum fluids, J. Stat. Phys. 134(5) (2009) 1133–1172. +[7] J. Lukkarinen and H. Spohn, Weakly nonlinear Schr¨odinger equation with random initial data, Invent. +Math. 183(1) (2011) 79–188. +[8] Yu Deng and Zaher Hani, Full derivation of the wave kinetic equation, arXiv:2201.07169v1 [math.AP]. +[9] D. V. Semikoz and I. I. Tkachev, Condensation of bosons in the kinetic regime, Phys. Rev. D 55(2) +(1997) 489–502. +[10] M. Escobedo, S. Mischler, and J. J. L. Vel´azquez, Singular solutions for the Uehling-Uhlenbeck equation, +Proc. Roy. Soc. Edinburgh Sect. A 138(1) (2008) 67–107. +[11] M. Escobedo, Classical approximation of a linearized three waves kinetic equation, J. Funct. Anal., +282(8) (2022) 109390. +[12] M. Escobedo, On the linearized system of equations for the condensate-normal fluid interaction near +the critical temperature, arXiv:2201.07169v1 [math.AP]. +[13] E. Cortes and M. Escobedo, On a system of equations for the normal fluid-condensate interaction in a +Bose gas, J. Funct. Anal., 278(2) (2020) 108315. +92 + +[14] S. Dyachenko, A. C. Newell, A. Pushkarev, and V. E. Zakharov, Optical turbulence: weak turbulence, +condensates and collapsing filaments in the nonlinear Schr¨odinger equation, Physica D 57(1-2) (1992) +96–160. +[15] X. Lu, On isotropic distributional solutions to the Boltzmann equation for Bose-Einstein particles, J. +Stat. Phys. 116(5) (2004) 1597–1649. +[16] X. Lu, The Boltzmann equation for Bose-Einstein particles: Velocity concentration and convergence to +equilibrium, J. Stat. Phys. 119(5-6) (2005) 1027–1067. +[17] H. Spohn, Kinetics of the Bose-Einstein condensation, Physica D 239(10) (2010) 627–634. +[18] W. Rudin, Functional Analysis. Tata McGraw-Hill, New Delhi, 1974. +[19] T. Kato, Perturbation Theory for Linear Operators Springer, 1980 +[20] F. Riesz, B. Sz.-Nagy, Functional Analysis Dover Publications Inc., New York, 1953 +93 + diff --git a/YdE2T4oBgHgl3EQfEQZV/content/tmp_files/load_file.txt b/YdE2T4oBgHgl3EQfEQZV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0405fd9e252607be2555fa9dedfeabcefbe73df9 --- /dev/null +++ b/YdE2T4oBgHgl3EQfEQZV/content/tmp_files/load_file.txt @@ -0,0 +1,3477 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf,len=3476 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='03633v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='AP] 9 Jan 2023 Smoothing Properties of a Linearization of the Three Waves Collision Operator in the bosonic Boltzmann–Nordheim Equation Jogia Bandyopadhyay∗ and Jani Lukkarinen †1 1University of Helsinki, Department of Mathematics and Statistics January 11, 2023 Abstract We consider the kinetic theory of a three-dimensional fluid of weakly interacting bosons in a non- equilibrium state which includes both normal fluid and a condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' More precisely, we look at the previously postulated nonlinear Boltzmann–Nordheim equations for such systems, in a spatially homo- geneous state which has an isotropic momentum distribution, and we linearize the equation around an equilibrium state which has a condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We study the most singular part of the linearized operator com- ing from the three waves collision operator for supercritical initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The operator has two types of singularities, one of which is similar to the marginally smoothing operator defined by the symbol ln(1+p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Our main result in this context is that for initial data in a certain Banach space of functions satisfying a H¨older type condition, at least for some finite time, evolution determined by the linearized operator improves the H¨older regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The main difficulty in this problem arises from the combination of a point singularity and a line singularity present in the linear operator, and we have to use some fine-tuned function spaces in order to carry out our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 1 Introduction An experimentally observed and also widely theoretically studied phenomenon of cold quantum fluids is Bose condensation: a macroscopic number of fluid particles form a condensate whose mechanical properties are very different from the properties of the same fluid at higher temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For instance, the resistance of the condensate can vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Many books have been written on the topic, for example, there is a recent review of results including physics of partial condensation by Griffin, Nikuni and Zaremba [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Mathematically, to study Bose condensation is a challenge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' even the definition of the corresponding equilibrium states for ideal Bose gas requires some effort (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' in [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For interacting Bose gases, rigorous results on condensation have only recently started to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' They mainly concern the case of total condensation, or zero temperature, where all of the available particles lie in the condensate [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In the mean-field limit, the system can then be well-described by a factored state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', the particles are not correlated) determined by a single wave-function whose dynamics follow the Gross-Pitaevskii equation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ∗jogiab@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='com †jani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='lukkarinen@helsinki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='fi 1 As far as we are aware, there are no rigorous results about the dynamics of non-equilibrium states in other than the mean-field limit with total condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In condensed matter physics, one commonly used tool in the study of the time-evolution of bosonic quantum fluids is the bosonic Boltzmann-Nordheim equation (also called Uehling-Uhlenbeck equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It describes the evolution of the “phase space” density of the particles, f(r, v, t) ≥ 0, with r ∈ R3 denoting position, v ∈ R3 velocity, and t ≥ 0 time, such that ∂tf(r, v, t) + v · ∇rf(r, v, t) = C4[f(r, ·, t)](v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) The collision operator is given by C4[h](v0) = 4π � (R3)3dv1dv2dv3 δ(v0 + v1 − v2 − v3)δ(ω0 + ω1 − ω2 − ω3) � ˜h0˜h1h2h3 − h0h1˜h2˜h3 � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2) where δ(·) denote Dirac δ-distributions: the first one is simply a shorthand notation for a convolution integral, and the second enforces conservation of kinetic energy in the “collisions”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' we have also used the shorthand notations hj = h(vj), ˜hj = 1 + hj, ωj = Ekin j = 1 2v2 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3) As we discuss in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 and Appendix A, the main contribution to the evolution of a suitably scaled perturbation ψt of an equilibrium solution containing a condensate density ¯n > 0 is obtained by linearization of a certain three wave collision operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This results in an evolution equation d dtψt = −¯nL3ψt, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4) where the operator L3 has the following explicit form L3ψ(x) = � ∞ 0 dy K3(x, y)(ψ(x) − ψ(y)), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5) K3(x, y) = 4¯h(x)2xye− min(x,y)fBE(|x − y|) ˜fBE (max(x, y)) ˜fBE(x + y) � 1 + e− max(x,y)� , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6) ¯h(x) = � x5/2fBE(x) ˜fBE(x) �− 1 2 , fBE(x) := 1 ex − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7) The integral kernel K3 has two types of singularities: there is boundary “point singularity” when x → 0, and a 1/|x−y| -type “line singularity” as x → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The defining integral is absolutely convergent for C(1)-functions with sufficiently fast decay at x → 0 and x → ∞, but it is not clear if such properties could be preserved by solutions to the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In this paper, we clarify two issues about the evolution equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We first define the action of the operator, L3ψ, by the integral (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5) for H¨older continuous functions ψ, with a weight ensuring that the integral is absolutely convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We then show that this operator is non-negative in a certain weighted L2-space, and we prove that it has a unique Friedrichs extension into a self-adjoint, non-negative operator on this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The kernel of the operator is one-dimensional, and we prove that it has a spectral gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Hence, the semigroup generated by the self-adjoint operator is contractive in the orthocomplement of the zero subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We then study pointwise solutions to the evolution equation which the difference functions ∆t(x, y) := ψt(x) − ψt(y) would satisfy if they were sufficiently regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For initial data, the sufficient regularity is guaranteed by picking them from a certain weighted Banach space of functions satisfying a H¨older-type condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We obtain unique solutions for the difference equation to these initial data in this Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 2 However, in order to prove the smoothing, we then have to show that the solutions obtained thus, are in fact differences of the form ψt(x) − ψt(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In order to do this, we study a regularized version of the original evolution equation for similar H¨older initial data, obtain unique solutions that are also difference functions, take a limit to remove the regularization, and prove that in this limit the difference function becomes identical to the ∆t solution obtained earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Our result proved for the ∆t solutions then imply that H¨older regularity is improved by the time evolution almost everywhere for some finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Finally, we show that the pointwise solutions coincide with the solutions given by the L2-semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This implies that also the semigroup is smoothing, at least for sufficiently regular initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The main technical difficulty in the study of the linearized operator mentioned above is that the evolution equation has two competing singularities, where the singularity connected with smoothing is marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus we have to exercise considerable care in defining function spaces that remain invariant under the evolution at least for some finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' To illustrate the point, let us first consider the semigroup generated by −L0, where L0 denotes the positive operator on L2(Rd) corresponding to multiplication with ln(1+p2) in the Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' An explicit computation shows that the operator then acts on Schwarz functions as � dy K(x, y)(ψ(x) − ψ(y)) where the integral kernel K has the same line singularity, 1/|x − y|, as L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then for t ≥ 0 the semigroup operator e−tL0 is given by multiplication with (1 + p2)−t in the Fourier space, and thus the semigroup provides slow smoothing of solutions: it maps the Sobolev space Hs to Hs+2t for any s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This behavior is different from the standard case of semigroup of the Laplacian which has the symbol e−tp2, and thus immediately produces smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In fact, it will become apparent later that the linearized operator, after a change of variables to make it act on a weighted L2(R) space, in the present case closely resembles V0L0, where V0 denotes multiplication with e−u/2, u ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' However, this space is very large for purposes of using it directly to study the full nonlinear problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For example, it contains unphysical solutions of infinite mass and some regularity is needed to make sense of the nonlinear collision integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Our original motivation for studying the problem was to complete a nonlinear perturbation argument, and it was clear there that some smoothing property of the linearized semigroup would be needed to control the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Indeed, this programme has been taken up the Escobedo, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', in a series of papers and preprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' An explicitly treatable asymptotic version of the linearized operator was considered in [11] where its solutions were generated via a Green’s function method, yielding an integral formula with an controllable kernel acting on sufficiently regular initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The control of the kernel leads to estimates which are consistent with the smoothing which we prove here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The full linearized operator has recently been studied in the preprint [12] where solutions to the evolution equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4) are generated by a perturbation argument on the kernel function derived in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Uniqueness and possible semigroup properties are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Therefore, the present works gives at least an partial answer, in the sense of L2 spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since the explicit form of the linearized operator is slightly different from the one in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5), we have explained in Appendix A how they nevertheless can be connected by a change of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us now briefly describe how the rest of this paper is organized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Section 2 contains an analysis of the problem in a weighted L2 space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' we define Friedrichs extensions for L3 and a sequence of approximating operators denoted by Lε 3, prove some properties for the corresponding semigroup solutions and then prove the most important result of this section, namely Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5, which shows that the semigroup solutions corresponding to the approximating operators converge to the L2 solution for our original operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Section 3 contains all of our Banach space results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The H¨older regularity is obtained in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 as a property in a certain Banach space of functions of two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We then show that these functions can be identified with differences of the L2 solutions obtained in Section 2, at least for some finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In order to do this we have to first obtain results such as Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5 for the approximating 3 operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The main smoothing result Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 follows as a straightforward consequence of our Banach space results and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Although the most important theorems, namely Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4, are proved in these Banach spaces, we have to rely on solutions in the weighted L2 space in order to connect these theorems and arrive at the main smoothing result of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus, all our results in the weighted L2 space are presented in Section 2 while Section 3 is reserved for the Banach space theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We devote the rest of this section to a discussion of the physical connection and derivation of the linearized three waves collision operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The necessary computational details of this derivation can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 Physical motivation of the nonlinear problem and the proposed linearization The Bolztmann–Nordheim equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2) has not yet been rigorously derived from evolution of a bosonic quantum system but the following conjecture can be generalized from the discussion in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Consider a translation invariant quasi-free initial state on the bosonic Fock space determined by a two-point correlation function g0(r1 −r2) where g0 is a rapidly decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Suppose that the N-particle dynamics is given by a Hamiltonian with weak pair interactions, HN = �N i=1 1 2p2 i + λ 1 2 � i̸=j V (ri − rj), 0 < λ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' If V is well-localized and � R3dr V (r) = 1, then up to times O(λ−2) the state should be translation invariant and quasi-free apart from small corrections, and the following limit of the Fourier transform of the time-evolved two-point function should exist: �gt(v)|t=λ−2˜t → W(v, ˜t) as λ → 0+ for any, not too large, ˜t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In addition, this limit should be well approximated by solutions to the equation ∂tW(v, t) = C4[W(·, t)](v), with initial data W(v, 0) = �g0(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (In [6], for technical reasons, only a discretized version of this problem is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The above conjecture is a generalization of Conjecture 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 in the bosonic case “θ = 1” there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=') Although the conjecture remains unproven at the moment, it relies on a perturbative expansion which has been successfully applied to a related problem of equilibrium time-correlations for a discrete nonlinear Schr¨odinger equation [7] (the connection to the above conjecture is outlined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 there) and, more recently, also for its continuum version [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The proof in [7] uses quite heavily a property analogous to supv |�g0(v)| < ∞ which in the above setup would be a consequence of the assumed sufficently fast decay of correlations for large spatial separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For a critical ideal Bose fluid at equilibrium one has W(v) equal to the critical Bose-Einstein distribution which blows up as |v|−2 near v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus this condition is violated in a critical system, and one can justifiably question the validity of the perturbative derivation for any system with critical and supercritical densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' If W is homogeneous and isotropic, it depends only on x = ω(v) = 1 2v2, and the evolution equation for f(x, t) := W( √ 2xˆe1, t) can be rewritten for x, t ≥ 0 as ∂tf(x, t) = C4[f(·, t)](x) after (with a slight abuse of notation and neglecting an overall numerical constant) we define C4[f](x0) := 1 √x0 � R2 + dx2dx3 1(x1 ≥ 0) min j=0,1,2,3 √xj � ˜f0 ˜f1f2f3 − f0f1 ˜f2 ˜f3 � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='8) where x1 = x2 + x3 − x0, and fi = f(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (To show the connection is not completely straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It has been done in Appendix A of [9] for W which are Schwartz functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=') The numerical solutions to this equation were studied by Semikoz and Tkachev in [9], and they found a finite time blowup at x = 0 for smooth, but supercritical, initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In light of the doubts about the perturbative derivation for singular cases, it is not clear how the solutions should be continued beyond the formation of the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' One possibility is that the perturbative kinetic argument simply becomes inapplicable, and one has to go back to the original time-evolution in Fock space to resolve the issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Another possibility is that nothing very special happens, and the equation continues to hold in its original (pointwise) sense, only restricted 4 to x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The equation ∂tf = C4[f], x > 0, with f(x, 0) ∼ x−7/6, was studied by Escobedo, Mischler, and Vel´azquez in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (The value 7 6 is related to Kolmogorov theory of wave turbulence [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=') They show the existence of solutions locally in time, preserving the x−7/6 singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' However, these solutions do not conserve total mass of particles (which is obviously conserved by the microscopic dynamics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' If one thinks that the extra mass is exchanged with the condensate, this way of “extending” the solution would correspond to adding the condensate mass as an extra degree of freedom with no backreaction to the normal fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' A different extension was considered by Lu in [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' He considers weak solutions, positive measures µt(dx) such that t �→ µt is weakly continuously differentiable and ∂tµt = C4[µt] in the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The existence of such solutions is proven in [15] wherein the precise meaning of how the measures are solutions is defined on page 1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In [16] he also proves that the solutions can be chosen so that they conserve both mass and energy and converge to the physically expected equilibrium distribution as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This occurs even in the supercritical case, and it is shown that then a portion of the total mass condenses to x = 0 (at least asymptotically as t → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' All of these results are deduced from subsequences of approximating solutions to a regularized problem, and as such leave open the uniqueness of these solutions, and are not amenable for numerical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In [9], Semikoz and Tkachev proposed a different method of continuing the solution after a condensate has formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' On physical grounds, they postulated that the solution would be a positive measure of the form ftot(x, t)√xdx = freg(x, t)√xdx + n(t)δ(x)dx, which corresponds to putting a mass n(t) ≥ 0 into a δ-distribution at the origin v = 0 ∈ R3 and allowing only for a regular distribution for |v| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (The square roots are explained by the identity v2d|v| = √ 2xdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=') With this ansatz they arrived at coupled equations of the form ∂tfreg(t) = C4[freg(t)] + n(t)C3[freg(t)] , d dtn(t) = −n(t)ρ[C3[freg(t)]] , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='9) where C3 is a new collision operator and ρ[f] denotes the mass functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since these equations involve n(t), their solutions do not coincide with the singular solutions studied in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The equations were again solved numerically, and convergence towards the expected equilibrium was found in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' But even this set of equations is somewhat problematic from the physical point of view: it does not answer how a condensate can be generated (if n(0) = 0, it remains zero for all times), and for regular functions f one can prove that ρ[C3[f]] ≥ 0, so it seems that n(t) can only decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' A more detailed analysis of the ansatz was made by Spohn in [17], and among other things, he showed that if freg(x, t) ≃ a(t)x−1, as would be the case for a critical equilibrium distribution, then the second equation is equal to d dtn(t) = −n(t)(2ρ[√xfreg(x, t)] − c0a(t)2) , with c0 = 1 3π2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='10) It follows that the critical freg are stationary solutions, and there can be an exchange of mass with either sign between the regular fluid and the condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This computation and those in [10] clearly illustrate that, for singular data, the meaning of the Boltzmann equation has to be carefully specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In this paper we look at a slight modification of the above evolution equations in which, at least in principle, condensate can be freely created and destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We use the same equations as before for the regular part, but do not try to form a differential equation for n(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Instead, as motivated by Lu’s results, we impose a strict conservation of mass for all times and use this as a definition of n(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Our aim is to study the linearization of the three waves collision operator C3 around a stationary solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' To this end, we consider the full distribution function for a Bose fluid at time t, in the presence of a condensate, given by the measure µtot t (dx) = freg(x, t)√xdx + n(t)δ0(dx) on R+ := [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Here δ0 denotes the unit measure concentrated at x = 0, and freg denotes a function on R+ with finite mass and energy, respectively defined 5 by the functionals ρ[f] := � ∞ 0 dx√xf(x), e[f] := � ∞ 0 dx√xxf(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Including the contribution from the condensate, the total mass and energy are then defined as M(t) = ρ[freg(t)] + n(t) , E(t) = e[freg(t)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' To summarize the conventions made so far: to get the “density” in the original 3-dimensional Boltzmann- Nordheim equation, we use “f(r, v, t)dv” = √ 2µtot t (d(v2/2))dΩ, where dΩ denotes the standard integration over the angular variables of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The presence of the scaling factor √ 2 here guarantees that “ ˜f(r, v, t)dv” = √ 2˜µtot(d(v2/2))dΩ with ˜µtot(dx) := (1 + freg(x, t))√xdx + n(t)δ0(dx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Suppose that the initial data is determined by n(0) = n0 ≥ 0 and freg(x, 0) = f0(x), where f0 ≥ 0 is suitably regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In particular, we assume that e0 := e[f0], ρ0 := ρ[f0] are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' If e0 = 0, then f0 = 0 almost everywhere, and defining freg(x, t) = 0, n(t) = n0, will yield a solution, corresponding to total condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This case is not of interest here, let us only point out that it is in line with the previous results on total condensation: the homogeneous solutions to the Gross-Pitaevskii equation are wave-functions with a constant magnitude, thus leading to particle densities which do not vary in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us thus assume that e0 > 0 when also ρ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For such initial data, and with weak interactions, one would physically expect the system to relax to an equilibrium distribution which is very close to that of free particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' As discussed in the introduction, the Boltzmann-Nordheim equation is believed to arise in a scaling limit with the strength of the interaction going to zero, thus its stationary solutions should be given by the ideal gas equilibrium distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For subcritical initial data these depend on two parameters: an inverse temperature β > 0 and the chemical potential µ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' A particular case is when µ = 0 and the corresponding distribution is called the critical Bose-Einstein distribution, fβ,0(x) = fBE(βx) , where fBE(x) := 1 ex − 1 , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='11) and the subcritical distributions are given by fβ,µ(x) := fBE(β(x − µ)), µ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since e0 > 0, there obviously is a unique β > 0 such that e0 = e[fβ,0], and this is determined by β := �e[fBE] e0 � 2 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12) We assume now that the initial state is supercritical, M(0) > ρ[fβ,0], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', n0 + ρ0 e3/5 0 > ρ[fBE] e[fBE] 3 5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' If this is true, there are no Bose-Einstein distributions which would have the right energy and particle densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' These results can be found in many references, for instance, see Section 2 in [17] and Section 6 in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' On physical grounds, it is expected that the normal fluid relaxes towards the corresponding critical distribution and that the additional particles are forming the condensate: since the particles in the condensate 6 do not contribute to energy density, this allows having the same mass and energy in the initial and the equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Motivated by this physical discussion, we define the equilibrium condensate density by ¯n := M(0) − ρ[fβ,0] = n0 + ρ0 − ρ[fβ,0] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='13) With these definitions, the assumption of a supercritical initial state is equivalent to assuming ¯n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We postulate that the time-evolution of the system conserves total mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', that n(t) = M(0) − ρ[freg(t)] = ¯n − ρ[freg(t) − fβ,0] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='14) so the evolution equation is d dtfreg(x, t) = C4[freg(·, t)](x) + n(t)C3[freg(·, t)](x), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='15) which becomes a closed equation for freg, after we insert the mass conservation law in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Here the first collision operator is given by C4[f](x0) = 1 √x0 � R2 + dx2dx3 1(x1 ≥ 0)I(x) � ˜f0 ˜f1f2f3 − f0f1 ˜f2 ˜f3 � x1=x2+x3−x0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='16) where fi := f(xi), ˜fi = 1 + fi, and I(x) := min j=0,1,2,3 √xj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We have also ignored an overall explicit constant which can be recovered by rescaling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' From this form, a formal substitution of the ansatz yields for the interaction term with the condensate C3[f](x) = 2 √x � x 0 dy � ˜f(x)f(x − y)f(y) − f(x) ˜f(x − y) ˜f(y) � − 4 √x � ∞ x dy � ˜f(y)f(y − x)f(x) − f(y) ˜f(y − x) ˜f(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='17) As shown in [17], for f0 = fβ,0 one has C4[f0] = 0 = C3[f0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus f(x, t) = fβ,0(x) yields a stationary solution of this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (Note that then by definition also n(t) = ¯n = n0 is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=') Here we inspect if “small” perturbations of such states lead to solutions which relax towards a state of the same type (the parameters of the new state do not need to be the same as those of the original one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' A convenient way to define the perturbation is to consider ψt(x) := f(x, t) − fβ,0(x) Rβ(x) , where Rβ(x) = βxfβ,0(x)(1 + fβ,0(x)) = R1(βx) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' As here R1 ≃ x−1, for solutions of the type considered by Spohn, f(x, t) ≃ a(t)x−1, one would have a(t) = (ψt(0)+1)/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus if continuous solutions with such asymptotics exist, then ψt would be continuous also at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Now we can solve the dependence on β, by scaling x′ = β−1x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then C4 gains a factor of β2 and C3 a factor of β1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus if we have a solution f1(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ¯n) for β = 1, a solution to the generic case is given by f(x, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ¯n, β) := f1(βx, β−2t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' β−3/2¯n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Therefore, it suffices to consider the case β = 1, when fβ,0 = fBE, and ψt(x) = f(x, t) − fBE(x) R(x) , where R(x) = R1(x) = xfBE(x) ˜fBE(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='18) 7 Here and in the following, we employ a shorthand notation ˜fBE := 1 + fBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The time-evolution of ψt(x) is thus determined by d dtψt(x) = 1 R(x) � C4[fBE + Rψt](x) + (¯n − ρ[Rψt]) C3[fBE + Rψt](x) � , Now let −Li denote the linearization of Ci around fBE, and Qi the corresponding remainder: Qi[h] := Ci[fBE + h] + Lih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since Ci[fBE] = 0, Qi[h] is quadratic in h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus the evolution equation for ψt can be written in the form d dtψt = −Lψt + Q[ψt] , where L = L4 + ¯nL3 , Li = R−1LiR , Q[ψ] = Q4[ψ] + ¯nQ3[ψ] − ρ[Rψ]Q3[ψ] + ρ[Rψ]L3ψ , Qi[ψ] := R−1Qi[Rψ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It turns out that the linearized three waves collision operator is most singular and can thus be thought of as dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Following the computations presented in Appendix A, we arrive at the evolution equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4) with the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5) for the operator L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Although it is not shown in this paper, our main result can be extended without difficulty to cover also the subdominant operator L4, leading to a smoothing result for the full linearized operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Acknowledgements We are deeply grateful to Antti Kupiainen for many illuminating discussions on this problem as well as valuable suggestions and comments on this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' During the many years over which we have worked on this project, we have benefited also from discussions with several other colleagues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In particular, we would like to thank Cl´ement Mouhot, Herbert Spohn, and Juan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Vel´azquez for their comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We are also thankful to Miguel Escobedo for correspondence about their newest work on the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The research has been supported by the Academy of Finland, via an Academy project (project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 339228), the Finnish Centre of Excellence in Randomness and Structures (project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 346306) and ERC Advanced Investigator Grants 741487 and 227772.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' J Bandyopadhyay’s work in this paper is intended as a small tribute to their father Raghab Bandyopadhyay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 2 Solutions in a Weighted L2-space In this section we consider, in a certain weighted L2 space of functions, the linear operator appearing in the evolution equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4) as well as a certain sequence of related linear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We first change variables x → u = ln(ex − 1), x being the energy variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus, while the old variable x was in R+, our new variable u ∈ R, and we look at (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4) in a weighted L2 space of functions on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Also, we will henceforth include the prefactor ¯n in the definition of L3 (while continuing to use the same symbol for the operator), so that the right hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4) now reads (−L3ψt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' As discussed in more detail in Appendix A, the resulting operator then naturally acts on a weighted L2 space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The weight function ν is ν(dw) = ν(w)dw = e−w (ln(1 + ew)) 5 2 dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 8 In this space the linearized operator acts on any suitably regular function ψ (say, compactly supported and smooth) in its domain D(L3) as follows: (L3ψ)(u) = � R ν(dv)K3(u, v) � ψ(u) − ψ(v) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) K3(u, v) = 4¯n [(ln(1 + eu))(ln(1 + ev))]− 3 2 e−|u−v| 1 + e− min(u,v) + e− max(u,v) 2 + emax(u,v) 1 − e−|u−v| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The associated sesquilinear form is given by ˜Q � φ, ψ � = 1 2 � R2(ν × ν)(du, dv)K3(u, v) � φ(u) − φ(v) �∗� ψ(u) − ψ(v) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ˜Q is evidently symmetric and non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We now extend the form domain D( ˜Q) to cover all ψ for which ˜Q � ψ, ψ � < ∞ in the sense the above integral is convergent (note that in this case the integrand is real and non-negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The Cauchy–Schwartz inequality then implies that ˜Q(φ, ψ) is defined for all φ, ψ ∈ D( ˜Q) as an absolutely convergent integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will also look at a sequence of regularized linear operators Lε 3, defined for all ε0 > 0 in the following way: (Lε 3ψ)(u) = � R ν(dv)K ε 3(u, v) � ψ(u) − ψ(v) � , ∀ψ ∈ D(Lε 3), where, K ε 3(u, v) = K3(u, v)1 − e− min(ε(u,v),|u−v|) 1 − e−ε(u,v) , ε(u, v) = ε(max(u, v)) = ε0 exp � − µ′ γ0 max(a1, max(u, v)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2) Here µ′, γ0, and a1 are positive-valued parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For results connected with only the L2-solutions, the values of these parameters in R+ do not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' However, for our main result, proved in a certain Banach space, these parameters have to be restricted to certain intervals in R+, as described in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4) and in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The sesquilinear form ˜Qε corresponding to the regularized linear operator is again non-negative and symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In the rest of this section we will construct Friedrichs extensions of the operators L3 and Lε 3 in L2(ν), show that these generate contractive semigroups in L2(ν), and finally, we prove that the semigroup solutions associated with the Friedrichs extension of Lε 3 approximate those for the extension of L3 as ε0 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 Friedrichs Extensions and Their Properties We will describe in detail the construction of the Friedrichs extension for L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The extensions for Lε 3, ∀ε > 0 can be constructed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Starting from the sesquilinear form ˜Q, we first define the following inner product Q(φ, ψ) = ˜Q(φ, ψ) + (φ, ψ) , on the earlier defined domain D(Q) = D( ˜Q) = {ψ ∈ L2(ν) : Q(ψ, ψ) < ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This domain is in fact already complete, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', suitable for the Friedrichs extension, as the next result shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' D(Q) is complete under the inner product Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In order to prove completeness of D(Q), consider an arbitrary sequence ψn ∈ D(Q), such that ψn is Cauchy under Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', Q(ψn − ψm, ψn − ψm) → 0 as m, n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since the assumptions imply that (ψn) is also Cauchy in L2(ν), there is ψ ∈ L2(ν) such that ψn → ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus we only need to show that i) ψ ∈ D(Q), and, ii) Q(ψn − ψ, ψn − ψ) → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us first note that, for any r > 0, we can define the bounded kernel Kr(u, v) = 1(K3(u, v) < 1/r)K3(u, v), so that, Kr ≤ K3 and Kr converges pointwise to K3 as r → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then the corresponding sesquilinear forms satisfy lim r→0+ Qr(φ, ψ) = Q(φ, ψ), ∀φ, ψ ∈ D(Q), by dominated convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Also, for any r > 0, there exist positive constants C, C1 < ∞, such that |Qr(φ, ψ)| ≤ |(φ, ψ)| + � R2(ν × ν)(du, dv)Kr(u, v)| � φ(u) − φ(v) � || � ψ(u) − ψ(v) � | ≤ � 1 + C r � ∥ψ∥L2∥φ∥L2, and, ∥ψ∥2 L2 ≤ Qr(ψ, ψ) ≤ � 1 + C1 r � ∥ψ∥2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The above inequalities imply that D(Qr) = L2(ν) and that the norm defined by Qr is equivalent to the L2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This means ψn → ψ ∈ D(Qr) and Qr(ψn, ψn) → Qr(ψ, ψ) as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since ψn ∈ D(Q) is Q−Cauchy, it is Q−bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus there exists some positive constant C′ < ∞, such that Qr(ψn, ψn) ≤ Q(ψn, ψn) ≤ C′ for all n, and for all r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This means Qr(ψ, ψ) ≤ C′, for all r > 0 and Qr(ψ, ψ) ր Q(ψ, ψ) ≤ C′ by monotone convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus ψ ∈ D(Q), and condition i) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We can now prove ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since ψn L2 −→ ψ ∈ D(Q), the sequence φn = ψn −ψ is such that φn ∈ D(Q) for all n and φn → 0 ∈ L2(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Choose ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then, since the sequence φn is Cauchy in D(Q), there exists n0 such that Q(φn − φm, φn − φm) < ε2, for all m, n > n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Now Qr(φn, φn) ր Q(φn, φn), and thus for ε > 0 and n > n0, there exists r = r(n, ε) such that 0 ≤ Q(φn, φn) − Qr(φn, φn) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then we have, for all m, n > n0, the following: Qr(φn, φn) = Qr(φn − φm, φn) − Qr(φm, φm − φn) + Qr(φm, φm) ≤ |Qr(φn − φm, φn)| + |Qr(φm, φm − φn)| + Qr(φm, φm) ≤ 4 √ C′ � Q(φn − φm, φn − φm) �1/2 + Qr(φm, φm) ≤ 4 √ C′ε + Qr(φm, φm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We can now let m → ∞, so that Qr(φm, φm) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus Q(φn, φn) < Qr(φn, φn) + ε ≤ (4 √ C′ + 2)ε, which means Q(φn, φn) → 0 as n → ∞, proving ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In order to construct the Friedrichs extension we start from a simpler version of the operator L3, defined on the space C0,α c of compactly supported α-H¨older continuous functions on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Clearly this is a subspace of D(Q) and dense in L2(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then for ψ ∈ C0,α c (R), we define (LR 3 ψ)(u) = � R ν(dv)K3(u, v) � ψ(u) − ψ(v) � , where the integral is absolutely convergent for all u, and it yields a function in L2(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We can use Fubini’s theorem to conclude that for all ψ ∈ C0,α c (R) and φ ∈ D(Q) (φ, LR 3 ψ) = Q(φ, ψ) − (φ, ψ), 10 so the form domain of LR 3 is contained in D(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We can then conclude that the Friedrichs extension L3 of LR 3 is given by (φ, L3ψ) = Q(φ, ψ) − (φ, ψ), ∀ψ ∈ D(L3), φ ∈ D(Q), D(L3) = {ψ ∈ D(Q) | ∃C < ∞ such that |Q(φ, ψ)| ≤ C∥φ∥L2, ∀φ ∈ D(Q)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We refer to chapter VIII, pages 329-334, [20] and chapter VI, pages 322-326, [19] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Our main results are proved in the Banach space X of continuous functions on R satisfying a H¨older-type condition (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' From the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4) of the weight Γ0 characterizing X, it is obvious that ∀ψ ∈ X, Q(ψ, ψ) < ∞, since K3(u, v) � Γ0(u, v) �2is absolutely integrable under ν × ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus for all φ ∈ D(Q), for all ψ ∈ X, the integral defining Q(φ, ψ) is absolutely convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It is easy to check that G ∈ L2(ν), where G(u) = � R ν(dv)K3(u, v)|ψ(u) − ψ(v)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Therefore, we may find a constant C > 0 such that |Q(φ, ψ)| ≤ C∥φ∥L2∥G∥L2, so ψ ∈ D(L3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus X ⊂ D(L3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In Section 3, where our main results are written, we drop the bar and simply denote the extended operator by L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us point out that, by a similar argument as used above, if ψ is bounded, measurable and satisfies supu � R ν(dv)K3(u, v)|ψ(u) − ψ(v)| < ∞, then ψ ∈ D(L3) and L3ψ is given by the absolutely convergent integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The Friedrichs extension L ε 3 of the operator Lε 3 is constructed in an exactly similar manner as above and it is easily seen that Xε ⊆ D(L ε 3), where Xε is the corresponding Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We now prove the following result about these extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The linear operators L3 and L ε 3 are non-negative and their zero subspace is spanned by the constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In addition, all have spectral gaps in L2(ν) and generate contraction semigroups in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We first prove the claim for L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The proof for L ε 3 is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' If ψ ∈ D(L3), we have the following lower bound for the corresponding quadratic form (ψ, L3ψ) = 1 2 � R2(ν × ν)(du, dv)K3(u, v)|ψ(u) − ψ(v)|2 ≥ (ψ, L′ψ), where (L′ψ)(u) = V ′(u)ψ(u) − � R ν(dv)K′(u, v)ψ(v), V ′(u) = � R ν(dv)K′(u, v), and, K′(u, v) = 2K′ 0(u, v) + K′ 1(u, v) � 1(v > u/2) + 1(v < 2u) � + K′ 2(u, v) � 1(v > 3u/2) + 1(v < 2u/3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Here K′ 0(u, v) = ¯n 1(− ln 2 ≤ u ≤ ln 2)1(− ln 2 ≤ v ≤ ln 2) � ln(1 + eu) ln(1 + ev) �−3/2 , K′ 1(u, v) = ¯n1(u < − ln 2)1(v < − ln 2) � ln(1 + eu) ln(1 + ev) �−3/2 emin(u,v)e−|u−v|, 11 and, K′ 2(u, v) = ¯n 1(u > ln 2)1(v > ln 2) � ln(1 + eu) ln(1 + ev) �−3/2 emax(u,v)e−|u−v|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then it is easy to see that there exists C′ > 0, such that the potential V ′(u) has the following lower bound: V ′(u) ≥ C′¯n �� 1 − 2e 1 2u� 1(u < − ln 4) + e 3 2 u 1(u < − ln 2) + e− 3 2u 1(− ln 2 ≤ u ≤ ln 2) + u− 1 2 e− 1 2u 1(u > ln 2) + u− 3 2 � �2 3u �2 − (ln 2)2� 1(u > 3 2 ln 2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Therefore, V ′(u) is strictly positive and there exists a∗ > 0 such that σ(V ) ⊂ (a∗, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It is also easily checked that the integral operator associated with the kernel K′ is Hilbert-Schmidt on L2(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This means, firstly, that L′ is seld-adjoint (see theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3, chapter IV, [19]) and secondly, that L′ has the same essential spectrum as V ′ (see theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='35, chapter IV, [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The first implication tells us that σ(L′) ⊂ [0, ∞) (the associated quadratic form being non-negative) and the second implication means that σ(L′)∪[0, a∗) contains only discrete semi-simple eigenvalues, so that a∗ is the only possible accumulation point of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Clearly then, L′ and hence L3 have spectral gaps in L2(ν), and thus generate contraction semigroups in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' If ψ is a constant function, it belongs to the domain of L3 and L3ψ is given by the convergent integral which yields zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' On the other hand, if ψ ∈ D(L3) is not constant almost everywhere, then (ψ, L3ψ) > 0 and thus also L3ψ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Therefore, the zero subspace of L3 is given by constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' From the definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2) of the kernel function K ε 3 it is clear that, for all ψ ∈ D(L ε 3), we again have the following lower bound: (ψ, L ε 3ψ) = 1 2 � R2(ν × ν)(du, dv)K ε 3(u, v)|ψ(u) − ψ(v)|2 ≥ (ψ, L′ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then the exact same argument as before leads us to the conclusion that L ε 3 has a spectral gap in L2(ν), it generates a contraction semigroup there, its zero subspace is given by constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 Approximating the Semigroup Solution generated in L2(ν) by L3 For the results proved in this subsection we will use certain lemmas and theorems proved in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The proofs of these lemmas/theorems from Section 3 rely on Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 but are independent of the results proved in this subsection, so there is no circularity of argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In what follows, we will mention explicitly whenever we use any result from the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We start with initial data ψ0 ∈ X, where X is the Banach space of continuous functions ψ on R such that ∥ψ∥X = sup v∈R |ψ(v)| ˜Γ(v) + sup (v,r)∈R×R+ |ψ(v) − ψ(v − r)| Γ0(v, r) < ∞, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3) where ˜Γ(v) = f(v) exp[µ max(a, c0v)], where f(v) = max � (ln(1 + ev))−α, (ln 2)−α� and, Γ0(v, r) = (f(v) + f(v − r)) exp[µ max(a, c0v, v − r)]g0(v, r), with g0(v, r) = � 1 − e−κr�γ0 , where 0 < α < 1/6, µ < 1 2 − 3 8α, µc0 ∈ (α, 1/4), a ≥ 9, γ0 ∈ (0, 1/8], and κ ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4) The parameters α, µ etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' appearing above do not have much bearing on the results in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' However, choosing the correct admissible values for them is critical for our results in the Banach spaces in 12 Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The intervals to which these parameters are restricted are largely determined by computational convenience in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For our computations in this paper, we choose 1/9 < α < 1/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5, µ ∈ [1/3, 7/16] and c0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' These choices obviously satisfy the conditions in the last line of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In Appendix B we describe in some detail how the weight function Γ0 and the choices for the above parameters are arrived at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us also mention here that the parameter µ′ appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2) is such that µ′ > µ, while a1 = a/c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We denote by Y , the Banach space of continuous functions on R × R+, bounded with respect to the weight function Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Now ∀(u, v) ∈ R2, ψ ∈ L2(ν), |ψ(u) − ψ(v)| = |ψ(max(u, v)) − ψ(max(u, v) − r)|, where r = |u − v|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In order to keep the notation simple, we will use, whenever convenient, the following equivalent definition for the weight Γ0 without changing the symbol (and follow a similar convention for the weights Γε defined subsequently) : Γ0(u, v) = (f(u) + f(v)) exp[µ max(a, c0 max(u, v), min(u, v))]g0(u, v), with g0(u, v) = � 1 − e−κ|u−v|�γ0 , ∀(u, v) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Evidently X is contained in both D(L3) and D(L ε 3), and the results in the previous subsection guarantee the existence of solutions e−tL3ψ0 and e−tL ε 3ψ0, unique in L2(ν), for the Cauchy problems associated with L3 and L ε 3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In this subsection we will prove that e−tL3ψ0 is actually the limit function that e−tL ε 3ψ0 converges to, as ε0 → 0 for times t ∈ [0, T ∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The time T ∗ > 0 comes from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5, and it has no dependence on ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Given the initial data described above, let us define for any ε0 > 0, ϕt = e−tL ε 3ψ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5, there exists T ∗ > 0 such that |Dϕt| ≤ A1Γε, for all t ∈ [0, T ∗] and some constant A1 < ∞, where Dϕt(v, r) = ϕt(v)−ϕt(v−r), for all (v, r) ∈ R×R+, both A1 and T ∗ being independent of ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The weight function Γε is defined as: Γε(v, r) = (f(v) + f(v − r)) exp[µ max(a, c0v, v − r)]g(v, r), g(v, r) = � 1 − e−κ(r+ε(v))�γ0 , γ0 = γ0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5) Let us now consider a sequence εn for the regularization parameter, such that εn → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For any εn > 0 taking the place of ε0 appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2), we write the following for simplicity’s sake, by a slight abuse of notation: K εn 3 (u, v) = K3(u, v)1 − e− min(εn(u,v),|u−v|) 1 − e−εn(u,v) , εn(u, v) = εn exp � − µ′ γ0 max(a1, max(u, v)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We also write Γεn(v, r) = (f(v) + f(v − r)) exp[µ max(a, c0v, v − r)]gn(v, r), gn(v, r) = � 1 − e−κ(r+εn(v))�γ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The corresponding L2-solution is e−tL εn 3 ψ0 and we write ϕn t = e−tL εn 3 ψ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then our results in the next section (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4) imply that for all ψ0 ∈ X, we have a unique solution ¯ϕn t of the associated Duhamel-integrated Cauchy problem in the Banach space Xεn, such that ∥ ¯ϕn t ∥Xεn = sup v∈R | ¯ϕn t (v)| ˜Γ(v) + sup (v,r)∈R×R+ | ¯ϕn t (v) − ¯ϕn t (v − r)| Γεn(v, r) < ∞, ∀εn > 0, 13 and ¯ϕn t = ϕn t , ν-almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We denote by Y εn the Banach space of continuous functions on R × R+ bounded with respect to the weight Γεn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that X ⊂ Xεn ⊂ D(L εn 3 ), for all εn > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For the rest of this section we will use some results, a couple of which have already been mentioned and all of which are easily obtained via short, straightforward computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since these are essential for the main theorem of this section, we collect them in the following lemma for ready reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Given initial data ¯ϕ0 ∈ X and εn > 0, the following are true: i) There exists A1 < ∞, T ∗ > 0, both independent of εn, such that |D ¯ϕn t | ≤ A1∥D ¯ϕ0∥Y Γεn, for all t ∈ [0, T ∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ii) L εn 3 ¯ϕn t ∈ L2(ν) for all t ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' iii) There exists a positive, measurable function �Γ on R2 such that � R2(ν × ν)(du, dv)K3(u, v)(�Γ(u, v))2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' iv) There exists a positive constant C < ∞, such that K εn 3 (u, v) � Γ εn(u, v) �2 ≤ CK3(u, v) ��Γ(u, v) �2, ∀(u, v) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' i) This estimate follows from the following upper bound obtained in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5: |D ¯ϕn t − ∆t| ≤ Q0 � Mεn �p ln � min(M, ε−1 n ) � ∥∆∥Y , ∀t ∈ [0, T ∗], where the constants Q0 and p > 0 are independent of εn, and ∆0 = D ¯ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ii) From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4 we know that ¯ϕn ∈ Xεn, ∀εn > 0, ∀t > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', sup t>0 (v,r)∈R×R+ |D ¯ϕn t (v, r)| Γεn(v, r) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It is straightforward to verify that G ∈ L2(ν), where G(u) = � R ν(dv)K εn 3 (u, v)Γεn(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It then follows naturally that L εn 3 ¯ϕn t ∈ L2(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' iii) Define �Γ(u, v) = (f(u) + f(v)) exp[µ max(a, c0 max(u, v), min(u, v))]¯g0(u, v), with ¯g0(u, v) = � 1 − e−κ|u−v|�γ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then it is easy to check that � R2(ν × ν)(du, dv)K3(u, v) ��Γ(u, v) �2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' iv) For all εn > 0, and r > 0, there exist C1 > 0, C2 > 0, depending on κ, such that the following is true: � 1 − e−κ(r+εn)�2γ0 1 − e− max(r,εn) ≤ C1 � 1 − e−κ(r+εn)�2γ0−1 ≤ C1 � 1 − e−κr�2γ0−1 ≤ C2 � 1 − e−κr�2γ0 1 − e−r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 14 It then follows directly from the definitions of the kernel functions and the weight functions that K εn 3 (u, v) � Γ εn(u, v) �2 ≤ CK3(u, v) ��Γ(u, v) �2, ∀(u, v) ∈ R2, for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Given initial data ψ0 ∈ X, let us consider again the sequence εn of regularization parameters, with εn → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Recall that corresponding to every εn, the regularized evolution equation (see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) has a solution ¯ϕn ∈ Xεn for all times by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then we can prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let {εn} be a sequence of regularization parameters such that εn → 0 and ψ0 ∈ X be the initial datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then the corresponding sequence { ¯ϕn} is Cauchy in C � [0, T ∗], L2(ν) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Consider m, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For the sake of simplicity, let us assume min(εm, εn) = εm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then we know that ¯ϕm ∈ Xεm and ¯ϕn ∈ Xεn and from the definitions of Γεm and Γεn it is clear that Xεm ⊂ Xεn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let ¯ϕn,m t = ¯ϕn t − ¯ϕm t , then ¯ϕn,m t ∈ Xεn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The function ¯ϕn,m t satisfies ∂t ¯ϕn,m t = −L εn 3 � ¯ϕn t − ¯ϕm t + ¯ϕm t � + L εm 3 ¯ϕm t = −L εn 3 ¯ϕn,m t + gn,m t , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6) where gn,m t (u) = L εm 3 ¯ϕm t (u) − L εn 3 ¯ϕm t (u) = � R ν(dv) � K εm 3 (u, v) − K εn 3 (u, v) �� ¯ϕm t (u) − ¯ϕm t (v) � = � R ν(dv)S εm,εn(u, v) � ¯ϕm t (u) − ¯ϕm t (v) � , with S εm,εn(u, v) = K εm 3 (u, v) − K εn 3 (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Now by part i) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 we can use: �� ¯ϕm t (u) − ¯ϕm t (v) �� ≤ A1∥D ¯ϕ0∥Y Γεm ≤ A1∥ψ0∥XΓεm, to obtain the following estimate: |gn,m t (u)| = ��� � R ν(dv)S εm,εn(u, v) � ¯ϕm t (u) − ¯ϕm t (v) ���� ≤ Fn,m(u), where Fn,m(u) = C0∥ψ0∥X � εn − εm �γ0eµ max(a,u)f(u) � ln(1 + eu) �− 1 2e− µ′ γ0 γ0 max(a1,u), C0 being some positive contant independent of εm and εn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Evidenly then, ∥Fn,m∥L2 → 0, as m, n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Looking back at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6) we can now write the following for all t ∈ [0, T ∗]: ∂t∥ ¯ϕn,m t ∥2 L2 = 2 Re � ¯ϕn,m t , ∂t ¯ϕn,m t � = 2 Re � ¯ϕn,m t , gn,m t � − 2 Re � ¯ϕn,m t , L εn 3 ¯ϕn,m t � < 2 Re � ¯ϕn,m t , gn,m t � by the positivity of L εn 3 , ≤ 2∥ ¯ϕn,m t ∥L2∥gn,m t ∥L2 ≤ 4∥ψ0∥L2∥Fn,m∥L2, 15 where in the last line we have used the fact that e−tL εn 3 ψ0 is the L2-representative of ¯ϕn t (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This means sup t∈[0,T ∗] ∥ ¯ϕn,m t ∥L2 ≤ � 4T ∗∥ψ0∥L2∥Fn,m∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since the right-hand side of the above equation goes to zero as m, n → ∞, we conclude that the sequence { ¯ϕn} is Cauchy in C � [0, T ∗], L2(ν) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since { ¯ϕn} is Cauchy, there exists a limit function in C � [0, T ∗], L2(ν) � that this sequence converges to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us call it ˜ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In what follows, we prove that this limiting function is independent of the choice of the sequence (εn) above, and we indeed then have limε→0 supt ∥ ¯ϕε t − ˜ψt∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For notational convenience, we denote already from the start ¯ϕε → ˜ψ as ε → 0, although strictly speaking these first refer to ¯ϕn → ˜ψ for some fixed choice of sequence εn → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We use ε to parametrize the regularized linear operator, ¯ϕε denotes the Banach space solution associated with the operator L ε 3, the corresponding Banach space is denoted by Xε and we are interested in the limit ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We are now in a position to prove the main result of this subsection, which states that this limit function ˜ψt is in fact the L2-solution associated with the original linearized operator L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Given initial datum ψ0 ∈ X, let ˜ψt = limε→0 ¯ϕε t in L2(ν) for t ∈ [0, T ∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then ˜ψt = e−tL3ψ0, ν-almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will first fix t ∈ [0, T ∗] and show that ˜ψt ∈ D(L3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us begin by demonstrating that ˜ψt ∈ D(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' There is now a subsequence of the original sequence (εn) such that along the subsequence ¯ϕε t → ˜ψt a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since limε→0 K ε 3(u, v) = K3(u, v) for all (u, v) ∈ R2, we have (along the subsequence) K ε 3(u, v) �� ¯ϕε t(u) − ¯ϕε t(v) ��2 −→ K3(u, v) �� ˜ψt(u) − ˜ψt(v) ��2 as ε → 0, except on a set that has measure zero under ν × ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then, by parts i) and iv) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3, there exist constants A and C < ∞ such that K ε 3(u, v) �� ¯ϕε t(u) − ¯ϕε t(v) ��2 ≤ A(∥ ¯ϕ0∥X)2K ε 3(u, v) � Γε(u, v) �2 ≤ CK3(u, v) ��Γ(u, v) �2, which implies the following by part iii) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3, and the dominated convergence theorem: lim ε→0 � R2(ν × ν)(du, dv)K ε 3(u, v) �� ¯ϕε t(u) − ¯ϕε t(v) ��2 = � R2(ν × ν)(du, dv)K3(u, v) �� ˜ψt(u) − ˜ψt(v) ��2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus, ˜ψt ∈ D(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We then return to the original sequence (εn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us now consider test functions φ from the space C0,α0 c (R) of compactly supported α0-H¨older continuous functions on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This is a dense subspace of L2(ν) and a subspace of D(L ε 3) for all ε > 0, as well as of D(L3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then, since K ε 3(u, v) ≤ K3(u, v) for all (u, v) ∈ R2 and � R ν(dv)K3(u, v) ��φ(u) − φ(v) �� is integrable, we have, by dominated convergence, lim ε→0 � L ε 3φ � (u) = � L3φ � (u), pointwise as well as in L2(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Also, by part i) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3, we have, for all t ∈ [0, T ∗], the following: ��� L ε 3 ¯ϕε t � (u) �� ≤ C1 � R ν(dv)K ε 3(u, v)Γ ε(u, v) ≤ C1H(u), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7) 16 where the constant C1 is independent of ε, H(u) = sup ε>0 � R ν(dv)K ε 3(u, v)Γ ε(u, v), and H ∈ L2(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then the following limit holds: lim ε→0 � φ, L ε 3 ¯ϕε t � = lim ε→0 � L ε 3φ, ¯ϕε t � = � L3φ, ˜ψt � = ˜Q(φ, ˜ψt), where we have used the self-adjointness of L ε 3, the limits obtained earlier and the fact that ˜ψt ∈ D(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' However, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7), we have the following: ��� φ, L ε 3 ¯ϕε t ��� ≤ C2∥φ∥L2∥H∥L2, where C2 is a constant independent of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This means that �� ˜Q(φ, ˜ψt) �� ≤ C2∥φ∥L2∥H∥L2, ∀φ ∈ C0,α0 c (R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus the map φ �→ Q(φ, ˜ψt) has a unique, bounded extension from the dense subspace containing our test functions to L2(ν) and we conclude that ˜ψt ∈ D(L3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then by self-adjointness of L3, ˜Q(φ, ˜ψt) = � L3φ, ˜ψt � = � φ, L3 ˜ψt � for any φ ∈ D(L3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The above bounds, uniform in ε, allow us to conclude that lim ε→0 � φ′, L ε 3 ¯ϕε t � = � φ′, L3 ˜ψt � , ∀φ′ ∈ L2(ν), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', L ε 3 ¯ϕε t−→L3 ˜ψt weakly in L2(ν), ∀t ∈ [0, T ∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Now for any �ψ ∈ D(L3) and 0 ≤ s ≤ t < T ∗, we have the following: ∂s � e−(t−s)L3 �ψ, ¯ϕε s � = � L3e−(t−s)L3 �ψ, ¯ϕε s � − � e−(t−s)L3 �ψ, L ε 3 ¯ϕε s � = � L3e−(t−s)L3 �ψ, ¯ϕε s − ˜ψs � + � e−(t−s)L3 �ψ, L3 ˜ψs − L ε 3 ¯ϕε s � −→ 0, as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Continuity in s and uniform boundedness in ε for the terms on the right hand side above then implies that we can apply the dominated convergence theorem to obtain � t 0 ds ∂s � e−(t−s)L3 �ψ, ¯ϕε s � = � �ψ, ¯ϕε t � − � e−tL3 �ψ, ψ0 � −→ 0 as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since we know that � �ψ, ¯ϕε t � −→ � �ψ, ˜ψt � as ε → 0, this means � �ψ, ˜ψt � = � �ψ, e−tL3ψ0 � for any �ψ ∈ D(L3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Therefore, ˜ψt = e−tL3ψ0, ν-almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 3 Smoothing Solutions in a weighted Banach Space of H¨older-continuous functions In this section we study the linearized three waves collision operator described in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) in a certain Banach space, and prove that the corresponding time evolution has a smoothing property in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The proof of this smoothing theorem relies on a few other results, proved in slightly different but related Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This section is organized as follows: first we state and briefly explain the main smoothing result as well as the theorems leading to it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' this is followed by two subsections containing the details of the proofs of these theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In the first subsection we prove the existence and uniqueness of solutions of an initial value 17 problem derived from our original evolution equation in a weighted Banach space characterized by a time- dependent H¨older-type condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In the second subsection we consider a regularized evolution equation which enables us to show that this solution is in fact identical to the one obtained in a space of differences of functions, with the same initial value, so that the result proved in the first subsection implies the smoothing of solutions of the original time evolution equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We begin by looking at the original evolution equation: ∂tψt(v) = −(L3ψt)(v) = � R dw K3(v, w) � ψt(v) − ψt(w) � , where the operator L3 is now written in terms of the kernel function in the flat space without the weight ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We keep in mind that L3, which we will analyze in a Banach space introduced in the previous section, is the Friedrichs-extended operator constructed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then the difference function ψt(v) − ψt(v − r) evolves in time as ∂t [ψt(v) − ψt(v − r)] = − [(L3ψt)(v) − (L3ψt)(v − r)] = − �� ∞ −∞ dw K3(v, w) (ψt(v) − ψt(w)) − � ∞ −∞ dw K3(v − r, w) (ψt(v − r) − ψt(w)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) Let us now write down the formulae defining K3 explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' i) When v > w, K3(v, w) = 4¯n (ln(1 + ev))− 3 2 ln(1 + ew) ew + 2e−(v−w) 1 + ew + e−(v−w) 1 1 − e−(v−w) , and, ii) when w > v, K3(v, w) = 4¯n (ln(1 + ev))− 3 2 ln(1 + ew) ev + 2e−(w−v) 1 + ev + e−(w−v) e−(w−v) 1 − e−(w−v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For our subsequent computations we will split the kernel function K3 as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This splitting is done in order to separate out some terms that do not contain the line-singularity and consequently have different asymptotic behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' When v > w, K3(v, w) = K1 3(v, w) + K2 3(v, w), where K1 3(v, w) = 4¯n (ln(1 + ev))− 3 2 ln(1 + ew) ew + 2e−(v−w) 1 + ew + e−(v−w) e−(v−w) 1 − e−(v−w) , and K2 3(v, w) = 4¯n (ln(1 + ev))− 3 2 ln(1 + ew) ew + 2e−(v−w) 1 + ew + e−(v−w) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For the region w > v we will use the following splitting of K3(v, w) in some parts of our computation: K3(v, w) = K3 1(v, w) + K3 2(v, w), where K3 2(v, w) = K3(v, w)e− 1 2(w−v), and K3 1(v, w) = K3(v, w) � 1 − e− 1 2 (w−v)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that the part K2 3 does not contain any line-singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Also, the integral of K2 3(v, w) with respect to the variable w yields a square-root-function-like growth for large, positive values of v, while K1 3(v, w) exhibits no such behavior due to the extra exponential decay in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us also note that in the absence of the point-singularity (when v does not assume arbitrarily large negative values), K3 1(v, w) leads to bounded integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will use this splitting of the kernel function for w > v, to cut out certain bounded parts later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The factor 1/2 appearing above has been chosen arbitrarily, but once fixed, this determines the behavior of the weight function Γ(v, r) for large, positive values of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We now write down a crucial but easily derived property of the kernel functions in the form of a lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 18 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The kernel functions satisfy the following inequalities: when v − r > w, K1 3(v − r, w) ≥ K1 3(v, w) and K2 3(v − r, w) ≥ K2 3(v, w) ∀r ≥ 0, and when w > v, K3(v, w) ≥ K3(v − r, w), as well as, K3 2(v, w) ≥ K3 2(v − r, w) ∀r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' When v−r > w, the inequalities in the first line are obvious from the formulae for the kernel functions K1 3 and K2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' When w > v, we can write the following: ∂ ∂v K3(v, w) = K3(v, w)b(v, w), where b(v, w) = −3 2 ev (1 + ev) ln(1 + ev) + 1 + 1 ew−v − 1 + ew−v 1 + ew + ew−v ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Therefore K3(v, w) ≥ K3(v − r, w), ∀r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Similarly, K3 2(v, w) ≥ K3 2(v − r, w), ∀r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 allows us to write a part of the right hand side of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) as the combination of a multiplication operator and a positivity-preserving operator, as we will see shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' we just write the following: [(L3ψt)(v) − (L3ψt)(v − r)] = �� v−r −∞ dw K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v) − ψt(w)) + � v v−r dw K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v) − ψt(w)) − � ∞ v dw K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(w) − ψt(v)) − � v−r −∞ dw K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v − r) − ψt(w)) + � v v−r dw K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(w) − ψt(v − r)) + � ∞ v dw K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(w) − ψt(v − r)) � + �� v−r −∞ dw K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v) − ψt(w)) + � v v−r dw K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v) − ψt(w)) − � v−r −∞ dw K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v − r) − ψt(w)) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2) where the last two lines do not contain any line singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We need to analyze the above operator in order to prove the main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Before proceeding to set the stage for that analysis, let us define the time-dependent weight function Γt(v, r), in terms of which our main theorem is stated: Γt(v, r) = �� f(v − r) + f(v) � exp � µ max (a, c0v, v − r) �� gt(v, r), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3) with the following H¨older-type time-dependent part gt : gt(v, r) = � 1 − e−κr�γt(v−r) , γt(v, r) = γ0 + a(t) 1 1 + eβ(v−r) , a(t) = 1 8 min(1, ¯n) t 1 + t, κ ≥ 7, 0 < γ0 ≤ 1/8, 1 ≤ β ≤ κ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4) 19 The choices for the parameters β, κ and γ0 relating to the time-dependent H¨older-type condition are ex- plained in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that at t = 0 this is just the weight Γ0 characterizing the Banach spaces Y and X defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4), with exponent γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The main result of this paper is then the following theorem: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Given initial datum ψ0 ∈ X, let us define ∆0(v, r) = ψ0(v)−ψ0(v−r) for all (v, r) ∈ R×R+, and ψt = e−tL3ψ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then there exists T ∗ > 0 such that for all t ∈ [0, T ∗] the following bound holds: ��ψt(v) − ψt(v − r) �� ≤ CΓt(v, r)∥∆0∥Y ν-almost everywhere on R × R+, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5) where C is a constant depending on the parameters appearing in the weight function Γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 is the obvious consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5 and three other results which are stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Before writing down the statements of these three theorems we briefly describe the evolution equations considered in them as well as the function spaces in which these results are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will just sketch out the schemes here, reserving the details of the constructions for later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The first theorem deals with an evolution equation derived from the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The main idea here is to introduce suitable cut-off functions δ1(v, r) and δ2(v, r), as well as cut-off parameters m0 and b0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' and then split the linear operator into an unbounded part Lu which is the sum of a potential function and a positivity-preserving operator, a bounded part Lb and a perturbation Lδ (see Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For t ≥ 0 and (v, r) ∈ R × R+, we define the variable ∆t(v, r) = ψt(v) − ψt(v − r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) is recast as the following evolution equation for the ∆-variable: ∂t∆t = −L∆t = −Lu∆t − Lb∆t − Lδ∆t = −Vu∆t + Ku∆t − Lb∆t − Lδ∆t, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6) where (Lu∆t)(v, r) = Vu(v, r)∆t(v, r) − (Ku∆t)(v, r) and the multiplication operator Vu is defined as Vu(v, r) = � v−r−δ1 −∞ dw K1 3(v − r, w) + � ∞ v+δ2 dw K3(v, w) + � v v−r+δ1 dw K3(v − r, w) + � v−δ2 v−r dw K1 3(v, w) + � v v−r dw K2 3(v, w) + � v−r −∞ dw K2 3(v − r, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7) We consider (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6) in the following Duhamel-integrated form: ∆t = e−tVu∆0 + � t 0 ds e−(t−s)VuKu[∆s] − � t 0 ds e−(t−s)VuLb[∆s] − � t 0 ds e−(t−s)VuLδ[∆s], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='8) in the Banach space Y of functions in C ([0, T] × (R × R+)), for some T > 0, bounded with respect to Γt as follows: ∥∆∥Y := sup t∈[0,T] sup (v,r)∈R×R+ |∆t(v, r)| Γt(v, r) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We then prove the following existence-uniqueness result for solutions of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 20 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' There exists T ∗ > 0 depending on the cut-off parameters used in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6), and the parameters α, µ and c0 appearing in the weight Γt, such that the following is true: Given initial value ∆0 ∈ Y , a solution ∆ of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='8) exists for all t ∈ [0, T ∗], such that ∥∆∥Y = sup t∈[0,T ∗] sup (v,r)∈R×R+ |∆t(v, r)| Γt(v, r) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This solution is unique in Y and given by ∆t = (1 − F)−1 e−tVu∆0, where F is the linear operator defined as F[∆t] = � t 0 ds e−(t−s)VuKu[∆s] − � t 0 ds e−(t−s)VuLb[∆s] − � t 0 ds e−(t−s)VuLδ[∆s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='9) In the theorem above, the time T ∗ is chosen so that �� � t 0 ds e−(t−s)VuLb[∆s] �� is small enough to guarantee the invertibility of the operator (1−F) in Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Observe that, ψ0 ∈ X (like in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2) implies that ∆0 ∈ Y , with ∆0(v, r) = ψ0(v)−ψ0(v−r), Y being the Banach space of functions in C (R × R+) bounded with respect to the weight function Γ0 defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' However, the above theorem does not guarantee that, given initial datum ∆0(v, r) = ψ0(v) − ψ0(v − r) in Y , the unique solution ∆t of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='8) obtained via Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 is still a difference of the ψ-variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' That this is indeed the case, is proved by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Before stating these theorems, let us briefly describe the evolution equations and their solutions consid- ered in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We start from the regularized evolution equation associated with the operator in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2), ∂tψε t (v) = −(Lε 3ψ)(u) = � R dv Kε 3(u, v) � ψε t (u) − ψε t (v) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='10) where Lε 3 is now written in terms of the kernel function Kε 3 in the flat space without the weight ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We tag solutions of this equation with ε in order to avoid confusion later, when we compare the solutions associated with different operators such as L3 and Lε 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We consider (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='10) for initial data ψε 0 ∈ X, where X = � h ∈ C(R) : ∥h∥X < ∞ � , ∥h∥X = sup v∈R |h(v)| ˜Γ(v) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='11) where ˜Γ is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12 guarantees, for all t ≥ 0, the existence of a unique solution ψε t ∈ X, of the following Duhamel-integrated form of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='10): ψε t =e−tV εψε 0 + � t 0 ds e−(t−s)V εKε u[ψε s] + � t 0 ds e−(t−s)V εKε b [ψε t ], where V ε is a multiplication operator, Kε b is a bounded operator on L2(ν), and a splitting akin to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6) recasts the operator Lε 3 in the following form: Lε 3ψε t = V εψε t − Kε uψε t − Kε b ψε t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The existence of such a solution ψε t implies the existence of the difference variable solution Dψε t , with Dψε t (v, r) = ψε t (v) − ψε t (v − r), of the following evolution equation of differences derived from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='10): ∂tDψε t (v, r) = −( ˜LεDψε t )(v, r) = −( ˜Lε uDψε t )(v, r) − ( ˜Lε sDψε t )(v, r) − Kε b[ψε t ](v, r), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12) 21 where again the linear operator ˜Lε has been split, `a la (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6), into an unbounded part ˜Lε u, a perturbation ˜Lε s, and a part Kε b which can be bounded in terms of the L2(ν)-norm and the X-norm of ψε t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Like before, the unbounded part ˜Lε u can be written as the sum ˜Lε u = ˜Vε − ˜Kε u, of the multiplication operator ˜Vε and a positivity-preserving integral operator ˜Kε u, leading to the following Duhamel-integrated form of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12): Dψε t = e−t˜VεDψε 0 + � t 0 ds e−(t−s)˜Vε ˜Kε uDψε s − � t 0 ds e−(t−s)˜Vε ˜Lε sDψε s − � t 0 ds e−(t−s)˜Vε ˜Kε b[ψε s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='13) Then the next theorem shows the existence and uniqueness of solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Given ε > 0, consider initial datum ψε 0 ∈ X such that ∥Dψε 0∥Y ε = sup (v,r)∈R×R+ |ψε 0(v) − ψε 0(v − r)| Γε(v, r) < ∞, Γε being the weight function defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then there exists for all t > 0, a unique solution Dψε t of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='13) with initial value ψε 0, such that ∥Dψε t ∥Y ε < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The next theorem connects the solutions Dψε t with the solutions ∆t obtained in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 fot t ∈ [0, T ∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Before stating the theorem let us define the variable whose time-evolution is dealt with in this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This variable is the following difference function, defined for all t ∈ [0, T ∗], Dε t = Dψε t − ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='14) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12) we see (see Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 for details) that for t ∈ [0, T ∗], the time evolution of Dt is governed by the equation ∂tDε t = − ˜LεDε t − ˜Lε 0[∆t], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='15) where the operator ˜Lε is defined exactly like ˜L, but with the kernel function Kε 3 replacing the original kernel function K3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' the operator ˜Lε 0 also has the same structure as ˜L, but with K3 − Kε 3 replacing K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Explicit expressions of these operators are given in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then the next result is Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Consider initial datum ψ0 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This means Dψ0 = ∆0 ∈ Y and D0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then there exists T ∗ > 0 and a unique Duhamel-integrated solution Dε t of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='15), such that Dε t ∈ Y ε for all t ∈ [0, T ∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Moreover, there exists some positive constant Q < ∞ depending on the parameters α, κ and γ0, such that |Dt| is bounded above as follows: |Dε t | ≤ Q (Mε0)p ln � min(M, ε−1 0 ) � ∥∆∥Y Γε, where p = p(γ0) > 0 and the positive constant M < ∞ appears as a cut-off parameter in the evolution equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6) for ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The smoothing result of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 can now be obtained in a very straightforward manner, as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Given initial datum ψ0 ∈ X, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5 imply the following ∀t ∈ [0, T ∗]: ��� e−tL3ψ0 � (v) − � e−tL3� ψ0(v − r) �� = lim ε0→0 |Dψε t (v, r)| ν-almost everywhere 22 ≤ |∆t(v, r)| ≤ C∥∆0∥Y Γt(v, r), where we have used Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 in the last line, C = ∥(1 − F)−1∥Y < ∞ and it is easy to see that ∥e−tVu∆0∥Y ≤ ∥∆0∥Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us make an important remark here about the time T ∗ mentioned in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This upper bound has its origin in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3, where the condition t ≤ T ∗ is used to make �� � t 0 ds e−(t−s)VuLb[∆s] �� < A(b0, m0, µ, c0)T ∗∥∆∥Y small enough to guarantee the invertibility of (1 − F) (see the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Now this result can be extended in time as follows: we first prove the theorem for t ∈ [0, T ∗], then we can choose an initial time 0 < t0 < T ∗, and prove the existence of a unique solution of the corresponding Duhamel-integrated equation for t ∈ [t0, t0 + T ∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Each such extension in time results in an extra factor of the form (1 − F1)−1 (here F1 represents the operator corresponding to F in the new Duhamel equation) in the formula for ∆t obtained in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3, and this process can be continued as long as the norm of the product is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This in turn means that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5 and finally, our main smoothing result Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2, all of which inherit the dependence on T ∗ from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3, can also be extended in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The subsections that follow are devoted to the derivations of the evolution equations for ∆t, ψε t and Dε t, and the proofs of Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Of these, the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 is the most delicate and the other proofs rely on estimates that are similar to those employed there, so we use the next subsection to first explain in detail how (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6) is arrived at and then prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 Existence and Uniqueness of Solutions in a Banach space with a Time-dependent H¨older-type Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 Derivation of Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6) As mentioned before, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6) is obtained via the introduction of suitable cut-off functions δ1(v, r) and δ2(v, r) in the original evolution equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) for the differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Before defining these cut-off functions, let us explain the idea behind the recasting of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Recall that eventually we want to solve a Duhamel-integrated form of the evolution equation, where a potential function is used to control the rest of the terms (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' the multiplication operator Vu in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' When we split our linear operator into a potential Vu, a positivity- preserving part Ku, a perturbation Lδ and the bounded part Lb, we thus want to make Vu as large as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We start from the original equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) for the evolution of the differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The right hand side is then rewritten (without the minus sign) as (L3ψt)(v) − (L3ψt)(v − r) = � v−r−δ1 −∞ dw K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v) − ψt(v − r)) − � v−r−δ1 −∞ dw � K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � (ψt(v) − ψt(w)) + � ∞ v+δ1 dw K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v) − ψt(v − r)) − � ∞ v+δ1 dw (K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)) (ψt(w) − ψt(v − r)) + � v+δ1 v+δ2 dw K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v) − ψt(v − r)) − � v+δ1 v+δ2 dw K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(w) − ψt(v − r)) + � v−δ2 v−δ1 dw K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v) − ψt(v − r)) − � v−δ2 v−δ1 dw K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(w) − ψt(v − r)) + � v−δ1 v−r dw K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v) − ψt(v − r)) − � v−δ1 v−r dw K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(w) − ψt(v − r)) 23 + � v v−r+δ1 dw K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v) − ψt(v − r)) − � v v−r+δ1 dw K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v) − ψt(w)) + �� v−r −∞ dw K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v) − ψt(v − r)) − � v−r −∞ dw � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � (ψt(v) − ψt(w)) + � v v−r dw K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v) − ψt(v − r)) − � v v−r dw K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(w) − ψt(v − r)) � + �� v−r v−r−δ1 dw K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v) − ψt(w)) − � v−r v−r−δ1 dw K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v − r) − ψt(w)) + � v+δ1 v dw K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(w) − ψt(v − r)) − � v+δ2 v dw K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(w) − ψt(v)) + � v v−δ2 dw K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(v) − ψt(w)) + � v−r+δ1 v−r dw K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψt(w) − ψt(v − r)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The last three lines within square brackets constitute the Lδ-part;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' it is only when these terms with the line-singularity are separated that the rest of the operator can be written in a (V − K)-form, with K being positivity-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We now move over to the ∆-variable, with ∆t(v, r) = ψt(v) − ψt(v − r) for all (v, r) ∈ R × R+, and write the above equation, now in terms of ∆t, as ∂t∆t = −L∆t, L∆t = Lpp∆t + Lδ∆t, where Lδ∆t(v, r) = � v−r v−r−δ1 dw K1 3(v, w)∆t(v, v − w) + � v+δ1 v dw K3(v − r, w)∆t(w, w − v + r) + � v−r+δ1 v−r dw K3(v − r, w)∆t(w, w − v + r) − � v−r v−r−δ1 dw K1 3(v − r, w)∆t(v − r, v − r − w) + � v v−δ2 dw K1 3(v, w)∆t(v, v − w) − � v+δ2 v dw K3(v, w)∆t(w, w − v), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='16) and, Lpp∆t(v, r) = Vu(v, r)∆t(v, r) − � v−r−δ1 −∞ dw � K1 3(v − r, w) − K1 3(v, w) � ∆t(v, v − w) − � ∞ v+δ1 dw (K3(v, w) − K3(v − r, w)) ∆t(w, w − v + r) − � v−r −∞ dw � K2 3(v − r, w) − K2 3(v, w) � ∆t(v, v − w) − � v−δ1 v−r dw K1 3(v, w)∆t(w, w − v + r) − � v v−r+δ1 dw K3(v − r, w)∆t(v, v − w) − � v v−r dw K2 3(v, w)∆t(w, w − v + r) − � v+δ1 v+δ2 dw K3(v, w)∆t(w, w − v + r) − � v−δ2 v−δ1 dw K1 3(v, w)∆t(w, w − v + r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='17) The potential Vu has already been defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The next step is then to carve out a bounded piece Lb from Lpp as follows: Lpp∆t(v, r) = Vu(v, r)∆t(v, r) − Ku∆t(v, r) − Lb∆t(v, r), where, Lb[∆t](v, r) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='18) 24 = − 1(v < −b0) � ∞ a1 dw � K3 1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K3 1(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � ∆t(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) − 1(v ≥ b0) � ∞ v+δ1 dw � K3 1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K3 1(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � ∆t(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) − 1(v − r < −b0)1(v > 0) � v 0 dw K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) − 1(v − r ≥ −b0) � v v−r+δ1 dw K3 1(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) − 1(v < −m0) � v−min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='−v−b0) v−r dw K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) − 1(−m0 < v < m0) � v v−r dw K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) − 1(v − r ≤ −m0) � v−r−˜r −∞ dw � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � ∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) − 1(−m0 < v − r < m0) � v−r −∞ dw � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � ∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) − 1(v − r ≥ m0) � 1(v ≤ 3r) � 0 −∞ dw � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � ∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) + 1(v > 3r) � v−r c0v dw � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � ∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) � − 1(v ≤ −m0)1(v + r > −b0) � 2v+b0 v−r dw K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) − 1(v ≥ m0) � 1(v ≤ r) � v v−r dw K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) + 1(0 < v − r < c0v) � v c−1 0 (v−r) dw K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='19) where a1 = a/c0, and ˜r appearing in line 7 of the formula for Lb, is defined as: ˜r = max(−v − b0, r), ∀v < −b0, = max(−b0 − v + r, 0), ∀v ≥ −b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='20) For our subsequent computations we choose b0 ≥ 10 and m0 ≥ max(2a1, b0 + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that whenever v ≥ −b0, ˜r = r − b0 − v, since this cut-off is used only when v − r ≤ −m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We also observe that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 guarantees that Ku, as well as Lb, is positivity-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The above formulae hold true for all positive functions δ1 and δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The choice of these functions is then determined by the requirement that |Lδ∆s| be made “small” compared to ∥∆∥Y VuΓs (see the estimate in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us write down the explicit forms of these functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' δ1(v, r) = (1 − e−αr) 4 γ0 M exp( µ′ γ0 max(a1, (v − r))) δ2(v, r) = (1 − e−αr) 4 γ0 M exp( µ′ γ0 max(a1, v)) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='21) 25 where µ′ > µ and M is used as a sort of “tuning parameter” to make these cut-off functions “small”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This “smallness” is inherited by Lδ[∆t](v, r), which consists of singular integrals over intervals of length δ1 and δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We do not care much about the actual value of the constant M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It is quite large and that such a choice 0 < M < ∞ can be made is enough for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Before we conclude this part about the derivation of Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6), some remarks about the necessity for using two different cut-off functions are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For the moment let us use δ as a stand-in for the cut-off functions δ1 and δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us take the term � v+δ v dw K3(v, w)∆t(w, w − v) from Lδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Observe that, for large, positive values of the variable v the expression � v+δ v dw K3(v, w)Γt(w, w − v) = � δ 0 dr′ K3(v, v + r′) � f(v) + f(v + r′) � eµ max(a,v,c0(v+r′))gt(v + r′, r′), contains an extra factor of eµ(1−c0)v relative to (VuΓt)(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus the only way this term can be controlled by VuΓt is by putting in a countervailing v-dependence in the definition of the cut-off function δ used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This explains the v-dependence in the definition of δ2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that with this definition of the cut-off function the extra factor of eµ(1−c0)v gets cancelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us now note that the potential Vu(v, r) defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7) behaves as Vu(v, r) ≃ ¯n � � ln(1 + ev−r) �−1/2 ln δ−1 1 + (ln(1 + ev))−1/2 ln δ−1 2 + � max(1, v) � For large, positive values of v, ln δ−1 2 has a part that behaves like ln v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In order for the main estimate (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6) used in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 to be true, Lu[Γt](v, r) = Vu(v, r)Γt(v, r) − Ku[Γt](v, r) must have the same asymptotic behavior as Vu(v, r)Γt(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It is clear from the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6 that the argument appearing in the point-singularity must be the same as that in the ln δ−1 multiplying it, otherwise we would end up with terms of the form e− 1 2(v−r) ln v, which would cause our estimate to fail in the region v ≫ 0, v − r ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus δ1 and δ2 have to be two different functions, as defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 Solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6): The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 This part of subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 is planned as follows: we first state three lemmas on which the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 rests;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' then we write down the proof of this theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' finally, we prove the aforementioned lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' These three lemmas concern the control of the three operators Ku, Lδ and Lb appearing in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Without further ado, we state the lemmas below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' There exists q0 > 0 depending on the parameters α, c0, µ′, γ0, such that the following bound holds: LuΓs + ˙Γs = VuΓs − KuΓs + ˙Γs ≥ q0 ln M VuΓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' There exists q1 > 0 depending on the parameters γ0 and κ, such that the following bound holds: |Lδ∆s| ≤ q1(κ, γ0) Mγ0 ln M ∥∆∥Y � VuΓs + ˙Γs � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The bounded linear operator Lb satisfies |Lb∆s| ≤ A(b0, m0, µ, c0)∥∆∥Y Γs, for some A(b0, m0, µ, c0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 26 Let us now write down the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 on the basis of the above lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Given initial datum ∆0 ∈ X, let us recall the Duhamel-integrated form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='8) for the evolution equation for ∆t: ∆t =e−tVu∆0 + F[∆t] =e−tVu∆0 + � t 0 ds e−(t−s)VuKu[∆s] − � t 0 ds e−(t−s)VuLb[∆s] − � t 0 ds e−(t−s)VuLδ[∆s], We observe that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6 implies the following upper bound: KuΓs ≤ � 1 − q0 ln M � � VuΓs + ˙Γs � , which means we can write ����� � t 0 ds e−(t−s)VuKu[∆s] ����� ≤ ∥∆∥Y � t 0 ds e−(t−s)VuKuΓs ≤ � 1 − q0 ln M � ∥∆∥Y � t 0 ds ∂s � e−(t−s)VuΓs � ≤ � 1 − q0 ln M � ∥∆∥Y Γt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='22) where in the first line we have used the fact that Ku is positivity-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Similarly, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7 yields the following estimate: ����� � t 0 ds e−(t−s)VuLδ[∆s] ����� ≤ q1 Mγ0 ln M ∥∆∥Y Γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='23) Finally let us observe that for all s′ > 0, VuΓs′ + ˙Γs′ > 0, which means � t s ds′∂s′ � e−(t−s′)VuΓs′ � > 0, ∀s ∈ (0, t), implying Γt > e−(t−s)VuΓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='8 then implies ����� � t 0 ds e−(t−s)VuLb[∆s] ����� ≤ A(b0, m0, µ, c0)T∥∆∥Y , ∀t ∈ [0, T], T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='24) Putting together (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='22), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='23) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='24), it is easy to see that |F∆t| ≤ ∥∆∥Y Γt � 1 − q0 ln M + AT + q1 Mγ0 ln M � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='25) where q0 and q1 have no dependence on M and it is evident that M < ∞ can be chosen large enough such that the following is true: 0 < T ∗ < 1 A ln M � q0 − q1 Mγ0 � , for some T ∗ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Now that M is chosen in the above manner, clearly we have, for 0 < t ≤ T ∗, |F∆t| < ∥∆∥Y Γt,, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', ∥F∥Y < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The corresponding Neumann series then converges and we obtain the following unique solution of our Duhamel-integrated equation: ∆t = (1 − F)−1 e−tVu∆0, ∀0 < t ≤ T ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 27 Note that, as we have mentioned already before the beginning of Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1, the above estimate can be extended in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We now move on to the proofs of the lemmas stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will prove Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='8 first and save the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6, which is much more involved, for last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The time-derivative of Γs is ˙Γs(v, r) = −1 8 min(1, ¯n) 1 1 + eβ(v−r) · 1 (1 + s)2 ln(1 − e−κr)−1Γs(v, r), s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='26) From the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7) of Vu, it is easy to see that Vu(v, r) ≥ 2¯n � ln(1 + ev−r) �−1/2 � ln M + ln(1 − e−r)−1� , so that, Vu(v, r)Γs(v, r) + ˙Γs(v, r) > ¯n � ln(1 + ev−r) �−1/2 � ln M + ln(1 − e−r)−1� Γs(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='27) By definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='16) we have, Lδ∆s(v, r) = � δ1 0 dr′ K1 3(v, v − r − r′)∆s(v, r + r′) + � δ1 0 dr′ K3(v − r, v + r′)∆s(v + r′, r + r′) + � δ1 0 dr′ K3(v − r, v − r + r′)∆s(v − r + r′, r′) − � δ1 0 dr′ K1 3(v − r, v − r − r′)∆s(v − r, r′) + � δ2 0 dr′ K1 3(v, v − r′)∆s(v, r′) − � δ2 0 dr′ K3(v, v + r′)∆s(v + r′, r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will write estimates for two of these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The rest can be estimated in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' i) � δ1 0 dr′ |K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)∆s(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′)| ≤ ∥∆∥Y � δ1 0 dr′ K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γs(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) ≤ 4¯n∥∆∥Y eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r) (f(v − r) + f(v)) κ (ln(1 + ev))− 1 2 × × � δ1 0 dr′ e−r−r′ � 1 − e−r−r′�γs(v−r)−1 ≤ 6κ¯n γ0 ∥∆∥Y Γs(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) (ln(1 + ev))− 1 2 δγs(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r) 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ii) � δ2 0 dr′ |K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)∆s(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′)| ≤ ∥∆∥Y � δ2 0 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γs(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) ≤ 4¯n∥∆∥Y (ln(1 + ev))− 1 2 κ (f(v − r) + f(v)) × × � δ2 0 dr′ e−r′ � 1 − e−r′�γs(v)−1 eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v+r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v) ≤ 6κ¯n γ0 ∥∆∥Y eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) (f(v − r) + f(v)) (ln(1 + ev))− 1 2 δγ0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Estimating the rest of the terms in a similar manner, we can write the following upper bound: |Lδ[∆s](v, r)| ≤ ∥∆∥Y p1κ γ0Mγ0 ¯n � ln(1 + ev−r) �− 1 2 (1 − e−αr)3 Γs(v, r), where p1 is a numerical constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Obviously then, |Lδ[∆s](v, r)| ≤ ∥∆∥Y q1(κ, γ0) Mγ0 ln M � VuΓs(v, r) + ˙Γs(v, r) � , ∀(v, r) ∈ R × R+, with q1 = 2p1κ γ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 28 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='8 is quite obvious from the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='18) of the bounded operator Lb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It is clear from definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='18) that Lb∆s(v, r) does not contain any point or line singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In addition, the exponential decay in the integrands ensures the finiteness of the integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The bound in the lemma is then obtained by straightforward computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We now come to the most crucial estimate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' the one contained in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The proof of this lemma relies quite heavily on certain properties of the H¨older-type condition used in the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3) of Γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' These properties are listed (and proved) in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since the computations are quite involved, we will try to give a general idea of the scheme of the proof before writing down the proof formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The main idea of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6 is to establish that (LuΓs)(v, r) has the same asymptotic behavior as Vu(v, r)Γs(v, r), for all (v, r) ∈ R × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us first observe that the potential Vu(v, r) grows exponentially at (−∞), has a line singularity at r = 0 and grows like the square-root function at +∞, somewhat like the function V ′(v, r) ≃ ¯n � e− 1 2 (v−r) ln(2 + 1/r) + � max(1, v) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In order to show that LuΓs has the same behavior, we will first split it into different parts and then show how each of these parts produces the correct asymptotic behavior in different regions of R × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We write Lu[Γt](v, r) = I1[Γt](v, r) + I2[Γt](v, r) + I3[Γt](v, r) + I4[Γt](v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='28) As the explicit formulae for the Ii’s (i ∈ {1, 2, 3, 4}) written below show, these terms consist of integrals grouped together on the basis of intervals of integration and the type of singularities they contain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For example, the terms in I3 do not contain any line singularity, unlike terms in I1 and I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' On the other hand, the integrals in I4 contain an extra “smallness” because they are integrated over intervals of length at most δ1, while in I2 the integrals are taken over intervals of length at most r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' As will become apparent shortly, the three groups I1, I2 and I3 produce different kinds of asymptotic behavior in LuΓt: I1 yields a term with point singularity which behaves like (ln(1+ev−r))−1/2 ln(1−e−r)−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', dominates when r ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' From I2 one gets a term with the point singularity, which becomes dominant for r ≫ 1, behaving like (ln(1 + ev−r))−1/2(1 − e−r)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Finally, the group I3 contributes to LuΓt the necessary √v growth, when v ≫ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The Ii[Γt]’s are given by the following formulae: I1[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = � ∞ r+δ1 dr′ � K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + 1(v < −b0) � � ∞ r+δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − 1(a1 − v ≤ r + δ1) �� ∞ r+δ1 dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − � r+a1−v r+δ1 dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − � ∞ r+a1−v dr′ K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � + 1(a1 − v > r + δ1) �� a1−v r+δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) + � ∞ a1−v dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − � r+a1−v r+δ1 dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − � ∞ r+a−v dr′ K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � 29 + 1(v ≥ −b0) � � ∞ r+δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � ∞ r+δ1 dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) + � ∞ r+δ1 dr′ K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='29) I2[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = � r δ1 dr′ K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Γt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � + � r δ1 dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + � r δ1 dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � 1(v − r < −b0) � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r−v) δ1 dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) + 1(v − r ≥ −b0) � r δ1 dr′ K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � + � r δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − 1(v < −b0) � 1(r + δ1 ≤ a1 − v) � r δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) + 1(a1 − v < r + δ1) �� min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='a1−v) δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) + 1(a1 − v < r) � r a1−v dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) �� − 1(v ≥ −b0) � r δ1 dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='30) I3[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = � ∞ 0 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − 1(v − r ≤ −m0) � ˜r 0 � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − 1(v − r ≥ m0) � 1(v ≤ 3r) � v−r 0 dr′ � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) + 1(v > 3r) � ∞ v−r−c0v dr′ {K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)}Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � − 1(v ≤ −m0) � 1(v + r < −b0) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) + 1(v + r ≥ −˜b0) � −v−b0 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � − 1(v ≥ m0) � 1(0 < v − r < c0v) � r v−c−1 0 (v−r) dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) + 1(v − r ≥ c0v) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � = I(1) 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I(2) 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='31) 30 where I(1) 3 includes all the terms with K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) in the integrand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' while I(2) 3 includes those with K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' and finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' I4[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = � r+δ1 r dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + � r+δ1 r dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − 1(v < −b0) � 1(r ≤ a1 − v < r + δ1) �� a1−v r dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) + � r+δ1 a1−v dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + + 1(a1 − v < r) � r+δ1 r dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) + 1(a1 − v ≥ r + δ1) � r+δ1 r dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � − 1(v ≥ −b0) � r+δ1 r dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) + � δ1 δ2 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Γt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + � δ1 δ2 dr′ K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='32) Let us now proceed to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6, keeping in mind that the computational details referred to in this proof are to be found in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The proof of this lemma hinges on obtaining a suitable lower bound for LuΓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will outline the general scheme of the estimates leading to this lower bound, writing down explicitly only those terms that lead to the correct asymptotic behavior in different regions and refer to Appendix D for all other details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We write Γt = Γ1 t + Γ2 t, where Γ1 t (v, r) = f(v − r) exp (µ max(a, c0v, v − r)) gt(v, r), and Γ2 t (v, r) = f(v) exp (µ max(a, c0v, v − r)) gt(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will now look at each of the Ii[Γt]’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' i) I1[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) : It is quite easy to obtain the following lower bound for I1[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r): I1[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = I1[Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I1[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) ≥ J0[Γ1 t + Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + J1[Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + J2[Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I[Γ1 t + Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where J0 denotes the dominant term close to the diagonal and is defined as J0[Γ1 t + Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = f(v − r) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r) � ∞ r+δ1 dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + � ∞ r+δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)e− 1 2 r′eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v+r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − gt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � � + f(v)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r) � ∞ r+δ1 dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + � ∞ r+δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)e− 1 2r′f(v + r′)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v+r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) −gt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='33) 31 J1, J2 and I contain only such integrals which do not contain the factor � 1 − e−r′�−1 in the integrand and are thus sub-dominant to J0 close to the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Some of these integrals are negative and have to controlled by the other parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The explicit formulae for these tems as well as the computations showing how they are controlled are in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 from Appendix C tells us: � ∞ r+δ1 dr′ K1 3(v − r, v − r − r′) � gt(v, r) + gt(v, r′) − gt(v, r + r′) � ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4gt(v, r) � ∞ r+δ1 dr′K1 3(v − r, v − r − r′), which yields a (ln(1 + ev−r))−1/2 ln(1 − e−r)−1 term typifying the correct asymptotic behavior for r ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ii) I2[Γt](v, r) : We write I2[Γt](v, r) = I2[Γ1 t ](v, r)+I2[Γ2 t](v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It is the lower bound for I2[Γ1 t ](v, r) that produces the correct asymptotic behavior in the region away from the diagonal when the point singularity dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This bound is: I2[Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) ≥ 1(v − r < −b0)eµ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v} � r δ1 dr′ � K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) � f(v − r) − (ln(1 + ev−r+r′))−α� − K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � f(v − r − r′) − f(v − r) � � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v < −b0)1(r > a1 − v)f(v − r) � r a1−v dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � eµa − eµc0ve−( 1 2−µc0)r′� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v ≥ −b0)1(v − r < −b0) � f(v − r) � r δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � eµ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v} − e− 1 2r′eµ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v+r′)}� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � r min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r−v) dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)(ln(1 + ev−r+r′))−α � 2gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � � + I− 2 [Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='34) where I− 2 [Γ1 t](v, r) includes all the negative parts coming from I2[Γ1 t ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' the integrals in I− 2 [Γ1 t ] contain differ- ences of the form (gt(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', r + r′) − gt(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', r)), which generate an extra exponential decay of e−κr, as explained in Appendix D, and hence are “small” in the region away from the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The relevant dominating behavior with point singularity for r ≫ 1, comes from the first term of the lower bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='34) for I2[Γt](v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We call this term C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then C3[Γ1 t ](v, r) = 1(v − r < −b0)eµ max{a,c0v}gt(v, r) � r δ1 dr′ � K3(v − r, v − r + r′) � f(v − r) − (ln(1 + ev−r+r′))−α� − K1 3(v − r, v − r − r′) � f(v − r − r′) − f(v − r) � � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='35) ≥ 1(v − r < −b0)4¯nΓ1 t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � ln(1 + ev−r) �− 1 2 � r δ1 dr′ ev−r + 2e−r′ 1 + ev−r + e−r′ e−r′ � ln(1 + ev−r−r′) ln(1 + ev−r) �1−α × × � ln(1 + ev−r+r′) ln(1 + ev−r−r′) �1−2α � 1 − e− 248 125 (1−2α)r′ 1 − e−r′ � � 1 − � ln(1 + ev−r−r′) ln(1 + ev−r) �α� iii) I3[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) : The square-root-like growth for large,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' positive values of v comes from I3[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) as follows: 1(v − r ≥ −m0)I1 3[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v ≥ m0)I2 3[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) 32 ≥ 1(v ≥ m0)¯nΓt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � 1(v − r ≤ 0)(ln(1 + ev)) 1 2 � 1 − � ln 2 ln(1 + ev) �2 � + 1(0 < v − r < c0v)3 2(ln(1 + ev)) 1 2 � 1 − �v − r c0v �2 � + 1(v − r ≥ m0) � 1(v − r ≤ 2 3v) 4 3 (ln(1 + ev)) 1 2 �ln(1 + ev−r) ln(1 + ev) �2 + 1(v − r > 2 3v) 2 � ln(1 + ev−r) � 1 2 � 1 − � ln(1 + ec0v) ln(1 + ev−r) �2� �� + 1(−m0 < v − r < m0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)f(v − r)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � ∞ 0 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) The computations in Appendix D lead us to the following lower bound for LuΓs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' for a suitable choice of the constant M : LuΓt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = I1[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I2[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I(1) 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I(2) 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I4[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) ≥ G[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where G[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = ¯nΓt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � 1(v − r ≤ −b0)¯b1(α) � ln(1 + ev−r) �− 1 2 � 1 − e−3αr�3 + 1 2 � ln(1 + ev−r) �− 1 2 ln � 1 − e− 7 2 (r+δ1)�−1 + 1(−m0 < v − r < m0)1 4 � ln(1 + ev−r) �− 1 2 + 1(0 < v − r < m0)1 8 � ln(1 + ev−r) � 1 2 + 1(0 < v < m0)1 2 (ln(1 + ev)) 1 2 � 1 − �ln(1 + ev−r) ln(1 + ev) �2� + 1(v ≥ m0)b3(c0) (ln(1 + ev)) 1 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where ¯b1(α) and b3(c0) are positive numbers bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then using formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='26) for ˙Γs(v, r) we can conclude that LuΓs(v, r) + ˙Γs(v, r) ≥ G[Γs](v, r) + ˙Γs(v, r) ≥ G[Γs](v, r), where G[Γs](v, r) = ¯nΓs(v, r) � � ln(1 + ev−r) �−1/2 �3 8 ln � 1 − e− 7 2(r+δ1)�−1 + 1(−m0 < v − r < 0)1 4 + 1(v − r ≤ −b0)b1(α) � 1 − e−3αr�3� + 1(0 ≤ v − r < m0)1 8 � � ln(1 + ev−r) �−1/2 + � ln(1 + ev−r) �1/2 � + 1(v > 0) (ln(1 + ev)) 1 2 � 1(0 < v < m0)1 2 � 1 − �ln(1 + ev−r) ln(1 + ev) �2� + 1(v ≥ m0)b3(c0) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' From the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7) of the potential Vu, it is easy to see that there exist positive numbers C1 = C1(µ′, γ0, a1) and C1 = C1(µ′, γ0, a1), such that the following is true for all (v, r) ∈ R × R+: ¯nC1 � (ln(1 + ev)) 1 2 + � ln(1 + ev−r) �− 1 2 ln � 2M(1 − e−αr)−1�� 33 ≤ Vu(v, r) ≤ ¯nC1 � (ln(1 + ev)) 1 2 + � ln(1 + ev−r) �− 1 2 ln � 2M(1 − e−αr)−1�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then it is clear that there exists q0 > 0, depending on α, c0, µ′, γ0, b0, such that: G[Γs](v, r) ≥ q0 ln M Vu(v, r)Γs(v, r), ∀(v, r) ∈ R × R+, which implies the lower bound claimed in this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 Solutions of the Regularized Evolution Equation: Let us recall the regularized evolution equation ∂tψt(v) = −(Lε 3ψt)(v) = − � R dw Kε 3(v, w) � ψt(v) − ψt(w) � , where Kε 3(v, w) = Kε(max(v,w)) 3 (v, w) = K3(v, w)1 − e− min(ε(v,w),|v−w|) 1 − e−ε(v,w) , ε(v, w) = ε(max(v, w)) = ε0 exp � − µ′ γ0 max(a1, max(v, w)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Recall that µ′ > µ and the admissible values for the parameters µ, a1 = a/c0 and γ0 have already been defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In this subsection our goal is to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For our subsequent computations it is useful to split the kernel function just like in the beginning of Section 2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' When v > w, Kε 3(v, w) = K1,ε(v) 3 (v, w) + K2,ε(v) 3 (v, w), where K1,ε(v) 3 (v, w) = K1 3(v, w)1 − e− min(ε(v),v−w) 1 − e−ε(v) , K2,ε(v) 3 (v, w) = K2 3(v, w)1 − e− min(ε(v),v−w) 1 − e−ε(v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Similarly in the region w > v, Kε 3(v, w) = K3 1,ε(w)(v, w) + K3 2,ε(w)(v, w), where K3 2,ε(w)(v, w) = K3 2(v, w)1 − e− min(ε(w),w−v) 1 − e−ε(w) , and K3 1,ε(w)(v, w) = K3 1(v, w)1 − e− min(ε(w),w−v) 1 − e−ε(w) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Just like Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1, we have the following result for the regularized case: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The kernel functions satisfy the following inequalities for all r > 0: i) For all w < v − r, K1,ε(v−r) 3 (v − r, w) > K1,ε(v) 3 (v, w), whenever either v ≤ a1 or v − w ≥ ε(v − r), and K2,ε(v−r) 3 (v − r, w) > K2,ε(v) 3 (v, w), whenever v − r − w ≥ ε(v − r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ii) For all w > v, Kε(w) 3 (v, w) > Kε(w) 3 (v − r, w), K3 2,ε(w)(v, w) > K3 2,ε(w)(v − r, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' i) When w < v − r let us define r′ = v − r − w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then: K1,ε(v−r) 3 (v − r, v − r − r′) − K1,ε(v) 3 (v, v − r − r′) ≥ 4¯n(ln(1 + ev−r))−3/2 ln(1 + ev−r−r′) ev−r−r′ + 2e−r′ 1 + ev−r−r” + e−r′ � e−r′ 1 − e− max(r′,ε(v−r)) − e−r−r′ 1 − e− max(r+r′,ε(v)) � Note that each of the two conditions, v ≤ a1 (which means ε(v) = ε(v − r)) and r + r′ ≥ ε(v − r), implies that max(r +r′, ε(v)) ≥ max(r′, ε(v −r)), and consequently K1,ε(v−r) 3 (v −r, v −r −r′) > K1,ε(v) 3 (v, v −r −r′) whenever either of these conditions is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 34 Whenever v − r − w ≥ ε(v − r), K2,ε(v−r) 3 (v − r, w) = K2 3(v − r, w) and K2,ε(v) 3 (v, w) = K2 3(v, w), so the corresponding inequality is proved by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ii) For w > v we have: Kε(w) 3 (v, w) − Kε(w) 3 (v − r, w) ≥ 1 1 − emax(ε(w),w−v) � f(v, w) − f(v − r, w) � ln(1 + ew), where f(v, w) = (ln(1 + ev))−3/2 ev + 2e−(w−v) 1 + ev + e−(w−v) e−(w−v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then ∂ ∂v f(v, w) ≥ f(v, w) � −3 2 ev (1 + ev) ln(1 + ev) + 3 + ev 2 + ev � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The last inequality then is an obvious consequence of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us remember that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4 deals with the difference variable Dψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In order to show the existence of this difference variable, we will first prove the existence of a unique solution of the ψ-equation written above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' As mentioned before, we will tag solutions of the regularized equation with ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In what follows, we will first show, given a suitable initial value ψε 0, the existence of a unique Duhamel- integrated solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then we will go on to prove a similar existence-uniqueness result, namely Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4, for the difference variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The final result of this subsection will be Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The proofs in this subsection follow closely the proofs of Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='8 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 Regularized Solution ψε t Let us consider the time-evolution, according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='10), of initial datum ψε 0 in the Banach space X, defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that, since X ⊂ D(L ε 3) for all ε > 0, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 guarantees the existence of a unique (in L2(ν)) solution, ϕt = e−tL ε 3ψε 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The evolution equation is first re-written as follows: ∂tψε t (v) = − � � ∞ 0 dr′ Kε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) + � ∞ 0 dr′ Kε(v+r′) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � ψε t (v) + � ∞ 0 dr′ Kε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)ψε t (v − r′) + � ∞ 0 dr′ Kε(v+r′) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)ψε t (v + r′) = −V ε(v)ψε t (v) + (Kε uψε t )(v) + (Kε b ψε t )(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where V ε(v) = � ∞ 0 dr′ Kε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) + � ∞ 0 dr′ Kε(v+r′) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Kε u consists of unbounded parts of the kernel function and Kε b is L2-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The explicit formulae for them are given in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will prove that there exists a unique Duhamel-integrated solution of the above evolution equation in X, for all t > 0, given by ψε t = e−tV εψε 0 + � t 0 ds e−(t−s)V εKε u[ψε s] + � t 0 ds e−(t−s)V εKε b[ψε s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='36) Before proving (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='36), we will consider the following equation, which differs from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='36) only in that the function ψε s in the L2-bounded part Kb is replaced by the L2-solution ϕs : ψε t = e−tV εψε 0 + � t 0 ds e−(t−s)V εKε u[ψε s] + � t 0 ds e−(t−s)V εKε b [e−sLε 3ψε 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='37) 35 Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='37) is just the Duhamel-integrated form of the evolution equation: ∂tψε t (v) = −V ε(v)ψε t (v) + (Kε uψε t )(v) + Kε b [e−tLε 3ψε 0](v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Given initial value ψε 0 ∈ X, suppose ψε t ∈ X solves equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='37) for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then ψε t = e−tLε 3ψε 0, ν-almost evrywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ψε t ∈ X ⊂ D(Lε 3), for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This means, given any t > 0, we have, for all s ∈ (0, t) ∂s � e−(t−s)Lε 3ψε s � = e−(t−s)Lε 3 [Lε 3ψε s + ∂sψε s] = e−(t−s)Lε 3 [(V εψε s − Kε uψε s − Kε bψε s) − (V εψε s − Kε uψε s − Kε b ϕs)] = e−(t−s)Lε 3Kε b [ϕs − ψε s], where ϕs = e−sLε 3ψε 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Integrating the above, we have ψε t − ϕt = � t 0 ds e−(t−s)Lε 3Kε b [ϕs − ψε s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Therefore by Minkowski’s integral inequality, ∥ψε t − ϕt∥L2 ≤ � t 0 ds ∥Kε b ∥L2∥ψε s − ϕs∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then by Gr¨onwall’s inequality ∥ψε t − ϕt∥L2 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', ψε t = ϕt ν-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ∀t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' There exists a number s0 > 0, depending on the parameters α, µ′, γ0, such that: V ε˜Γ − Kε u˜Γ ≥ s0 ln ε−1 0 V ε˜Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' From the definition of the potential V ε, it is easy to see that there exist positive numbers C2 = C2(µ′, γ0) > 1 and C2 = C2(µ′, γ0), such that the following is true: ¯n �1 4 (ln(1 + ev))− 1 2 ln ε−1 0 + C2 (ln(1 + ev)) 1 2 � ≤ V ε(v) ≤ C2¯n � (ln(1 + ev))− 1 2 ln ε−1 0 + (ln(1 + ev)) 1 2 � , ∀v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' From E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 in Appendix E we get the following lower bound: V ε(v)˜Γ(v) − (Kε u˜Γ)(v) ≥ ˜Γ(v) � 1(v ≤ 0)p2(α) (ln(1 + ev))− 1 2 + 1(v > 0)p3 (ln(1 + ev)) 1 2 � , where p2 and p3 are positive constants bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It is then obvious that for some s0 = s0(α, µ′, γ0) > 0 we can write V ε(v)˜Γ(v) − (Kε u˜Γ)(v) ≥ s0 ln ε−1 0 V ε(v)˜Γ(v), ∀v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 36 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Given initial datum ψε 0 ∈ X, there exists, for all t > 0, a unique solution ψε t of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='36), such that: ∥ψε∥′ = sup t∈R+ ∥ψε t ∥X = sup t∈R+ sup v∈R |ψε t (v)| ˜Γ(v) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In order to prove the statement of this theorem, we will first show that given initial datum ψε 0 ∈ X, there exists a unique solution ψε t of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='37), such that supt∈R+ ∥ψε t ∥X < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Given such a solution, the statement of this theorem is implied by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let F1[ψt] = � t 0 ds e−(t−s)V εKε u[ψε s], and, Bt[ψε 0] = � t 0 ds e−(t−s)V εKε b[e−sLε 3ψε 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then we can write (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='37) as ψε t =e−tV εψε 0 + F1[ψε t ] + Bt[ψε 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='11 and the fact that Kε u is positivity-preserving, we have: |F1[ψε t ](v)| ≤ ∥ψε∥′ � 1 − s0 ln ε−1 0 � � t 0 ds ∂s � e−(t−s)V ε ˜Γ � (v) ≤ ∥ψε∥′ � 1 − s0 ln ε−1 0 � ˜Γ(v), ∀v ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus ∥F1∥′ < 1, and the corresponding Neumann series converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The invertibility of F1 means we can write ψε t = (1 − F1)−1 � e−tV εψε 0 + Bt[ψε 0] � , which is clearly the unique solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='37) corresponding to the initial value ψε 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='10, ψε t = e−tLε 3ψε 0, for all t ≥ 0, ν-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', so that we can replace e−tLε 3ψε 0 by ψε t in the L2-bounded integrals and write � t 0 ds e−(t−s)V εKε b[e−sLε 3ψε 0] = � t 0 ds e−(t−s)V εKε b[ψε s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus ψε t solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The uniqueness of this solution is straightforward since any such solution is also a solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 Regularized Evolution Equation for the Difference Variable Dψε t and the Existence of a Unique Solution Our results above guarantee the existence of the difference variable Dψε, where Dψε t (v, r) = ψε t (v)−ψε t (v−r), in the Banach space Z of continuous functions defined on R+ × (R × R+), such that ∥Dψε∥Z = sup t>0 (v,r)∈R×R+ |Dψε t (v, r)| Γ(v, r) < ∞, where Γ(v, r) = (f(v) + f(v − r)) eµ max(a,c0v,v−r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 37 Given ψε 0 ∈ X, this difference variable Dψε t provides us with a solution of the following Cauchy problem (derived from the corresponding problem for ψε t ): ∂tDψε t (v, r) = −( ˜LεDψε t )(v, r) = − � wv dw Kε 3(v, w)Dψε t (w, w − v) + � wv−r dw Kε 3(v − r, w)Dψε t (w, w − v + r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='38) Given ψε 0, ψε t is uniquely determined in X by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12, and we have the following trivial bound for Dψε t : |Dψε t (v, r)| = ����(1 − F1)−1 � e−tV εψ0 + Bt[ψ0] � (v) − (1 − F1)−1 � e−tV εψε 0 + Bt[ψε 0] � (v − r) ���� ≤ A0(α, µ′, γ0, b0, m0) � ln ε−1 0 � (∥ψε 0∥X + ∥ψε 0∥L2) Γ(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='39) Observe that although our results in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 guarantee the existence of Dψε t evolving according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='38), this solution may not be unique in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will now show that equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='38) has a unique Duhamel-integrated solution in a certain subspace Yε of Z, containing functions h bounded in the following norm: ∥h∥Yε = sup t>0 (v,r)∈R×R+ |ht(v, r)| Γε(v, r) < ∞, where Γε(v, r) = (f(v) + f(v − r)) eµ max(a,c0v,v−r)g(v, r), g(v, r) = � 1 − e−κ(r+ε(v))�γ0 , γ0 = γ0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' If we can prove that the Duhamel-integrated form of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='38) has a unique solution in Z and also a solution in Yε ⊂ Z, then we will have proved that this unique solution corresponding to the given initial datum, satisfies a certain H¨older-like condition (see the definition of g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This result will be crucial in the final step of the proof of the main smoothing result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In our computations to prove the existence-uniqueness result in both Z and Yε, we will use the following generic weight function which covers both cases: Γ ′ ε(v, r) = (f(v) + f(v − r)) eµ max(a,c0v,v−r)˜g(v, r), ˜g(v, r) = � 1 − e−κ(r+ε(v))�˜γ0 , where ˜γ0 ∈ {0, γ0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The reason for using this weight is that almost all of our computations work for both Γ and Γε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' So our results are proved for the generic Γ ′ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Whenever there is a difference, we point it out in the relevant computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We denote by ˜Yε the Banach space corresponding to the weight Γ ′ ε, which means ˜Yε is either Z or Yε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The analysis of the evolution equation for Dψε t is done along the same lines as the analysis of the ∆t- equation in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus, here too the linear operator governing the time evolution is split into a potential, a positivity-preserving part, a bounded part and a perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12), where this splitting has already been effected and the equation has been cast in a form amenable to our subsequent computations: ∂tDψε t (v, r) = −( ˜LεDψε t )(v, r) = −( ˜Lε uDψε t )(v, r) − ( ˜Lε sDψε t )(v, r) − ˜Kε b[ψε t ](v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 38 Here ( ˜Lε sDψε t )(v, r) = − 1(v ≥ a1) � v−r −∞ dw � K1,ε(v−r) 3 (v − r, w) − K1,ε(v) 3 (v, w) � Dψε t (v, v − w)1(v − w ≤ ε(v − r)) − � v−r v−r−ε(v−r) dw � K2,ε(v−r) 3 (v − r, w) − K2,ε(v) 3 (v, w) � Dψε t (v, v − w), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='40) where ˜r has already been defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Also, as before, the unbounded operator ˜Lε u can be written as ( ˜Lε uDψε t )(v, r) = ˜Vε(v, r)Dψε t (v, r) − ( ˜Kε uDψε t )(v, r), with ˜Vε(v, r) = � v−r −∞ dw Kε(v−r) 3 (v − r, w) + � v v−r dw Kε(v) 3 (v, w) + � v v−r dw Kε(w) 3 (v − r, w) + � ∞ v Kε(w) 3 (v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' A useful bound for the Kε b-part (see Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2) is obtained in a simple, straightforward way and is hence stated without proof in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Given initial datum ψε 0 ∈ X the following bound holds: �� ˜Kε bψε t �� ≤ M0(ln ε−1 0 ) (∥ψε 0∥X + ∥ψε 0∥L2) Γ ′ ε, where M0 depends on the parameters α, µ, µ′, c0, a1, b0, γ0 and m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' There exists q2 > 0, depending on the parameters γ0, µ, µ′ and κ, such that the following bound holds: | ˜Lε sDψε t | ≤ q2(κ, µ, µ′, γ0)ε0 ln ε−1 0 ∥Dψε∥ ˜ Yε ˜VεΓ ′ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' There are two possible cases: case 1: When we have ˜γ0 = 0 and Γ ′ ε = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It is quite easy to see there exist C3 = C3(µ′, γ0) > 0 and C4 = C4(µ′, γ0) > 0 such that: | ˜Lε sDψε(v, r)| ≤ ∥Dψε∥ZΓ(v, r) � C3ε(v − r) � ln(1 + ev−r) �− 1 2 + C4 1(v ≥ a1)1(r ≤ ε(v − r))(ε(v − r))2 (ln(1 + ev))− 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' case 2: When ˜Y = Y ε, we have ˜γ0 = γ0 and Γ ′ ε = Γε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It is quite easy to see there exist C3 = C3(µ′, κ, µ, γ0) > 0 and C4 = C4(µ′, κ, µ, γ0) > 0 such that: | ˜Lε sDψε(v, r)| ≤ ∥Dψε∥YεΓε(v, r) � C3ε(v − r) � ln(1 + ev−r) �− 1 2 + C4 1(v ≥ a1)1(r ≤ ε(v − r))(ε(v − r))2 (ln(1 + ev))− 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Putting these together and using the lower bound of ˜Vε ( see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='15), we obtain the required bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' There exists a number σ > 0, depending on the parameters α, c0, µ′, γ0, such that: ˜Lε uΓ ′ ε = ˜VεΓ ′ ε − ˜Kε uΓ ′ ε ≥ σ ln ε−1 0 ˜VεΓ ′ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 39 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It is easy to see that there exist positive numbers C3 = C3(µ′, γ0, a1) > 1 and C3 = C3(µ′, γ0, a1) such that the following bound holds: ¯nC3(µ′, γ0, a1) � � ln(1 + ev−r) �− 1 2 ln ε−1 0 + (ln(1 + ev)) 1 2 � ≤ ˜Vε(v, r) ≤ ¯nC3(µ′, γ0, a1) � � ln(1 + ev−r) �− 1 2 ln ε−1 0 + (ln(1 + ev)) 1 2 � , ∀(v, r) ∈ R × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' By computations outlined in Appendix E we have ˜Lε uΓ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) ≥ ¯nΓ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � ln(1 + ev−r) �− 1 2 � 1(v − r < −b0)¯b1(α) � 1 − e−3αr�2 � 1 − e−3α max(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r−ε(v−r))� + 1 2 ln � 1 − e− 7 2 max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r))�−1 � + 1(−m0 < v − r < m0)¯nΓ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � 1 6 � ln(1 + ev−r) �− 1 2 + 1(v − r > ln 2)1 9 � ln(1 + ev−r) � 1 2 � + 1(ln 2 < v < m0) ¯n 2Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) (ln(1 + ev)) 1 2 � 1 − � ln(1 + ev−r) ln(1 + ev−min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v))) �2 � + 1(v ≥ m0)b4(c0)¯nΓ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) (ln(1 + ev)) 1 2 = G0[Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' From the upper bound on ˜Vε u it is clearly seen that there exists σ > 0, depending on the parameters α, µ′, γ0, a1 and c0, such that G0[Γ ′ ε](v, r) ≥ σ ln ε−1 0 ˜Vε(v, r)Γ ′ ε(v, r), ∀(v, r) ∈ R × R+, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We are now in a position to prove the main result about the solution Dψε t , namely, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us begin by recalling that, for initial datum ψε 0 ∈ X, the Duhamel-integrated form of the difference variable Dψε t is given by equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We can write Dψε t = e−t˜VεDψε 0 + � t 0 ds e−(t−s)˜Vε ˜Kε uDψε s − � t 0 ds e−(t−s)˜Vε ˜Lε sDψε s − � t 0 ds e−(t−s)˜Vε ˜Kε b[ψε s] = e−t˜VεDψε 0 + F[Dψε t ] − Bt[ψε 0], where F[Dψε t ] = � t 0 ds e−(t−s)˜Vε ˜Kε uDψε s − � t 0 ds e−(t−s)˜Vε ˜Lε sDψε s, and, Bt[ψε 0] = � t 0 ds e−(t−s)˜Vε ˜Kε b[ψε s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 40 Then by Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='15 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='14, we arrive at the following estimate: |F[Dψε t ]| ≤ ∥Dψε∥ ˜YεΓ ′ ε � 1 − σ ln ε−1 0 + q2ε0 ln ε−1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since σ and q2 have no dependence on ε0, we can choose the regularization parameter ε0 > 0 small enough so that the following is true: σ ln ε−1 0 − q2ε0 ln ε−1 0 > 0, which implies ∥F∥ ˜Yε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The corresponding Neumann series then converges and we have the following unique solution for the Duhamel-integrated equation: Dψε t = (1 − F)−1 � e−t˜VεDψε 0 − Bt[ψε 0] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Finally, it is easily seen that there exists A1 = A1(µ, µ′, c0, a1, b0, m0) > 0 such that |Bt[ψε 0](v, r)| ≤ A1(ln ε−1 0 )Γ ′ ε(v, r) � ∥ψε 0∥X0 + ∥ψε 0∥L2 � , ∀(v, r) ∈ R × R+, which completes the proof of our assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus, given ψε 0 ∈ X, we obtain a unique solution Dψε ∈ Yε of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='13), ∀t > 0, satisfying |Dψε t (v, r)| ≤ C (f(v) + f(v − r)) eµ max(a,c0v,v−r) � 1 − e−κ(r+ε(v))�γ0 , ∀(v, r) ∈ R × R+ and some C < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 Connecting the Variables ∆t and Dψε t : Evolution Equation for Dt = Dψε t − ∆t Recalling the original evolution equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6) for ∆t, it is easily seen that we can write ∂t∆t = −Lε∆t − Lε 0∆t, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='41) where the part Lε contains Kε and the part Lε 0 has exactly the same structure as Lε, but with K − Kε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The explicit expressions are given in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us now look back at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It is easily checked that ˜LεDψε t (v, r) = LεDψε t (v, r), ∀(v, r) ∈ R × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us define the following difference function, for all t ∈ [0, T ∗]: Dt = Dψε t − ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Given ψε 0 ∈ X and ∆0 ∈ Y , such that Dψε 0 = ∆0, we now look at the Cauchy problem for the variable Dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='41) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12), we can write ∂tDt = −LεDt − Lε 0[∆t], D0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='42) Let us define the Banach space Y ∗ ε , analogous to Yε, but defined only for t ∈ [0, T ∗], via the following norm: ∥h∥Y ∗ ε = sup t∈[0,T ∗] (v,r)∈R×R+ |ht(v, r)| Γε(v, r) < ∞, 41 Since Y ⊆ Y ∗ ε , and Dψε ∈ Y ∗ ε , it follows that D ∈ Y ∗ ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Our goal here is to show that Dt can be made arbitrarily small by choosing ε0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This will be done by proving that a unique solution of the Duhamel-integrated form of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='42) exists and can be made arbitrarily small as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The analysis of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='42) is done in the same manner as already seen for the evolution equations of the ∆t-variable and the Dψε t -variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus, we split the operator into three parts, namely i) Lε u, which can be written in the form of (Vε u − Kε u), Kε u being positivity-preserving, ii) Lε δ, which is to be controlled by the potential as a “perturbation”, and, iii) a bounded part Lε b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' These operators and the expressions that appear subsequently are only slightly different from the ones we have seen already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' So we will dispense with writing out most of the terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will mention only those terms for which the differences from analogous expressions encountered in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 become evident in the final estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We write ∂tDt = −Lε uDt − Lε δDt − Lε bDt − Lε 0[∆t] = −Vε uDt + Kε uDt − Lε δDt − Lε bDt − Lε 0[∆t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The Duhamel-integrated form that we will be analyzing is: Dt =e−tVε uD0 + � t 0 ds e−(t−s)Vε uKε u[Ds] − � t 0 ds e−(t−s)Vε uLε b[Ds] − � t 0 ds e−(t−s)Vε uLε δ[Ds] − � t 0 ds e−(t−s)Vε uLε 0[∆s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='43) The operator Lε δ is somewhat different from Lδ, in the sense that it contains extra terms coming from the perturbation ˜Lε s, so we write it down below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lε δDt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = � � v−r v−r−δ1 dw K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) − � v−r v−r−δ1 dw K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dt(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − w) + � v+δ1 v dw Kε(w) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dt(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) − � v+δ2 v dw Kε(w) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dt(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v) + � v v−δ2 dw K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) + � v−r+δ1 v−r dw Kε(w) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dt(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) � − 1(v ≥ a1) � v−r −∞ dw � K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � Dt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w)1(v − w ≤ ε(v − r)) − � v−r v−r−ε(v−r) dw � K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � Dt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='44) The following lemmas are easy to obtain via simple and straightforward computations, so we state them without proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' There exists q1 > 0, depending on the parameters a1, γ0 and κ, such that the following bound holds: |Lε δDs| ≤ q1(κ, γ0) Mγ0 ln � min � M, ε−1 0 ��∥D∥Y ∗ ε Vε uΓε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' and, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The bounded linear operator Lb satisfies: |Lε bDs| ≤ A(b0, m0, µ, c0)∥D∥Y ∗ ε Γε, for some A(b0, m0, µ, c0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 42 We now state and prove the lemma which is the key to proving the existence of a unique solution Dt: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' There exists a number σ > 0, depending on the parameters α, c0, µ′, γ0, such that: Lε uΓε = Vε uΓε − Kε uΓε ≥ σ ln � min � M, ε−1 0 ��Vε uΓε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' As before, it is readily seen that there exist C4 = C4(µ′, γ0, a1) > 0 and C4 = C4(µ′, γ0, a1) > 0 such that the following bound holds: ¯nC4(µ′, γ0, a1) � � ln(1 + ev−r) �− 1 2 � ln � 1 − e−αr�− 4 γ0 + ln min � M, ε−1 0 � � + (ln(1 + ev)) 1 2 � ≤ ˜Vε u(v, r) ≤ ¯nC4(µ′, γ0, a1) � � ln(1 + ev−r) �− 1 2 � ln � 1 − e−αr�− 4 γ0 + ln min � M, ε−1 0 � � + (ln(1 + ev)) 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Computations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' with the like of which we have become quite familiar by now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' also reveal the following: Lε uΓε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) ≥ ¯nΓε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � ln(1 + ev−r) �− 1 2 � 1(v − r < −b0)b2(α) � 1 − e−3αr�2 � 1 − e−3α max(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r−ε(v−r))� + 1 2 ln � 1 − e− 7 2 max(r+δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r))�−1 � + 1(−m0 < v − r < m0)¯nΓε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � 1 6 (ln(1 + ev))− 1 2 + 1(v − r > ln 2)1 9 � ln(1 + ev−r) � 1 2 � + 1(ln 2 < v < m0) ¯n 2 Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) (ln(1 + ev)) 1 2 � 1 − � ln(1 + ev−r) ln(1 + ev−min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v))) �2 � + 1(v ≥ m0)b4(c0)¯nΓε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) (ln(1 + ev)) 1 2 = G0[Γε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This makes it clear that there exists some σ > 0, depending on α, µ′, c0, γ0, such that G0[Γε](v, r) ≥ σ ln � min � M, ε−1 0 �� ˜Vε u(v, r)Γε(v, r), ∀(v, r) ∈ R × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Before the next step let us note that henceforth we will assume Mε0 < 1 since M will be fixed while ε0 will eventually be taken to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We note the following estimate for Lε 0[∆t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' There exist some positive constant C′ < ∞, depending κ, γ0 and α, and, p = p(γ0) > 0 such that the following bound is true: ���Lε 0[∆t](v, r) ��� ≤ C′(κ, γ0, α)(Mε0)p∥∆∥Y Γε(v, r)Vε u(v, r), ∀(v, r) ∈ R × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 43 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We note that in many of the terms in Lε 0[∆], an extra “smallness” comes from the fact that the integrand contains a term like ∆(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', r′), where r′ is the variable of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' However, there are some terms for which this is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will take one such term below and show what makes such a term still “small” (compared to the potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We keep in mind that now we can assume Mε0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Consider the term 1(δ1 < ε(v − r)) � ε(v−r) δ1 dr′ K1 3(v − r, v − r − r′)e−r′ − e−ε(v−r) 1 − eε(v−r) ∆t(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' δ1 < ε(v − r) =⇒ 1 M � 1 − e−αr�4/γ0 < ε0 =⇒ r < 1 α ln � 1 − (Mε0)γ0/4�−1 =⇒ � 1 − e−κr� < 1 − � 1 − (Mε0)γ0/4�κ/α =⇒ � 1 − e−κr�γ0−γ0 < �κ α �γ0−γ0 (Mε0) γ0 4 (γ0−γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then we can write the following: ����� 1(δ1 < ε(v − r)) � ε(v−r) δ1 dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)e−r′ − e−ε(v−r) 1 − eε(v−r) ∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) ����� ≤ 1(δ1 < ε(v − r))∥∆∥Y � ln(1 + ev−r) �− 1 2 eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r) (f(v − r) + f(v)) (1 − eκr)γ0 � ε(v−r) δ1 dr′ e−r′ 1 − e−r′ ≤ 1(δ1 < ε(v − r))∥∆∥Y Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � ln(1 + ev−r) �− 1 2 (1 − eκr)γ0−γ0 ln � ε0 1 M (1 − e−αr)4/γ0 � ≤ 1(δ1 < ε(v − r))∥∆∥Y Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � ln(1 + ev−r) �− 1 2 (1 − eκr)γ0−γ0 4 γ0 ln � 1 − e−αr�−1 ≤ 1(δ1 < ε(v − r))∥∆∥Y Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � ln(1 + ev−r) �− 1 2 4 γ0 �κ α � 1 4γ0 (Mε0) 1 16 γ0 � (1 − eκr) 1 4 γ0 ln � 1 − e−αr�−1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where we have used the fact that γ0 = γ0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Following similar computations it is easy to see that there exist p > 0 depending on γ0 and C′ = C′(κ, γ0, α) > 0, such that ���Lε 0[∆t](v, r) ��� ≤ C′(κ, γ0, α)(Mε0)p∥∆∥Y Γε(v, r)Vε u(v, r), (v, r) ∈ R × R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Finally, we are in a position to prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We first write the Duhamel-integrated equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='43) as Dt = e−tVε uD0 + ˜F[Dt] − � t 0 ds e−(t−s)Vε uLε 0[∆s], where ˜F[Dt] = � t 0 ds e−(t−s)Vε uKε u[Ds] − � t 0 ds e−(t−s)Vε uLε b[Ds] − � t 0 ds e−(t−s)Vε uLε δ[Ds].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then, as before, Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='18, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='16 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='17 mean that, for all t ≤ T ∗: | ˜FDt| ≤ ∥D∥Y ∗ ε Γε � 1 − σ ln � min � M, ε−1 0 �� + AT ∗ + q1 Mγ0 ln � min � M, ε−1 0 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='45) 44 Obviously then, we can choose M < ∞ large enough and ε−1 0 > M, so that we have, for some T ∗, 0 < T ∗ < 1 A ln M � σ − q1 Mγ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This guarantees that ∥ ˜F∥Y ∗ ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The convergence of the relevant Neumann series then gives us the following unique solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='43): Dt = − � 1 − ˜F �−1 � t 0 ds e−(t−s)Vε uLε 0[∆s], ∀0 < t ≤ T ∗, since the initial conditions are such that D0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Finally, from lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='19 it is clear that for some Q > 0, p > 0 depending on the parameters of the weight functions, we have |Dt| ≤ Q ln � min(M, ε−1 0 ) � (Mε0)p ∥∆∥Y Γε, ∀t ∈ [0, T ∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that the time T ∗ above may be different from the time that appears in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3, although the same symbol has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This does not cause any problem because it is enough to choose the minimum of these two times, name it our new T ∗, and understand that this minimum is the T ∗ that appears in our main result Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 as well as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' A Linearization of the Three-wave Collision Operator C3 In what follows, we describe briefly how the expressions for the kernel functions appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) are obtained from the linearization of the three waves collision operator C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The linearized operator is obtained first in terms of energy variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We reserve the letters x and y (x, y are in R+) for these variables, so that the kernel function is written, by a slight abuse of notation, as K3(x, y) in the flat metric (in the weighted L2 space we use the notation K3(x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then we change variables to u and v, which take values in R, and obtain the functions K3(u, v) and K3(u, v) appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We also show that our linearized operator is identical, upto a numerical factor, to the operator considered in [11] and [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' As mentioned in the introduction, the operator C3 is linearized around the equilibrium distribution fBE, by considering perturbations fper of the form fper(x, t) = xfBE(x) ˜fBE(x)ψt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Considering the action of C3 on ftot = fBE + fper and keeping in mind that C3[fBE] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' the following evolution equation for the variable ψ in the linearized model is easily obtained: ∂tψt(x) = −L3ψt(x) = − 2¯n √xgBE(x)˜gBE(x) �� x 0 dy x ˜fBE(x)fBE(y)fBE(x − y) � xψt(x) − yψt(y) − (x − y)ψt(x − y) � +2 � ∞ x dy x ˜fBE(x)fBE(y) ˜fBE(y − x) � xψt(x) − yψt(y) + (y − x)ψt(y − x) �� = −4¯n(¯h(x))2 �� ∞ 0 dy H(min(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' |x − y|)(ψt(x) − ψt(y)) |x − y| − � ∞ 0 dy H(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' y)(ψt(x) − ψt(y)) x + y � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) where H(x, y) = xfBE(x)yfBE(y)(x + y) ˜fBE(x + y), and ¯h(x) = �√xgBE(x)˜gBE(x) �− 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 45 It is also easy to see that one can then write L3ψt(x) = � ∞ 0 dy ˜H(x, y) (ψt(x) − ψt(y)) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2) where ˜H(x, y) = 4¯n¯h(x)2xye− min(x,y)fBE(|x − y|) ˜fBE (max(x, y)) ˜fBE(x + y) � 1 + e− max(x,y)� , and we have used the following identity: 1 − emin(x,y)fBE(x + y) fBE(|x − y|) = fBE(x + y) fBE(min(x, y)) � 1 + emax(x,y)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' From formula (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2) it is evident that the operator L3 has a non-negative, symmetric sesquilinear form on a dense domain of a weighted L2-space characterized by the weight ν(dx) = ¯h(x)−2dx, and that L3 is conveniently defined as an unbounded operator in L2(ν) as follows: L3ψt(x) = � ∞ 0 ν(dy)K3(x, y) (ψt(x) − ψt(y)) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3) where K3(x, y) = 4¯n¯h(x)2¯h(y)2xye− min(x,y)fBE(|x − y|) ˜fBE (max(x, y)) ˜fBE(x + y) � 1 + e− max(x,y)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This yields (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We now change variables from x, y to u, v, where u = ln(ex − 1) and v = ln(ey − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For the convenience of writing, we indulge in a slight abuse of notation and continue to use the same symbols ν and ψ as before for the L2-weight and in the perturbed density respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then it is obvious that in terms of the new variable the weight in our L2-space can be written as ν(dv) = e−v (ln(1 + ev)) 5 2 dv and that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) is just (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3) written in terms of the new variables in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In [11] and [12] the same linearized model has been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that in these papers the model is described in terms of the variable |p|/ √ 2, p being the momentum, while we have used the energy variable here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' To establish the equivalence, let us agree to use letters x, y with primes for these new variables, so that x and y in Equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='9) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12) in [12] are now replaced by x′ and y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then Equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='9) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12) in [12] read: ∂u ∂t = pc(t) � ∞ 0 dy′ � u(t, y′) − u(t, x′) � M(x′, y′), and M(x′, y′) = � 1 sinh |x′2 − y′2| − 1 sinh(x′2 + y′2) � y′3 sinh x′2 x′3 sinh y′2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The variable x′ is related to our energy variable x via x = 2x′2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Changing variables x′ → x, retaining the same symbol for the perturbation u, we get ∂u ∂t = 1 4pc(t) � ∞ 0 dy (u(t, y) − u(t, x)) M(x, y), where M(x, y) = M(x′, y′) y′ ����� x=2x′2,y=2y′2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Using the fact that sinh x = 1 2 exp(−x) fBE(2x) , we easily see that M(x, y) = 2 √ 2 √x y x fBE(y) fBE(x)e− x−y 2 � e 1 2 |x−y|fBE(|x − y|) − e 1 2(x+y)fBE(x + y) � = 2 √ 2¯h(x)2xyfBE(x)fBE(y)ex+ye− min(x,y) � fBE(|x − y|) − emin(x,y)fBE(x + y) � = √ 2 2¯n ˜H(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 46 B Parameters α, c0 and µ appearing in the Weight Functions for the Banach Spaces As is evident from the arguments used in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3, the choice of the weight function Γ is motivated by the requirement that Lu[Γt](v, r) = Vu(v, r)Γt(v, r) − Ku[Γt](v, r) mimics the asymptotic behavior of Vu(v, r)Γt(v, r), so that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6 is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus we want to cut out as much as possible of the bounded part of Lpp (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='16)), and try to ensure the asymptotic dominance of KuΓ by VuΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We know that the potential Vu(v, r) grows exponentially at (−∞), has a line singularity at r = 0 and grows like the square-root function at +∞, somewhat like V (v, r) = ¯n � e− 1 2(v−r) ln(2 + 1/r) + � max(1, v) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The line singularity appearing in the kernel functions contributes the factor gt(v, r) = (1−e−κr)γt(v,r) to the norm Γt(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The nature of the dependence of the smoothing exponent γ on (v − r) is determined by the way ˙Γt has to behave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The other factors in the norm are determined by the “desired” behavior of Lu[Γt](v, r) for asymptotically large positive and negative values of the argument v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' These considerations are simple to understand when we look at the behavior of K3(v, v − r′) and K3(v, v + r′) in the regions v ≪ 0 and v ≫ 0 respectively, as we explain below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' a) Behavior when v → −∞: When v ≪ 0, we need to consider the case when the point singularity becomes dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' To see how it affects the behavior of our linearized operator (note that, since we are not looking at the line singularity now, it makes sense to think about this in the space of ψ-variables instead of the differences), it is enough to consider the following toy model: T(v, v − r′) = e− 1 2 ve−2r′ T(v, v + r′) = e− 1 2 ve−r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The important thing to note here is the relative tilt (extra factor of e−r′) on the left side of the diagonal with respect to the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' A quick back-of-the-envelope calculation then tells us the following: the relative tilt in the kernel function means that the weight function for the ψ-variable should behave as e−αv asymptotically for some α > 0, which in turn implies that, the asymptotic behavior of our Γt(v, r) should be like e−α(v−r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' A cheap upper bound is already imposed on α by the conservation of energy in the BEC problem, so 0 < α < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Coming back to our original linear operator Lu in the space of differences now, we observe that the choice of a negative α generates some extra negative terms coming from the left side of the diagonal and these need to be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' An easy way out is to add a sub-dominant factor e−αv to Γt(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This generates some extra positive terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Finally, these singular factors are needed only when we are dealing with large, negative values of the argument, so we include a cut-off in the relevant terms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' we use max(e−α(v−r), (ln 2)α) instead of the bare e−α(v−r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The inclusion of this cut-off makes our computations simpler for large positive values of the arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' b) Behavior when v → +∞: At +∞ the behavior is dominated by the part K2 3 of the kernel function (this is in fact the motivation behind the splitting of K3(v, v − r′) into K1 3(v, v − r′) and K2 3(v, v − r′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that in this case there is no relative tilt unlike for the region of large, negative values of the arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We have already mentioned that 47 for asymptotically large values of v, the potential grows like √v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Actually, the relevant behavior in this case comes from the last two terms appearing in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7) defining Vu, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', the terms � v v−r dw K2 3(v, w) and � v−r −∞ dw K2 3(v − r, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The latter term becomes important when v − r ≫ 0, especially when v and v − r are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us then begin by comparing the contributions from K2 3(v, v − r′) and K2 3(v − r, v − r − r′) to the potential term for v − r ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' V1(v, r) = � r 0 dr′ K2 3(v, v − r′) ∼ 4¯n√v � 1 − �v − r v �2� , V2(v, r) = � ∞ 0 dr′ K2 3(v − r, v − r − r′) ∼ 4¯n √ v − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Now define x = v−r v , and consider the quantity V2−V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let g(x) = x2+√x−1, so that V2−V1 = √vg(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that 0 < x < 1, for all v − r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Evidently, g′(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus, for all x ∈ (0, 1), g(x) has one and only one zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us call it c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then g(c0) = 0, and it is easy to ascertain that c0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus x < c0 implies V1 > V2, and x > c0 implies V2 > V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' A similar growth for Lu can only come from the part I3[Γt](v, r), defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We obtain this behavior is by making sure Lu[Γt](v, r) inherits as much as possible of the potential term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This motivates us to look for a norm which renders the following terms bounded in certain regions for large values of (v − r) and v: � ∞ 0 dr′ K2 3(v − r, v − r − r′)Γt(v, r + r′) and � r 0 dr′ K2 3(v, v − r′)Γt(v − r′, r − r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that ∀ v − r − r′ ≫ 0, K2 3(v − r, v − r − r′) ∼ 4¯n(v − r)− 3 2 (v − r − r′), and, ∀ v − r′ ≫ 0, K2 3(v, v − r′) ∼ 4¯nv− 3 2(v − r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The corresponding integrals can be rendered bounded if we choose the weight function Γt in such a way that Γt(v, r +r′) grows exponentially as v −r −r′ → (+∞), and Γt(v −r′, r −r′) has a similar exponential growth as v − r′ → (+∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Clearly, we must include a factor h(v, r) in the weight function that behaves as follows: h(v, r) = exp [µ max(v − r, c0v)] , for some suitably chosen c0 ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then our little computation above, comparing V1 and V2, gives us an idea about how the relative weight c0 can be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We choose c0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We obtain upper bounds on permissible values of α and µ in the course of our computations described in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' A brief description of how these bounds are arrived at is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Look at the combination of terms denoted by Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The positivity of this combination rests on the fact that α has a certain upper bound less than 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Essentially we do the following: The term � −(K2 3(v − r, v − r − r′) − K2 3(v, v − r − r′))(f(v − r − r′) − f(v − r)) � is controlled by the term K 1 3(v, v + r′)f(v + r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We are in the region v − r < −m0, v < −b0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus it makes sense to look at the toy model without 48 the line singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then the lower bound K 1 3(v, v + r′)f(v + r′) > (K2 3(v − r, v − r − r′) − K2 3(v, v − r − r′))(f(v − r − r′) − f(v − r)) holds true for all − v − b0 > r′ > r, for all r > 0, if e−( 1 2+α)ve−(1+α)r > e−( 1 2 +α)(v−r)e−(2−α)r, for all r > 0, or equivalently, if α < 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Our computations for the lower bound for I2[Γt](v, r) leads to upper bounds on µ and µc0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We choose µc0 ≤ 1 4, so that the following is true: eµ max(˜a,v−r,c0(v−r′)) − e− 1 2r′eµ max(˜a,v−r,c0(v+r′)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The other upper bound chosen for computational convenience while controlling I2[Γt](v, r) is the fol- lowing: µ < 1 2 − 3 8α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Finally, our computation for an estimate on I4[Γt](v, r) relies on the choice µc0 > α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' C Properties of the H¨older-type Conditions C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 The Time-dependent H¨older-type Condition Let us recall the definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4) of the time-dependent H¨older-type condition gt(v, r): gt(v, r) = � 1 − e−κr�γt(v,r) , γt(v, r) = γt(v − r) = γ0 + a(t) 1 1 + eβ(v−r) , a(t) = 1 8 min(1, ¯n) t 1 + t, κ ≥ 7, 0 < γ0 ≤ 1/8, 1 ≤ β ≤ κ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will now prove a few bounds for certain combinations of gt, which are used in the computations detailed in Appendix D and which will also elucidate the above bounds on β, γ0 and κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all 0 < r′ ≤ r, the following are true: i) gt(v, r) − gt(v, r − r′) ≥ 0, gt(v, r + r′) − gt(v, r) ≥ 0, ii) gt(v, r) − gt(v − r′, r − r′) ≥ 0, gt(v + r′, r + r′) − gt(v, r) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For part i) it is enough to notice that, for all r′′ ≥ 0, ∂gt(v, r′′) ∂r′′ ≥ (κ − β)gt(v, r′′) a(t) 1 + eβ(v−r′′) e−κr′′ 1 − e−κr′′ > 0, ∀κ > β, so that gt(v, r′′) is an increasing function in the second variable r′′, which implies the inequalities in part i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The inequalities in part ii) are obvious from the definition of the H¨older condition gt(v, r), since in this case the H¨older exponent is γt(v, r) for all the terms involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all r′ > r, for all (v, r) ∈ R × R+, the following inequality holds: gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′) > � 2 − (1 + e−κr)γt(v,r)� gt(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 49 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us define a function b0 as follows: b0(v, r, r′) = gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then ∂ ∂r′ b0 = κγt(v, r) � e−κr′ � 1 − e−κr′�γt(v,r)−1 − e−κ(r+r′) � 1 − e−κ(r+r′)�γt(v,r)−1 � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then we can write the following: b0(v, r, r′) > b0(v, r, r) = 2gt(v, r) − gt(v + r, 2r) = 2 � 1 − e−κr�γt(v,r) − � 1 − e−2κr�γt(v,r) = � 2 − � 1 + e−κr�γt(v,r) � gt(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all r′ > r, for all (v, r) ∈ R × R+, the following inequality holds: gt(v, r) + gt(v, r′) − gt(v, r + r′) ≥ � 1 − e−κr′ (1 + e−κr)1−γt(v,r) � gt(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that: gt(v, r + r′) − gt(v, r′) = � 1 − e−κ(r+r′)�γt(v,r+r′) − � 1 − e−κr′�γt(v,r′) ≤ � 1 − e−κ(r+r′)�γt(v,r′) − � 1 − e−κr′�γt(v,r′) ≤ � 1 − e−κ(r+r′)�γt(v,r′) � 1 − � 1 − e−κr′ 1 − e−κr 1 − e−κ(r+r′) �γt(v,r′)� ≤ � 1 − e−κ(r+r′)�γt(v,r)−1 e−κr′(1 − e−κr) ≤ � 1 − e−2κr�γt(v,r)−1 e−κr′(1 − e−κr) ≤ � 1 − e−κr�γt(v,r) e−κr′ (1 + e−κr)1−γt(v,r) , which leads to the bound stated in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' As an aside, let us observe that, for all r′ > r: e−κr′ (1 + e−κr)1−γt(v,r) ≤ f1(r) = e−κr (1 + e−κr)3/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then it is easy to check that f1 is a decreasing function, so that the following estimate holds: e−κr′ (1 + e−κr)1−γt(v,r) ≤ f1(r) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6, and consequently, gt(v, r) + gt(v, r′) − gt(v, r + r′) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4gt(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all 0 ≤ r′ < r, for all (v, r) ∈ R × R+, the following inequality holds: 2gt(v, r) − gt(v, r − r′) − gt(v, r + r′) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 50 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let b2(v, r, r′) = −gt(v, r − r′) − gt(v, r + r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then ∂b2 ∂r′ = γ0κ � e−κ(r−r′) � 1 − e−κ(r−r′)�γt(v−r+r′)−1 − e−κ(r+r′) � 1 − e−κ(r+r′)�γt(v−r−r′)−1� + a(t) � H(v, r − r′) − H(v, r + r′) � , where H(v, r′) = 1 1 + eβ(v−r′) � 1 − e−κr′�γt(v−r′) � κe−κr′ 1 − e−κr′ + βeβ(v−r′) 1 + eβ(v−r′) ln(1 − e−κr′) � = 1 1 + eβ(v−r′) � 1 − e−κr′�γt(v−r′) F2(v, r′), with F2(v, r′) = κe−κr′ 1 − e−κr′ + βeβ(v−r′) 1 + eβ(v−r′) ln(1 − e−κr′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then it is not difficult to see that: ∂ ∂r′ H(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) = 1 1 + eβ(v−r′) � 1 − e−κr′�γt(v−r′) � − � κ2 e−κr′ (1 − e−κr′)2 − β eβ(v−r′) 1 + eβ(v−r′) � κe−κr′ 1 − e−κr′ − β ln(1 − e−κr′) 1 + eβ(v−r′) �� + � κγ0 e−κr′ 1 − e−κr′ + β eβ(v−r′) 1 + eβ(v−r′) � F2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + a(t) 1 + eβ(v−r′) (F2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′))2 � ≤ gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) 1 + eβ(v−r′) � − κ2 e−κr′ (1 − e−κr′)2 � 1 − β κ − �β κ �2� + a(t) 1 + eβ(v−r′) κ2e−2κr′ (1 − e−κr′)2 + κ e−κr′ 1 − e−κr′ � β eβ(v−r′) 1 + eβ(v−r′) + κγ0 e−κr′ 1 − e−κr′ � � < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The upper bound in the last line holds because our parameters have been chosen so as to guarantee the following inequality: 1 − γ0 − a(t) > 2β κ + �β κ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The above inequality means that H is a strictly decreasing function of r′ and so, H(v, r − r′) ≥ H(v, r + r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' On the other hand, note that γt(v − r + r′) ≤ γt(v − r − r′), and it is easily seen that e−κ(r−r′) � 1 − e−κ(r−r′)�γt(v−r+r′)−1 ≥ e−κ(r+r′) � 1 − e−κ(r+r′)�γt(v−r−r′)−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus, ∂ ∂r′ b2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This means we have the following inequality: 2gt(v, r) − gt(v, r − r′) − gt(v, r + r′) = 2gt(v, r) + b2(v, r, r′) ≥ 2gt(v, r) + b2(v, r, r′) �� r′=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 51 Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all 0 ≤ r′ < r, for all (v, r) ∈ R × R+, the following inequality holds: 2gt(v, r) − gt(v − r′, r − r′) − gt(v + r′, r + r′) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Taking the partial derivative with respect to r′, we see that: ∂ ∂r′ � 2gt(v, r) − gt(v − r′, r − r′) − gt(v + r′, r + r′) � = κγt(v, r) � e−κ(r−r′) � 1 − e−κ(r−r′)�γt(v−r)−1 − e−κ(r+r′) � 1 − e−κ(r+r′)�γt(v−r)−1� ≥ 0, so that, 2gt(v, r) − gt(v − r′, r − r′) − gt(v + r′, r + r′) ≥ � 2gt(v, r) − gt(v − r′, r − r′) − gt(v + r′, r + r′) � ���� r′=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all r′ > r, for all (v, r) ∈ R × R+, the following inequality holds: 2 � gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′) � − gt(v, r) > � 1 − e−r� 1 2 gt(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' From Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2, we have, for all r′ > r: 2 � gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′) � − gt(v, r) > � 2 � 2 − � 1 + e−κr�γt(v,r)� − 1 � gt(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all r < (ln 2)/3, it is obvious that 2 � 2 − (1 + e−κr)γt(v,r)� − 1 > (1 − e−r) 1 2, since γt(v, r) < 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all r ≥ (ln 2)/3, let us define f2(r) = 2 � 2 − (1 + e−κr)1/4� − 1 − (1 − e−r)1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then it is easy to see that, d drf2 < 0, so that, f2(r) > 0, which implies � 2 � 2 − (1 + e−κr)γt(v,r)� − 1 � gt(v, r) > (1 − e−r)1/2 gt(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all r′ ≥ r, for all p > γt(v, r), and for all ν ≤ 1, the following is true: � 1 − e−νr�p gt(v, r′) ≤ � 1 − e−νr�p gt(v − r + r′, r′) ≤ � 1 − e−νr′�p gt(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let h1(r) = (1 − e−νr)(1 − e−κr)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then it is easily computed that d dr′ h1 ≥ 0, so that, h1(r′) ≥ h1(r), ∀r′ ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This means 1 − e−νr 1 − e−νr′ ≤ 1 − e−κr 1 − e−κr′ < 1, 52 which implies, � 1 − e−νr 1 − e−νr′ �p ≤ � 1 − e−κr 1 − e−κr′ �p ≤ � 1 − e−κr 1 − e−κr′ �γt(v,r) , ∀p > γt(v, r), that is, � 1 − e−νr�p � 1 − e−κr′�γt(v,r) ≤ � 1 − e−νr′�p � 1 − e−κr�γt(v,r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since γt(v, r) is, by definition, an increasing function of the radial variable r, (1 − e−κr′)γt(v,r′) ≤ (1 − e−κr′)γt(v,r), and this, together with the inequality obtained above, proves the statement of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 The Regularized, Time-independent H¨older-type Condition In this case, the H¨older exponent, denoted here by ˜γ0, is independent of time, and the results below hold for each of the two values ˜γ0 is allowed to take in our computations for the regularized linear operator, namely 0 and γ0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The H¨older-type condition ˜g(v, r) depends on the first variable v only through the inclusion of the regularization parameter ε(v) in the argument as follows: ˜g(v, r) = � 1 − e−κ(r+ε(v))�˜γ0 , ˜γ0 ∈ {0, γ0/2}, and, ε(v) = ε0e− µ γ0 max(a1,v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then the function ˜g has properties similar to those proved in lemmas C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 -C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Most of the proofs are also very similar to those recorded above, therefore we will write down the statements of the lemmas while omitting some of the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all 0 < r′ ≤ r, the following are true: i) ˜gt(v, r) − ˜gt(v, r − r′) ≥ 0, ˜gt(v, r + r′) − ˜gt(v, r) ≥ 0, ii) ˜gt(v, r) − ˜gt(v − r′, r − r′) ≥ 0, ˜gt(v + r′, r + r′) − ˜gt(v, r) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The inequalities contained in part i) are obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For part ii), let us observe that 0 ≤ ε(v − r′) − ε(v) < r′, and 0 ≤ ε(v) − ε(v + r′) < r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' So: ˜g(v − r′, r − r′) = � 1 − e−κ(r+(ε(v−r′)−r′))�˜γ0 ≤ � 1 − e−κ(r+ε(v))�˜γ0 = ˜g(v, r), ˜g(v + r′, r + r′) = � 1 − e−κ(r+(ε(v+r′)+r′))�˜γ0 ≥ � 1 − e−κ(r+ε(v))�˜γ0 = ˜g(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all r′ > r, for all (v, r) ∈ R × R+, the following inequality holds: ˜g(v, r) + ˜g(v, r′) − ˜g(v, r + r′) ≥ � 2 − (1 + e−κ(r+ε(v)))˜γ0� ˜g(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' A straightforward differentiation reveals ∂ ∂r′ � ˜g(v, r′) − ˜g(v, r + r′) � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' So, ˜g(v, r) + ˜g(v, r′) − ˜g(v, r + r′) > ˜g(v, r) + � ˜g(v, r′) − ˜g(v, r + r′) � ���� r′=r ≥ � 2 − (1 + e−κ(r+ε(v)))˜γ0 � ˜g(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 53 Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all r′ > r, for all (v, r) ∈ R × R+, the following inequality holds: ˜g(v, r) + ˜g(v − r + r′, r′) − ˜g(v + r′, r + r′) ≥ � 2 − (1 + e−κ(r+ε(v)))˜γ0� ˜g(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In the region v + r′ ≤ a1, the statement of this lemma is the same as the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' When v + r′ > a1, ˜g(v, r) + ˜g(v − r + r′, r′) − ˜g(v + r′, r + r′) > ˜g(v, r) + ˜g(v + r′, r′) − ˜g(v + r′, r + r′) > ˜g(v, r) + � ˜g(v + r′, r′) − ˜g(v + r′, r + r′) � ���� r′=r , by a simple differentiation, ≥ � 2 − (1 + e−κ(r+ε(v)))˜γ0� ˜g(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all 0 ≤ r′ < r, for all (v, r) ∈ R × R+, the following inequality holds: 2˜g(v, r) − ˜g(v, r − r′) − ˜g(v, r + r′) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This lemma is obtained by following the proof of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4 exactly and keeping in mind that the exponent ˜γ0 is time-independent (so a(t) = 0) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all 0 ≤ r′ < r, for all (v, r) ∈ R × R+, the following inequality holds: 2˜g(v, r) − ˜g(v − r′, r − r′) − ˜g(v + r′, r + r′) ≥ − � � 1 − e−κ(r−r′+ε(v−r′))�˜γ0 − � 1 − e−κ(r−r′+ε(v))�˜γ0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The inequality of this lemma is obtained by the following simple observation: 2˜g(v, r) − ˜g(v − r′, r − r′) − ˜g(v + r′, r + r′) ≥ 2˜g(v, r) − ˜g(v, r − r′) − ˜g(v, r + r′) − �� 1 − e−κ(r−r′+ε(v−r′))�˜γ0 − � 1 − e−κ(r−r′+ε(v))�˜γ0� ≥ − �� 1 − e−κ(r−r′+ε(v−r′))�˜γ0 − � 1 − e−κ(r−r′+ε(v))�˜γ0� , by Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all r′ > r, for all (v, r) ∈ R × R+, the following inequality holds: 2 � ˜g(v, r) + ˜g(v − r + r′, r′) − ˜g(v + r′, r + r′) � − ˜g(v, r) > � 1 − e−r� 1 2 ˜g(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The proof of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='13 follows the corresponding proof for C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all r′ ≥ r, for all p > ˜γ0, for all ν ≤ 1, the following is true: � 1 − e−νr�p ˜g(v, r′) ≤ � 1 − e−νr′�p ˜g(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The proof is the same as that of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 54 D Computations for Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 Some Useful Inequalities involving the Kernel Function In the computations for Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6 we use certain properties of the kernel function K3 and the weight function Γt, collected in the lemmas below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all v − r < −b0, for all 0 < r′ ≤ r the following inequality holds: K3(v − r, v − r + r′) � f(v − r) − (ln(1 + ev−r+r′))−α� − K1 3(v − r, v − r − r′) � f(v − r − r′) − f(v − r) � ≥ K3(v − r, v − r + r′) � ln(1 + ev−r+r′) �−α � ln(1 + ev−r−r′) ln(1 + ev−r+r′) �α \uf8f1 \uf8f2 \uf8f31 − � ln(1 + ev−r−r′) ln(1 + ev−r+r′) �1−2α\uf8fc \uf8fd \uf8fe × × � 1 − � ln(1 + ev−r−r′) ln(1 + ev−r) �α� Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that: K3(v − r, v − r + r′) � f(v − r) − (ln(1 + ev−r+r′))−α� − K1 3(v − r, v − r − r′) � f(v − r − r′) − f(v − r) � ≥ 4¯n � ln(1 + ev−r) �−3/2 e−r′ 1 − e−r′ ev−r + 2e−r′ 1 + ev−r + e−r′ � ln(1 + ev−r+r′)(ln(1 + ev−r))−α � 1 − � ln(1 + ev−r) ln(1 + ev−r+r′) �α� −(ln(1 + ev−r−r′))1−α � 1 − � ln(1 + ev−r−r′) ln(1 + ev−r) �α�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Now, ln(1 + ev−r+r′)(ln(1 + ev−r))−α � 1 − � ln(1 + ev−r) ln(1 + ev−r+r′) �α� − (ln(1 + ev−r−r′))1−α � 1 − � ln(1 + ev−r−r′) ln(1 + ev−r) �α� = (ln(1 + ev−r+r′))1−2α � (ln(1 + ev−r+r′))α �� ln(1 + ev−r+r′) ln(1 + ev−r) �α − 1 �� − (ln(1 + ev−r−r′))1−2α � (ln(1 + ev−r−r′))α � 1 − � ln(1 + ev−r−r′) ln(1 + ev−r) �α�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) Now for all r′ ≥ 0, let us define the function h as follows: (ln(1 + ev−r))αh(v − r, r′) = (ln(1 + ev−r+r′))α �� ln(1 + ev−r+r′) ln(1 + ev−r) �α − 1 � − (ln(1 + ev−r−r′))α � 1 − � ln(1 + ev−r−r′) ln(1 + ev−r) �α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then ∂ ∂r′ h(v − r, r′) = α(ln(1 + ev−r+r′))α−1 ev−r+r′ 1 + ev−r+r′ � 2(ln(1 + ev−r+r′))α − (ln(1 + ev−r))α� 55 − α(ln(1 + ev−r−r′))α−1 ev−r−r′ 1 + ev−r−r′ � 2(ln(1 + ev−r−r′))α − (ln(1 + ev−r))α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Observe that for all v − r < −b0, e−r′ < ln(1+ev−r−r′) ln(1+ev−r) < e− 24 25 r′, since b0 ≥ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then for all r′ > 0 such thtat � ln(1 + ev−r−r′) ln(1 + ev−r) �α < 1 2, it is obvious from the definition that ∂ ∂r′ h(v − r, r′) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' On the other hand, for all r′ > 0 such that � ln(1 + ev−r−r′) ln(1 + ev−r) �α ≥ 1 2, we have e− 24 25 αr′ > 1 2, which, as one can easily check, implies v − r + r′ < −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5, for all v − r < −b0, and this in turn means, (ln(1 + ev−r+r′))α−1 ev−r+r′ 1 + ev−r+r′ > (ln(1 + ev−r−r′))α−1 ev−r−r′ 1 + ev−r−r′ , because ew 1+ew (ln(1 + ew))α−1 is a strictly increasing function for all w < −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus we have, again, ∂ ∂r′ h(v − r, r′) > 0 in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since we have proved h(v − r, r′) to be a strictly increasing function of r′, the following inequality holds: (ln(1 + ev−r+r′))α �� ln(1 + ev−r+r′) ln(1 + ev−r) �α − 1 � > (ln(1 + ev−r−r′))α � 1 − � ln(1 + ev−r−r′) ln(1 + ev−r) �α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Using the above we can refer to equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) and write: (ln(1 + ev−r+r′))1−2α � (ln(1 + ev−r+r′))α �� ln(1 + ev−r+r′) ln(1 + ev−r) �α − 1 �� − (ln(1 + ev−r−r′))1−2α � (ln(1 + ev−r−r′))α � 1 − � ln(1 + ev−r−r′) ln(1 + ev−r) �α�� > � ln(1 + ev−r+r′) �1−α � ln(1 + ev−r−r′) ln(1 + ev−r+r′) �α \uf8f1 \uf8f2 \uf8f31 − � ln(1 + ev−r−r′) ln(1 + ev−r+r′) �1−2α\uf8fc \uf8fd \uf8fe × × � 1 − � ln(1 + ev−r−r′) ln(1 + ev−r) �α� , which obviously implies the inequality stated in this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all v < −b0, for all 0 < r′ ≤ r and v + r < 0 the following inequality holds: � r δ1 dr′K3(v, v + r′) � f(v) − f(v + r′) � gt(v, r) ≥ � r δ1 dr′K1 3(v, v − r′) � f(v − r′) − f(v) � gt(v − r′, r − r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 56 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us define h2(v, r′) = ln(1+ev+r′) � (ln(1 + ev))−α − (ln(1 + ev+r′))−α� −ln(1+ev−r′) � (ln(1 + ev−r′))−α − (ln(1 + ev))−α� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then ∂ ∂r′ h2 = ev+r′ 1 + ev+r′ � (ln(1 + ev))−α − (1 − α)(ln(1 + ev+r′))−α� − ev−r′ 1 + ev−r′ � (ln(1 + ev))−α − (1 − α)(ln(1 + ev−r′))−α� > 0, since (ln(1 + ev−r′))−α > (ln(1 + ev+r′))−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus h2(v, r′) > h2(v, 0) = 0, for all r′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then the inequality stated in the lemma is obvious from the definition of the kernel functions and the fact that gt(v, r) > gt(v − r′, r − r′), for all r′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For all v + r′ < 0, for all r′ > r the following is true: K3(v, v + r′)f(v + r′) − K1 3(v − r, v − r − r′)f(v − r − r′) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′) − K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)f(v − r − r′) ≥ 4¯n e−r′ 1 − e−r′ ev + 2e−r′ 1 + ev + e−r′ (ln(1 + ev))− 3 2(ln(1 + ev+r′))1−α \uf8ee \uf8f01 − � ln(1 + ev) ln(1 + ev−r) � 3 2 � ln(1 + ev−r−r′) ln(1 + ev+r′) �1−α\uf8f9 \uf8fb ≥ 4¯n e−r′ 1 − e−r′ ev + 2e−r′ 1 + ev + e−r′ (ln(1 + ev))− 3 2(ln(1 + ev+r′))1−α \uf8ee \uf8f01 − � ln(1 + ev−r−r′) ln(1 + ev−r) �1−α � ln(1 + ev) ln(1 + ev−r) � 1 2+α \uf8f9 \uf8fb > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where we have used the fact that 1 − α > 1 2 + α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' for all α < 1/4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' and that ln(1 + ev−r) ln(1 + ev−r−r′) > ln(1 + ev) ln(1 + ev−r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 Lower Bounds for Ii[Γt](v, r), i ∈ {1, 2, 3, 4} D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 Estimating I1[Γt](v, r): We start with the following lower bound, which can be derived quite easily by using Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7 proved above: I1[Γt](v, r) = I1[Γ1 t](v, r) + I1[Γ2 t](v, r) ≥ J0[Γ1 t + Γ2 t](v, r) + J1[Γ1 t](v, r) + J2[Γ2 t](v, r) + I[Γ1 t + Γ2 t ](v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2) J0 has already been defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We now write down explicitly the other terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' As mentioned before, these terms are sub-dominant to J0 close to the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' J1[Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) 57 = f(v − r) � 1(v < −b0) � eµa � r+a1−v r+δ1 dr′ � (K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) − K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′))gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − (K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′))gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � + � ∞ r+a1−v dr′ � (K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) − K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′))e− 1 2 r′eµc0(v−r+r′)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − (K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′))eµagt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) �� + 1(v ≥ −b0) � ∞ r+δ1 dr′ � � K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) − K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � e− 1 2 r′eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r) � K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3) J2[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) =1(v < −b0) � 1(r ≤ a1 − v ≤ r + δ1) � � ∞ r+δ1 dr′ � � K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) − K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � e− 1 2r′ × × exp � µ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0(v − r + r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r} � f(v − r + r′)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − � K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � exp � µ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′} � f(v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) �� + 1(r + δ1 < a1 − v) � � a1−v r+δ1 dr′ � � K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) − K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � × × exp � µ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0(v − r + r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r} � f(v + r′)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − � K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � exp � µ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′} � f(v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � + � ∞ a1−v dr′ � � K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) − K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � e− 1 2r′ × × exp � µ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0(v − r + r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r} � f(v + r′)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − � K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � exp � µ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′} � f(v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) �� + 1(r > a1 − v) � � r+a1−v r+δ1 dr′ � � K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) − K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � × × exp � µ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0(v − r + r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r} � f(v + r′)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − � K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � exp � µ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′} � f(v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � + � ∞ r+a1−v dr′ � � K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) − K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � e− 1 2r′ × × exp � µ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0(v − r + r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r} � f(v + r′)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − � K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � exp � µ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′} � f(v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) ��� 58 + 1(v ≥ −b0) � � ∞ r+δ1 dr′ � � K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) − K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � e− 1 2 r′ × × exp � µ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0(v − r + r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r} � f(v + r′)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − � K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � exp � µ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′} � f(v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4) and finally, I[Γ1 t + Γ2 t ](v, r) = − eµ max(a,c0v) � ∞ r+δ1 dr′ K1 3(v, v − r′) � f(v − r − r′) − f(v − r′) � gt(v, r′) − eµ max(a,c0v) � ∞ r+δ1 dr′ � K1 3(v − r, v − r − r′) − K1 3(v, v − r′) � � f(v − r − r′) − f(v − r) � gt(v, r′) − eµ max(a,c0v) � ∞ r+δ1 dr′ K1 3(v − r, v − r − r′) � f(v − r − r′) − f(v − r) � � gt(v, r + r′) − gt(v, r′) � + I(1)[Γ1 t ](v, r) + I(2)[Γ2 t ](v, r) + I(3)[Γ1 t + Γ2 t](v, r), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5) with I(1)[Γ1 t ](v, r) = 1(v < −b0)f(v − r) � � r+a1−v r+δ1 dr′ K3(v, v + r′) � eµa − e− 1 2 r′eµ max{a,v−r,c0(v+r′)}� × × � gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′) � + 1(a1 − v ≥ r + δ1) � r+a1−v a1−v dr′ K3(v, v + r′) � eµa − eµc0ve−( 1 2−µc0)r′� gt(v + r′, r + r′) + 1(a1 − v < r + δ1) � r+a1−v r+δ1 dr′ K3(v, v + r′) � eµa − eµc0ve−( 1 2 −µc0)r′� gt(v + r′, r + r′) � , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6) I(2)[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v < −b0) � � ∞ max(r+δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='a1−v) dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′) (eµa − eµc0v) gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � ∞ r+δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � f(v) − f(v + r′) � eµagt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(r ≤ a1 − v ≤ r + δ1) � � r+a1−v r+δ1 dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′) � eµa − eµc0(v−r+r′)� gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + � ∞ r+δ1 dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � f(v − r + r′) − f(v + r′) � eµ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′)}gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + � ∞ r+δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′)eµc0v � 1 − 2e−( 1 2−µc0)r′ + e−2( 1 2 −µc0)r′� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � r+a1−v r+δ1 dr′ K3 1(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)f(v − r + r′)eµagt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � 59 + 1(a1 − v > r + δ1) � � ∞ r+δ1 dr′ K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) � f(v − r + r′) − f(v + r′) � × × eµ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′)}gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + � r+a1−v r+δ1 dr′ � K3 1(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) − K3 1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � � f(v − r + r′) − f(v + r′) � × × eµ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′)}gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + � r+a1−v a1−v dr′ � K3 1(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) − K3 1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � f(v + r′)eµ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′)}gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + eµa � a1−v r+δ1 dr′ K3 1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − gt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + � r+a1−v a1−v dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′) � eµc0v � 1 − e−( 1 2−µc0)r′� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r)+ + � eµa − eµc0ve−( 1 2−µc0)r′� gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � + eµa � r+a1−v r+δ1 dr′ K3 1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � f(v − r + r′) − f(v + r′) � gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � + 1(a1 − v < r) � � r+a1−v r+δ1 dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′) � eµa − eµc0(v−r+r′)� gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + eµa � r+a1−v r+δ1 dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � f(v − r + r′) − f(v + r′) � gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + � r+a1−v r+δ1 dr′ K3 1(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)eµ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′)} � f(v − r + r′) − f(v + r′) � gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + � ∞ r+δ1 dr′ � K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) − K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � � f(v − r + r′) − f(v + r′) � × × eµ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′)}gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7) and finally, I(3)[Γ2 t](v, r) = 1(v ≥ −b0) � ∞ r+δ1 dr′ K3 2(v − r, v − r + r′) � f(v − r + r′) − f(v + r′) � eµ max(a,v−r,c0(v−r+r′))gt(v − r + r′, r′) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='8) The parts J1 and J2 yield some negative terms that will need to be controlled, as we will see shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 Estimating I2[Γt](v, r) I2[Γt](v, r) = I2[Γ1 t ](v, r) + I2[Γ2 t ](v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' A Lower Bound for I2[Γ1 t](v, r) Recall from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='30) I2[Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) ≥ 1(v − r < −b0)eµ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v} � r δ1 dr′ � K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) � f(v − r) − (ln(1 + ev−r+r′))−α� − K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � f(v − r − r′) − f(v − r) � � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v < −b0)1(r > a − v)f(v − r) � r a1−v dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � eµa − eµc0ve−( 1 2 −µc0)r′� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v ≥ −b0)1(v − r < −b0) � f(v − r) � r δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � eµ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v} − e− 1 2r′eµ max{a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v+r′)}� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � r min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r−v) dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)(ln(1 + ev−r+r′))−α � 2gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � � + I− 2 [Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where I− 2 [Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = − f(v − r)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � r δ1 dr′ � K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) − K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � � gt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � − 1(v < −b0)eµa� 4¯n � ln(1 + ev−r) �− 3 2 � r δ1 dr′ � ln(1 + ev−r+r′) �1−α e−r′ 1 − e−r′ × × � 1 − � ln(1 + ev−r−r′) ln(1 + ev−r+r′) �α� � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � + 4¯nf(v − r) (ln(1 + ev))− 3 2 � r δ1 dr′ ln(1 + ev+r′) ev−r′(1 − e−r′)(1 − e− 5 4 r′) (1 + ev−r + e−r′)(1 + ev + e−r′)× × � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � � − 1(v ≥ −b0) � 1(v − r < −b0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)� 4¯n � ln(1 + ev−r) �− 3 2 � r δ1 dr′ � ln(1 + ev−r−r′) �1−α × × e−r′ 1 − e−r′ � 1 − � ln(1 + ev−r−r′) ln(1 + ev−r+r′) �α� � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � + 4¯nf(v − r) (ln(1 + ev))− 3 2 � r δ1 dr′ ln(1 + ev+r′)e−r′(1 − e−r−2r′) (1 + ev−r + e−r′) × � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � � + 1(v − r ≥ −b0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)� 4¯nf(v − r) (ln(1 + ev))− 3 2 � r δ1 dr′ ln(1 + ev+r′) e−r′(1 − e−2r′) (1 + ev−r + e−r′) × × � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � + 4¯nf(v − r) (ln(1 + ev))− 3 2 � r δ1 dr′ ln(1 + ev−r′) ev−r′(1 − e−r′) (1 + ev−r′ + e−r′)(1 + ev + e−r′)× × � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � �� (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='9) 61 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' A Lower Bound for I2[Γ2 t](v, r): I2[Γ2 t](v, r) ≥ 1(v < −b0) � eµa � r δ1 dr′ K3(v, v + r′) � f(v) − f(v + r′) � gt(v, r) + 1(r > a1 − v) � r δ1 dr′ K3(v, v + r′)f(v + r′) � eµa − eµc0ve−( 1 2−µc0)r′� gt(v, r) � + 1(v ≥ −b0) � r δ1 dr′ K3(v, v + r′)f(v) � eµ max(a,v−r,c0v) − e− 1 2 r′eµ max(a,v−r,c0(v+r′)� gt(v, r) + I− 2 [Γ2 t](v, r), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='10) where I− 2 [Γ2 t](v, r) = − eµ max(a,v−r,c0v)f(v) � r δ1 dr′ � K3(v, v + r′) − K1 3(v, v − r′) � � gt(v + r′, r + r′) − gt(v, r + r′) � − 4¯n eµ max(a,v−r,c0v)f(v) (ln(1 + ev))− 3 2 � r δ1 dr′ ln(1 + ev+r′) e−r′(1 − e−2r′) 1 + ev−r + e−r′ � gt(v, r + r′) − gt(v, r) � − 4¯n eµ max(a,v−r,c0v)f(v) (ln(1 + ev))− 3 2 � r δ1 dr′ ln(1 + ev−r′) ev−r′(1 − e−r′) (1 + ev−r′ + e−r′)(1 + ev + e−r′)× × � gt(v, r + r′) − gt(v, r) � D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 Lower Bound for I3[Γt](v, r): I3[Γt](v, r) = I(1) 3 [Γt](v, r) + I(2) 3 [Γt](v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' For I3 we will just write out the relevant definitions explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Expression for I(1) 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) : a) I(1) 3 [Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v − r ≤ −m0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)f(v − r) � � ∞ ˜r dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � ˜r 0 dr′ � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − (K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′))gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) �� + 1(−m0 < v − r < m0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)f(v − r) � ∞ 0 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v − r ≥ m0)f(v − r) � 1(v ≤ 3r) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � ∞ v−r dr′K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � v−r 0 dr′ � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) �� + 1(v > 3r) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � v−r−c0v 0 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � ∞ v−r−c0v dr′ � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) 62 − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) ��� + A− 1 [Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + A− 2 [Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + A− 3 [Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + A− 4 [Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='11) where A− 1 [Γ1 t ](v, r) = − 1(v − r ≤ −m0)1(v < −b0)1(˜r > r + δ1) � ˜r r+δ1 dr′ � K2 3(v − r, v − r − r′) − K2 3(v, v − r − r′) � × × eµ max(a,c0v) � f(v − r − r′) − f(v − r) � gt(v, r + r′) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12) A− 2 [Γ1 t](v, r) = − 1(v − r ≤ −m0) � min(r,˜r) min(δ1,˜r) dr′ � K2 3(v − r, v − r − r′) − K2 3(v, v − r − r′) � × × eµ max(a,c0v) � f(v − r − r′) − f(v − r) � gt(v, r + r′) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='13) A− 3 [Γ1 t](v, r) = − 1(v − r ≥ m0)1(v > 3r) � ∞ v−r dr′ � K2 3(v − r, v − r − r′) − K2 3(v, v − r − r′) � × × eµ max(a,c0v) � f(v − r − r′) − f(v − r) � gt(v, r + r′) (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='14) A− 4 [Γ1 t](v, r) = − 1(v − r ≤ −m0)eµ max(a,c0v)� � min(δ1,˜r) 0 dr′ � K2 3(v − r, v − r − r′) − K2 3(v, v − r − r′) � × × � f(v − r − r′) − f(v − r) � gt(v, r + r′) + 1(v < −b0) � min(˜r,r+δ1) r dr′ � K2 3(v − r, v − r − r′) − K2 3(v, v − r − r′) � × × � f(v − r − r′) − f(v − r) � gt(v, r + r′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='15) b) I(1) 3 [Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) =1(v − r ≤ −m0)f(v)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � � ∞ ˜r dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � ˜r 0 dr′ � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) �� +1(−m0 < v − r < m0)f(v)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � ∞ 0 dr′K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) +1(v − r ≥ m0)f(v) � 1(v ≤ 3r) � � v−r 0 dr′ � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)� × × K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) 63 + � v−r 0 dr′ eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)� K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − − � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r) � ∞ v−r dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � + 1(v > 3r) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r) � v−r−c0v 0 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � ∞ v−r−c0v dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′)� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � ∞ v−r−c0v dr′ eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)� K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='16) Expression for I(2) 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) : a) I(2) 3 [Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v ≤ −m0)eµaf(v − r) � 1(v + r < −b0) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − gt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � + 1(v + r ≥ −b0) � −v−b0 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − gt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � + 1(v + r ≥ −b0) � r −v−b0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � + 1(−m0 < v < m0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)f(v − r) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v ≥ m0)f(v − r) � 1(v ≤ r)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(0 < v − r < c0v) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � r v−c−1 0 (v−r) dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r′))gt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � + 1(v − r ≥ c0v) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r′))gt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) �� (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='17) b) I(2) 3 [Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v ≤ −m0)eµa� − 1(v + r < −b0) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � f(v − r′) − f(v) � gt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) − 1(v + r ≥ −b0) � −v−b0 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � f(v − r′) − f(v) � gt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) 64 + 1(v + r < −b0)f(v) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − gt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � + 1(v + r ≥ −b0)f(v) � � r −v−b0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � −v−b0 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − gt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � �� + 1(−m0 < v < m0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)f(v) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v ≥ m0) � 1(v ≤ r)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)f(v) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(0 < v − r < c0v) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)f(v) � v−c−1 0 (v−r) 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � r v−c−1 0 (v−r) dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)f(v) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r′))gt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) �� + 1(v − r ≥ c0v)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)f(v) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − gt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='18) Let us recall that the correct asymptotic behavior of LuΓt for large, positive values of v comes from 1(v−r ≥ m0)I(1) 3 [Γt](v, r) (when v and (v − r) are comparable) and 1(v ≥ m0)I(2) 3 [Γt](v, r) (when v is much bigger than (v − r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will look at I4 later because we already know it inherits a degree of “smallness” from the length of the interval of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3 Combining the Estimates to arrive at the Main Lower Bound: In this section we combine the terms written out in the previous subsection, in a way which will enable us to find a useful lower bound on LuΓt(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' As mentioned already in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6, J0[Γt](v, r) from the lower bound for I1[Γt](v, r) and C3[Γ1 t](v, r), defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='35), from the lower bound for I2[Γ1 t](v, r), contribute to LuΓt(v, r) the correct asymptotic behavior with point singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Our main objective now is to tackle the negative terms obtained in the estimates above by using some suitable combinations of integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We enumerate these combinations as Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1, Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' These com- binations yield useful lower bounds for sums of the Ii’s, which eventually lead to our main estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In what follows these lower bounds are denoted by LB1, LB2 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In other words, for ease of referencing, we will denote the most important estimates as LB1, LB2 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', the particular combinations of terms leading to these estimates by Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1, Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' and reserve the usual numbering by section for everything else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that terms of the form (gt(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', r + r′) − gt(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', r)) generate an extra exponential decay of e−κr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This additional “smallness” makes it easy for us to deal with such negative terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' So we will focus first on those negative terms which do not contain differences of the form (gt(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', r + r′) − gt(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will first look at the first two negative terms in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We combine the first negative term with terms from I(2)[Γ2 t](v, r) and I1 3[Γ2 t ](v, r) as follows: − eµ max(a,c0v) � ∞ r+δ1 dr′ K1 3(v, v − r′) � f(v − r − r′) − f(v − r′) � gt(v, r′) (Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1) + 1(v − r ≤ −m0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)f(v) � ∞ ˜r dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) 65 + 1(−m0 < v − r < m0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)f(v) � ∞ r+δ1 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v < −b0)1(r + δ1 < a1 − v) � ∞ r+δ1 dr′ K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) � f(v − r + r′) − f(v + r′) � × × eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′))gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) ≥ − 1(v ≥ 0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � 2r+δ1 r+δ1 dr′ K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � f(v − r − r′) − f(v − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − 1(v − r ≥ m0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � ∞ 2r+δ1 dr′ K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � f(v − r − r′) − f(v − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + B+ 1 [Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where B+ 1 [Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v < −b0)eµaf(v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � 1(v − r < −m0) � ∞ max(−v−b0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r) dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) × × � 1 − e−( 122 250 −2α)r′ + e−( 122 250 −α)re−r′ 1 − e−(1−α)r′ 1 − e−r′ � + 1(v − r ≥ −m0) � ∞ r+δ1 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)× × � 1 − e−( 122 250 −2α)r′ + e−( 122 250 −α)re−r′ 1 − e−(1−α)r′ 1 − e−r′ �� + 1(v ≥ −b0) � eµaf(v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r)1(−b0 ≤ v < 0) � ∞ r+δ1 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � 1 − e−2( 1 3−α)r′e−( 1 3+α)(r′−r) + e−(r′−r)e−( 2 3 −α)r 1 − e−(1−α)r′ 1 − e−r′ � + 1(v ≥ 0)1(v − r < m0)f(v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � ∞ r+δ1 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) × × � 1 − �ln(1 + ev−r) ln(1 + ev) � 3 2 e−(1−2α)re−(1−α)r′� + 4¯n eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) (ln(1 + ev))− 3 2 � ∞ r+δ1 dr′ ln(1 + ev−r−r′) ev−r−r′ + 2e−r′ 1 + ev−r−r′ + e−r′ 1 − e−(1−α)(r+r′) 1 − e−r−r′ e−r′e−(1−α)r + � ∞ r+δ1 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)��� (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='19) Now we look at the second term in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We combine this with a term from I(2)[Γ2 t](v, r) and B+ 1 [Γ2 t](v, r) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This is the second combination Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' − eµ max(a,c0v) � ∞ r+δ1 dr′ � K1 3(v − r, v − r − r′) − K1 3(v, v − r′) � � f(v − r − r′) − f(v − r) � gt(v, r′) (Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2) + eµa 1(v < −b0)1(a1 − v > r + δ1) � a1−v r+δ1 dr′ K3 1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′)× 66 × � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − gt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + B+ 1 [Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) ≥ − 1(v − r ≥ m0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � ∞ r+δ1 dr′ � K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � × × � f(v − r − r′) − f(v − r) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + E1[Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + E2[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where E1[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v < −b0)eµa � 1(−v − b0 > r + δ1) � � a1−v −v−b0 dr′ K3 1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′) � 2 − (1 + e−κr)γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r)� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � −v−b0 r+δ1 dr′ K3 1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′) � 1 − e− 124 125 (1−α)r′� � 1 − e−κr�2γ1 + f(v) � ∞ −v−b0 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � � 1 − e−( 122 250 −2α)r′� � 1 − e−(1−2α)r′� + e− 116 750 r′e−(1−2α)r′(1 − e−αr′)1 − e− 634 750 r′ 1 − e−r′ � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � + 1(−v − b0 ≤ r + δ1)eµa � 1(r + δ1 ≤ a1 − v) � a1−v r+δ1 dr′ K3 1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′) × × � 2 − (1 + e−κr)γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r)� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + f(v) � ∞ r+δ1 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � 1 − e−( 122 250 −2α)r′� � 1 − e−(1−2α)r′� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � + eµaf(v) � ∞ max(−v−b0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r+δ1) dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)e−r′e−( 122 250 −α)r 1 − e−(1−α)r′ 1 − e−r′ gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='20) and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' E2[Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v ≥ −b0) � 1(−b0 ≤ v < 0)eµaf(v) � ∞ r+δ1 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � � 1 − e−2( 1 3 −α)r′� � 1 − e−(1−2α)r′� + e−(1−α)r′ � 1 − e−( 2 3−3α)r′� + e−(r′−r)e−( 2 3−α)r 1 − e−(1−α)r′ 1 − e−r′ � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v ≥ 0)1(v − r < m0)f(v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) �ln(1 + ev−r) ln(1 + ev) � 3 2 � ∞ r+δ1 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)× × e−r′e−(1−α)r 1 − e−(1−α)(r+r′) 1 − e−(r+r′) + 1(v − r ≤ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0v)) � ∞ r+δ1 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � � 1 − e−(1−2α)r′� � 1 − e−(1−α)r′e−(1−2α)r� 67 + e−(1−α)r′ � 1 − e−(1−2α)(r+r′)� � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) + 1(v − r > max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0v)) � eµ(v−r) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)� � ∞ r+δ1 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � 1 − e−(1−2α)re−(1−α)r′� + 1(v − 2r − δ1 < max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0v) < v − r) � ∞ r+δ1 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)e−αr′ � 1 − e−αr +e−αr(1 − e−(1−3α)(r+r′)) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) + 1(v − 2r − δ1 ≥ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' c0v)) � ∞ r+δ1 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � � 1 − e−(1−2α)r′� � 1 − e−(1−α)r′e−(1−2α)r� + e−(1−α)r′ � 1 − e−(1−2α)(r+r′)� � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) �� (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='21) Clearly, what the combinations Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 and Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 do to the two negative terms of I, is push them away from the point singularities into regions where v − r > 0 and v > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We now combine a term from E1[Γ2 t ](v, r) with A−1 1 [Γ1 t](v, r) as follows, in Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3: A− 1 [Γ1 t](v, r) (Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3) + 1(v < −b0)1(−v − b0 > r + δ1)eµa � −v−b0 r+δ1 dr′ K3 1(v, v + r′)f(v + r′) � 1 − e− 124 125 (1−α)r′� � 1 − e−κr�2γ1 ≥ 1(r + δ1 < −v − b0) � 1(v − r < −m0)4¯n (ln(1 + ev))− 3 2 � −v−b0 r+δ1 dr′ ev−r′(ln(1 + ev+r′))1−α (1 + ev + e−r′)(1 + ev−r−r′ + e−r′) × × (1 − e− 1 2r′)(1 − e−(r′+r)) � 1 − e− 124 125 (1−α)r′� (1 − e−κr)2γ1 + 1(v − r ≥ −m0) � −v−b0 r+δ1 dr′ K3 1(v, v + r′)f(v + r′) � 1 − e− 124 125 (1−α)r′� (1 − e−κr)2γ1 � eµa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us define: E′ 1[Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = E1[Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − 1(−v − b0 > r + δ1)eµa � −v−b0 r+δ1 dr′ K3 1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′) � 1 − e− 124 125 (1−α)r′� � 1 − e−κr�2γ1 + 1(−v − b0 > r + δ1) � 1(v − r < −m0)4¯n (ln(1 + ev))− 3 2 � −v−b0 r+δ1 dr′ ev−r′(ln(1 + ev+r′))1−α (1 + ev + e−r′)(1 + ev−r−r′ + e−r′) × × (1 − e− 1 2r′)(1 − e−(r′+r)) � 1 − e− 124 125 (1−α)r′� (1 − e−κr)2γ1 + 1(v − r ≥ −m0) � −v−b0 r+δ1 dr′ K3 1(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′) � 1 − e− 124 125 (1−α)r′� (1 − e−κr)2γ1 � eµa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='22) 68 This means we can put together the combinations (Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1), (Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2) and (Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' and obtain the following lower bound: I[Γ1 t + Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I(1) 3 [Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + A− 1 [Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) (LB1) ≥ B− 1 [Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + B2[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I(1)[Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + B3[Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + E′ 1[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + E2[Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I(3)[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where B− 1 [Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = − 1(v − r ≥ m0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)� � ∞ 2r+δ1 dr′ K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � f(v − r − r′) − f(v − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + � ∞ r+δ1 dr′ � K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � � f(v − r − r′) − f(v − r) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � − 1(v ≥ 0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � 2r+δ1 r+δ1 dr′ K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � f(v − r − r′) − f(v − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � ∞ r+δ1 dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � f(v − r − r′) − f(v − r) � (gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='23) B2[Γ2 t](v, r) = I(1) 3 [Γ2 t](v, r) − 1(v − r ≤ −m0)eµ max(a,c0v)f(v) � ∞ ˜r dr′ K2 3(v − r, v − r − r′)gt(v, r) − 1(−m0 < v − r < m0)eµ max(a,v−r,c0v)f(v) � ∞ r+δ1 dr′ K2 3(v − r, v − r − r′)gt(v, r), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='24) and, B3[Γ2 t ](v, r) = I(2)[Γ2 t](v, r) − 1(v < −b0)1(r + δ1 < a1 − v) � � ∞ r+δ1 dr′ K3 2(v − r, v − r + r′) � f(v − r + r′) − f(v + r′) � × × eµ max(a,c0(v−r+r′))gt(v − r + r′, r′) + eµa � a1−v r+δ1 dr′ K3 1(v, v + r′)f(v + r′) � gt(v, r) + gt(v − r + r′, r′) − gt(v + r′, r + r′) � � (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='25) We will now turn to J1[Γ1 t](v, r) and J2[Γ2 t](v, r), defined in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It is not difficult to obtain the following lower bound on the sum of these terms: J1[]Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + J2[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) ≥ 1(v ≤ −b0) 4¯n � − eµaf(v − r) � r+a1−v r+δ1 dr′ (ln(1 + ev))− 3 2 ev−r′(1 − e−r) ln(1 + ev+r′) (1 + ev + e−r′)(1 + ev−r + e−r′)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − f(v − r) � ∞ r+a1−v dr′ (ln(1 + ev))− 3 2 eµc0(v−r+r′) ev− 3 2r′(1 − e−r) ln(1 + ev+r′) (1 + ev + e−r′)(1 + ev−r + e−r′)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − 1(r ≤ a1 − v ≤ r + δ1) � ∞ r+δ1 dr′ (ln(1 + ev))− 3 2 ev− 3 2 r′(1 − e−r) ln(1 + ev+r′) (1 + ev + e−r′)(1 + ev−r + e−r′)× 69 × � f(v − r + r′) − f(v + r′) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′))gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − � ∞ r+δ1 dr′ (ln(1 + ev))− 3 2 ev− 3 2 r′(1 − e−r) ln(1 + ev+r′) (1 + ev + e−r′)(1 + ev−r + e−r′)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′))f(v + r′)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − 1(r + δ1 ≤ a1 − v) � a1−v r+δ1 dr′ (ln(1 + ev))− 3 2 ev−r′(1 − e− 1 2r′)(1 − e−r) ln(1 + ev+r′) (1 + ev + e−r′)(1 + ev−r + e−r′) × × eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′))f(v + r′)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + 2eµa � r+a1−v r+δ1 dr′ � ln(1 + ev−r) �− 3 2 e−2r′ ln(1 + ev−r+r′) (1 − e−r′)(1 + ev + e−r′) � 1 − ln(1 + ev−r−r′) ln(1 + ev−r+r′) � × × � 1 − �ln(1 + ev−r) ln(1 + ev) � 3 2 ln(1 + ev+r′) ln(1 + ev−r+r′) � f(v − r)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + 2 � ∞ r+a1−v dr′ � ln(1 + ev−r) �− 3 2 e− 5 2r′ ln(1 + ev−r+r′) (1 − e−r′)(1 + ev + e−r′) � 1 − e 1 2 r′ ln(1 + ev−r−r′) ln(1 + ev−r+r′) � × × � 1 − �ln(1 + ev−r) ln(1 + ev) � 3 2 ln(1 + ev+r′) ln(1 + ev−r+r′) � eµc0(v−r+r′)f(v − r)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + 1(r > a1 − v) � r+a1−v r+δ1 dr′ K3 1(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)f(v + r′)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′))gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � − 1(v > −b0) � 4¯n (ln(1 + ev))− 3 2 � ∞ r+δ1 dr′ � f(v − r) + f(v + r′) � ev− 3 2r′(1 − e−r) ln(1 + ev+r′) (1 + ev + e−r′)(1 + ev−r + e−r′)× × eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′))gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + 1(v − r ≥ 0) eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � ∞ r+δ1 dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) (f(v − r) + f(v)) × × (1 − e− 1 2 r′)(1 − e− 1 2r)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � We now combine J0[Γ1 t + Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' I(1)[Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) and E′ 1[Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) with the lower bound for J1[Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + J2[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) written above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' to arrive at the following estimate denoted by Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' J1[Γ1 t](v, r) + J2[Γ2 t](v, r) + J0[Γ1 t + Γ2 t](v, r) + I(1)[Γ1 t ](v, r) + E′ 1[Γ2 t](v, r) (Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4) ≥ J 0[Γ1 t + Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + E[Γ1 t + Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 70 where J 0[Γ1 t + Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = − 1(v < −b0)1(r ≤ a1 − v ≤ r + δ1)4¯n (ln(1 + ev))− 3 2 � r−v r+δ1 dr′ ev− 3 2r′(1 − e− 1 2r′)(1 − e−r) (1 + ev + e−r′)(1 + ev−r + e−r′)× × eµa ln(1 + ev+r′) � f(v − r + r′) − f(v + r′) � gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + 1(v < −b0)eµa (f(v − r) + f(v)) � ∞ r+δ1 dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + 1(v ≥ −b0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r) (f(v − r) + f(v)) � 1(v − r < 0) � ∞ r+δ1 dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) × × � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + 1(v − r ≥ 0) � ∞ r+δ1 dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)e− 1 2r′ � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � � + 4¯n (ln(1 + ev))− 3 2 � ∞ r+δ1 dr′ ln(1 + ev+r′) ev + 2 1 + ev + e−r′ e− 5 2 r′ 1 − e−r′ eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v+r′))× × � f(v − r) + f(v + r′) � � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − gt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + 1(v ≥ −b0)4¯n (ln(1 + ev))− 3 2 � ∞ r+δ1 dr′ ln(1 + ev+r′) ev−r + e−r′ 1 + ev−r + e−r′ ev− 3 2r′ 1 + ev + e−r′ × × � f(v − r) + f(v + r′) � � 2 − (1 + e−κr)γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r)� eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v+r′))gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='26) and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' E[Γ1 t + Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v < −b0) � 4¯nf(v − r) (ln(1 + ev))− 3 2 � r+a1−v r+δ1 dr′ ev + 2 1 + ev + e−r′ e−2r′ 1 − e−r′ ln(1 + ev+r′) × × � eµa − e− 1 2r′eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v+r′))� � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − gt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + eµaf(v) � ∞ max(−v−b0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r+δ1) dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)e−r′e−( 122 250 −α)r � 1 − e−(1−α)r′ 1 − e−r′ � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(r + δ1 < −v − b0) � � ∞ r+δ1 dr′ K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) � f(v − r + r′) − f(v + r′) � × × � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′)) − eµa� gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + eµaf(v) � ∞ −v−b0 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � � 1 − e−( 122 250 −2α)r′� � 1 − e−(1−2α)r′� + e− 116 750 r′e−(1−2α)r′(1 − e−αr′) � 1 − e− 634 750 r′ 1 − e−r′ � � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � + 1(r + δ1 > −v − b0)eµaf(v) � ∞ max(−v−b0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r) dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � 1 − e−( 122 250 −2α)r′� × × � 1 − e−(1−2α)r′� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) 71 + 1(r > a1 − v) � r+a1−v r+δ1 dr′ K3 1(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)f(v + r′)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′))gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + 1(r + δ1 > a1 − v)f(v − r) � r+a1−v r+δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � eµa − eµc0ve−( 1 2 −µc0)r′� gt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) + 1(a1 − v ≥ r + δ1)f(v − r) � r+a1−v a1−v dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � eµa − eµc0ve−( 1 2 −µc0)r′� gt(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='27) This leads us to consolidate (Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4) and (LB1) into the second lower bound (LB2) I1[Γ1 t + Γ2 t ](v, r) + I(1) 3 [Γ2 t ](v, r) + A− 1 [Γ1 t ](v, r) (LB2) ≥ B− 1 [Γ1 t ](v, r) + B2[Γ2 t ](v, r) + B3[Γ2 t](v, r) + E2[Γ2 t](v, r) + I(3)[Γ2 t ](v, r) + J 0[Γ1 t + Γ2 t ](v, r) + E[Γ1 t + Γ2 t](v, r), Recall now the lower bound on I2[Γ1 t](v, r) given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='34) and the following definition (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='35)): C3[Γ1 t](v, r) = 1(v − r < −b0)eµ max{a,c0v}gt(v, r) � r δ1 dr′ � K3(v − r, v − r + r′) � f(v − r) − (ln(1 + ev−r+r′))−α� − K1 3(v − r, v − r − r′) � f(v − r − r′) − f(v − r) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' This term now has to be used to offset the negative term A− 2 [Γ1 t](v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' By virtue of Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1, this combination yields the following inequality: C3[Γ1 t](v, r) + A− 2 [Γ1 t](v, r) (Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='5) ≥ 1(−m0 < v − r < −b0)C3[Γ1 t ](v, r) + 1(v − r ≤ −m0)eµ max(a,c0v)4¯n � ln(1 + ev−r) �− 3 2 � min(r,˜r) δ1 dr′ ev−r−r′ + 2e−2r′ 1 + ev−r + e−r′ � ln(1 + ev−r−r′) �1−α × × � ln(1 + ev−r+r′) ln(1 + ev−r−r′) �1−2α �� 1 − e− 248 125 (1−2α)r′ 1 − e−r′ � − er′e− 248 125 (1−2α)r′ � � 1 − � ln(1 + ev−r−r′) ln(1 + ev−r) �α� gt(v, r) ≥ 1(−m0 < v − r < −b0)C3[Γ1 t ](v, r) + 1(v − r ≤ −m0)¯n � ln(1 + ev−r) �− 1 2 Γ1 t (v, r)b1(α) � 1 − e−3αr�3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us define ˜B[Γ1 t](v, r) = 1(v − r ≤ −m0)eµ max(a,c0v)f(v − r)b1(α)¯n � ln(1 + ev−r) �− 1 2 � 1 − e−3αr�3 gt(v, r), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='28) where b1 is positive and bounded away from 0, for all α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that ˜B has the requisite singular behavior like (ln(1 + ev−r))−1/2 for r ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then we combine the rest of I2[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) with E[Γ1 t + Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) and J 0[Γ1 t + Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' and obtain the following lower bound: I1[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I2[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I(1) 3 [Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + A− 1 [Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + A− 2 [Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) (LB3) ≥ B− 1 [Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(−m0 < v − r < −b0)C3[Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + ˜B[Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + B2[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + Bf 3 [Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) 72 + E2[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I(3)[Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + J f 0[Γ1 t + Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + E f[Γ1 t + Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where B2[Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = B2[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v < −b0)eµa � r δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)(f(v) − f(v + r′))gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v ≥ −b0) � r δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) − e− 1 2 r′eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v+r′))� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='29) J f 0[Γ1 t + Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v < −b0)eµa (f(v − r) + f(v)) � ∞ r+δ1 dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + 1(v ≥ −b0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r) (f(v − r) + f(v)) � 1(v − r < 0) � ∞ r+δ1 dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) × × � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + 1(v − r ≥ 0) � ∞ r+δ1 dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)e− 1 2r′ � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � � + 4¯n (ln(1 + ev))− 3 2 � ∞ 2r dr′ ln(1 + ev+r′) e− 5 2 r′ 1 − e−r′ × × eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)f(v + r′) � 2 − (1 + e−κr)γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r)� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v ≥ −b0)4¯n (ln(1 + ev))− 3 2 � ∞ 2r dr′ ln(1 + ev+r′) e− 5 2r′ 1 − e−r′ × × eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)f(v − r) � 2 − (1 + e−κr)γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r)� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='30) E f[Γ1 t + Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v < −b0) � eµaf(v) � ∞ max(−v−b0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r+δ1) dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)e−r′e−( 122 250 −α)r � 1 − e−(1−α)r′ 1 − e−r′ � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(r + δ1 < −v − b0) � � ∞ r+δ1 dr′ K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) � f(v − r + r′) − f(v + r′) � × × � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′)) − eµa� gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + eµaf(v) � ∞ −v−b0 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � � 1 − e−( 122 250 −2α)r′� � 1 − e−(1−2α)r′� + e− 116 750 r′e−(1−2α)r′(1 − e−αr′) � 1 − e− 634 750 r′ 1 − e−r′ � � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � + 1(r + δ1 ≥ −v − b0)eµaf(v) � ∞ max(−v−b0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r) dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � 1 − e−( 122 250 −2α)r′� × 73 × � 1 − e−(1−2α)r′� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + 1(r > a1 − v) � r+a1−v r+δ1 dr′ K3 1(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)f(v + r′)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v−r+r′))gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='31) and Bf 3 [Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v < −b0) � � ∞ max(r+δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='a1−v) dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′) (eµa − eµc0v) gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � ∞ r+δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � f(v) − f(v + r′) � eµagt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(r ≤ a1 − v ≤ r + δ1) � � r+a1−v r+δ1 dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′) � eµa − eµc0(v−r+r′)� gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + � r−v r+δ1 dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)e− 1 2r′ � f(v − r + r′) − f(v + r′) � eµagt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + � ∞ r+δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′)eµc0v � 1 − 2e−( 1 2−µc0)r′ + e−2( 1 2−µc0)r′� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � r+a−v r+δ1 dr′ K3 1(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)f(v − r + r′)eµagt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � + 1(a1 − v > r + δ1) � � a1−v r+δ1 dr′ K3 1(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) � f(v − r + r′) − f(v + r′) � eµagt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + � r+a1−v a1−v dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v + r′) � eµa − eµc0ve−( 1 2−µc0)r′� gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + 8¯n eµc0v � r+a1−v a1−v dr′ (ln(1 + ev))− 3 2 � ln(1 + ev+r′) �1−α e−2r′(1 − e− 1 2 r′)(1 − e−( 1 2−µc0)r′) (1 − e−r′)(1 + ev + e−r′) gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 8¯n eµa � r+a1−v a1−v dr′ � ln(1 + ev−r) �− 3 2 � ln(1 + ev−r+r′) �1−α e−2r′(1 − e− 1 2 r′)2 (1 − e−r′)(1 + ev + e−r′)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � + 1(a1 − v < r) � � r+a1−v r+δ1 dr′ K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) � f(v − r + r′) − f(v + r′) � eµagt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + � r+a1−v r+δ1 dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � eµa − eµc0(v−r+r′)� f(v + r′)gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) + � r+a1−v r+δ1 dr′ K3 1(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)eµa � f(v − r + r′) − f(v + r′) � gt(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us now turn to the negative terms in 1(v < −m0)I(2) 3 [Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) and observe that we can use a term from B2[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) to control these negative terms as follows: 1(v ≤ −m0)eµa� − 1(v + r < b0) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � f(v − r′) − f(v) � gt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) 74 − 1(v + r ≥ b0) � −v−b0 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � f(v − r′) − f(v) � gt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � + 1(v < −b0)eµa � r δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)(f(v) − f(v + r′))gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) ≥ 1(−m0 < v < −b0)eµa � r δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)(f(v) − f(v + r′))gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' so that we can write B2[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I(2) 3 [Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) (Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='6) ≥ B + 2 [Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v − r ≥ m0)I(1) 3 [Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v ≥ m0)I(2) 3 [Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' + 1(−m0 < v < m0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)f(v) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where B + 2 [Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v − r ≤ −m0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)f(v) � ˜r 0 dr′ � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + 1(v ≤ −m0)eµaf(v) � 1(v + r < b0) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − gt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � + 1(v + r ≥ −b0)f(v) � � r −v−b0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � −v−b0 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − gt(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � �� + 1(−m0 < v < −b0)eµa � r δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)(f(v) − f(v + r′))gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v ≥ −b0) � r δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)f(v) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) − e− 1 2 r′eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0(v+r′))� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='32) We control the negative term A− 3 [Γ1 t ] by a term from I(1) 3 [Γ2 t ] as follows: 1(v − r ≥ m0)1(v > 3r)f(v) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � ∞ v−r−c0v dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) + A− 3 [Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) > 1(v − r ≥ m0)1(v > 3r)4¯n � ln(1 + ev−r) �− 3 2 � ∞ v−r dr′ ev−r−r′ + 2e−r′ 1 + ev−r−r′ + e−r′ × × � ln(1 + ev−r−r′) �1−α e− 2 3r � 1 − e−(κ− 2 3)r� eµc0vgt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' which allows us to arrive at the following estimate: 1(v − r ≥ m0) � I(1) 3 [Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I(1) 3 [Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � 75 ≥ B4[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v − r ≥ m0)I 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='+ 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where B4[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v − r ≥ m0)1(v > 3r)4¯n � ln(1 + ev−r) �− 3 2 � ∞ v−r dr′ ev−r−r′ + 2e−r′ 1 + ev−r−r′ + e−r′ × × � ln(1 + ev−r−r′) �1−α e− 2 3 r � 1 − e−(κ− 2 3)r� eµc0vgt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) + 1(v − r ≥ m0) � 1(v ≤ 3r) (f(v − r) + f(v)) � ¯n(ln 2)2 � ln(1 + ev−r) �− 3 2 eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � v−r 0 dr′ � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � + 1(v > 3r) � f(v − r) � ∞ v−r−c0v dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v)� gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + (f(v − r) + f(v)) eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � ∞ v−r−c0v dr′ � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='33) and I 1,+ 3 [Γt](v, r) = 1(v − r ≥ m0)Γt(v, r) � 1(v ≤ 3r) 4 3 ¯n (ln(1 + ev)) 1 2 �ln(1 + ev−r) ln(1 + ev) �2 + 1(v > 3r) 2¯n � ln(1 + ev−r) � 1 2 � 1 − � ln(1 + ec0v) ln(1 + ev−r) �2� � (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='34) I + 3 contributes the correct asymptotic behavior for large, positive values of v − r, when v and v − r are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Putting the above estimates together,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' we can write the following lower bound: I1[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I2[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I(1) 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I(2) 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) (LB4) ≥ B− 1 [Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + A− 4 [Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(−m0 < v − r < −b0)C3[Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + ˜B[Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + B + 2 [Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + Bf 3 [Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + E2[Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + J f 0[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + E f[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + B4[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(−m0 < v − r < m0)I(1) 3 [Γ1 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(−m0 < v < m0)I(2) 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='+ 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v ≥ m0)I2 3[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I(3)[Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us reflect for a moment on the asymptotic behavior for large, positive values of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that v ≤ 3r ⇐⇒ v − r ≤ 2 3v and v > 3r ⇐⇒ v − r > 2 3v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The desired behavior mimicking that of the potential Vu(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) 76 comes from I 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='+ 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 1(−m0 < v − r < m0)I(1) 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) and 1(v ≥ m0)I2 3[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' as is evident from the following lower bound: I 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='+ 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(−m0 < v − r < m0)I(1) 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v ≥ m0)I2 3[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) ≥ 1(v ≥ m0) � 1(v − r ≤ 0)¯n(ln(1 + ev)) 1 2 � 1 − � ln 2 ln(1 + ev) �2 � Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(−m0 < v − r < m0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)f(v − r)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � ∞ 0 dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) + 1(0 < v − r < c0v)3 2 ¯n(ln(1 + ev)) 1 2 � 1 − �v − r c0v �2 � Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v − r ≥ m0)Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � 1(v − r ≤ 2 3v) 4 3 ¯n (ln(1 + ev)) 1 2 �ln(1 + ev−r) ln(1 + ev) �2 + 1(v − r > 2 3v) 2¯n � ln(1 + ev−r) � 1 2 � 1 − � ln(1 + ec0v) ln(1 + ev−r) �2� �� (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='35) In the lower bound (LB4) the first two terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=', B− 1 [Γ1 t ] and A− 4 [Γ1 t ], are the only non-positive ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Between these two, A4 has a δ1-smallness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' while the term may have a point singularity, it also contains a factor of δ1, which we can choose to be arbitrarily small), as is evident from (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Our next step is to combine some suitable positive terms with B− 1 , so that we are left with a negative term which has a δ1-smallness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus, at the end of the next step all the negative terms will have this kind of smallness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We combine some terms from B + 2 [Γ2 t ], 1(−m0 < v < m0)I(2) 3 [Γ2 t ] and J f 0[Γt] to control the negative term B− 1 [Γ1 t] as follows: B− 1 [Γ1 t](v, r) + 1(v ≥ −b0)f(v) � r δ1 dr′ K3(v, v + r′) � eµ max(a,c0v,v−r) − e− 1 2 r′eµ max(a,c0(v+r′),v−r)� gt(v, r) (Comb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='7) + eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)f(v) � ∞ r+δ1 dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)e− 1 2r′ � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + 1(0 ≤ v < m0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)f(v) � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) ≥ B− 1 [Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + ˜B+ 1 [Γ2 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where B− 1 [Γ1 t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = − 1(v ≥ 0) � 2r+δ1 2r dr′ K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v) � f(v − r − r′) − f(v − r′) � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' and ˜B+ 1 [Γ2 t](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(0 ≤ v < m0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)f(v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) 1 1 + e− 1 2r � r 0 dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) + 1 2eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)f(v) � ∞ r+δ1 dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)e− 1 2 r′ � gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + 1 2 � 2 − 2γ0 − 1 2 � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)f(v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � ∞ r+δ1 dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)e− 1 2 r′(1 − e−(κ−1)r′) 77 + 1(v ≥ 0)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)f(v)gt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � r δ1 dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � 1 − e−( 1 2−µc0)r′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' At this stage, all the negative terms we are left with, are integrals over intervals of lengths δ1 and (δ1−δ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us denote the sum of these negative terms by S− δ1,δ2[Γt](v, r), as follows: S− δ1,δ2[Γt](v, r) = B− 1 [Γ1 t ](v, r) + I4[Γt](v, r) + A− 4 [Γ1 t](v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Since S− δ1,δ2 has a δ1-smallness, the constant M in the definition of the cut-off functions (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='21)) may be chosen large enough,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' so that the positive terms appearing in the estimate (LB4) can be used to offset this negative term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' leading to the following lower bound: I1[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I2[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I(1) 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I(2) 3 [Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I4[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) ≥ G[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (LB5) where G[Γt](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v − r ≤ −b0)¯b1(α)¯nΓt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � ln(1 + ev−r) �− 1 2 � 1 − e−3αr�3 + ¯n 2 � ln(1 + ev−r) �− 1 2 ln � 1 − e− 7 2(r+δ1)�−1 Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(−m0 < v − r < m0) ¯n 4 � ln(1 + ev−r) �− 1 2 Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(0 < v < m0) ¯n 2 (ln(1 + ev)) 1 2 � 1 − �ln(1 + ev−r) ln(1 + ev) �2� Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(0 < v − r < m0) ¯n 8 � ln(1 + ev−r) � 1 2 Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v ≥ m0)b3(c0)¯n (ln(1 + ev)) 1 2 Γt(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='36) where ¯b1(α) and b3(c0) are positive numbers bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' It is important to note here that, for controlling S− δ1,δ2, we need to put a suitable upper bound on the constant M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us describe briefly how that comes about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The bound comes from the fact that ˜B[Γ1 t](v, r) is used to control A− 4 [Γ1 t ](v, r) as follows: A− 4 [Γ1 t](v, r) ≥ − eµ max(a,c0v)f(v − r)gt(v, r)¯n � ln(1 + ev−r) �− 1 2 1(v − r ≤ −m0) � 3 4 (min(δ1, ˜r))2 + 1(v < −b0)1(r < −b0 − v)2e−rδ1 � , and we choose M such that 1 M2 + 1 M ≤ b1(α) 10 , so that ˜B[Γ1 t ](v, r) (see (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='28)) dominates A− 4 [Γ1 t](v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' E Computations pertaining to the Regularized Problem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 Formulae and Computations Relating to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='12 The the unbounded operator Ku and the L2-bounded part Kb are defined as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' via cut-off parameters b′ 0 and m′ 0: (Kε uψε t )(v) 78 = 1(v < −b′ 0) � � −v−b′ 0 0 dr′ K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)ψε t (v − r′) + � ∞ −v−b′ 0 dr′ K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)e−r′ψε t (v − r′) + � −v−b′ 0 0 dr′ Kε(v+r′) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)ψε t (v + r′) + � ∞ −v−b′ 0 dr′ Kε(v+r′) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)e−r′ψε t (v + r′) � + 1(v ≥ −b′ 0) � 1(−b′ 0 ≤ v ≤ m′ 0) � ∞ 0 dr′ K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)e−r′ψε t (v − r′) + 1(v > m′ 0) � v 0 dr′ K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)ψε t (v − r′) + 1(v ≤ m′ 0) � ∞ 0 dr′ Kε(v+r′) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)e−r′ψε t (v + r′) + 1(v > m′ 0) � ∞ 0 dr′ Kε(v+r′) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)ψε t (v + r′) � + 1(v < −m′ 0) � r0 0 dr′ K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)ψε t (v − r′) + 1(v > m′ 0) � v−a1 0 dr′ K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)ψε t (v − r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' and (Kε b ψε t )(v) =1(v < −b′ 0) � � ∞ −v−b′ 0 dr′ K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)(1 − e−r′)ψε t (v − r′) + � ∞ −v−b′ 0 dr′ Kε(v+r′) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)(1 − e−r′)ψε t (v + r′) � + 1(v ≥ −b′ 0) � 1(v ≤ m′ 0) � ∞ 0 dr′ K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)(1 − e−r′)ψε t (v − r′) + 1(v > m′ 0) � ∞ v dr′ K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)ψε t (v − r′) + 1(−b′ 0 ≤ v ≤ m0) � ∞ 0 dr′ Kε(v+r′) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)(1 − e−r′)ψε t (v + r′) � + 1(v < −m′ 0) � ∞ r0 dr′ K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)ψε t (v − r′) + 1(−m′ 0 ≤ v ≤ m′ 0) � ∞ 0 dr′ K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)ψε t (v − r′) + 1(v > m′ 0) � ∞ v−a1 dr′ K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)ψε t (v − r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' In the formulae above r0 = −v − b′ 0 plays the same role as ˜r in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' m′ 0 > max(b′ 0, 2a1) has to be chosen large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The computations for Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='11 follow the same scheme as employed in Appendix D and are quite straightforward, so we skip them and write the resulting estimate, for some positive constants p2 and p3 bounded away from zero: V ε(v)˜Γ(v) − (Kε u˜Γ)(v) ≥ ˜Γ(v) � 1(v ≤ 0)p2(α) (ln(1 + ev))− 1 2 + 1(v > 0)p3 (ln(1 + ev)) 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 Formulae and Computations Relating to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4 The computations for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='4 are almost the same as those described in Appendix D (which lead to a similar result for the ∆-variable, namely Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='3), so there is nothing to be gained by repeating those arguments and estimates here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will only write down the explicit forms of the operators, since the limits 79 of the integrals are slightly different now (owing to the difference between the cut-off functions δ and ε in these two cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Recall that the solution we are seeking proves the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The operator ˜Lε s has already been defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We now write down the expressions for the other operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ( ˜Lε uDψε t )(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v < a1) � � v−r −∞ dw K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � v−r −∞ dw � K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) � + 1(v ≥ a1) � � v−r −∞ dw K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � v−r −∞ dw � K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w)1(v − w > ε(v − r)) � + � ∞ v dw Kε(w) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � 1(v < −b0) � � a1 v dw � Kε(w) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − Kε(w) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � Dψε t (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) + � ∞ a1 dw � K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(w)(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(w)(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � Dψε t (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) � + 1(v ≥ −b0) � ∞ v dw � K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(w)(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(w)(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � Dψε t (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) � + � v v−r dw Kε(w) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � 1(v − r < −b0) � 1(v > 0) � 0 v−r dw Kε(w) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) + 1(v ≤ 0) � v v−r dw Kε(w) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) � + 1(v − r ≥ −b0) � v v−r dwK3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(w)(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) � + � v v−r dw K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − 1(v < −m0) � v v−r dw 1(v − w < min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' −v − b0))K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dψε t (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) − 1(v ≥ −m0) � v v−r dw K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)e−(v−w)Dψε t (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) + � v−r −∞ dw K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − 1(v − r ≤ −m0) � v−r−ε(v−r) v−r−max(˜r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dw � K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) − 1(v − r ≥ m0) � 1(v ≤ 3r) � v−r−ε(v−r) 0 dw � K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) + 1(v > 3r) � c0v −∞ dw � K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) � + � v v−r dw K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − 1(v ≤ −m0) � 1(v + r < −b0) � v v−r dw K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dψε t (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) 80 + 1(v + r ≥ −b0) � v 2v+b0 dw K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dψε t (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) � − 1(v ≥ m0) � 1(0 < v − r < c0v) � c−1 0 (v−r) v−r dw K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dψε t (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) + 1(v − r ≥ c0v) � v v−r dw K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)Dψε t (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' which means we can write ( ˜Lε uDψε t )(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = ˜Vε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r)Dψε t (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − ( ˜Kε uDψε t )(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' the definitions of ˜Vε and ˜Kε u being obvious from the formula for ˜Lε u written above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Kε b[ψε t ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = − 1(v < −b0) � ∞ a1 dw � K3 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(w)(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K3 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(w)(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � (ψε t (w) − ψε t (v − r)) − 1(v ≥ −b0) � ∞ v dw � K3 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(w)(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K3 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(w)(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � (ψε t (w) − ψε t (v − r)) − 1(v − r < −b0)1(v > 0) � v 0 dw Kε(w) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψε t (v) − ψε t (w)) − 1(v − r ≥ −b0) � v v−r dw K3 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(w)(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψε t (v) − ψε t (w)) − 1(v < −m0) � v−min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='−v−b0) v−r dw K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψε t (w) − ψε t (v − r)) − 1(v − r ≤ −m0) � v−r−max(˜r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) −∞ dw � K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � (ψε t (v) − ψε t (w)) − 1(−m0 < v − r < m0) � v−r−ε(v−r) −∞ dw � K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � (ψε t (v) − ψε t (w)) − 1(v − r ≥ m0) � 1(v ≤ 3r) � 0 −∞ dw � K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � (ψε t (v) − ψε t (w)) + 1(v > 3r) � v−r−ε(v−r) c0v dw � K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � (ψε t (v) − ψε t (w)) � − 1(v ≤ −m0)1(v + r ≥ −b0) � 2v+b0 v−r dw K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψε t (w) − ψε t (v − r)) − 1(−m0 < v < m0) � v v−r dw K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψε t (w) − ψε t (v − r)) − 1(v ≥ m0) � 1(0 < v − r < c0v) � v c−1 0 (v−r) dw K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψε t (w) − ψε t (v − r)) + 1(v − r ≤ 0) � v v−r dw K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (ψε t (w) − ψε t (v − r)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The results for Dψε are proved via methods which are completely analogous to those employed in section 1 for the variable ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Thus it is critical to establish that ( ˜Lε uΓ)(v, r) has the same asymptotic behavior as 81 ˜Vε(v, r)Γ(v, r), ∀(v, r) ∈ R×R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Like in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='28)), we will now split ˜Lε uΓ into several terms as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We use the symbol Ii (recall Ii from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='2) for the main integral terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Then it is easy to see: ( ˜Lε uΓ ′ ε)(v, r) =I1[Γ](v, r) + I2[Γ ′ ε](v, r) + I3[Γ ′ ε](v, r) + 8 � i=1 ei[Γ ′ ε](v, r) + I ε 4[Γ ′ ε](v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Let us first define I1, I2 and I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 1) I1[Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = ∞ � max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′� K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + 1(v < −b0) � ∞ � max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − 1(r ≤ a1 − v) � a1−v � max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γ ′ ε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) + � ∞ a1−v dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γ ′ ε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − r+a1−v � max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γ ′ ε(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − ∞ � r+a1−v dr′ K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γ ′ ε(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � − 1(r > a1 − v) � ∞ � max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γ ′ ε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − r+a1−v � max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γ ′ ε(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − ∞ � r+a1−v dr′ K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γ ′ ε(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) �� + 1(v ≥ −b0) ∞ � max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ � K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γ ′ ε(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γ ′ ε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 2) I2[Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v − r < −b0) � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r−v) � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) � Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � + r � min(r−v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r) dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � � 82 + 1(v − r ≥ −b0) � r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) � Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − e− 1 2r′Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � − r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � � + 1(v < −b0) � r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Γ ′ ε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � − min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='a1−v) � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � Γ ′ ε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � + r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='a1−v) dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � e− 1 2 r′Γ ′ ε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � � + 1(v ≥ −b0) � r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Γ ′ ε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � − r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � e− 1 2r′Γ ′ ε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � � + 1(v < −m0) r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='−v−b0) dr′K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γ ′ ε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) + 1(v ≥ −m0) r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)(1 − e−r′)Γ ′ ε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' and 3) I3[Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = I (1) 3 [Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + I (2) 3 [Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where a) I (1) 3 [Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v − r ≤ −m0) � 1(v < −b0) � 1(˜r < max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ε(v − r))) ∞ � ε(v−r) dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(˜r ≥ max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ε(v − r))) � 1(r > ε(v − r)) r � ε(v−r) dr′eµa(f(v − r) + f(v)) � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)˜g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � ˜g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � 83 + ˜r � max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′eµa(f(v − r) + f(v)) � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)˜g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � ˜g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + � ∞ ˜r dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) �� + 1(v ≥ −b0) � ∞ � max(˜r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + ˜r � min(˜r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′)(f(v) + f(v − r))˜g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) ��� + 1(−m0 < v − r < m0) � ∞ ε(v−r) dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v − r ≥ m0) � 1(v ≤ 3r) � � ∞ v−r dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � v−r ε(v−r) (f(v) + f(v − r)) � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)˜g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′) � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � ˜g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) �� + 1(v > 3r) � v−r−c0v � ε(v−r) dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + ∞ � v−r−c0v (f(v) + f(v − r)) � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)˜g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′) � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � ˜g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) �� + A − 1 [Γ ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 ε ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + A − 2 [Γ ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 ε ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + A − 3 [Γ ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 ε ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' with A − 1 [Γ ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 ε ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = − 1(v − r ≤ −m0)1(v < −b0)1(˜r > max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ε(v − r)))eµa× × ˜r � max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′� K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � � f(v − r − r′) − f(v − r) � ˜g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 84 A − 2 [Γ ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 ε ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = − 1(v − r ≤ −m0) � 1(v < −b0)1(r > ε(v − r))eµa × × r � ε(v−r) dr′� K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � � f(v − r − r′) − f(v − r) � ˜g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) + 1(v ≥ −b0)1(˜r > ε(v − r)) ˜r � ε(v−r) dr′� K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � × × � f(v − r − r′) − f(v − r) � ˜g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' A − 3 [Γ ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='1 ε ](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = − 1(v − r > m0)1(v > 3r) ∞ � v−r dr′� K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � × × eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′) � f(v − r − r′) − f(v − r) � ˜g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' and b) I (2) 3 [Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − 1(v ≤ −m0) � 1(v + r < −b0) r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γ ′ ε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) + 1(v + r ≥ −b0) −v−b0 � ε(v−r) dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γ ′ ε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � − 1(v ≥ m0) � 1(0 < v − r < c0v) r � max(v−c−1 0 (v−r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γ ′ ε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) + 1(v − r ≥ c0v) r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γ ′ ε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The other terms are “small”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' as seen from the definitions below: 4) I ε 4[Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v ≥ a1)1(r ≤ ε(v − r)) � ε(v−r) � r dr′ K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − ε(v−r) � max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)−r) dr′ K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � 85 + 1(v ≥ −b0) � ∞ 0 dr′ � K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r+r′)(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) − K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v+r′)(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) � Γ ′ ε(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 5) e1[Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) =1(v < a1) max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) � r dr′ � K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 6) e2[Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) =1(v < −b0) max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) � r dr′ � Kε(v+r′) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Kε(v+r′) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γ ′ ε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) +Kε(v+r′) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γ ′ ε(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 7) e3[Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) =1(v ≥ −b0) max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) � r dr′ � Kε(v+r′) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v+r′)(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γ ′ ε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) +K3 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v+r′)(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γ ′ ε(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 8) e4[Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v − r < −b0) min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) � 0 dr′ Kε(v−r+r′) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) � Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � + 1(v − r ≥ −b0) min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) � 0 dr′ Kε(v−r+r′) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) � Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − e− 1 2r′Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � − 1(v < a1) min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) � 0 dr′ K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � − 1(v ≥ a1) min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) � 0 dr′ 1(r + r′ > ε(v − r))K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � + 1(v ≥ a1) min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) � 0 dr′ 1(r + r′ ≤ ε(v − r))K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 9) e5[Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) � 0 dr′ K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Γ ′ ε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � 86 − min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) � 0 dr′ Kε(v+r′) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � Γ ′ ε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � + 1(v ≥ −b0) min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) � 0 dr′ Kε(v+r′) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)(1 − e− 1 2 r′)Γ ′ ε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 10) e6[Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v ≥ −m0) min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) � 0 dr′ K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)(1 − e−r′)Γ ′ ε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 11) e7[Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = � ε(v−r) 0 dr′ Kv−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′ 3 Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' and 12) e8[Γ ′ ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) � 0 dr′K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γ ′ ε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − 1(v ≤ −m0) � 1(v + r < −b0) min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) � 0 dr′K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γ ′ ε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) + 1(v + r ≥ −b0) � ε(v−r) 0 dr′ K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γ ′ ε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � − 1(v ≥ m0) � 1(0 < v − r < c0v) max(ε(v−r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−c−1 0 (v−r))) � v−c−1 0 (v−r) dr′ K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γ ′ ε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) + 1(v − r ≥ c0v) min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) � 0 dr′ K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γ ′ ε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' F Computations Relating to the Evolution of Dt = Dψε t − ∆t In the evolution equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='41) the operator Lε is given by: Lε∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = � v−r−δ1 −∞ dw K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � v−r−δ1 −∞ dw � K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � ∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) + � ∞ v+δ1 dw Kε(w) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � ∞ v+δ1 dw � Kε(w) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − Kε(w) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � ∆t(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) + � v−δ1 v−r dw K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − ∆t(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r)) + � v v−r+δ1 dw Kε(w) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − ∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w)) + � v+δ1 v+δ2 dw Kε(w) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − ∆t(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r)) + � v−δ2 v−δ1 dw K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − ∆t(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r)) 87 + � v v−r dw K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) (∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − ∆t(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r)) + � v−r −∞ dw K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − � v−r −∞ dw � K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) − K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) � ∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w) + � � v−r v−r−δ1 dw K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) − � v−r v−r−δ1 dw K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − w) + � v+δ1 v dw Kε(w) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) − � v+δ2 v dw Kε(w) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v) + � v v−δ2 dw K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v) 3 (v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − w) + � v−r+δ1 v−r dw Kε(w) 3 (v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w)∆t(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' w − v + r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Note that the last six terms in square brackets contribute towards our new “Lδ∆t(v, r)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The other part Lε 0 is defined exactly as above, but with Kε( , ) 3 ( , ) substituted by K3( , ) − Kε( , ) 3 ( , ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' Like before, Lε is separated into three parts as Lε = Lε u + Lε δ + Lε b and then Lε uΓε is written as: Lε uΓε(v, r) =˜I1[Γε](v, r) + ˜I2[Γε](v, r) + ˜I3[Γε](v, r) + 8 � i=1 ˜ei[Γε](v, r) + ˜Iε 4[Γε](v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' We will write the definitions for ˜I1, ˜I2 and ˜I3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' These terms determine the asymptotic behavior of Lε uΓε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' The lower limits of the relevant integrals are different from similar terms seen before, owing to the new interplay between the two kinds of “smallness”-parameters, namely the δ-functions and the ε-functions, so we write below their complete expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ˜I4 and all the ˜ei’s have “smallness” coming from either ε or δ, and we have already seen how such terms are controlled, so we will skip writing down the explicit formulae for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 1) ˜I1[Γε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = ∞ � max(r+δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′� K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + 1(v < −b0) � ∞ � max(r+δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − 1(r + δ1 ≤ a1 − v) � a1−v � max(r+δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) + � ∞ a1−v dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − r+a1−v � max(r+δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γε(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − ∞ � r+a1−v dr′ K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γε(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) � − 1(r + δ1 > a1 − v) � ∞ � max(r+δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) 88 − r+a1−v � max(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γε(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − ∞ � r+a1−v dr′ K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γε(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) �� + 1(v ≥ −b0) ∞ � max(r+δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ � K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + K3 2(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γε(v − r + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r′) − K3 2(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′)Γε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 2) ˜I2[Γε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v − r < −b0) � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r−v) � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='max(δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r))) dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) � Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � + r � min(r−v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r) dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′)Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='max(δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r))) dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � � +1(v − r ≥ −b0) � r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='max(δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r))) dr′ K3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r + r′) � Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − e− 1 2r′Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � − r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='max(δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r))) dr′ K1 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � � +1(v < −b0) � r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='max(δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r))) dr′ K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Γε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � − min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='a1−v) � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='max(δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r))) dr′K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � Γε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � + r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='a1−v) dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � e− 1 2 r′Γε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � � +1(v ≥ −b0) � r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='max(δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r))) dr′ K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Γε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � − r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='max(δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r))) dr′ K3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v + r′) � e− 1 2r′Γε(v + r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) − Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � � 89 + 1(v < −m0) r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='−v−b0) dr′K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) + 1(v ≥ −m0) r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='max(δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r))) dr′K1 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)(1 − e−r′)Γε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' 3) ˜I3[Γε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = ˜I(1) 3 [Γε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + ˜I(2) 3 [Γε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' where a) ˜I(1) 3 [Γε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v − r ≤ −m0) � 1(v < −b0) � ∞ � max(˜r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(r > ε(v − r)) r � max(δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ eµa(f(v − r) + f(v)) � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − (K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′))g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � + 1(˜r > max(r + δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ε(v − r))) ˜r � max(r+δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′eµa(f(v − r) + f(v)) � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − (K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′))g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) �� + 1(v ≥ −b0) � ∞ � ˜r dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(˜r > max(δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ε(v − r))) ˜r � max(δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − (K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′))eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′)(f(v) + f(v − r))g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) ��� + 1(−m0 < v − r < m0) � ∞ ε(v−r) dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + 1(v − r ≥ m0) � 1(v ≤ 3r) � � ∞ v−r dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + � v−r ε(v−r) (f(v) + f(v − r)) � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′)(K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′))g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � � 90 + 1(v > 3r) � v−r−c0v � ε(v−r) dr′ K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + ∞ � v−r−c0v (f(v) + f(v − r)) � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′)eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r)g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′)(K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′))g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � � + ˜A− 1 [Γ 1 ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + ˜A− 2 [Γ 1 ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) + ˜A− 3 [Γ 1 ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' with ˜A− 1 [Γ 1 ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = − 1(v − r ≤ −m0)1(v < −b0)1(˜r > max(r + δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ε(v − r)))eµa× × ˜r � max(r+δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ � K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � � f(v − r − r′) − f(v − r) � g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ˜A− 2 [Γ 1 ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = − 1(v − r ≤ −m0) � 1(v < −b0)1(r > ε(v − r))eµa × × r � max(δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′� K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � � f(v − r − r′) − f(v − r) � g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) + 1(v ≥ −b0)1(˜r > max(δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ε(v − r))) ˜r � min(δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′� K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � × × � f(v − r − r′) − f(v − r) � g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' ˜A− 3 [Γ 1 ε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = − 1(v − r > m0)1(v > 3r) ∞ � v−r dr′� K2 3(v − r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) − K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r − r′) � × × eµ max(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='c0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−r−r′) � f(v − r − r′) − f(v − r) � g(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r + r′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' and b) ˜I(2) 3 [Γε](v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) = 1(v ≤ −m0) � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='−v−b0) � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='max(ε(v−r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='δ1)) dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Γε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � − r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='−v−b0) dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � + 1(−m0 < v < m0) � r min(ε(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='r) dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) 91 + 1(v ≥ m0) � 1(0 < v − r < c0v) � r � max(v−c−1 0 (v−r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='ε(v−r)) dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′) � Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) − Γε(v − r′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r − r′) � + v−c−1 0 (v−r) � min(ε(v−r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='v−c−1 0 (v−r)) dr′ K2 3(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' v − r′)Γε(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content=' r) � + 1(v − r ≥ c0v) r � min(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE2T4oBgHgl3EQfEQZV/content/2301.03633v1.pdf'} +page_content='max(δ1,' metadata={'source': 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b/Z9FJT4oBgHgl3EQf7y3s/content/tmp_files/2301.11680v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..54b81035aa515258e6e9a4480f4e08bb46f0af55 --- /dev/null +++ b/Z9FJT4oBgHgl3EQf7y3s/content/tmp_files/2301.11680v1.pdf.txt @@ -0,0 +1,1407 @@ +arXiv:2301.11680v1 [cs.IT] 27 Jan 2023 +Codes for Correcting Asymmetric Adjacent +Transpositions and Deletions +Shuche Wang∗, Van Khu Vu§, and Vincent Y. F. Tan†‡∗ +∗ Institute of Operations Research and Analytics, National University of Singapore, Singapore +† Department of Mathematics, National University of Singapore, Singapore +‡ Department of Electrical and Computer Engineering, National University of Singapore, Singapore +§ Department of Industrial Systems Engineering and Management, National University of Singapore, Singapore +Emails: shuche.wang@u.nus.edu, isevvk@nus.edu.sg, vtan@nus.edu.sg +Abstract +Owing to the vast applications in DNA-based data storage, Gabrys, Yaakobi, and Milenkovic recently proposed to study codes +in the Damerau–Levenshtein metric, where both deletion and adjacent transposition errors occur. In particular, they designed a +code correcting a single deletion and s adjacent transpositions with at most (1 + 2s) log n bits of redundancy. In this work, we +consider a new setting where both asymmetric adjacent transpositions (also known as right-shifts or left-shifts) and deletions occur. +We present several constructions of the codes correcting these errors in various cases. In particular, we design a code correcting +a single deletion, s+ right-shift, and s− left-shift errors with at most (1 + s) log(n + s + 1) + 1 bits of redundancy where +s = s+ + s−. In addition, we investigate codes correcting t 0-deletions and s adjacent transpositions with both unique decoding +and list-decoding algorithms. Our main contribution here is a construction of a list-decodable code with list-size O(nmin{s+1,t}) +and has at most (max{t, s + 1}) log n + O(1) bits of redundancy. Finally, we provide both non-systematic and systematic codes +for correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions. +I. INTRODUCTION +The Levenshtein (edit) distance of two different strings is the smallest number of operations (including deletions, insertions, +and substitutions) required to transform one string into the other. This metric has a long history and has attracted a lot of research +in computer science in the past as well as recently [2]–[4]. Codes in the Levenshtein metric have been investigated extensively +recently due to theoretical interests and their numerous applications, including racetrack memory [5]–[7] and DNA-based data +storage [8]–[10]. +This paper was presented in part at the 2022 IEEE Information Theory Workshop (ITW) [1]. + +1 +In some channels, such as DNA-based data storage ones, we observe that, besides deletions, insertions, and substitutions, +there are also adjacent transpositions. Hence, there exists some recent work concerning the Damerau–Levenshtein distance +which is motivated by applications to DNA-based data storage. The distance is a generalization of the well-known Levenshtein +distance taking into account adjacent transpositions. More precisely, the Damerau–Levenshtein metric is the smallest number +of operations (including deletions, insertions, substitutions, and adjacent transpositions) required to transform one string into +another. We note that it is possible to compute the exact Damerau–Levenshtein distance of two strings in polynomial time [11] +but it is not known if we can compute the distance in linear time. Recently, Gabrys, Yaaboki, and Milenkovic [12] proposed +to study codes in the Damerau–Levenshtein distance. They provided several constructions of codes correcting both deletions +and adjacent transpositions. However, these codes are not optimal in general. For example, to correct a single deletion and +at most s adjacent transpositions, the authors require (1 + 2s) log n bits of redundancy. Designing an optimal code correcting +both deletions and multiple adjacent transpositions has turned out to be a formidable challenge for coding theorists in recent +times. +The problem of constructing codes for correcting synchronization errors, including deletions and insertions, was first +investigated by Levenshtein [13] and Ullman [14], [15]. Sticky deletions/insertions and duplication deletions can be considered +as asymmetric deletions/insertions via the Gray mapping [16]. Owing to various applications, such as in flash memories [17], +[18], racetrack memories [6], and DNA data storage systems [19], [20], codes for correcting asymmetric deletions/insertions +have garnered significant attention recently. Tallini et al. [16], [21]–[24] provided a series of theories and code designs for +correcting these kinds of errors. Especially, Mahdavifar and Vardy [18] provided some efficient encoding/decoding algorithms +for an optimal code correcting sticky-insertion and thus for an optimal code correcting 0-deletion. +Codes correcting adjacent transposition errors have been investigated for a long time as codes for shift errors [25]–[27]. +Codes correcting asymmetric shift errors have also been studied recently [28]. In this work, we are interested in codes correcting +a combination of both asymmetric adjacent transposition errors and deletion errors. We aim to obtain some optimal codes with +simple efficient encoding/decoding algorithms. +We note that codes correcting substitutions, deletions, and their combinations have attracted a lot of research recently [29], +[30]. However, there are only a few code constructions that correct a combination of adjacent transposition and other kinds of +errors. Klove [31] proposed a class of perfect constant-weight codes capable of correcting a single deletion, a single insertion or +an adjacent transposition. Gabrys, Yaakobi, and Milenkovic [12] presented several codes correcting a combination of deletions +and adjacent transpositions. If there is a single adjacent transposition or a single deletion, there exist codes correcting the error +with at most log n + O(log log n) bits of redundancy [32]. The best-known codes correcting a single deletion and at most s + +2 +adjacent transpositions require (1 + 2s) log n bits of redundancy [12]. In this work, we design several new families of codes +in numerous cases. We provide our main contributions as follows. +Our first contribution in this work is Construction 1, which presents a construction of an optimal code correcting a single +adjacent transposition or a single 0-deletion. Analyzing the size of our code, we obtain the following result. +Theorem 1. There is a code correcting a single 0-deletion or a single adjacent transposition with at most log n + 2 bits of +redundancy. +Next, we construct a code correcting t 0-deletions and s adjacent transpositions with at most (t + 2s) log n + o((t + +2s) log n) bits of redundancy. The constructed code is the best known that corrects multiple 0-deletions and multiple adjacent +transpositions. See Theorem 7 for the detail. +Theorem2. There is a code correcting t 0-deletions and s adjacent transpositions with at most (t+2s) log n+o((t+2s) log n) +bits of redundancy. +Further, we construct an optimal code for correcting a single deletion, s+ right-shift and s− left-shift errors. Throughout +this paper, we denote the adjacent transposition as 01 → 10 or 10 → 01, right-shift of 0 as 01 → 10 and left-shift of 0 as +10 → 01. See Construction 2 and Theorem 8 for the detail. +Theorem 3. There is a code correcting a single deletion, s+ right-shift and s− left-shift errors with at most (1 + s) log(n + +s + 1) + 1 bits of redundancy where s = s+ + s−. +Compare the results in [12], where the code for correcting a single deletion and s adjacent transpositions needs at most +(1 + 2s) log(n + 2s + 1) redundancy. If we know the direction of these s adjacent transpositions containing s+ right-shifts +of 0 and s− left-shifts of 0, the redundancy of the code can be further reduced to at most (1 + s) log(n + s + 1) + 1 where +s = s+ + s−. +We also investigate list-decodable codes of small list-size and construct a list-decodable code for at most t 0-deletions and +s adjacent transpositions. See the proof of Theorem 9 for the construction. Our results are the first known list-decodable codes +for the asymmetric Damerau–Levenshtein distance. +Theorem 4. There is a list-decodable code that can correct t 0-deletions and s adjacent transpositions with list size +O(nmin(t,s+1)) and has max(t, s + 1) log n + O(1) bits of redundancy. +Finally, we construct both non-systematic and systematic codes for correcting t blocks of 0-deletions with ℓ-limited-magnitude +and s adjacent transpositions. See the proof of Theorem 10 for the construction. + +3 +Theorem5. There is a code capable of correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions +with at most ⌈2(t + 2s)(1 − 1/p)⌉ log(n + 1) + O(1) bits of redundancy, where p is the smallest prime larger than tℓ + 2. +The rest of this paper is organized as follows. Section II provides the notation and preliminaries. Section III presents three +uniquely-decodable codes for correcting asymmetric deletions and adjacent transpositions. Section IV proposes list-decodable +codes for correcting asymmetric deletions and adjacent transpositions with low redundancy. In Section V, we construct codes +both non-systematic and systematic codes are capable of correcting t blocks of 0-deletions with ℓ-limited-magnitude and s +adjacent transpositions. Finally, Section VI concludes this paper. +II. NOTATION AND PRELIMINARIES +We now describe the notations used throughout this paper. Σq denotes the finite alphabet of size q and Σn +q represents the set +of all sequences of length n over Σq. Without loss of generality, we assume Σq = {0, 1, . . ., q − 1}. For two integers i < j, +let [i, j] denote the set {i, i + 1, i + 2, . . . , j}. The size of a binary code C ⊆ Σn +2 is denoted |C| and its redundancy is defined +as n − log |C|, where all logarithms without a base in this paper are to the base 2. +We write sequences with bold letters, such as x and their elements with plain letters, e.g., x = x1 · · · xn for x ∈ Σn +q . The +length of the sequence x is denoted |x|. The weight wt(x) of a sequence x represents the number of non-zero symbols in it. +A run is a maximal substring consisting of identical symbols and nr(x) denotes the number of runs of the sequence x. For +functions, if the output is a sequence, we also write them with bold letters, such as φ(x). The ith position in φ(x) is denoted +φ(x)i. In addition, for a sequence u ∈ Σn +q , denote (u mod a) = (u1 mod a, u2 mod a, . . . , un mod a), where a < q. +For a binary sequence x ∈ Σn +2, we can uniquely write it as x = 0u110u210u3 . . . 10uw+1, where w = wt(x). +Definition 1. Define function φ +: +Σn +2 +→ +Σw+1 and φ(x) +def= +(u1, u2, u3, . . . , uw+1) +∈ +Σw+1, where x += +0u110u210u3 . . . 10uw+1 with w = wt(x). +Example 1. Suppose x = (0, 1, 1, 1, 0, 1, 0, 1, 0, 0). Then, φ(x) = (1, 0, 0, 1, 1, 2). +Definition 2. Define function ψ : Σn +2 → Σn +2 such that ψ(x) = (x1, x1 + x2, . . . , x1 + x2 + · · · + xn). +Definition 3. The Lee weight of an element xi ∈ Σq is defined by +wL(xi) = + + + + + + + + + +xi, +if 0 ≤ xi ≤ q/2 +q − xi, +otherwise +For a sequence x ∈ Σn +q , the Lee weight of x is +wL(x) = +n +� +i=1 +wL(xi). + +4 +Define the Lee distance of two sequences x, x′ ∈ Σn +q as +dL(x, x′) = wL(x − x′). +Example 2. Suppose x ∈ Σ7 +6 = (1, 4, 0, 5, 2, 3, 4). Then, wL(x) = 1 + 2 + 0 + 1 + 2 + 3 + 2 = 11. +Example 3. Suppose x ∈ Σ7 +6 = (1, 4, 0, 5, 2, 3, 4) and x′ ∈ Σ7 +6 = (0, 3, 0, 5, 3, 3, 3). Then, x − x′ = (1, 1, 0, 0, 5, 0, 1) and +dL(x, x′) = wL(x − x′) = 4. +For any x ∈ Σn +2, denote Bt,s(x) as the error ball of x under t 0-deletions and s adjacent transpositions. The code Ct,s(n) is +a unique-decodable code for correcting t 0-deletions and s adjacent transpositions, for which holds that Bt,s(c1)∩Bt,s(c2) = ∅ +for all c1, c2 ∈ Ct,s(n). The code CList +t,s (n) is a list-decodable code for correcting t 0-deletions and s adjacent transpositions +with list size L such that for any corrupted sequence x′ ∈ Σn−t +2 +there exist at most L codewords in CList +t,s (n) that can be +obtained by t 0-deletions and s adjacent transpositions. +Example 4. Suppose x = (0, 1, 1, 1, 0, 1, 0, 1, 0, 0), the first and last 0 bits are deleted and two pairs of ((4th, 5th) and (7th, +8th)) adjacent bits are transposed in x = (❆0, 1, 1, 1, 0, 1, 0, 1, 0, ❆0). Then, x′ = (1, 1, 0, 1, 1, 1, 0, 0) ∈ B2,2(x). +Proposition 1. Once a 0-deletion occurs in x and we receive x′, there is an index i such that φ(x)i − 1 = φ(x′)i. +Proposition 2. Suppose an adjacent transposition occurs in x at the ith 1, the corresponding changes in φ(x) can be shown +as follows: +1) 10 → 01: (φ(x)′ +i, φ(x)′ +i+1) = (φ(x)i + 1, φ(x)i+1 − 1). +2) 01 → 10: (φ(x)′ +i, φ(x)′ +i+1) = (φ(x)i − 1, φ(x)i+1 + 1). +Example 5. Suppose x = (0, 1, 1, 1, 0, 1, 0, 1, 0, 0), φ(x) = (1, 0, 0, 1, 1, 2) and the adjacent transposition is occurred in +the 4-th bit 1 and the following bit 0 in x. Then, x′ = (0, 1, 1, 1, 0, 0, 1, 1, 0, 0) and φ(x′) = (1, 0, 0, 2, 0, 2), where +(φ(x′)4, φ(x′)5) = (φ(x)4 + 1, φ(x)5 − 1). +The well-known Varshamov–Tenengol’ts (VT) code will be use of in this paper, and we will introduce the following lemma. +For x ∈ Σn +2, we define the syndrome of VT code as VT(x) = �n +i=1 ixi. +Lemma 1 (Varshamov-Tenengol’ts (VT) code [33]). For integers n and a ∈ [0, n], +VTa(n) = {x ∈ Σn +2 : VT(x) ≡ a mod (n + 1)} +is capable of correcting a single deletion. +Define Mt,s(n) as maximal size of binary codes for correcting t deletions and s adjacent transpositions. + +5 +Lemma 2 (cf. Levenstein [2]). For enough large n, Mt,s(n) ≤ (s + t)! 2n +ns+t . +Proof. t deletions and s adjacent transpositions in x can be considered as t deletions and s substitutions in ψ(x). An asymptotic +bound for the size of any codes is capable of correcting up to t deletions, insertions and substitutions have been shown in [2], +which is (t! · 2n)/nt. Since the function ψ is a one-to-one mapping function, an upper bound of binary codes for correcting t +deletions and s adjacent transpositions can be derived. +From Lemma 2, we can obtain a lower bound of the minimal redundancy of the code for correcting t 0-deletions and s +adjacent transpositions. +Corollary1. A lower bound of the minimal redundancy of binary codes for correcting t 0-deletions and s adjacent transpositions +is (t + s) log n − O(1).1 +III. UNIQUELY-DECODABLE CODES FOR ASYMMETRIC DELETIONS AND ADJACENT TRANSPOSITIONS +In this section, we will present three uniquely-decodable codes for correcting asymmetric deletions and adjacent transposi- +tions, that is, once there are some errors, we can correct these errors to recover the original codeword uniquely. +A. Codes for correcting a single 0-deletion or a single adjacent transposition +In this subsection, we present the first construction of an optimal code correcting a single 0-deletion or a single adjacent +transposition. +Construction 1. The code C1(n, a; p) is defined as the set of all x ∈ Σn +2 such that the syndrome +S(x) = +w+1 +� +i=1 +i2φ(x)i ≡ a mod p +where w = wt(x) and p is a prime such that p > 2n. +Theorem 6. The code C1(n, a; p) in Construction 1 can correct a single 0-deletion or a single adjacent transposition. +Proof. Let x = (x1, . . . , xn) ∈ Σn +2 be the original vector and x′ be the received vector after a single 0-deletion or a single +adjacent transposition. +If x′ ∈ Σn−1 +2 +, that is the length of x′ is n−1, then there is a single 0 deletion. In this case, we compute the vector φ(x′) and +a′ < p such that a′ = S(x′) mod p. We note that dL(φ(x), φ(x′)) = 1 and there is an index i such that φ(x)i − 1 = φ(x′)i. +Hence, S(x) − S(x′) = i2. That is, a − a′ = i2 mod p. Since i2 − j2 ̸= 0 mod p for all i ̸= j, i, j < n < p/2, we can +determine the unique index i such that a − a′ = i2 mod p. And thus, we locate the error and can correct it. +1The difference between the lower bound of the redundancy for correcting general t deletions and t 0-deletions is only O(1). [17] + +6 +If x′ ∈ Σn +2, that is the length of x′ is n, then there is no 0 deletion and at most a single adjacent transposition. Similar to +the previous case, we also compute the vector φ(x′) and a′ < p such that a′ = S(x′) mod p. Once an adjacent transposition +occurs, there are two types of errors: a symbol 0 moves to the left and a symbol 0 moves to the right. If a symbol 0 moves +to the left, there exists 0 ≤ j ≤ n − 1 such that a − a′ = 2j + 1 mod p. Otherwise, if a symbol 0 moves to the right, +there is 0 ≤ j ≤ n − 1 such that a − a′ = −2j − 1 mod p. Since p > 2n, for all i, j < n < p/2 and i ̸= j, these four +values, {2i + 1, −2i − 1, 2j + 1, −2j − 1} are distinct. Hence, we can determine the type of error and the unique j such that +a − a′ = 2j + 1 mod p or a − a′ = −2j − 1 mod p. And thus, we can correct the error. +In conclusion, either a 0 deletion occurs or an adjacent transposition occurs, we always can correct the error and recover +the original vector. The theorem is proven. +From the well-known Bertrand–Chebyshev theorem, there exists a prime p such that 2n < p < 4n. Hence, by the pigeonhole +principle, there exists a code C1(n, a; p) of size at least 2n/(4n). That is, it is possible to construct the code C1(n, a; p) at most +log n + 2 redundancy. Therefore, we can conclude that we can correct a single 0-deletion or a single adjacent transposition +with at most log n + 2 redundancy. +B. Codes for correcting t 0-deletions and s adjacent transpositions +In this subsection, we explore the general case in the asymmetric Damerau–Levenshtein distance scheme. We investigate a +code correcting at most t 0-deletions and s adjacent transpositions, given constants t and s. +We observe that the asymmetric Damerau–Levenshtein distance between two vectors x and y is closely related to Lee +distance between φ(x) and φ(y). Indeed, once an adjacent transposition occurs in x, the Lee weight of x is changed by two +based on Proposition 2 and once a 0-deletion occurs in x, the Lee weight of x is changed by one. Hence, if there are at most +s adjacent transpositions and t 0-deletions, the Lee weight of x is changed by at most t + 2s. Now, we present a well-known +BCH code in the Lee distance. +Lemma 3. ( [18], [34]) The systematic BCH code CBCH(n, t + 1; p) : x ∈ Σm +2 → E(x) ∈ Σn +p with the lower bound of +minimum Lee distance +dL(CBCH(n, t + 1; p)) ≥ + + + + + + + + + +2(t + 1), +if t ≤ (p − 3)/2 +p, +if (p − 1)/2 ≤ t ≤ p +can correct errors up to t Lee weight with redundancy t log n + o(t log n), where p is a prime. +Furthermore, Mahdavifar and Vardy [18] used the above code to construct a code C(n, r) of length n correcting r 0 +insertions with at most r log n + o(r log n) bits of redundancy. It is known that for any two words c1, c2 ∈ C(n, r), we + +7 +have dL(φ(c1), φ(c2)) ≥ 2(r + 1) by Lemma 3. Hence, we can use the code C(n, r) to correct t 0-deletions and s adjacent +transpositions. +Theorem 7. The code C(n, r) can correct at most t 0-deletions and s adjacent transpositions, given t + 2s = r. +Proof. Let x = (x1, . . . , xn) ∈ Σn +2 be the original vector and x′ ∈ Σn−t +2 +be the received vector after t 0-deletions and s +adjacent transpositions. Hence, we obtain the vector y′ = φ(x′). We consider two vectors φ(x) and φ(x′). We observe that +once an adjacent transposition occurs in x, the Lee weight of x is changed by at most two based on Proposition 2 and once +a 0-deletion occurs in x, the Lee weight of x is changed by one. Hence, if there are at most s adjacent transpositions and t +0-deletions, the Lee weight of x is changed by at most t + 2s. That is, the Lee distance between two vectors φ(x) and φ(x′) +is dL(φ(x), φ(x′)) ≤ t + 2s. Therefore, we set r = t + 2s and then the code C(n, r) can correct at most t 0-deletions and s +adjacent transpositions with redundancy (t + 2s) log n + o((t + 2s) log n). +C. Codes for correcting a single deletion and multiple right-shifts +In previous two subsections, we focus on the error type of 0-deletions and arbitrary adjacent transposition (both 01 → 10 +and 10 → 01 can occur) in the asymmetric Damerau-Levenshtein distance. In this subsection, we propose an optimal code for +correcting a single deletion and s right-shifts of 0. We denote the adjacent transposition as 01 → 10 or 10 → 01, right-shift +of 0 as 01 → 10 and left-shift of 0 as 10 → 01 throughout this subsection. +Construction 2. The code C(n, a, b) is defined as follows. +C(n, a, b) = {x ∈ Σn +2 : VT(x) ≡ a mod (n + s + 1), +n +� +i=1 +xi ≡ b mod 2, ψ(x) ∈ CH(n, 2s + 1)}, +where CH(n, 2s + 1) is a linear binary code capable of correcting errors with 2s + 1 distance. +Proposition 3. (cf. [12]) A single adjacent transposition (01 → 10 or 10 → 01) in x is equivalent to a single substitution in +ψ(x). +Proposition 4. Suppose there are s right-shifts of 0 occurs in x, we have VT(x) − VT(x′) = s. +Proof. Suppose a right-shift of 0 (01 → 10) occurs at the i-th 1 in x. The index of this 1 in x′ will be i − 1. Thus, for +a single right-shift of 0, the change of the VT syndrome will be 1. If there are s right-shifts of 0 occurs in x, we have +VT(x) − VT(x′) = s. +Lemma 4. The following statements are true: + +8 +• Suppose a 0 is deleted before p-th 1 in x, and insert a 0 before (p + v)-th 1 to get ˆx. x can be obtained from ˆx by v +adjacent transpositions. +• Suppose a 1 is deleted after p-th 0 in x, and insert a 1 after (p − v)-th 0 to get ˆx. x can be obtained from ˆx by v +adjacent transpositions. +Proof. Denote the indexes of p-th 1, (p + 1)-th 1, . . . , (p + v − 1)-th 1 in x as ip, ip+1, . . . , ip+v−1. Then, we can see that +the indexes of these 1s in ˆx should be ip − 1, ip+1 − 1, . . . , ip+v−1 − 1. Since 0 is inserted before (p + v)-th 1, we can swap +the (ip+v−1 − 1)-th and ip+v−1-th bits and hence ˆx[ip+v−1,ip+v] = x[ip+v−1,ip+v]. Continuing this process, we can see that x +can be recovered from ˆx by v adjacent transpositions. The case of deleting 1 is the same deleting 0, hence we can have the +above two statements. +Theorem 8. For all a ∈ [0, n + s] and b ∈ [0, 1], the code C(n, a, b) can correct a single deletion and s right-shifts of 0 with +redundancy at most (1 + s) log(n + s + 1) + 1. +Proof. Denote the retrieved sequence as x′ ∈ Σ2 through a single deletion and at most s right-shifts of 0. We first use the VT +syndrome to correct the deletion and then apply the CH(n, 2s + 1) on ψ(x) to correct the right-shifts of 0. +Further, let ∆ = VT(x) − VT(x′), w be the weight of x′ and p be the index of deletion. Then, let L0 be the number of 0s +on the left of the deleted bits in x′ and R0 on its left. Similarly, denote L1, R1. We have the following cases when recover x +by x′: +• If x′ = Σn +2, it means no deletion occurs in x and there are at most s right-shifts of 0. Based on Proposition 3, there are +at most s substitutions in ψ(x). Hence we can recover ψ(x) by ψ(x′) since ψ(x) ∈ CH(n, 2s + 1), and then recover x. +• If x′ = Σn−1 +2 +and suppose a 0 is deleted. From Proposition 4, then ∆ = R1 + k, where k is the actual number of +right-shifts of 0s. We can first recover ˆx by inserting 0 in the rightmost index of (∆ − s) 1s. Since ∆ = R1 + k and we +insert 0 in the rightmost index of (R1 + k − s) 1s. Based on the Case 1 of Lemma 4, we can have that there are at least +(s − k) adjacent transpositions between ˆx and x. In addition, there are also k right-shifts of 0s occur in x. Therefore, x +can be obtained from ˆx by total s adjacent transpositions. Hence, we can recover ψ(x) by ψ(ˆx) and then x. +• If x′ = Σn−1 +2 +and suppose a 1 is deleted. From Proposition 4, then ∆ = p + R1 + k = w + L0 + k + 1. We recover ˆx +by inserting 1 in the leftmost index of (∆ − w − s − 1) 0s. Similar as Case 2, since ∆ = w + L0 + k + 1 and we insert +1 in the leftmost index of (L0 + k − s) 0s. Based on the Case 2 of Lemma 4, we can have that there are at least (s − k) +adjacent transpositions between ˆx and x. Similarly, x can be obtained from ˆx by total s adjacent transpositions. Hence, +we can recover ψ(x) by ψ(ˆx) and then x. + +9 +It is worth noticing that Case 1 and Case 2, 3 can be distinguished by the length of the retrieved sequence x′. Case 2 and +Case 3 can distinguished based on the constraint of �n +i=1 xi ≡ b mod 2, from where we can know the deleted bit is 0 or 1. +There are three constraints on the sequence x ∈ C(n, a, b) including a VT code, a parity check bit and a linear binary +(n, 2s + 1)-code. It can be easily shown that the redundancy of the code C(n, a, b) is log(n + s + 1) + s log n + 1. Thus, the +redundancy of the code C(n, a, b) is at most (1 + s) log(n + s + 1) + 1. +The decoding algorithm of the code C(n, a, b) for correcting a single deletion and s right-shifts of 0 is summarized in +Algorithm 1. +Algorithm 1: Decoding procedure of C(n, a, b) +Input: Corrupted Sequence x′ +Output: Original Sequence x ∈ C(n, a, b) +∆ = VT(x) − VT(x′), b = �n +i=1 xi − �|x′| +i=1 x′ +i and w = wt(x′). +if |x′| = n then +Recover ψ(x) by ψ(x′) and then x. +else +if b = 0 then +Insert a 0 in the rightmost index of (∆ − s) 1s to get ˆx. Recover ψ(x) by ψ(ˆx) and then x. +else +Insert a 1 in the leftmost index of (∆ − w − s − 1) 0s to get ˆx. Recover ψ(x) by ψ(ˆx) and then x. +end +end +Further, Construction 2 and Theorem 8 can be naturally extended to construct codes for correcting a single deletion, s+ +right-shifts of 0 and s− left-shifts of 0 with s = s+ + s−. +Corollary 2. For all a ∈ [0, n + s] and b ∈ [0, 1], the code C2(n, a, b) such that +C2(n, a, b) = {x ∈ Σn +2 : VT(x) ≡ a mod (n + s + 1), +n +� +i=1 +xi ≡ b mod 2, ψ(x) ∈ CH(n, 2s + 1)}. +can correct a single deletion, s+ right-shifts of 0 and s− left-shifts of 0 with redundancy at most (1 + s) log(n + s + 1) + 1, +where s = s+ + s−. +Proof. Similar as Proposition 4, suppose there are at most s− left-shifts of 0s, the change of VT syndrome is VT(x) − +VT(x′) = −s−. Suppose a 0 is deleted, and the same as the proof of Theorem 8 with the same notations, we can also have +∆ = R1 + k+ − k−, where k+ and k− are actual number of right-shifts and left-shifts of 0 occur. Also, we still insert a 0 in + +10 +the index of rightmost of (∆ − s+ + s−) 1s to obtain ˆx. Based on the Case 1 of Lemma 4, we can have that there are at least +((s+ − s−) − (k+ − k−)) adjacent transpositions between ˆx and x and there are k+ + k− adjacent transpositions occur in x. +Therefore, the total number of adjacent transpositions that x can be obtained from ˆx is at most +(s+ − s−) − (k+ − k−) + (k+ + k−) = s+ − s− + 2k− ≤ s+ + s− = s +Hence, we can recover ψ(x) by ψ(ˆx) since there are at most s substitutions and then x. Also, the analysis of redundancy is +the same as the proof of Theorem 8. +Compare the results in [12], where the code for correcting a single deletion and s adjacent transpositions needs at most +(1 + 2s) log(n + 2s + 1) redundancy. If we know the direction of these s adjacent transpositions containing s+ right-shifts +of 0 and s− left-shifts of 0, the redundancy of the code can be further reduced to at most (1 + s) log(n + s + 1) + 1 where +s = s+ + s−. +IV. LIST-DECODABLE CODES FOR CORRECTING ASYMMETRIC DELETIONS AND ADJACENT TRANSPOSITIONS +In this section, we aim to construct List-Decodable codes with low redundancy. For correcting t 0-deletions without s +adjacent transpositions, Dolecek and Anatharam [17] proposed a well-known construction with optimal redundancy t log n. +Inspired by this, we have the following construction: +Construction 3. The construction CList +t,s (n, K, a; p) is defined as the set of all x ∈ Σn +2 such that +w+1 +� +i=1 +imφ(x)i ≡ am mod p, ∀m ∈ {1, . . . , K}. +where the prime p such that p > 2n and a = (a1, a2, . . . , aK). +Let x = (x1, . . . , xn) ∈ Σn +2 be the original vector and x′ ∈ Σn−t +2 +be the received vector after t 0-deletions and s adjacent +transpositions. Hence, we obtain the vector φ(x′) and the corresponding a′ at the receiver. Let a′ +m = �w+1 +i=1 imφ(x′)i and +a′′ +m = am − a′ +m, ∀m ∈ {1, . . . , K}. +Proposition 5. Suppose there is only a single adjacent transposition occurs in x at the position of j-th 1, the change of +syndrome a′′ +m can be shown as follows: +1) 10 → 01: +a′′ +m = (j + 1)m − jm mod p = +m−1 +� +i=0 +�m +i +� +ji mod p +2) 01 → 10: +a′′ +m = jm − (j + 1)m mod p = − +m−1 +� +i=0 +�m +i +� +ji mod p + +11 +Then, suppose t 0-deletions occur in the 0-run before the (d1, d2, . . . , dt)-th 1, respectively, where d1 ≤ d2 ≤ · · · ≤ dt. +Also, ℓ (10 → 01) adjacent transpositions occur in (j1, j2, . . . , jℓ)-th 1 and r (01 → 10) adjacent transpositions occur in +(k1, k2, . . . , kr)-th 1, respectively. +Based on Proposition 5, considering all t 0-deletions and s adjacent transpositions and set K = t + s, we have a set of +equations showing the change of syndromes for all m ∈ {1, . . ., t + s} as follows: +a′′ +m ≡ +t +� +u=1 +dm +u + +m−1 +� +i=0 +��m +i +�� +ℓ +� +v=1 +ji +v − +r +� +w=1 +ki +w +�� +mod p. +(1) +If there are only t 0-deletions without s adjacent transpositions, Dolecek and Anantharam [17] showed that the following +system of equations has the unique solution. +Lemma 5 (Dolecek and Anatharam [17]). Without s adjacent transpositions, (1) can be rewritten as the following set of +constraints with t equations such that + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +a′′ +1 ≡ d1 + d2 + . . . + dt mod p, +a′′ +2 ≡ d2 +1 + d2 +2 + . . . + d2 +t mod p, +... +a′′ +t ≡ dt +1 + dt +2 + . . . + dt +t mod p. +(2) +which can uniquely determine the solution set {d1, d2, . . . , dt}, where p is a prime such that p > 2n and d1 ≤ d2 ≤ · · · ≤ dt. +Following the technique in [17], if we can determine uniquely the solution set {d1, . . . , dt, j1, . . . , jℓ, k1, . . . , kr} of (1), we +also can correct t 0-deletions and s adjacent transpositions with at most (t + s) log n bits of redundancy. However, the result +is not known to us and is still open for future work. +In this section, we focus on List-Decodable code CList +t,s (n, κ, a; p) for correcting t 0-deletions and s adjacent transpositions. +Set K = κ in Construction 3, where κ = max(t, s + 1) and p is a prime such that p > 2n. For the following system of +equations, we can determine the solution set uniquely. +Lemma 6. A set of constraints with s equations such that + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +b′′ +1 ≡ �ℓ +v=1 j1 +v − �r +w=1 k1 +w mod p, +b′′ +2 ≡ �ℓ +v=1 j2 +v − �r +w=1 k2 +w mod p, +... +b′′ +s ≡ �ℓ +v=1 js +v − �r +w=1 ks +w mod p. +(3) + +12 +is capable of uniquely determining the solution set {j1, . . . , jℓ, k1, . . . , kr}, where p is a prime such that p > 2n. Also, ℓ+r ≤ s, +j1 < j2 < · · · < jℓ, k1 < k2 < · · · < kr and jv ̸= kw, ∀v ∈ {1, . . . , ℓ}, w ∈ {1, . . . , r}. +We note that Lemma 6 is similar to Lemma 5. The only difference is that the coefficients of all terms in Lemma 5 are +positive while the coefficients of all terms in Lemma 6 can be either positive or negative. Hence, we can use the same technique +in Lemma 5 to prove Lemma 6. +Proof. Define the polynomials +σ+(x) = +ℓ +� +v=1 +(1 − jvx) +and +σ−(x) = +r +� +w=1 +(1 − kwx). +Let σ(x) = �s +m=0 σmxm be defined by +σ(x) = σ+(x)/σ−(x) mod xs +Then, we define σ∗(x) = σ(x) mod p. +We also define +S∗(x) = +∞ +� +m=1 +� +ℓ +� +v=1 +jm +v − +r +� +w=1 +km +w +� +xm. +and S∗ +m = �ℓ +v=1 jm +v − �r +w=1 km +w mod p. +Then, we have Newton’s identities over GF(p) as follows +σ∗(x)S∗(x) + x(σ∗(x))′ = 0 +u−1 +� +m=0 +σ∗ +mS∗ +u−m + uσ∗ +u = 0, u ≥ 1. +(4) +where (σ∗(x))′ is derivative of σ∗(x). (see [35, Lemma 10.3] for details) +Using the similar technique as the proof of Lemma 5, from (4), σ∗ +m can be recursively obtained by {S∗ +1, . . . , S∗ +m} and +{σ∗ +1, . . . , σ∗ +m−1}, where {S∗ +1, . . . , S∗ +m} = {b′′ +1, . . . , b′′ +m}, which follows that all the coefficients of the polynomial σ∗(x) = +�s +m=0 σ∗ +mxm mod p are known. Further, we know that the polynomial σ∗(x) has at most s solutions by Lagrange Theorem. +Denote I0 = {j1, . . . , jℓ, k1, . . . , kr} with the value of each element in I0 is less than p and let Im = {j1 + mp, . . . , jℓ + +mp, k1+mp, . . . , kr+mp} be one of the incongruent solution sets of I0. We can have I0∩Im = ∅ due to p > 2n, which follows +that all incongruent solutions are distinguishable. Therefore, we can conclude that the solution set {j1, . . . , jℓ, k1, . . . , kr} is +unique. +Theorem 9. The list-decodable code CList +t,s (n, κ, a; p) has redundancy κ log n, where κ = max(t, s + 1) and prime p > 2n. If +there are at most t 0-deletions and s adjacent transpositions, we can do list-decoding with list size O(nmin(t,s+1)). + +13 +Proof. Let x = (x1, . . . , xn) ∈ Σn +2 be the original vector and x′ be the received vector after t 0-deletions and s single adjacent +transpositions. Hence, we can compute φ(x′) and a′ from x′. Also, we can obtain a′′ = a′ − a, where a′′ = {a′′ +1, . . . , a′′ +κ}. +Suppose t ≥ s + 1 and expand (1). We have the following set of equations with κ = t: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +a′′ +1 ≡ �t +u=1 du + (ℓ − r) mod p, +a′′ +2 ≡ �t +u=1 d2 +u + (ℓ − r) + 2(�ℓ +v=1 j1 +v − �r +w=1 k1 +w) mod p, +... +a′′ +t ≡ �t +u=1 dt +u + (ℓ − r) + t(�ℓ +v=1 j1 +v − �r +w=1 k1 +w) ++ · · · + t(�ℓ +v=1 jt−1 +v +− �r +w=1 kt−1 +w +) mod p. +(5) +Recall that we can decode uniquely if we can determine the unique solution set of (5). However, the method to solve (5) +uniquely is not known to us. We know that, given e = {e1, . . . , es+1}, we can solve the following equations uniquely. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +e1 ≡ ℓ − r mod p, +e2 ≡ (ℓ − r) + 2(�ℓ +v=1 j1 +v − �r +w=1 k1 +w) mod p, +... +es+1 ≡ (ℓ − r) + (s + 1)(�ℓ +v=1 j1 +v − �r +w=1 k1 +w) ++ · · · + (s + 1)(�ℓ +v=1 js +v − �r +w=1 ks +w) mod p. +(6) +Indeed, denote e′ = {e′ +1, . . . , e′ +s+1} with me′ +m = em − �m−1 +i=1 +�� m +i−1 +� +e′ +i +� +for all m ∈ {2, . . ., s + 1} and e′ +1 = e1, we can +rearrange (6) to be similar to Lemma 6 as follows. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +e′ +1 ≡ ℓ − r mod p, +e′ +2 ≡ �ℓ +v=1 j1 +v − �r +w=1 k1 +w mod p, +... +e′ +s+1 ≡ �ℓ +v=1 js +v − �r +w=1 ks +w mod p. +(7) +Therefore, based on Lemma 6, we can obtain the unique solution set {j1, . . . , jℓ, k1, . . . , kr} from (7). +Once the solution set {j1, . . . , jℓ, k1, . . . , kr} is obtained, we can compute the following values {es+2, . . . , et}. +em = +m−1 +� +i=0 +��m +i +�� +ℓ +� +v=1 +ji +v − +r +� +w=1 +ki +w +�� +mod p. +(8) +where m ∈ {s + 2, . . . , t}. + +14 +Denote a∗ = {a∗ +1, . . . , a∗ +t } with a∗ +m = a′′ +m − em, ∀m ∈ {1, . . ., t}. Substituting (6) and (8) into (5), we obtain the following +set of equations. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +a∗ +1 ≡ �t +u=1 du mod p, +a∗ +2 ≡ �t +u=1 d2 +u mod p, +... +a∗ +t ≡ �t +u=1 dt +u mod p. +(9) +The set of equations (9) provides the unique solution set {d1, . . . , dt} by Lemma 5. Therefore, the unique solution of all +positions of 0-deletions and adjacent transpositions {d1, . . . , dt, j1, . . . , jℓ, k1, . . . , kr} can be obtained. So, for each set of +s+1 values {e1, . . . , es+1}, we can obtain the set {d1, . . . , dt, j1, . . . , jℓ, k1, . . . , kr}. There are ps+1 sets of these values. One +of these sets corresponds to the true value of x and gives us the correct vector x. So, we can do list-decoding with the list +size O(ns+1) since p = O(n). Moreover, the size of the list-decodable code CList +t,s (n, κ, a; p) with κ = t is at least 2n/(4n)t, +that is, we need at most κ log n bits of redundancy to construct the code CList +t,s (n, κ, a; p). +When t < s + 1, we can do similarly to the case t ≤ s + 1. In this case, we can do list-decoding with the list-size O(nt). +The size of the code CList +t,s (n, κ, a; p) is at least 2n/(4n)s+1. +Then, we can conclude that the list-decodable code CList +t,s (n, κ, a; p) can correct t 0-deletions and s adjacent transpositions +with list size at most O(nmin(t,s+1)) and has redundancy κ log n+O(1), where both t, s are constant and κ = max(t, s+1). +The decoding algorithm of the list-decodable code CList +t,s (n, κ, a; p) for correcting t 0-deletions and s adjacent transpositions +is summarized in Algorithm 2, where t > s + 1. +Algorithm 2: List decoding procedure +Input: Corrupted Sequence x′ ∈ Σn−t +2 +Output: O(ns+1) possible sequences, including the original codeword x ∈ CList +t,s (n, κ, a; p) +Compute φ(x′) based on x′ and compute a′′ to obtain (5). +for e = (e1, . . . , es+1) such that ei ∈ {0, 1, . . ., p − 1}, ∀i ∈ {1, . . . , s + 1} do +Get the solution set {j1, . . . , jℓ, k1, . . . , kr} by (6) and (7). +Compute em from the solution set {j1, . . . , jℓ, k1, . . . , kr} using (8) for each s + 2 ≤ m ≤ t. Compute +a∗ +m = a′′ +m − em. Solve (9) to obtain the unique solution set {d1, . . . , dt}. +end +For each fixed e, we can recover φ(x) from φ(x′) by a set of error positions {d1, . . . , dt, j1, . . . , jℓ, k1, . . . , kr} and +then output x. + +15 +Next, we will present the result for a special case t = 1. +Corollary 3. The list-decodable code CList +1,s (n, s + 1, a; p) can correct a single 0-deletion and s adjacent transpositions with +list size at most 2s and has redundancy (s + 1) log n + O(1). +Proof. When t = 1, It can be noticed that when the deletion position is determined, means d is known. Since l, r ∈ {1, . . . , s} +and a′′ +1 ≡ d + (ℓ − r) mod p, hence there are 2s choice for d, which means that the list size of CList +1,s (n, s + 1, a; p) is at most +2s. +The above code CList +1,s (n, s + 1, a; p) is capable of correcting a single 0-deletion and s adjacent transpositions with constant +list size at most 2s and has redundancy (s + 1) log n + O(1). The list size is constant 2s, which is less than the list size O(n) +when we directly substitute t = 1 to Theorem 9. +V. CODES FOR CORRECTING LIMITED-MAGNITUDE BLOCKS OF 0-DELETIONS AND ADJACENT TRANSPOSITIONS +In this section, we focus on studying the error of t blocks of asymmetric deletions with ℓ-limited-magnitude and s adjacent +transpositions. t blocks of asymmetric deletions with ℓ-limited-magnitude denotes that there are at most t blocks of 0s are +deleted with the length of each block is at most ℓ. Therefore, at most tℓ 0s are deleted and these t blocks of 0-deletions may +occur in at most t 0 runs. +For the sake of convenience in the following paper, we append a bit 1 at the end of x and denote it as x1. Since the sequence +x1 always ends with 1, x1 can be always written as x1 = 0u110u210u3 . . . 0uw1, where w = wt(x1). In addition, we revisit +the definition of function φ : Σn +2 → Σw and φ(x) +def= (u1, u2, u3, . . . , uw) ∈ Σw. Then, combining with Proposition 2, we can +have that the length of each 0 run increase by at most 1 and decrease by at most tℓ + 1 through t blocks of 0-deletions with +ℓ-limited-magnitude and s adjacent transpositions. Then, the definition of t blocks of 0-deletions with ℓ-limited-magnitude and +s adjacent transpositions is provided as follows. +Definition 4. Define the error ball B(n, t, k+, k−) such that +B(n, t, k+, k−) = {u ∈ Σn +q : −k− ≤ ui ≤ k+, wt(u) ≤ t}. +where at most t entries increase by at most k+ and decrease by at most k− for a sequence with length n. +Definition 5. t blocks of asymmetric deletions with ℓ-limited-magnitude and s adjacent transpositions denote that given a +sequence x ∈ Σn +2 , the retrieved sequence x′ through this type of error can be written as φ(x′1) = φ(x1) + v, where +v ∈ B(w, t + 2s, 1, tℓ + 1) and w = wt(x′1) = wt(x1) + +16 +Example 6. Suppose we have x = 0100101001 ∈ Σ10 +2 +with ℓ = 2, t = 3 and s = 1, then φ(x1) = 12120. If the retrieved +sequence x′ = 0110110 ∈ Σ6 +2 and the corresponding φ(x′1) = 10101, by comparing φ(x1) and φ(x′1), we can see +v = (0, −2, 0, −2, 1) ∈ B(5, 5, 1, 7). +Denote Φ be the set of mapping Σn +2 by the function φ and Σn +2 is the set containing all binary sequences with length n. +Lemma 7. The cardinality of Φ is: +|Φ| = +n+1 +� +w=1 +� +n +w − 1 +� += 2n. +(10) +Proof. For a binary sequence x ∈ Σn +2, the corresponding sequence φ(x1) is with length w = w(x1) and wt(φ(x1)) = n+1−w. +Also, the cardinality of Φ can be considered the number of ways of arranging n + 1 − w indistinguishable objects in w +distinguishable boxes. Thus, we can get the cardinality of Φ as shown in Lemma 7. +On the other side, since the mapping function φ is a one-to-one mapping function, the cardinality of Φ should be the same +as |Σn +2| = 2n. +Proposition 6. (cf. [36]) The code C(n, t, ℓ, s) for correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent +transpositions is equivalent to a packing to Σw by the error ball B(w, t+2s, 1, tℓ+1), where w = wt(x) and x ∈ C(n, t, ℓ, s). +A. Non-systematic Code Construction +In this section, we will provide a non-systematic construction for the code capable of correcting t blocks of 0-deletions with +ℓ-limited-magnitude and s adjacent transpositions. Then, we present the decoding algorithm of this code and a lower bound +of the code size. +Construction 4. The code C(n, t, ℓ, s) is defined as +C(n, t, ℓ, s) = {x ∈ Σn +2 : φ(x1) mod p ∈ Cp, wt(φ(x1)) = n + 1 − w}, +where w = wt(x1) and Cp is a code over Σp with p is the smallest prime larger than tℓ + 2. +Lemma 8. C(n, t, ℓ, s) is capable of correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions +for x ∈ C(n, t, ℓ, s) if Cp is capable of correcting t + 2s symmetric errors for φ(x1). +Lemma 9. ( [37], Theorem 10 ) Let p be a prime such that the distance 2 ≤ d ≤ p⌈m/2⌉−1 and n = pm − 1. Then, there +exists a narrow-sense [n, k, d]-BCH code Cp over Σp with +n − k = ⌈(d − 1)(1 − 1/p)⌉m. + +17 +Theorem 10. Let p be the smallest prime such that p ≥ tℓ + 2, w = pm − 1, w = wt(x1) and Cp is a primitive narrow-sense +[w, k, 2(t + 2s) + 1]-BCH code with w − k = ⌈2(t + 2s)(1 − 1/p)⌉m, the code C(n, t, ℓ, s) such that +C(n, t, ℓ, s) = {x ∈ Σn +2 : φ(x1) mod p ∈ Cp, wt(φ(x1)) = n + 1 − w}. +is capable of correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions. +Proof. Let x ∈ C(n, t, ℓ, s) be a codeword, and x′ be the output through the channel that has t blocks of 0-deletions with +ℓ-limited-magnitude and s adjacent transpositions. Let z′ = φ(x′1) mod p, where p is the smallest prime larger than tℓ + 2. +Run the decoding algorithm of Cp on z′ and output z∗. Thus, z∗ is also a linear code in Cp and it can be shown that +z∗ = φ(x1) mod p. Denote ǫ′ = (z′ − z∗) mod p, we can have that +(φ(x′1) − φ(x1)) mod p = (z′ − z∗) mod p = ǫ′. +(11) +and the error vector ǫ satisfies +ǫi = + + + + + + + + + +ǫ′ +i, +if 0 ≤ ǫ′ +i ≤ 1 +ǫ′ +i − p, +otherwise +. +(12) +Hence, the output is φ(x1) = φ(x′1) − ǫ and then recover x from φ(x1). +The detailed decoding steps are shown in Algorithm 3. +Algorithm 3: Decoding Algorithm of C(n, t, ℓ, s) +Input: Retrieved sequence x′ +Output: Decoded sequence x ∈ C(n, t, ℓ, s). +Initialization: Let p be the smallest prime larger than tℓ + 2. Also, append 1 at the end of x′ and get φ(x′1). +Step 1: z′ = φ(x′1) mod p. Run the decoding algorithm of Cp on z′ to get the output z∗. +Step 2: ǫ′ = (z′ − z∗) mod p and then ǫ. φ(x1) = φ(x′1) − ǫ. +Step 3: Output x1 = φ−1(φ(x1)) and then x. +Example 7. Suppose x = 0100101001 and x′ = 0110110 ∈ Σ6 +2 with ℓ = 2, t = 3 and s = 1. Since the retrieved sequence +x′ = 0110110, then φ(x′1) = 10101 and z′ = φ(x′) mod 11 = 10101, where p = 11 is smallest prime such that p ≥ tℓ + 2. +Run the decoding algorithm of Cp on z′ ∈ Cp, we have the output sequence z∗ = 12120. Hence ǫ′ = (z − z∗) mod 11 = +(0, 9, 0, 9, 1) and ǫ = (0, −2, 0, −2, 1). Thus, the output of the decoding algorithm φ(x1) = φ(x′1) − ǫ = (1, 0, 1, 0, 1) − +(0, −2, 0, −2, 1) = (1, 2, 1, 2, 0). Finally, x1 = 01001010011 and x = 0100101001. +Next, we will present a lower bound of the size of C(n, t, ℓ, s). + +18 +Theorem 11. The size of the code C(n, t, ℓ, s) in Theorem 10 is bounded by +|C(n, t, ℓ, s)| ≥ +2n +p(n + 1)⌈2(t+2s)(1−1/p)⌉ . +where p is the smallest prime larger than tℓ + 2. +Proof. Denote z = φ(x1) mod p. φ(x1) can be written as φ(x1) → (z, a) such that φ(x1) = z + p · a, where a is a +vector with the same length as φ(x1) and z. Further, since z ∈ Cp and Cp is a linear code, the code Cp with length w can be +considered as a set which is obtained by Σw +p partitioned into pw−k classes. +Denote φ(x1)w as the φ(x1) with length w. Thus, for any fixed number of weight w, the cardinality of φ(x1)w such that +φ(x1)w mod p ∈ Cp with length w is: +|φ(x1)w| = +� +n +w−1 +� +pw−k . +Then, the size of the code C(n, t, ℓ, s) in Theorem 10 can be shown as: +|C(n, t, ℓ, s)| = +n+1 +� +w=1 +|φ(x1)w| = +n+1 +� +w=1 +�� n +w−1 +� +pw−k +� +≥ +�n+1 +w=1 +� +n +w−1 +� +pn+1−k += +2n +pn+1−k . +(13) +From Lemma 9 and Theorem 10, let d = 2(t + 2s) + 1 and m = logp(n + 1). +pn−k+1 = p⌈2(t+2s)(1−1/p)⌉·logp(n+1)+1 = p(n + 1)⌈2(t+2s)(1−1/p)⌉. +(14) +Therefore, from (13) and (14), the size of the code C(n, t, ℓ, s) in Theorem 10 is bounded by +|C(n, t, ℓ, s)| ≥ +2n +p(n + 1)⌈2(t+2s)(1−1/p)⌉ . +where p is the smallest prime larger than tℓ + 2. +B. Systematic Code Construction +In the previous subsection, we propose a non-systematic code C(n, t, ℓ, s) for correcting t blocks of 0-deletions with ℓ-limited- +magnitude and s adjacent transpositions. In this subsection, we will provide the efficient encoding and decoding function based +on the code C(n, t, ℓ, s) presented in Theorem 10. +1) Efficient Encoding: Before providing the efficient systematic encoding algorithm, we now introduce a useful lemma +proposed in [38] for encoding balanced sequences efficiently. The balanced sequence denotes the binary sequence with an +equal number of 0s and 1s, which will be used for distinguishing the boundary of redundancy. + +19 +Lemma 10. (cf. [38]) Given the input x ∈ Σk +2, let the function s′ : Σk +2 → Σn +2 such that s′(x) ∈ Σn +2 is a balanced sequence, +where n = k + log k. +Definition 6. Given the input x ∈ Σk +2, let the function s : Σk +2 → Σn′ +2 such that s(x) ∈ Σn′ +2 whose first bit is 1 and s(x)[2,n′] +is balanced sequence with (n′ − 1)/2 0s and (n′ − 1)/2 1s, where n′ = k + log k + 1. +An adjacent transposition can be considered as two substitutions, hence the maximum total number of deletions and +substitutions in the t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions is r = tℓ+2s. The following +lemma is used for correcting deletions, insertions and substitutions up to r = tℓ + 2s in a binary sequence. +Lemma 11. (cf. [39]) Let t, ℓ, s be constants with respect to k. There exist an integer a ≤ 22r log k+o(log k) and a labeling +function fr : Σk +2 → Σ2Rr(k), where Rr(k) = O(r4 log k) such that {(x, a, fr(x) mod a) : x ∈ Σk +2} can correct deletions, +insertions and substitutions up to r = tℓ + 2s. Let gr(x) = (a, fr(x) mod a) ∈ Σ4r log k+o(log k) +2 +for given x ∈ Σk +2. +Next, we define the mapping function from non-binary to binary. +Definition 7. Given the input x ∈ Σk +2, define the function b : Σk +p → Σn +2 such that b(u)[i·⌈log p⌉+1,(i+1)·⌈log p⌉] is the binary +form of ui, where n = k · ⌈log p⌉. +Given the parameters t, ℓ and s, let p be the smallest prime larger than tℓ + 2 and Cp in Lemma 9 be the p-ary primitive +narrow-sense [n, k, 2(t + 2s) + 1]-BCH codes. +Definition 8. Define the labeling function as g : Σk +p → Σn−k +p +such that (x, g(x)) is a p-ary primitive narrow-sense [n, k, 2(t + +2s) + 1]-BCH codes, where n = k + ⌈2(t + 2s)(1 − 1/p)⌉m and n = pm − 1. +Suppose the input sequence is c ∈ Σk +2, and we have φ(c1) with length rc = wt(c1). Then, let c′ = φ(c1) mod p ∈ Σrc +p , +where p is the smallest prime larger than tℓ + 2, and append 0k+1−rc at the end of c′. Hence, we denote ¯c ∈ Σk+1 +p += +(c′, 0k+1−rc). +Next, encode ¯c via the labeling function g of the p-ary primitive narrow-sense [n, k, 2(t + 2s) + 1]-BCH code and output +the redundancy part g(¯c). We map the redundancy part g(¯c) into binary sequence b(g(¯c)) and make b(g(¯c)) to the balanced +sequence s(b(g(¯c))). Then, we prepend two 1s as the protecting bits at the beginning of s(b(g(¯c))) and denote h1(¯c) = +(1, 1, s(b(g(¯c)))). +Further, we need to protect the redundancy part h1(¯c). The idea is to apply the code in Lemma 11 on h1(¯c) since the code +in Lemma 11 is capable of correcting at most tℓ + 2s deletions and substitutions. Then, we output gr(h1(¯c)). In addition, +make gr(h1(¯c)) to balanced sequence s(gr(h1(¯c))) and repeat its each bit 2tℓ + 3 times. Let h2(¯c) = Rep2tℓ+3s(gr(h1(¯c))), + +20 +where Repkx is the k-fold repetition of x. +Finally, we have the output Enc(c) = (c, h(c)), where h(c) = (h1(¯c), h2(¯c)). The detailed encoding steps are summarized +in the following Algorithm 4. +Algorithm 4: Encoding Algorithm +Input: c ∈ Σk +2 +Output: Encoded sequence Enc(c) ∈ ΣN +2 +Initialization: Let p be the smallest prime larger than tℓ + 2. +Step 1: Append 1 at the end of c and get φ(c1) with length rc = wt(c1). +Step 2: c′ = φ(c1) mod p ∈ Σrc +p . Append 0k+1−rc at the end of c′, then ¯c = (c′, 0k+1−rc). +Step 3: Encode ¯c via Cp and output g(¯c). Mapping g(¯c) to balanced binary sequence s(b(g(¯c))) and introduce +protecting bits h1(¯c) = (1, 1, s(b(g(¯c)))). +Step 4: Protect h1(¯c) via gr and obtain the total redundancy h(c) = (h1(¯c), h2(¯c)). +Step 5: Output Enc(c) = (c, h(c)) ∈ ΣN +2 . +Lemma 12. Given a sequence c ∈ Σk +2, Algorithm 4 outputs an encoded sequence Enc(c) ∈ ΣN +2 capable of correcting t blocks +of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions. +Therefore, the redundancy of the code h(c) = (h1(¯c), h2(¯c)) via this encoding process can be shown as follows. +Theorem 12. The total redundancy of the code Enc(c) ∈ ΣN +2 by given input c ∈ Σk +2 is +N − k = ⌈2(t + 2s)(1 − 1/p)⌉ · ⌈log p⌉ +log p +log(N + 1) + O(log log N). +where p is smallest prime such that p ≥ tℓ + 2. +Proof. Let m = logp(N + 1), hence N = pm − 1. The lengths of the redundancy parts are as follows: +• n′′ +1 is the length of g(¯c): n′′ +1 = ⌈2(t + 2s)(1 − 1/p)⌉m; +• n′ +1 is the length of b(g(¯c)): n′ +1 = n′′ +1 · ⌈log p⌉; +• n1 is the length of h1(¯c): n1 = n′ +1 + log n′ +1 + 3; +• n′′ +2 is the length of gr(h1(¯c)): n′′ +2 = 4(tℓ + 2s) log n1 + log n1; +• n′ +2 is the length of s(f0(h1(¯c))): n′ +2 = n′′ +2 + log n′′ +2 + 1; +• n2 is the length of h2(¯c): n2 = (2tℓ + 3)n′ +2; + +21 +Based on the above statement, we can see that N − k = n1 + n2, where +n′ +1 = (⌈2(t + 2s)(1 − 1/p)⌉m) · ⌈log p⌉ +with m = logp(N + 1). Hence, we have +n′ +1 = ⌈2(t + 2s)(1 − 1/p)⌉ · ⌈log p⌉ +log p +log(N + 1) +Since both t, p and s are constants, then log n′ +1 = O(log log N) and n2 = O(log log N). Therefore, the total redundancy of +the code Enc(c) ∈ ΣN +2 given the input c ∈ Σk +2 can be shown as the Theorem 12. +2) Decoding Algorithm: Without loss of generality, suppose the encoded sequence Enc(c) ∈ ΣN +2 is transmitted through the +t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions channel, and we have the retrieved sequence +d ∈ ΣN−tℓ +2 +. In this subsection, we will introduce the decoding algorithm for obtaining Dec(d) ∈ Σk +2 by given d ∈ ΣN−tℓ +2 +. +First, we need to distinguish where the redundancy part begins. Since the error type is at most t blocks of 0-deletions with +ℓ-limited-magnitude and s adjacent transpositions, the number of 1s in d is the same as that of in Enc(c). Thus, we can +count the number of 1s from the end of d to find the beginning of the redundancy since the redundancy part is the balanced +sequence. +Hence, we find the (n2 + 2tℓ + 3)/2-th 1 and (n1/2 + n2/2 + tℓ + 3)-th 1 from the end of d and denote their entries as +ir2 and ir1, respectively. For the subsequence d[ir2,N−tℓ], since there are at most tℓ 0s deletions and s adjacent transpositions +occur in Enc(c)[N−n2+1,N], the (2tℓ + 3)-fold repetition code can help recover s(gr(h1(¯c))). Further, we can obtain parity +bits gr(h1(¯c)). +For the subsequence d[ir1,ir2−1], there are also at most tℓ 0-deletions and 2s substitutions occur in Enc(c)[N−n1−n2+1,N−n2]. +The recovered parity bits gr(h1(¯c)) can help recover h1(¯c). Further, we remove the two 1 bits at the beginning of h1(¯c) and +get the g(¯c) from h1(¯c) = s(b(g(¯c))). +Finally, denote z = (φ(d[1,ir1−1], 1), 0k+1−rc) and z′ = z mod p, where rc is the length of φ(d[1,ir1−1], 1) and k = +N − n1 − n2. Then, the following decoding steps are the same as Algorithm 3 where z′ is the input of Step 1 of Algorithm 3. +The only difference is we need to first remove 0k+1−rc at the end before the last step of φ−1. Therefore, the main steps for +decoding d ∈ ΣN−tℓ +2 +is summerized in Algorithm 5. +3) Time Complexity: For the encoding algorithm, it can be easily shown that the time complexity is dominated by the p-ary +narrow-sense BCH code and the code in Lemma 11, which is O(tn log n + (log n)2(tℓ+2s)+1). +For the decoding algorithm, the time complexity is also dominated by the decoding of the p-ary narrow-sense BCH code +and decoding for the code in Lemma 11. Therefore, the total time complexity of decoding is O(tn + (log n)tℓ+2s+1). + +22 +Algorithm 5: Decoding Algorithm +Input: d ∈ ΣN−tℓ +2 +Output: Decoded sequence Dec(d) ∈ Σk +2 +Initialization: Let p be the smallest prime larger than tℓ + 2. +Step 1: Find the (n2 + 2tℓ + 3)/2-th 1 and (n1/2 + n2/2 + tℓ + 3)-th 1 from the end of d and denote their entries as +ir2 and ir1, respectively. +Step 2: Recover s(gr(h1(¯c))) from d[ir2,N−tℓ] and then get gr(h1(¯c)). +Step 3: Recover h1(¯c) via gr(h1(¯c)) and then obtain h1(¯c). +Step 4: Denote z′ = (φ(d[1,ir1−1], 1), 0k+1−rc) mod p. Input z′ to Step 1 of Algorithm 3 and run the remaining steps +of Algorithm 3. +Step 5: Output Dec(d). +VI. CONCLUSION +In this paper, motivated by the errors in the DNA data storage and flash memories, we presented codes for correcting +asymmetric deletions and adjacent transpositions. We first present three uniquely-decodable codes for different types of +asymmetric deletions and adjacent transpositions. 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Schwartz, “On lattice packings and coverings of asymmetric limited-magnitude balls,” IEEE Transactions on Information +Theory, vol. 67, no. 8, pp. 5104–5115, 2021. +[37] S. A. Aly, A. Klappenecker, and P. K. Sarvepalli, “On quantum and classical BCH codes,” IEEE Transactions on Information Theory, vol. 53, no. 3, +pp. 1183–1188, 2007. +[38] D. Knuth, “Efficient balanced codes,” IEEE Transactions on Information Theory, vol. 32, no. 1, pp. 51–53, 1986. +[39] J. Sima, R. Gabrys, and J. Bruck, “Optimal systematic t-deletion correcting codes,” in 2020 IEEE International Symposium on Information Theory (ISIT). +IEEE, 2020, pp. 769–774. + diff --git a/Z9FJT4oBgHgl3EQf7y3s/content/tmp_files/load_file.txt b/Z9FJT4oBgHgl3EQf7y3s/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..124da0b9aa17725204f4278680b4f50ad468d8fd --- /dev/null +++ b/Z9FJT4oBgHgl3EQf7y3s/content/tmp_files/load_file.txt @@ -0,0 +1,1068 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf,len=1067 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='11680v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='IT] 27 Jan 2023 Codes for Correcting Asymmetric Adjacent Transpositions and Deletions Shuche Wang∗, Van Khu Vu§, and Vincent Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Tan†‡∗ ∗ Institute of Operations Research and Analytics, National University of Singapore, Singapore † Department of Mathematics, National University of Singapore, Singapore ‡ Department of Electrical and Computer Engineering, National University of Singapore, Singapore § Department of Industrial Systems Engineering and Management, National University of Singapore, Singapore Emails: shuche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='wang@u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='edu, isevvk@nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='sg, vtan@nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='sg Abstract Owing to the vast applications in DNA-based data storage, Gabrys, Yaakobi, and Milenkovic recently proposed to study codes in the Damerau–Levenshtein metric, where both deletion and adjacent transposition errors occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In particular, they designed a code correcting a single deletion and s adjacent transpositions with at most (1 + 2s) log n bits of redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In this work, we consider a new setting where both asymmetric adjacent transpositions (also known as right-shifts or left-shifts) and deletions occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We present several constructions of the codes correcting these errors in various cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In particular, we design a code correcting a single deletion, s+ right-shift, and s− left-shift errors with at most (1 + s) log(n + s + 1) + 1 bits of redundancy where s = s+ + s−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In addition, we investigate codes correcting t 0-deletions and s adjacent transpositions with both unique decoding and list-decoding algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Our main contribution here is a construction of a list-decodable code with list-size O(nmin{s+1,t}) and has at most (max{t, s + 1}) log n + O(1) bits of redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Finally, we provide both non-systematic and systematic codes for correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' INTRODUCTION The Levenshtein (edit) distance of two different strings is the smallest number of operations (including deletions, insertions, and substitutions) required to transform one string into the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' This metric has a long history and has attracted a lot of research in computer science in the past as well as recently [2]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Codes in the Levenshtein metric have been investigated extensively recently due to theoretical interests and their numerous applications, including racetrack memory [5]–[7] and DNA-based data storage [8]–[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' This paper was presented in part at the 2022 IEEE Information Theory Workshop (ITW) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 1 In some channels, such as DNA-based data storage ones, we observe that, besides deletions, insertions, and substitutions, there are also adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, there exists some recent work concerning the Damerau–Levenshtein distance which is motivated by applications to DNA-based data storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The distance is a generalization of the well-known Levenshtein distance taking into account adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' More precisely, the Damerau–Levenshtein metric is the smallest number of operations (including deletions, insertions, substitutions, and adjacent transpositions) required to transform one string into another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We note that it is possible to compute the exact Damerau–Levenshtein distance of two strings in polynomial time [11] but it is not known if we can compute the distance in linear time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Recently, Gabrys, Yaaboki, and Milenkovic [12] proposed to study codes in the Damerau–Levenshtein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' They provided several constructions of codes correcting both deletions and adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' However, these codes are not optimal in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For example, to correct a single deletion and at most s adjacent transpositions, the authors require (1 + 2s) log n bits of redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Designing an optimal code correcting both deletions and multiple adjacent transpositions has turned out to be a formidable challenge for coding theorists in recent times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The problem of constructing codes for correcting synchronization errors, including deletions and insertions, was first investigated by Levenshtein [13] and Ullman [14], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Sticky deletions/insertions and duplication deletions can be considered as asymmetric deletions/insertions via the Gray mapping [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Owing to various applications, such as in flash memories [17], [18], racetrack memories [6], and DNA data storage systems [19], [20], codes for correcting asymmetric deletions/insertions have garnered significant attention recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Tallini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' [16], [21]–[24] provided a series of theories and code designs for correcting these kinds of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Especially, Mahdavifar and Vardy [18] provided some efficient encoding/decoding algorithms for an optimal code correcting sticky-insertion and thus for an optimal code correcting 0-deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Codes correcting adjacent transposition errors have been investigated for a long time as codes for shift errors [25]–[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Codes correcting asymmetric shift errors have also been studied recently [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In this work, we are interested in codes correcting a combination of both asymmetric adjacent transposition errors and deletion errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We aim to obtain some optimal codes with simple efficient encoding/decoding algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We note that codes correcting substitutions, deletions, and their combinations have attracted a lot of research recently [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' However, there are only a few code constructions that correct a combination of adjacent transposition and other kinds of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Klove [31] proposed a class of perfect constant-weight codes capable of correcting a single deletion, a single insertion or an adjacent transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Gabrys, Yaakobi, and Milenkovic [12] presented several codes correcting a combination of deletions and adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' If there is a single adjacent transposition or a single deletion, there exist codes correcting the error with at most log n + O(log log n) bits of redundancy [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The best-known codes correcting a single deletion and at most s 2 adjacent transpositions require (1 + 2s) log n bits of redundancy [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In this work, we design several new families of codes in numerous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We provide our main contributions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Our first contribution in this work is Construction 1, which presents a construction of an optimal code correcting a single adjacent transposition or a single 0-deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Analyzing the size of our code, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' There is a code correcting a single 0-deletion or a single adjacent transposition with at most log n + 2 bits of redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Next, we construct a code correcting t 0-deletions and s adjacent transpositions with at most (t + 2s) log n + o((t + 2s) log n) bits of redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The constructed code is the best known that corrects multiple 0-deletions and multiple adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' See Theorem 7 for the detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Theorem2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' There is a code correcting t 0-deletions and s adjacent transpositions with at most (t+2s) log n+o((t+2s) log n) bits of redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Further, we construct an optimal code for correcting a single deletion, s+ right-shift and s− left-shift errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Throughout this paper, we denote the adjacent transposition as 01 → 10 or 10 → 01, right-shift of 0 as 01 → 10 and left-shift of 0 as 10 → 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' See Construction 2 and Theorem 8 for the detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' There is a code correcting a single deletion, s+ right-shift and s− left-shift errors with at most (1 + s) log(n + s + 1) + 1 bits of redundancy where s = s+ + s−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Compare the results in [12], where the code for correcting a single deletion and s adjacent transpositions needs at most (1 + 2s) log(n + 2s + 1) redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' If we know the direction of these s adjacent transpositions containing s+ right-shifts of 0 and s− left-shifts of 0, the redundancy of the code can be further reduced to at most (1 + s) log(n + s + 1) + 1 where s = s+ + s−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We also investigate list-decodable codes of small list-size and construct a list-decodable code for at most t 0-deletions and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' See the proof of Theorem 9 for the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Our results are the first known list-decodable codes for the asymmetric Damerau–Levenshtein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' There is a list-decodable code that can correct t 0-deletions and s adjacent transpositions with list size O(nmin(t,s+1)) and has max(t, s + 1) log n + O(1) bits of redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Finally, we construct both non-systematic and systematic codes for correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' See the proof of Theorem 10 for the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 3 Theorem5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' There is a code capable of correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions with at most ⌈2(t + 2s)(1 − 1/p)⌉ log(n + 1) + O(1) bits of redundancy, where p is the smallest prime larger than tℓ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Section II provides the notation and preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Section III presents three uniquely-decodable codes for correcting asymmetric deletions and adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Section IV proposes list-decodable codes for correcting asymmetric deletions and adjacent transpositions with low redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In Section V, we construct codes both non-systematic and systematic codes are capable of correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Finally, Section VI concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' NOTATION AND PRELIMINARIES We now describe the notations used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Σq denotes the finite alphabet of size q and Σn q represents the set of all sequences of length n over Σq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Without loss of generality, we assume Σq = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=', q − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For two integers i < j, let [i, j] denote the set {i, i + 1, i + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The size of a binary code C ⊆ Σn 2 is denoted |C| and its redundancy is defined as n − log |C|, where all logarithms without a base in this paper are to the base 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We write sequences with bold letters, such as x and their elements with plain letters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=', x = x1 · · · xn for x ∈ Σn q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The length of the sequence x is denoted |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The weight wt(x) of a sequence x represents the number of non-zero symbols in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' A run is a maximal substring consisting of identical symbols and nr(x) denotes the number of runs of the sequence x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For functions, if the output is a sequence, we also write them with bold letters, such as φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The ith position in φ(x) is denoted φ(x)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In addition, for a sequence u ∈ Σn q , denote (u mod a) = (u1 mod a, u2 mod a, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , un mod a), where a < q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For a binary sequence x ∈ Σn 2, we can uniquely write it as x = 0u110u210u3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 10uw+1, where w = wt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Define function φ : Σn 2 → Σw+1 and φ(x) def= (u1, u2, u3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , uw+1) ∈ Σw+1, where x = 0u110u210u3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 10uw+1 with w = wt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Suppose x = (0, 1, 1, 1, 0, 1, 0, 1, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, φ(x) = (1, 0, 0, 1, 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Define function ψ : Σn 2 → Σn 2 such that ψ(x) = (x1, x1 + x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , x1 + x2 + · · · + xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The Lee weight of an element xi ∈ Σq is defined by wL(xi) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 xi, if 0 ≤ xi ≤ q/2 q − xi, otherwise For a sequence x ∈ Σn q , the Lee weight of x is wL(x) = n � i=1 wL(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 4 Define the Lee distance of two sequences x, x′ ∈ Σn q as dL(x, x′) = wL(x − x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Suppose x ∈ Σ7 6 = (1, 4, 0, 5, 2, 3, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, wL(x) = 1 + 2 + 0 + 1 + 2 + 3 + 2 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Suppose x ∈ Σ7 6 = (1, 4, 0, 5, 2, 3, 4) and x′ ∈ Σ7 6 = (0, 3, 0, 5, 3, 3, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, x − x′ = (1, 1, 0, 0, 5, 0, 1) and dL(x, x′) = wL(x − x′) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For any x ∈ Σn 2, denote Bt,s(x) as the error ball of x under t 0-deletions and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The code Ct,s(n) is a unique-decodable code for correcting t 0-deletions and s adjacent transpositions, for which holds that Bt,s(c1)∩Bt,s(c2) = ∅ for all c1, c2 ∈ Ct,s(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The code CList t,s (n) is a list-decodable code for correcting t 0-deletions and s adjacent transpositions with list size L such that for any corrupted sequence x′ ∈ Σn−t 2 there exist at most L codewords in CList t,s (n) that can be obtained by t 0-deletions and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Suppose x = (0, 1, 1, 1, 0, 1, 0, 1, 0, 0), the first and last 0 bits are deleted and two pairs of ((4th, 5th) and (7th, 8th)) adjacent bits are transposed in x = (❆0, 1, 1, 1, 0, 1, 0, 1, 0, ❆0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, x′ = (1, 1, 0, 1, 1, 1, 0, 0) ∈ B2,2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Once a 0-deletion occurs in x and we receive x′, there is an index i such that φ(x)i − 1 = φ(x′)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Suppose an adjacent transposition occurs in x at the ith 1, the corresponding changes in φ(x) can be shown as follows: 1) 10 → 01: (φ(x)′ i, φ(x)′ i+1) = (φ(x)i + 1, φ(x)i+1 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 2) 01 → 10: (φ(x)′ i, φ(x)′ i+1) = (φ(x)i − 1, φ(x)i+1 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Suppose x = (0, 1, 1, 1, 0, 1, 0, 1, 0, 0), φ(x) = (1, 0, 0, 1, 1, 2) and the adjacent transposition is occurred in the 4-th bit 1 and the following bit 0 in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, x′ = (0, 1, 1, 1, 0, 0, 1, 1, 0, 0) and φ(x′) = (1, 0, 0, 2, 0, 2), where (φ(x′)4, φ(x′)5) = (φ(x)4 + 1, φ(x)5 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The well-known Varshamov–Tenengol’ts (VT) code will be use of in this paper, and we will introduce the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For x ∈ Σn 2, we define the syndrome of VT code as VT(x) = �n i=1 ixi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Lemma 1 (Varshamov-Tenengol’ts (VT) code [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For integers n and a ∈ [0, n], VTa(n) = {x ∈ Σn 2 : VT(x) ≡ a mod (n + 1)} is capable of correcting a single deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Define Mt,s(n) as maximal size of binary codes for correcting t deletions and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 5 Lemma 2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Levenstein [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For enough large n, Mt,s(n) ≤ (s + t)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 2n ns+t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' t deletions and s adjacent transpositions in x can be considered as t deletions and s substitutions in ψ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' An asymptotic bound for the size of any codes is capable of correcting up to t deletions, insertions and substitutions have been shown in [2], which is (t!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' · 2n)/nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Since the function ψ is a one-to-one mapping function, an upper bound of binary codes for correcting t deletions and s adjacent transpositions can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' From Lemma 2, we can obtain a lower bound of the minimal redundancy of the code for correcting t 0-deletions and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Corollary1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' A lower bound of the minimal redundancy of binary codes for correcting t 0-deletions and s adjacent transpositions is (t + s) log n − O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='1 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' UNIQUELY-DECODABLE CODES FOR ASYMMETRIC DELETIONS AND ADJACENT TRANSPOSITIONS In this section, we will present three uniquely-decodable codes for correcting asymmetric deletions and adjacent transposi- tions, that is, once there are some errors, we can correct these errors to recover the original codeword uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Codes for correcting a single 0-deletion or a single adjacent transposition In this subsection, we present the first construction of an optimal code correcting a single 0-deletion or a single adjacent transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Construction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The code C1(n, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) is defined as the set of all x ∈ Σn 2 such that the syndrome S(x) = w+1 � i=1 i2φ(x)i ≡ a mod p where w = wt(x) and p is a prime such that p > 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The code C1(n, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) in Construction 1 can correct a single 0-deletion or a single adjacent transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Let x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , xn) ∈ Σn 2 be the original vector and x′ be the received vector after a single 0-deletion or a single adjacent transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' If x′ ∈ Σn−1 2 , that is the length of x′ is n−1, then there is a single 0 deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In this case, we compute the vector φ(x′) and a′ < p such that a′ = S(x′) mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We note that dL(φ(x), φ(x′)) = 1 and there is an index i such that φ(x)i − 1 = φ(x′)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, S(x) − S(x′) = i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' That is, a − a′ = i2 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Since i2 − j2 ̸= 0 mod p for all i ̸= j, i, j < n < p/2, we can determine the unique index i such that a − a′ = i2 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' And thus, we locate the error and can correct it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 1The difference between the lower bound of the redundancy for correcting general t deletions and t 0-deletions is only O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' [17] 6 If x′ ∈ Σn 2, that is the length of x′ is n, then there is no 0 deletion and at most a single adjacent transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Similar to the previous case, we also compute the vector φ(x′) and a′ < p such that a′ = S(x′) mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Once an adjacent transposition occurs, there are two types of errors: a symbol 0 moves to the left and a symbol 0 moves to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' If a symbol 0 moves to the left, there exists 0 ≤ j ≤ n − 1 such that a − a′ = 2j + 1 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Otherwise, if a symbol 0 moves to the right, there is 0 ≤ j ≤ n − 1 such that a − a′ = −2j − 1 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Since p > 2n, for all i, j < n < p/2 and i ̸= j, these four values, {2i + 1, −2i − 1, 2j + 1, −2j − 1} are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, we can determine the type of error and the unique j such that a − a′ = 2j + 1 mod p or a − a′ = −2j − 1 mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' And thus, we can correct the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In conclusion, either a 0 deletion occurs or an adjacent transposition occurs, we always can correct the error and recover the original vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The theorem is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' From the well-known Bertrand–Chebyshev theorem, there exists a prime p such that 2n < p < 4n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, by the pigeonhole principle, there exists a code C1(n, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) of size at least 2n/(4n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' That is, it is possible to construct the code C1(n, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) at most log n + 2 redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Therefore, we can conclude that we can correct a single 0-deletion or a single adjacent transposition with at most log n + 2 redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Codes for correcting t 0-deletions and s adjacent transpositions In this subsection, we explore the general case in the asymmetric Damerau–Levenshtein distance scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We investigate a code correcting at most t 0-deletions and s adjacent transpositions, given constants t and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We observe that the asymmetric Damerau–Levenshtein distance between two vectors x and y is closely related to Lee distance between φ(x) and φ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Indeed, once an adjacent transposition occurs in x, the Lee weight of x is changed by two based on Proposition 2 and once a 0-deletion occurs in x, the Lee weight of x is changed by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, if there are at most s adjacent transpositions and t 0-deletions, the Lee weight of x is changed by at most t + 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Now, we present a well-known BCH code in the Lee distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' ( [18], [34]) The systematic BCH code CBCH(n, t + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) : x ∈ Σm 2 → E(x) ∈ Σn p with the lower bound of minimum Lee distance dL(CBCH(n, t + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p)) ≥ \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 2(t + 1), if t ≤ (p − 3)/2 p, if (p − 1)/2 ≤ t ≤ p can correct errors up to t Lee weight with redundancy t log n + o(t log n), where p is a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Furthermore, Mahdavifar and Vardy [18] used the above code to construct a code C(n, r) of length n correcting r 0 insertions with at most r log n + o(r log n) bits of redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' It is known that for any two words c1, c2 ∈ C(n, r), we 7 have dL(φ(c1), φ(c2)) ≥ 2(r + 1) by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, we can use the code C(n, r) to correct t 0-deletions and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The code C(n, r) can correct at most t 0-deletions and s adjacent transpositions, given t + 2s = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Let x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , xn) ∈ Σn 2 be the original vector and x′ ∈ Σn−t 2 be the received vector after t 0-deletions and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, we obtain the vector y′ = φ(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We consider two vectors φ(x) and φ(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We observe that once an adjacent transposition occurs in x, the Lee weight of x is changed by at most two based on Proposition 2 and once a 0-deletion occurs in x, the Lee weight of x is changed by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, if there are at most s adjacent transpositions and t 0-deletions, the Lee weight of x is changed by at most t + 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' That is, the Lee distance between two vectors φ(x) and φ(x′) is dL(φ(x), φ(x′)) ≤ t + 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Therefore, we set r = t + 2s and then the code C(n, r) can correct at most t 0-deletions and s adjacent transpositions with redundancy (t + 2s) log n + o((t + 2s) log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Codes for correcting a single deletion and multiple right-shifts In previous two subsections, we focus on the error type of 0-deletions and arbitrary adjacent transposition (both 01 → 10 and 10 → 01 can occur) in the asymmetric Damerau-Levenshtein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In this subsection, we propose an optimal code for correcting a single deletion and s right-shifts of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We denote the adjacent transposition as 01 → 10 or 10 → 01, right-shift of 0 as 01 → 10 and left-shift of 0 as 10 → 01 throughout this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The code C(n, a, b) is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' C(n, a, b) = {x ∈ Σn 2 : VT(x) ≡ a mod (n + s + 1), n � i=1 xi ≡ b mod 2, ψ(x) ∈ CH(n, 2s + 1)}, where CH(n, 2s + 1) is a linear binary code capable of correcting errors with 2s + 1 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' [12]) A single adjacent transposition (01 → 10 or 10 → 01) in x is equivalent to a single substitution in ψ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Suppose there are s right-shifts of 0 occurs in x, we have VT(x) − VT(x′) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Suppose a right-shift of 0 (01 → 10) occurs at the i-th 1 in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The index of this 1 in x′ will be i − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Thus, for a single right-shift of 0, the change of the VT syndrome will be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' If there are s right-shifts of 0 occurs in x, we have VT(x) − VT(x′) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The following statements are true: 8 Suppose a 0 is deleted before p-th 1 in x, and insert a 0 before (p + v)-th 1 to get ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' x can be obtained from ˆx by v adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Suppose a 1 is deleted after p-th 0 in x, and insert a 1 after (p − v)-th 0 to get ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' x can be obtained from ˆx by v adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Denote the indexes of p-th 1, (p + 1)-th 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , (p + v − 1)-th 1 in x as ip, ip+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , ip+v−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, we can see that the indexes of these 1s in ˆx should be ip − 1, ip+1 − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , ip+v−1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Since 0 is inserted before (p + v)-th 1, we can swap the (ip+v−1 − 1)-th and ip+v−1-th bits and hence ˆx[ip+v−1,ip+v] = x[ip+v−1,ip+v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Continuing this process, we can see that x can be recovered from ˆx by v adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The case of deleting 1 is the same deleting 0, hence we can have the above two statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For all a ∈ [0, n + s] and b ∈ [0, 1], the code C(n, a, b) can correct a single deletion and s right-shifts of 0 with redundancy at most (1 + s) log(n + s + 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Denote the retrieved sequence as x′ ∈ Σ2 through a single deletion and at most s right-shifts of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We first use the VT syndrome to correct the deletion and then apply the CH(n, 2s + 1) on ψ(x) to correct the right-shifts of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Further, let ∆ = VT(x) − VT(x′), w be the weight of x′ and p be the index of deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, let L0 be the number of 0s on the left of the deleted bits in x′ and R0 on its left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Similarly, denote L1, R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We have the following cases when recover x by x′: If x′ = Σn 2, it means no deletion occurs in x and there are at most s right-shifts of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Based on Proposition 3, there are at most s substitutions in ψ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence we can recover ψ(x) by ψ(x′) since ψ(x) ∈ CH(n, 2s + 1), and then recover x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' If x′ = Σn−1 2 and suppose a 0 is deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' From Proposition 4, then ∆ = R1 + k, where k is the actual number of right-shifts of 0s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We can first recover ˆx by inserting 0 in the rightmost index of (∆ − s) 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Since ∆ = R1 + k and we insert 0 in the rightmost index of (R1 + k − s) 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Based on the Case 1 of Lemma 4, we can have that there are at least (s − k) adjacent transpositions between ˆx and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In addition, there are also k right-shifts of 0s occur in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Therefore, x can be obtained from ˆx by total s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, we can recover ψ(x) by ψ(ˆx) and then x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' If x′ = Σn−1 2 and suppose a 1 is deleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' From Proposition 4, then ∆ = p + R1 + k = w + L0 + k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We recover ˆx by inserting 1 in the leftmost index of (∆ − w − s − 1) 0s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Similar as Case 2, since ∆ = w + L0 + k + 1 and we insert 1 in the leftmost index of (L0 + k − s) 0s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Based on the Case 2 of Lemma 4, we can have that there are at least (s − k) adjacent transpositions between ˆx and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Similarly, x can be obtained from ˆx by total s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, we can recover ψ(x) by ψ(ˆx) and then x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 9 It is worth noticing that Case 1 and Case 2, 3 can be distinguished by the length of the retrieved sequence x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Case 2 and Case 3 can distinguished based on the constraint of �n i=1 xi ≡ b mod 2, from where we can know the deleted bit is 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' There are three constraints on the sequence x ∈ C(n, a, b) including a VT code, a parity check bit and a linear binary (n, 2s + 1)-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' It can be easily shown that the redundancy of the code C(n, a, b) is log(n + s + 1) + s log n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Thus, the redundancy of the code C(n, a, b) is at most (1 + s) log(n + s + 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The decoding algorithm of the code C(n, a, b) for correcting a single deletion and s right-shifts of 0 is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Algorithm 1: Decoding procedure of C(n, a, b) Input: Corrupted Sequence x′ Output: Original Sequence x ∈ C(n, a, b) ∆ = VT(x) − VT(x′), b = �n i=1 xi − �|x′| i=1 x′ i and w = wt(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' if |x′| = n then Recover ψ(x) by ψ(x′) and then x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' else if b = 0 then Insert a 0 in the rightmost index of (∆ − s) 1s to get ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Recover ψ(x) by ψ(ˆx) and then x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' else Insert a 1 in the leftmost index of (∆ − w − s − 1) 0s to get ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Recover ψ(x) by ψ(ˆx) and then x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' end end Further, Construction 2 and Theorem 8 can be naturally extended to construct codes for correcting a single deletion, s+ right-shifts of 0 and s− left-shifts of 0 with s = s+ + s−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For all a ∈ [0, n + s] and b ∈ [0, 1], the code C2(n, a, b) such that C2(n, a, b) = {x ∈ Σn 2 : VT(x) ≡ a mod (n + s + 1), n � i=1 xi ≡ b mod 2, ψ(x) ∈ CH(n, 2s + 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' can correct a single deletion, s+ right-shifts of 0 and s− left-shifts of 0 with redundancy at most (1 + s) log(n + s + 1) + 1, where s = s+ + s−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Similar as Proposition 4, suppose there are at most s− left-shifts of 0s, the change of VT syndrome is VT(x) − VT(x′) = −s−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Suppose a 0 is deleted, and the same as the proof of Theorem 8 with the same notations, we can also have ∆ = R1 + k+ − k−, where k+ and k− are actual number of right-shifts and left-shifts of 0 occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Also, we still insert a 0 in 10 the index of rightmost of (∆ − s+ + s−) 1s to obtain ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Based on the Case 1 of Lemma 4, we can have that there are at least ((s+ − s−) − (k+ − k−)) adjacent transpositions between ˆx and x and there are k+ + k− adjacent transpositions occur in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Therefore, the total number of adjacent transpositions that x can be obtained from ˆx is at most (s+ − s−) − (k+ − k−) + (k+ + k−) = s+ − s− + 2k− ≤ s+ + s− = s Hence, we can recover ψ(x) by ψ(ˆx) since there are at most s substitutions and then x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Also, the analysis of redundancy is the same as the proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Compare the results in [12], where the code for correcting a single deletion and s adjacent transpositions needs at most (1 + 2s) log(n + 2s + 1) redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' If we know the direction of these s adjacent transpositions containing s+ right-shifts of 0 and s− left-shifts of 0, the redundancy of the code can be further reduced to at most (1 + s) log(n + s + 1) + 1 where s = s+ + s−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' LIST-DECODABLE CODES FOR CORRECTING ASYMMETRIC DELETIONS AND ADJACENT TRANSPOSITIONS In this section, we aim to construct List-Decodable codes with low redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For correcting t 0-deletions without s adjacent transpositions, Dolecek and Anatharam [17] proposed a well-known construction with optimal redundancy t log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Inspired by this, we have the following construction: Construction 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The construction CList t,s (n, K, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) is defined as the set of all x ∈ Σn 2 such that w+1 � i=1 imφ(x)i ≡ am mod p, ∀m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' where the prime p such that p > 2n and a = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , aK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Let x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , xn) ∈ Σn 2 be the original vector and x′ ∈ Σn−t 2 be the received vector after t 0-deletions and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, we obtain the vector φ(x′) and the corresponding a′ at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Let a′ m = �w+1 i=1 imφ(x′)i and a′′ m = am − a′ m, ∀m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Suppose there is only a single adjacent transposition occurs in x at the position of j-th 1, the change of syndrome a′′ m can be shown as follows: 1) 10 → 01: a′′ m = (j + 1)m − jm mod p = m−1 � i=0 �m i � ji mod p 2) 01 → 10: a′′ m = jm − (j + 1)m mod p = − m−1 � i=0 �m i � ji mod p 11 Then, suppose t 0-deletions occur in the 0-run before the (d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , dt)-th 1, respectively, where d1 ≤ d2 ≤ · · · ≤ dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Also, ℓ (10 → 01) adjacent transpositions occur in (j1, j2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , jℓ)-th 1 and r (01 → 10) adjacent transpositions occur in (k1, k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , kr)-th 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Based on Proposition 5, considering all t 0-deletions and s adjacent transpositions and set K = t + s, we have a set of equations showing the change of syndromes for all m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=', t + s} as follows: a′′ m ≡ t � u=1 dm u + m−1 � i=0 ��m i �� ℓ � v=1 ji v − r � w=1 ki w �� mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (1) If there are only t 0-deletions without s adjacent transpositions, Dolecek and Anantharam [17] showed that the following system of equations has the unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Lemma 5 (Dolecek and Anatharam [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Without s adjacent transpositions, (1) can be rewritten as the following set of constraints with t equations such that \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 a′′ 1 ≡ d1 + d2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' + dt mod p, a′′ 2 ≡ d2 1 + d2 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' + d2 t mod p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' a′′ t ≡ dt 1 + dt 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' + dt t mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (2) which can uniquely determine the solution set {d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , dt}, where p is a prime such that p > 2n and d1 ≤ d2 ≤ · · · ≤ dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Following the technique in [17], if we can determine uniquely the solution set {d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , dt, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , jℓ, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , kr} of (1), we also can correct t 0-deletions and s adjacent transpositions with at most (t + s) log n bits of redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' However, the result is not known to us and is still open for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In this section, we focus on List-Decodable code CList t,s (n, κ, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) for correcting t 0-deletions and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Set K = κ in Construction 3, where κ = max(t, s + 1) and p is a prime such that p > 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For the following system of equations, we can determine the solution set uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' A set of constraints with s equations such that \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 b′′ 1 ≡ �ℓ v=1 j1 v − �r w=1 k1 w mod p, b′′ 2 ≡ �ℓ v=1 j2 v − �r w=1 k2 w mod p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' b′′ s ≡ �ℓ v=1 js v − �r w=1 ks w mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (3) 12 is capable of uniquely determining the solution set {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , jℓ, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , kr}, where p is a prime such that p > 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Also, ℓ+r ≤ s, j1 < j2 < · · · < jℓ, k1 < k2 < · · · < kr and jv ̸= kw, ∀v ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , ℓ}, w ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We note that Lemma 6 is similar to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The only difference is that the coefficients of all terms in Lemma 5 are positive while the coefficients of all terms in Lemma 6 can be either positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, we can use the same technique in Lemma 5 to prove Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Define the polynomials σ+(x) = ℓ � v=1 (1 − jvx) and σ−(x) = r � w=1 (1 − kwx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Let σ(x) = �s m=0 σmxm be defined by σ(x) = σ+(x)/σ−(x) mod xs Then, we define σ∗(x) = σ(x) mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We also define S∗(x) = ∞ � m=1 � ℓ � v=1 jm v − r � w=1 km w � xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' and S∗ m = �ℓ v=1 jm v − �r w=1 km w mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, we have Newton’s identities over GF(p) as follows σ∗(x)S∗(x) + x(σ∗(x))′ = 0 u−1 � m=0 σ∗ mS∗ u−m + uσ∗ u = 0, u ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (4) where (σ∗(x))′ is derivative of σ∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (see [35, Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='3] for details) Using the similar technique as the proof of Lemma 5, from (4), σ∗ m can be recursively obtained by {S∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , S∗ m} and {σ∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , σ∗ m−1}, where {S∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , S∗ m} = {b′′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , b′′ m}, which follows that all the coefficients of the polynomial σ∗(x) = �s m=0 σ∗ mxm mod p are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Further, we know that the polynomial σ∗(x) has at most s solutions by Lagrange Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Denote I0 = {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , jℓ, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , kr} with the value of each element in I0 is less than p and let Im = {j1 + mp, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , jℓ + mp, k1+mp, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , kr+mp} be one of the incongruent solution sets of I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We can have I0∩Im = ∅ due to p > 2n, which follows that all incongruent solutions are distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Therefore, we can conclude that the solution set {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , jℓ, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , kr} is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The list-decodable code CList t,s (n, κ, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) has redundancy κ log n, where κ = max(t, s + 1) and prime p > 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' If there are at most t 0-deletions and s adjacent transpositions, we can do list-decoding with list size O(nmin(t,s+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Let x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , xn) ∈ Σn 2 be the original vector and x′ be the received vector after t 0-deletions and s single adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, we can compute φ(x′) and a′ from x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Also, we can obtain a′′ = a′ − a, where a′′ = {a′′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , a′′ κ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Suppose t ≥ s + 1 and expand (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We have the following set of equations with κ = t: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 a′′ 1 ≡ �t u=1 du + (ℓ − r) mod p, a′′ 2 ≡ �t u=1 d2 u + (ℓ − r) + 2(�ℓ v=1 j1 v − �r w=1 k1 w) mod p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' a′′ t ≡ �t u=1 dt u + (ℓ − r) + t(�ℓ v=1 j1 v − �r w=1 k1 w) + · · · + t(�ℓ v=1 jt−1 v − �r w=1 kt−1 w ) mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (5) Recall that we can decode uniquely if we can determine the unique solution set of (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' However, the method to solve (5) uniquely is not known to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We know that, given e = {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , es+1}, we can solve the following equations uniquely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 e1 ≡ ℓ − r mod p, e2 ≡ (ℓ − r) + 2(�ℓ v=1 j1 v − �r w=1 k1 w) mod p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' es+1 ≡ (ℓ − r) + (s + 1)(�ℓ v=1 j1 v − �r w=1 k1 w) + · · · + (s + 1)(�ℓ v=1 js v − �r w=1 ks w) mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (6) Indeed, denote e′ = {e′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , e′ s+1} with me′ m = em − �m−1 i=1 �� m i−1 � e′ i � for all m ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=', s + 1} and e′ 1 = e1, we can rearrange (6) to be similar to Lemma 6 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 e′ 1 ≡ ℓ − r mod p, e′ 2 ≡ �ℓ v=1 j1 v − �r w=1 k1 w mod p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' e′ s+1 ≡ �ℓ v=1 js v − �r w=1 ks w mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (7) Therefore, based on Lemma 6, we can obtain the unique solution set {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , jℓ, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , kr} from (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Once the solution set {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , jℓ, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , kr} is obtained, we can compute the following values {es+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , et}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' em = m−1 � i=0 ��m i �� ℓ � v=1 ji v − r � w=1 ki w �� mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (8) where m ∈ {s + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 14 Denote a∗ = {a∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , a∗ t } with a∗ m = a′′ m − em, ∀m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=', t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Substituting (6) and (8) into (5), we obtain the following set of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 a∗ 1 ≡ �t u=1 du mod p, a∗ 2 ≡ �t u=1 d2 u mod p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' a∗ t ≡ �t u=1 dt u mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (9) The set of equations (9) provides the unique solution set {d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , dt} by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Therefore, the unique solution of all positions of 0-deletions and adjacent transpositions {d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , dt, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , jℓ, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , kr} can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' So, for each set of s+1 values {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , es+1}, we can obtain the set {d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , dt, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , jℓ, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , kr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' There are ps+1 sets of these values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' One of these sets corresponds to the true value of x and gives us the correct vector x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' So, we can do list-decoding with the list size O(ns+1) since p = O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Moreover, the size of the list-decodable code CList t,s (n, κ, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) with κ = t is at least 2n/(4n)t, that is, we need at most κ log n bits of redundancy to construct the code CList t,s (n, κ, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' When t < s + 1, we can do similarly to the case t ≤ s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In this case, we can do list-decoding with the list-size O(nt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The size of the code CList t,s (n, κ, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) is at least 2n/(4n)s+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, we can conclude that the list-decodable code CList t,s (n, κ, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) can correct t 0-deletions and s adjacent transpositions with list size at most O(nmin(t,s+1)) and has redundancy κ log n+O(1), where both t, s are constant and κ = max(t, s+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The decoding algorithm of the list-decodable code CList t,s (n, κ, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) for correcting t 0-deletions and s adjacent transpositions is summarized in Algorithm 2, where t > s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Algorithm 2: List decoding procedure Input: Corrupted Sequence x′ ∈ Σn−t 2 Output: O(ns+1) possible sequences, including the original codeword x ∈ CList t,s (n, κ, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) Compute φ(x′) based on x′ and compute a′′ to obtain (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' for e = (e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , es+1) such that ei ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=', p − 1}, ∀i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , s + 1} do Get the solution set {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , jℓ, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , kr} by (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Compute em from the solution set {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , jℓ, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , kr} using (8) for each s + 2 ≤ m ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Compute a∗ m = a′′ m − em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Solve (9) to obtain the unique solution set {d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , dt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' end For each fixed e, we can recover φ(x) from φ(x′) by a set of error positions {d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , dt, j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , jℓ, k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , kr} and then output x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 15 Next, we will present the result for a special case t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The list-decodable code CList 1,s (n, s + 1, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) can correct a single 0-deletion and s adjacent transpositions with list size at most 2s and has redundancy (s + 1) log n + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' When t = 1, It can be noticed that when the deletion position is determined, means d is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Since l, r ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , s} and a′′ 1 ≡ d + (ℓ − r) mod p, hence there are 2s choice for d, which means that the list size of CList 1,s (n, s + 1, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) is at most 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The above code CList 1,s (n, s + 1, a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' p) is capable of correcting a single 0-deletion and s adjacent transpositions with constant list size at most 2s and has redundancy (s + 1) log n + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The list size is constant 2s, which is less than the list size O(n) when we directly substitute t = 1 to Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' CODES FOR CORRECTING LIMITED-MAGNITUDE BLOCKS OF 0-DELETIONS AND ADJACENT TRANSPOSITIONS In this section, we focus on studying the error of t blocks of asymmetric deletions with ℓ-limited-magnitude and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' t blocks of asymmetric deletions with ℓ-limited-magnitude denotes that there are at most t blocks of 0s are deleted with the length of each block is at most ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Therefore, at most tℓ 0s are deleted and these t blocks of 0-deletions may occur in at most t 0 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For the sake of convenience in the following paper, we append a bit 1 at the end of x and denote it as x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Since the sequence x1 always ends with 1, x1 can be always written as x1 = 0u110u210u3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 0uw1, where w = wt(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In addition, we revisit the definition of function φ : Σn 2 → Σw and φ(x) def= (u1, u2, u3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' , uw) ∈ Σw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, combining with Proposition 2, we can have that the length of each 0 run increase by at most 1 and decrease by at most tℓ + 1 through t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, the definition of t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions is provided as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Define the error ball B(n, t, k+, k−) such that B(n, t, k+, k−) = {u ∈ Σn q : −k− ≤ ui ≤ k+, wt(u) ≤ t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' where at most t entries increase by at most k+ and decrease by at most k− for a sequence with length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' t blocks of asymmetric deletions with ℓ-limited-magnitude and s adjacent transpositions denote that given a sequence x ∈ Σn 2 , the retrieved sequence x′ through this type of error can be written as φ(x′1) = φ(x1) + v, where v ∈ B(w, t + 2s, 1, tℓ + 1) and w = wt(x′1) = wt(x1) 16 Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Suppose we have x = 0100101001 ∈ Σ10 2 with ℓ = 2, t = 3 and s = 1, then φ(x1) = 12120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' If the retrieved sequence x′ = 0110110 ∈ Σ6 2 and the corresponding φ(x′1) = 10101, by comparing φ(x1) and φ(x′1), we can see v = (0, −2, 0, −2, 1) ∈ B(5, 5, 1, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Denote Φ be the set of mapping Σn 2 by the function φ and Σn 2 is the set containing all binary sequences with length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The cardinality of Φ is: |Φ| = n+1 � w=1 � n w − 1 � = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For a binary sequence x ∈ Σn 2, the corresponding sequence φ(x1) is with length w = w(x1) and wt(φ(x1)) = n+1−w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Also, the cardinality of Φ can be considered the number of ways of arranging n + 1 − w indistinguishable objects in w distinguishable boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Thus, we can get the cardinality of Φ as shown in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' On the other side, since the mapping function φ is a one-to-one mapping function, the cardinality of Φ should be the same as |Σn 2| = 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' [36]) The code C(n, t, ℓ, s) for correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions is equivalent to a packing to Σw by the error ball B(w, t+2s, 1, tℓ+1), where w = wt(x) and x ∈ C(n, t, ℓ, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Non-systematic Code Construction In this section, we will provide a non-systematic construction for the code capable of correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, we present the decoding algorithm of this code and a lower bound of the code size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The code C(n, t, ℓ, s) is defined as C(n, t, ℓ, s) = {x ∈ Σn 2 : φ(x1) mod p ∈ Cp, wt(φ(x1)) = n + 1 − w}, where w = wt(x1) and Cp is a code over Σp with p is the smallest prime larger than tℓ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' C(n, t, ℓ, s) is capable of correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions for x ∈ C(n, t, ℓ, s) if Cp is capable of correcting t + 2s symmetric errors for φ(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' ( [37], Theorem 10 ) Let p be a prime such that the distance 2 ≤ d ≤ p⌈m/2⌉−1 and n = pm − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, there exists a narrow-sense [n, k, d]-BCH code Cp over Σp with n − k = ⌈(d − 1)(1 − 1/p)⌉m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 17 Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Let p be the smallest prime such that p ≥ tℓ + 2, w = pm − 1, w = wt(x1) and Cp is a primitive narrow-sense [w, k, 2(t + 2s) + 1]-BCH code with w − k = ⌈2(t + 2s)(1 − 1/p)⌉m, the code C(n, t, ℓ, s) such that C(n, t, ℓ, s) = {x ∈ Σn 2 : φ(x1) mod p ∈ Cp, wt(φ(x1)) = n + 1 − w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' is capable of correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Let x ∈ C(n, t, ℓ, s) be a codeword, and x′ be the output through the channel that has t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Let z′ = φ(x′1) mod p, where p is the smallest prime larger than tℓ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Run the decoding algorithm of Cp on z′ and output z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Thus, z∗ is also a linear code in Cp and it can be shown that z∗ = φ(x1) mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Denote ǫ′ = (z′ − z∗) mod p, we can have that (φ(x′1) − φ(x1)) mod p = (z′ − z∗) mod p = ǫ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (11) and the error vector ǫ satisfies ǫi = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ǫ′ i, if 0 ≤ ǫ′ i ≤ 1 ǫ′ i − p, otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (12) Hence, the output is φ(x1) = φ(x′1) − ǫ and then recover x from φ(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The detailed decoding steps are shown in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Algorithm 3: Decoding Algorithm of C(n, t, ℓ, s) Input: Retrieved sequence x′ Output: Decoded sequence x ∈ C(n, t, ℓ, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Initialization: Let p be the smallest prime larger than tℓ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Also, append 1 at the end of x′ and get φ(x′1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Step 1: z′ = φ(x′1) mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Run the decoding algorithm of Cp on z′ to get the output z∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Step 2: ǫ′ = (z′ − z∗) mod p and then ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' φ(x1) = φ(x′1) − ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Step 3: Output x1 = φ−1(φ(x1)) and then x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Suppose x = 0100101001 and x′ = 0110110 ∈ Σ6 2 with ℓ = 2, t = 3 and s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Since the retrieved sequence x′ = 0110110, then φ(x′1) = 10101 and z′ = φ(x′) mod 11 = 10101, where p = 11 is smallest prime such that p ≥ tℓ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Run the decoding algorithm of Cp on z′ ∈ Cp, we have the output sequence z∗ = 12120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence ǫ′ = (z − z∗) mod 11 = (0, 9, 0, 9, 1) and ǫ = (0, −2, 0, −2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Thus, the output of the decoding algorithm φ(x1) = φ(x′1) − ǫ = (1, 0, 1, 0, 1) − (0, −2, 0, −2, 1) = (1, 2, 1, 2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Finally, x1 = 01001010011 and x = 0100101001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Next, we will present a lower bound of the size of C(n, t, ℓ, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 18 Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The size of the code C(n, t, ℓ, s) in Theorem 10 is bounded by |C(n, t, ℓ, s)| ≥ 2n p(n + 1)⌈2(t+2s)(1−1/p)⌉ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' where p is the smallest prime larger than tℓ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Denote z = φ(x1) mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' φ(x1) can be written as φ(x1) → (z, a) such that φ(x1) = z + p · a, where a is a vector with the same length as φ(x1) and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Further, since z ∈ Cp and Cp is a linear code, the code Cp with length w can be considered as a set which is obtained by Σw p partitioned into pw−k classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Denote φ(x1)w as the φ(x1) with length w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Thus, for any fixed number of weight w, the cardinality of φ(x1)w such that φ(x1)w mod p ∈ Cp with length w is: |φ(x1)w| = � n w−1 � pw−k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, the size of the code C(n, t, ℓ, s) in Theorem 10 can be shown as: |C(n, t, ℓ, s)| = n+1 � w=1 |φ(x1)w| = n+1 � w=1 �� n w−1 � pw−k � ≥ �n+1 w=1 � n w−1 � pn+1−k = 2n pn+1−k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (13) From Lemma 9 and Theorem 10, let d = 2(t + 2s) + 1 and m = logp(n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' pn−k+1 = p⌈2(t+2s)(1−1/p)⌉·logp(n+1)+1 = p(n + 1)⌈2(t+2s)(1−1/p)⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (14) Therefore, from (13) and (14), the size of the code C(n, t, ℓ, s) in Theorem 10 is bounded by |C(n, t, ℓ, s)| ≥ 2n p(n + 1)⌈2(t+2s)(1−1/p)⌉ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' where p is the smallest prime larger than tℓ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Systematic Code Construction In the previous subsection, we propose a non-systematic code C(n, t, ℓ, s) for correcting t blocks of 0-deletions with ℓ-limited- magnitude and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In this subsection, we will provide the efficient encoding and decoding function based on the code C(n, t, ℓ, s) presented in Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 1) Efficient Encoding: Before providing the efficient systematic encoding algorithm, we now introduce a useful lemma proposed in [38] for encoding balanced sequences efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The balanced sequence denotes the binary sequence with an equal number of 0s and 1s, which will be used for distinguishing the boundary of redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 19 Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' [38]) Given the input x ∈ Σk 2, let the function s′ : Σk 2 → Σn 2 such that s′(x) ∈ Σn 2 is a balanced sequence, where n = k + log k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Given the input x ∈ Σk 2, let the function s : Σk 2 → Σn′ 2 such that s(x) ∈ Σn′ 2 whose first bit is 1 and s(x)[2,n′] is balanced sequence with (n′ − 1)/2 0s and (n′ − 1)/2 1s, where n′ = k + log k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' An adjacent transposition can be considered as two substitutions, hence the maximum total number of deletions and substitutions in the t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions is r = tℓ+2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The following lemma is used for correcting deletions, insertions and substitutions up to r = tℓ + 2s in a binary sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' [39]) Let t, ℓ, s be constants with respect to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' There exist an integer a ≤ 22r log k+o(log k) and a labeling function fr : Σk 2 → Σ2Rr(k), where Rr(k) = O(r4 log k) such that {(x, a, fr(x) mod a) : x ∈ Σk 2} can correct deletions, insertions and substitutions up to r = tℓ + 2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Let gr(x) = (a, fr(x) mod a) ∈ Σ4r log k+o(log k) 2 for given x ∈ Σk 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Next, we define the mapping function from non-binary to binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Given the input x ∈ Σk 2, define the function b : Σk p → Σn 2 such that b(u)[i·⌈log p⌉+1,(i+1)·⌈log p⌉] is the binary form of ui, where n = k · ⌈log p⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Given the parameters t, ℓ and s, let p be the smallest prime larger than tℓ + 2 and Cp in Lemma 9 be the p-ary primitive narrow-sense [n, k, 2(t + 2s) + 1]-BCH codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Define the labeling function as g : Σk p → Σn−k p such that (x, g(x)) is a p-ary primitive narrow-sense [n, k, 2(t + 2s) + 1]-BCH codes, where n = k + ⌈2(t + 2s)(1 − 1/p)⌉m and n = pm − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Suppose the input sequence is c ∈ Σk 2, and we have φ(c1) with length rc = wt(c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, let c′ = φ(c1) mod p ∈ Σrc p , where p is the smallest prime larger than tℓ + 2, and append 0k+1−rc at the end of c′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, we denote ¯c ∈ Σk+1 p = (c′, 0k+1−rc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Next, encode ¯c via the labeling function g of the p-ary primitive narrow-sense [n, k, 2(t + 2s) + 1]-BCH code and output the redundancy part g(¯c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We map the redundancy part g(¯c) into binary sequence b(g(¯c)) and make b(g(¯c)) to the balanced sequence s(b(g(¯c))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, we prepend two 1s as the protecting bits at the beginning of s(b(g(¯c))) and denote h1(¯c) = (1, 1, s(b(g(¯c)))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Further, we need to protect the redundancy part h1(¯c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The idea is to apply the code in Lemma 11 on h1(¯c) since the code in Lemma 11 is capable of correcting at most tℓ + 2s deletions and substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, we output gr(h1(¯c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In addition, make gr(h1(¯c)) to balanced sequence s(gr(h1(¯c))) and repeat its each bit 2tℓ + 3 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Let h2(¯c) = Rep2tℓ+3s(gr(h1(¯c))), 20 where Repkx is the k-fold repetition of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Finally, we have the output Enc(c) = (c, h(c)), where h(c) = (h1(¯c), h2(¯c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The detailed encoding steps are summarized in the following Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Algorithm 4: Encoding Algorithm Input: c ∈ Σk 2 Output: Encoded sequence Enc(c) ∈ ΣN 2 Initialization: Let p be the smallest prime larger than tℓ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Step 1: Append 1 at the end of c and get φ(c1) with length rc = wt(c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Step 2: c′ = φ(c1) mod p ∈ Σrc p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Append 0k+1−rc at the end of c′, then ¯c = (c′, 0k+1−rc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Step 3: Encode ¯c via Cp and output g(¯c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Mapping g(¯c) to balanced binary sequence s(b(g(¯c))) and introduce protecting bits h1(¯c) = (1, 1, s(b(g(¯c)))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Step 4: Protect h1(¯c) via gr and obtain the total redundancy h(c) = (h1(¯c), h2(¯c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Step 5: Output Enc(c) = (c, h(c)) ∈ ΣN 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Given a sequence c ∈ Σk 2, Algorithm 4 outputs an encoded sequence Enc(c) ∈ ΣN 2 capable of correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Therefore, the redundancy of the code h(c) = (h1(¯c), h2(¯c)) via this encoding process can be shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The total redundancy of the code Enc(c) ∈ ΣN 2 by given input c ∈ Σk 2 is N − k = ⌈2(t + 2s)(1 − 1/p)⌉ · ⌈log p⌉ log p log(N + 1) + O(log log N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' where p is smallest prime such that p ≥ tℓ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Let m = logp(N + 1), hence N = pm − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The lengths of the redundancy parts are as follows: n′′ 1 is the length of g(¯c): n′′ 1 = ⌈2(t + 2s)(1 − 1/p)⌉m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' n′ 1 is the length of b(g(¯c)): n′ 1 = n′′ 1 · ⌈log p⌉;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' n1 is the length of h1(¯c): n1 = n′ 1 + log n′ 1 + 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' n′′ 2 is the length of gr(h1(¯c)): n′′ 2 = 4(tℓ + 2s) log n1 + log n1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' n′ 2 is the length of s(f0(h1(¯c))): n′ 2 = n′′ 2 + log n′′ 2 + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' n2 is the length of h2(¯c): n2 = (2tℓ + 3)n′ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 21 Based on the above statement, we can see that N − k = n1 + n2, where n′ 1 = (⌈2(t + 2s)(1 − 1/p)⌉m) · ⌈log p⌉ with m = logp(N + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, we have n′ 1 = ⌈2(t + 2s)(1 − 1/p)⌉ · ⌈log p⌉ log p log(N + 1) Since both t, p and s are constants, then log n′ 1 = O(log log N) and n2 = O(log log N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Therefore, the total redundancy of the code Enc(c) ∈ ΣN 2 given the input c ∈ Σk 2 can be shown as the Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 2) Decoding Algorithm: Without loss of generality, suppose the encoded sequence Enc(c) ∈ ΣN 2 is transmitted through the t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions channel, and we have the retrieved sequence d ∈ ΣN−tℓ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' In this subsection, we will introduce the decoding algorithm for obtaining Dec(d) ∈ Σk 2 by given d ∈ ΣN−tℓ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' First, we need to distinguish where the redundancy part begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Since the error type is at most t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions, the number of 1s in d is the same as that of in Enc(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Thus, we can count the number of 1s from the end of d to find the beginning of the redundancy since the redundancy part is the balanced sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Hence, we find the (n2 + 2tℓ + 3)/2-th 1 and (n1/2 + n2/2 + tℓ + 3)-th 1 from the end of d and denote their entries as ir2 and ir1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For the subsequence d[ir2,N−tℓ], since there are at most tℓ 0s deletions and s adjacent transpositions occur in Enc(c)[N−n2+1,N], the (2tℓ + 3)-fold repetition code can help recover s(gr(h1(¯c))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Further, we can obtain parity bits gr(h1(¯c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For the subsequence d[ir1,ir2−1], there are also at most tℓ 0-deletions and 2s substitutions occur in Enc(c)[N−n1−n2+1,N−n2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The recovered parity bits gr(h1(¯c)) can help recover h1(¯c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Further, we remove the two 1 bits at the beginning of h1(¯c) and get the g(¯c) from h1(¯c) = s(b(g(¯c))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Finally, denote z = (φ(d[1,ir1−1], 1), 0k+1−rc) and z′ = z mod p, where rc is the length of φ(d[1,ir1−1], 1) and k = N − n1 − n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Then, the following decoding steps are the same as Algorithm 3 where z′ is the input of Step 1 of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' The only difference is we need to first remove 0k+1−rc at the end before the last step of φ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Therefore, the main steps for decoding d ∈ ΣN−tℓ 2 is summerized in Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 3) Time Complexity: For the encoding algorithm, it can be easily shown that the time complexity is dominated by the p-ary narrow-sense BCH code and the code in Lemma 11, which is O(tn log n + (log n)2(tℓ+2s)+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' For the decoding algorithm, the time complexity is also dominated by the decoding of the p-ary narrow-sense BCH code and decoding for the code in Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Therefore, the total time complexity of decoding is O(tn + (log n)tℓ+2s+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' 22 Algorithm 5: Decoding Algorithm Input: d ∈ ΣN−tℓ 2 Output: Decoded sequence Dec(d) ∈ Σk 2 Initialization: Let p be the smallest prime larger than tℓ + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Step 1: Find the (n2 + 2tℓ + 3)/2-th 1 and (n1/2 + n2/2 + tℓ + 3)-th 1 from the end of d and denote their entries as ir2 and ir1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Step 2: Recover s(gr(h1(¯c))) from d[ir2,N−tℓ] and then get gr(h1(¯c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Step 3: Recover h1(¯c) via gr(h1(¯c)) and then obtain h1(¯c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Step 4: Denote z′ = (φ(d[1,ir1−1], 1), 0k+1−rc) mod p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Input z′ to Step 1 of Algorithm 3 and run the remaining steps of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Step 5: Output Dec(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' CONCLUSION In this paper, motivated by the errors in the DNA data storage and flash memories, we presented codes for correcting asymmetric deletions and adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We first present three uniquely-decodable codes for different types of asymmetric deletions and adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' We then construct a list-decodable code for correcting asymmetric deletions and adjacent transpositions with low redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' At last, we present the code for correcting t blocks of 0-deletions with ℓ-limited-magnitude and s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' However, there still remain some interesting problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Construct codes that are capable of correcting symmetric t deletions and s adjacent transpositions with low redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Construct codes that are capable of correcting t deletions/insertions + k substitutions + s adjacent transpositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' Construct codes for Damerau-Levenshtein distance for larger number of errors, not only constant t and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9FJT4oBgHgl3EQf7y3s/content/2301.11680v1.pdf'} +page_content=' REFERENCES [1] S.' metadata={'source': 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0000000000000000000000000000000000000000..47eb131d50a8750f1df09a3feeb7ae39be0a1d2e --- /dev/null +++ b/_9AyT4oBgHgl3EQfRfam/content/tmp_files/2301.00068v1.pdf.txt @@ -0,0 +1,488 @@ +On the Inconsistencies of Conditionals Learned by Masked Language +Models +Tom Young +Yang You +School of Computing, +National University of Singapore +tomyoung@nus.edu.sg +youy@comp.nus.edu.sg +Abstract +Learning to predict masked tokens in a se- +quence has been shown to be a powerful +pretraining objective for large-scale language +models. After training, such masked language +models can provide distributions of tokens con- +ditioned on bidirectional context. +In this short draft, we show that such bidirec- +tional conditionals often demonstrate consider- +able inconsistencies, i.e., they can not be de- +rived from a coherent joint distribution when +considered together. We empirically quantify +such inconsistencies in the simple scenario of +bigrams for two common styles of masked lan- +guage models: T5-style and BERT-style 1. For +example, we show that T5 models often con- +fuse its own preference regarding two similar +bigrams. +Such inconsistencies may represent a theoreti- +cal pitfall for the research work on sampling +sequences based on the bidirectional condi- +tionals learned by BERT-style MLMs. This +phenomenon also means that T5-style MLMs +capable of infilling will generate discrepant +results depending on how much masking is +given, which may represent a particular trust +issue. +1 +Introduction +Pretraining objectives of large language models +can be roughly divided into two categories. First, +vanilla next token prediction (Brown et al., 2020) +aims to learn the distribution of the next token in a +sequence given the context to the left. Second, the +masked language modeling (MLM) objective (De- +vlin et al., 2018; Raffel et al., 2020), which masks +out a portion of the tokens in a sequence and asks +the model to predict them, aims to learn the distri- +bution of one or more tokens given bidirectional +context. +1https://github.com/tomyoung903/MLM_ +inconsistencies +While the major breakthrough, aka, GPT3 +(Brown et al., 2020) was demonstrated using +vanilla next token prediction, recent work (Tay +et al., 2022; Zeng et al., 2022; Bavarian et al., 2022) +has hinted that incorporating the masked language +modeling objective may be highly beneficial. In ad- +dition, (Tay et al., 2022) has demonstrated that such +bidirectional conditionals provide strong infilling +capabilities. +One may notice that, unlike the unidirectional +conditional distributions that vanilla next token pre- +diction learns, the bidirectional conditionals that +MLMs learn are overly abundant in terms of rep- +resenting a coherent joint distribution. Therefore, +they are not guaranteed to be self-consistent (see +Chapter 2). +A very simple example for such inconsisten- +cies is shown in Figure 1. In this example, we +obtain the bidirectional conditional distributions +that the T5 model learned using two input masked +sequences. The two similar sequences are designed +with a small difference, in order to examine if the +resulting conditionals satisfy a basic law of prob- +abilities (hold consistency). Results clearly show +otherwise. We design experiments to quantify such +inconsistencies in Chapter 3. +One interesting line of research in the litera- +ture focused on whether and how the bidirectional +conditionals that BERT-style MLMs provide can +be used to construct the joint probability of a se- +quence in a principled manner (Goyal et al., 2021; +Ghazvininejad et al., 2019; Wang et al., 2019), just +like vanilla next token prediction models. But the +numerous papers on this topic have overlooked +the concern of inconsistencies. (Yamakoshi et al., +2022) stated that “any deviations (supposedly) tend +to be negligible with large datasets”. The experi- +ments shown in Chapter 4 demonstrate that this is +not the case at all. We thus posit that addressing the +consistency issue should be treated as the first step +in modeling the joint distribution with BERT-style +arXiv:2301.00068v1 [cs.CL] 30 Dec 2022 + +MLMs. +2 +Why inconsistencies can occur in +MLMs +For a set of conditional distributions to be self- +consistent, they need to be able to be derived from +a single coherent joint distribution. +One essential reason for the inconsistencies to +occur among the conditionals provided by a trained +MLM is that the number of conditionals it can +calculate far exceeds the number of degrees of free- +dom of a joint distribution. +Consider a sequence of length L and with vo- +cabulary V , the joint distribution of the tokens in +such a sequence is defined by |V |L probabilities +that sum to 1. Therefore, the number of degrees of +freedom (D) of such a joint distribution is given +by: +Djoint = |V |L − 1, +(1) +Vanilla next token prediction models or MLMs +essentially learn conditionals that predict some +tokens in the sequence given others. Such con- +ditional probabilities and probabilities from the +joint distribution can be linearly derived from each +other. Therefore, each free conditional that the +language model is capable of specifying provides +an additional constraint on the joint distribution. +One can easily verify that a vanilla next token pre- +diction based language model provides |V |L − 1 +free conditionals 2 to just exactly determine the +joint distribution. +Therefore, a vanilla next to- +ken prediction model (no matter how it is trained, +or even untrained) would never suffer from self- +inconsistencies. +MLMs, which can provide distributions of +masked tokens given bidirectional context, could +specify far more free conditionals. +Even for the simplest case, where the MLM pre- +dicts the distribution of only 1 (masked) token +given L − 1 other (unmasked) tokens in the se- +quence, the total number of free conditionals (N) +is +Nmlm(1) = L × (|V |L − |V |L−1), +(2) +Just Nmlm(1) is already far larger than Djoint. +We leave the discussions for Nmlm(k) for later +2A single softmax operation over V essentially gives |V |− +1 free conditionals. Here we call conditionals free when they +can be assigned any values decided by an underlying neural +network. +work. This fact sets up room for there to be in- +consistencies among the conditionals an MLM pro- +vides. +We explain our strategies and quantification +methods for diagnosing T5-style and BERT-style +MLMs in the next 2 sections. +3 +Diagnosing T5-style MLMs +T5-style MLMs are capable of modeling the dis- +tribution of segments of variable length in a given +bidirectional context. Here we use the simple bi- +gram scenario to expose the inconsistencies that +exist among such distributions. Consider two bi- +grams x1x21 and x1x22 that share a same token x1 +in the first position, the conditional distributions +concerning such two bigrams should satisfy +p(x21|x1) +p(x22|x1) = p(x1x21) +p(x1x22) +(3) +The left hand side can be obtained by only mask- +ing the second token, leaving x1 in the context. +While the right hand side can be obtained by mask- +ing the whole bigram. For the example in Figure 1, +“chicken” corresponds to x1. “Salad” and “breast” +correspond to x21 and x22. +We automatically build such a dataset of bigram +pairs in a given context by running BART (Lewis +et al., 2019) on a portion of the C4 dataset (Raffel +et al., 2020) to generate another plausible bigram +alternative to an existing one. We then use the two +sequences to test T5’s inconsistencies regarding +Equation 3 3. +We can use relative difference (dr) of the left +and right hand side of Equation 3 to quantify the +inconsistency. +dr = |lhs(3) − rhs(3)| +lhs(3) +(4) +dr is expected to be 0 for a self-consistent MLM. +Table 1 shows that dr is typically very large for +the T5 family, although scaling up the model has a +markable effect on reducing it. +Another way to quantify the inconsistency re- +garding the two bigrams is to count how often a +severe case happens where the MLM disagrees +with itself on which bigram it prefers. I.e., some- +times lhs(3) > 1 and rhs(3) < 1, or lhs(3) < 1 +3We focus on plausible bigrams in this draft because they +are most relevant in practice but Equation 3 should hold for +all bigrams in all sentences in all corpora in a self-consistent +MLM. + +The +is a common choice of food. + +option +𝑝 +… +… +breast +0.030 +… +… +salad +0.024 +… +… +The +is a common choice of food. + +option +𝑝 +… +… +chicken salad +0.00028 +… +… +chicken breast +0.00017 +… +… +Basic law of probabilities +𝑝 salad chicken) +𝑝 breast chicken) = 𝑝(chicken salad) +𝑝(chicken breast) +𝑝 breast chicken) > 𝑝 salad chicken) +𝑝(chicken breast) < 𝑝(chicken salad) +chicken +Figure 1: A simple bigram example that exposes the inconsistencies in the T5 model. The conditional probabilities +that the model learned (quoted from t5-11b fed with the shown masked sequences) contradict each other greatly. +Not only are the ratios unbalanced, the model confuses its own preference of the two bigrams. +and rhs(3) > 1. +Figure 1 shows such a case, +where t5-11b prefers “chicken salad” over “chicken +breast” when considering the conditionals provided +in rhs(3), yet its preference flips when considering +lhs(3). Table 1 shows that disagreement on com- +parison happens with considerable frequency, but +scaling up models helps reduce it. +4 +Diagnosing BERT-style MLMs +Ever since the success of BERT (Devlin et al., +2018), there has been research effort (Goyal et al., +2021; Wang et al., 2019; Yamakoshi et al., 2022) +on sampling sequences from it by modeling its +implicitly specified joint distribution one way or +another. For example, (Goyal et al., 2021) views +it as an energy-based model defined using the +bidirectional conditionals of the masked tokens. +Such research effort is based on the intuition that +bidirectional conditionals could be more robust +than auto-regressive (unidirectional) conditionals +(Goyal, 2021). +This line of research operates based on the as- +sumption that the overly abundant bidirectional +conditionals that the BERT-style MLMs provide +are self-consistent. (Yamakoshi et al., 2022) based +on (Heckerman et al., 2000; Neville and Jensen, +2007) stated that “any deviations (supposedly) tend +to be negligible”. +We demonstrate in this section that this is not +the case at all. There are considerable inconsis- +tencies that exist among the bidirectional condi- +tionals that a trained BERT-style model provides. +Figure 2 demonstrates such an example. Again +we use bigrams as the simplest example to expose +the inconsistencies. Because BERT-style MLMs +cannot directly model the distribution of multiple +tokens together (local joint distribution), we con- +sider 4 bigrams this time: x11x21, x11x22, x12x21 +and x12x22. x11 and x12 are two possible tokens +that the first position can take. x21 and x22 the sec- +ond. One can easily verify 4 that the 8 conditional +distributions concerning such four bigrams should +theoretically satisfy +p(x21|x11) +p(x22|x11) × p(x11|x22) +p(x12|x22) = +p(x11|x21) +p(x12|x21) × p(x21|x12) +p(x22|x12) +(5) +One way to test the inconsistencies among the 8 +conditionals is to try to solve one using the other +7 and compare the solved conditional with the +original (inferred by model) one. We show the +solved conditionals in the example in Figure 2. It +clearly demonstrates that the probabilities given by +4Clue: converting each term to local joint distributions. + +I had +eggs + +I had +at lunch. + +𝑝 +Inferred +Solved +𝑝 duck eggs) +1.7 × 10−4 +9.0 × 10−6 +𝑝 chicken eggs) +1.1 × 10−3 +0.020 +𝑝 duck soup) +3.1 × 10−4 +5.8 × 10−3 +𝑝 chicken soup) +0.17 +9.2 × 10−3 +𝑝 eggs duck) +5.7 × 10−3 +0.11 +𝑝 soup duck) +0.23 +0.012 +𝑝 eggs chicken) +6.8 × 10−4 +3.7 × 10−5 +𝑝 soup chicken) +0.13 +2.31 +𝑝(eggs|duck) +𝑝(soup|duck) × +𝑝(duck|soup) +𝑝(chicken|soup) = +𝑝(duck|eggs) +𝑝(chicken|eggs) × 𝑝(eggs|chicken) +𝑝(soup|chicken) +at lunch. +I had +soup + +at lunch. +duck +I had +at lunch. + +chicken +Figure 2: An example of inconsistencies in the BERT-style MLM. Each “inferred” value refers to the probability +given by the MLM (RoBERTa-large in this figure). Each “solved” value is obtained by passing the other 7 “inferred” +values to the equation in the red square. We see that the difference between each inferred and solved value is +significant. And the solved value may even be larger than 1. +Metric +T5-base +T5-large +T5-3b +T5-11b +Relative difference (dr, median, %) +47.5 +45.8 +44.7 +42.0 +Disagreement on comparison (%) +9.64 +8.85 +7.53 +6.54 +Table 1: Inconsistencies in the T5 model tested on 19399 pairs of bigrams. We show the median value for relative +difference as it is resilient to outliers. +a BERT-style MLM can be in serious inconsisten- +cies with each other. We build a testing dataset +with 4 such plausible bigrams for each context and +quantify consistencies in using difference of log +probabilities: +| log psolved − log pinferred| +(6) +Table 2 shows the results. +5 +Summary +This draft demonstrates and naively quantifies the +inconsistencies that exist in large MLMs in the +simple scenario of bigrams. Such inconsistencies +originate from the fact that the number of bidirec- +tional conditionals MLMs can learn far exceeds +what is needed for constructing the joint distribu- +tion. Given the recent evidence that MLM-based +pretraining might be a powerful paradigm, we think +that resolving the its consistency issue could be a +necessary step for future work. +Acknowledgements +We would like to thank Fuzhao Xue for the useful +discussions. +References +Mohammad Bavarian, Heewoo Jun, Nikolas Tezak, +John Schulman, Christine McLeavey, Jerry Tworek, +and Mark Chen. 2022. +Efficient training of lan- +guage models to fill in the middle. arXiv preprint +arXiv:2207.14255. +Tom B Brown, Benjamin Mann, Nick Ryder, Melanie +Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind +Neelakantan, Pranav Shyam, Girish Sastry, Amanda +Askell, et al. 2020. Language models are few-shot +learners. arXiv preprint arXiv:2005.14165. +Jacob Devlin, Ming-Wei Chang, Kenton Lee, and +Kristina Toutanova. 2018. 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In Findings of the Association +for Computational Linguistics: ACL 2022, pages +3977–3992. +Aohan Zeng, Xiao Liu, Zhengxiao Du, Zihan Wang, +Hanyu Lai, Ming Ding, Zhuoyi Yang, Yifan Xu, +Wendi Zheng, Xiao Xia, et al. 2022. +Glm-130b: +An open bilingual pre-trained model. arXiv preprint +arXiv:2210.02414. + diff --git a/_9AyT4oBgHgl3EQfRfam/content/tmp_files/load_file.txt b/_9AyT4oBgHgl3EQfRfam/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3dac3cd53a714106896adfe9c459d2d83ac0708b --- /dev/null +++ b/_9AyT4oBgHgl3EQfRfam/content/tmp_files/load_file.txt @@ -0,0 +1,238 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf,len=237 +page_content='On the Inconsistencies of Conditionals Learned by Masked Language Models Tom Young Yang You School of Computing, National University of Singapore tomyoung@nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='sg youy@comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='sg Abstract Learning to predict masked tokens in a se- quence has been shown to be a powerful pretraining objective for large-scale language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' After training, such masked language models can provide distributions of tokens con- ditioned on bidirectional context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' In this short draft, we show that such bidirec- tional conditionals often demonstrate consider- able inconsistencies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', they can not be de- rived from a coherent joint distribution when considered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' We empirically quantify such inconsistencies in the simple scenario of bigrams for two common styles of masked lan- guage models: T5-style and BERT-style 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' For example, we show that T5 models often con- fuse its own preference regarding two similar bigrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Such inconsistencies may represent a theoreti- cal pitfall for the research work on sampling sequences based on the bidirectional condi- tionals learned by BERT-style MLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' This phenomenon also means that T5-style MLMs capable of infilling will generate discrepant results depending on how much masking is given, which may represent a particular trust issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 1 Introduction Pretraining objectives of large language models can be roughly divided into two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' First, vanilla next token prediction (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2020) aims to learn the distribution of the next token in a sequence given the context to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Second, the masked language modeling (MLM) objective (De- vlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2020), which masks out a portion of the tokens in a sequence and asks the model to predict them, aims to learn the distri- bution of one or more tokens given bidirectional context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='com/tomyoung903/MLM_ inconsistencies While the major breakthrough, aka, GPT3 (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2020) was demonstrated using vanilla next token prediction, recent work (Tay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Bavarian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2022) has hinted that incorporating the masked language modeling objective may be highly beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' In ad- dition, (Tay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2022) has demonstrated that such bidirectional conditionals provide strong infilling capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' One may notice that, unlike the unidirectional conditional distributions that vanilla next token pre- diction learns, the bidirectional conditionals that MLMs learn are overly abundant in terms of rep- resenting a coherent joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Therefore, they are not guaranteed to be self-consistent (see Chapter 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' A very simple example for such inconsisten- cies is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' In this example, we obtain the bidirectional conditional distributions that the T5 model learned using two input masked sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' The two similar sequences are designed with a small difference, in order to examine if the resulting conditionals satisfy a basic law of prob- abilities (hold consistency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Results clearly show otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' We design experiments to quantify such inconsistencies in Chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' One interesting line of research in the litera- ture focused on whether and how the bidirectional conditionals that BERT-style MLMs provide can be used to construct the joint probability of a se- quence in a principled manner (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Ghazvininejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2019), just like vanilla next token prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' But the numerous papers on this topic have overlooked the concern of inconsistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' (Yamakoshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2022) stated that “any deviations (supposedly) tend to be negligible with large datasets”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' The experi- ments shown in Chapter 4 demonstrate that this is not the case at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' We thus posit that addressing the consistency issue should be treated as the first step in modeling the joint distribution with BERT-style arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='00068v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='CL] 30 Dec 2022 MLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 2 Why inconsistencies can occur in MLMs For a set of conditional distributions to be self- consistent, they need to be able to be derived from a single coherent joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' One essential reason for the inconsistencies to occur among the conditionals provided by a trained MLM is that the number of conditionals it can calculate far exceeds the number of degrees of free- dom of a joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Consider a sequence of length L and with vo- cabulary V , the joint distribution of the tokens in such a sequence is defined by |V |L probabilities that sum to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Therefore, the number of degrees of freedom (D) of such a joint distribution is given by: Djoint = |V |L − 1, (1) Vanilla next token prediction models or MLMs essentially learn conditionals that predict some tokens in the sequence given others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Such con- ditional probabilities and probabilities from the joint distribution can be linearly derived from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Therefore, each free conditional that the language model is capable of specifying provides an additional constraint on the joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' One can easily verify that a vanilla next token pre- diction based language model provides |V |L − 1 free conditionals 2 to just exactly determine the joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Therefore, a vanilla next to- ken prediction model (no matter how it is trained, or even untrained) would never suffer from self- inconsistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' MLMs, which can provide distributions of masked tokens given bidirectional context, could specify far more free conditionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Even for the simplest case, where the MLM pre- dicts the distribution of only 1 (masked) token given L − 1 other (unmasked) tokens in the se- quence, the total number of free conditionals (N) is Nmlm(1) = L × (|V |L − |V |L−1), (2) Just Nmlm(1) is already far larger than Djoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' We leave the discussions for Nmlm(k) for later 2A single softmax operation over V essentially gives |V |− 1 free conditionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Here we call conditionals free when they can be assigned any values decided by an underlying neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' This fact sets up room for there to be in- consistencies among the conditionals an MLM pro- vides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' We explain our strategies and quantification methods for diagnosing T5-style and BERT-style MLMs in the next 2 sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 3 Diagnosing T5-style MLMs T5-style MLMs are capable of modeling the dis- tribution of segments of variable length in a given bidirectional context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Here we use the simple bi- gram scenario to expose the inconsistencies that exist among such distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Consider two bi- grams x1x21 and x1x22 that share a same token x1 in the first position, the conditional distributions concerning such two bigrams should satisfy p(x21|x1) p(x22|x1) = p(x1x21) p(x1x22) (3) The left hand side can be obtained by only mask- ing the second token, leaving x1 in the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' While the right hand side can be obtained by mask- ing the whole bigram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' For the example in Figure 1, “chicken” corresponds to x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' “Salad” and “breast” correspond to x21 and x22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' We automatically build such a dataset of bigram pairs in a given context by running BART (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2019) on a portion of the C4 dataset (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2020) to generate another plausible bigram alternative to an existing one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' We then use the two sequences to test T5’s inconsistencies regarding Equation 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' We can use relative difference (dr) of the left and right hand side of Equation 3 to quantify the inconsistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' dr = |lhs(3) − rhs(3)| lhs(3) (4) dr is expected to be 0 for a self-consistent MLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Table 1 shows that dr is typically very large for the T5 family, although scaling up the model has a markable effect on reducing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Another way to quantify the inconsistency re- garding the two bigrams is to count how often a severe case happens where the MLM disagrees with itself on which bigram it prefers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', some- times lhs(3) > 1 and rhs(3) < 1, or lhs(3) < 1 3We focus on plausible bigrams in this draft because they are most relevant in practice but Equation 3 should hold for all bigrams in all sentences in all corpora in a self-consistent MLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' The is a common choice of food.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' option 𝑝 … … breast 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='030 … … salad 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='024 … … The is a common choice of food.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' option 𝑝 … … chicken salad 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='00028 … … chicken breast 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='00017 … … Basic law of probabilities 𝑝 salad chicken) 𝑝 breast chicken) = 𝑝(chicken salad) 𝑝(chicken breast) 𝑝 breast chicken) > 𝑝 salad chicken) 𝑝(chicken breast) < 𝑝(chicken salad) chicken Figure 1: A simple bigram example that exposes the inconsistencies in the T5 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' The conditional probabilities that the model learned (quoted from t5-11b fed with the shown masked sequences) contradict each other greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Not only are the ratios unbalanced, the model confuses its own preference of the two bigrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' and rhs(3) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Figure 1 shows such a case, where t5-11b prefers “chicken salad” over “chicken breast” when considering the conditionals provided in rhs(3), yet its preference flips when considering lhs(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Table 1 shows that disagreement on com- parison happens with considerable frequency, but scaling up models helps reduce it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 4 Diagnosing BERT-style MLMs Ever since the success of BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2018), there has been research effort (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Yamakoshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2022) on sampling sequences from it by modeling its implicitly specified joint distribution one way or another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' For example, (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2021) views it as an energy-based model defined using the bidirectional conditionals of the masked tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Such research effort is based on the intuition that bidirectional conditionals could be more robust than auto-regressive (unidirectional) conditionals (Goyal, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' This line of research operates based on the as- sumption that the overly abundant bidirectional conditionals that the BERT-style MLMs provide are self-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' (Yamakoshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2022) based on (Heckerman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Neville and Jensen, 2007) stated that “any deviations (supposedly) tend to be negligible”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' We demonstrate in this section that this is not the case at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' There are considerable inconsis- tencies that exist among the bidirectional condi- tionals that a trained BERT-style model provides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Figure 2 demonstrates such an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Again we use bigrams as the simplest example to expose the inconsistencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Because BERT-style MLMs cannot directly model the distribution of multiple tokens together (local joint distribution), we con- sider 4 bigrams this time: x11x21, x11x22, x12x21 and x12x22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' x11 and x12 are two possible tokens that the first position can take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' x21 and x22 the sec- ond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' One can easily verify 4 that the 8 conditional distributions concerning such four bigrams should theoretically satisfy p(x21|x11) p(x22|x11) × p(x11|x22) p(x12|x22) = p(x11|x21) p(x12|x21) × p(x21|x12) p(x22|x12) (5) One way to test the inconsistencies among the 8 conditionals is to try to solve one using the other 7 and compare the solved conditional with the original (inferred by model) one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' We show the solved conditionals in the example in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' It clearly demonstrates that the probabilities given by 4Clue: converting each term to local joint distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' I had eggs I had at lunch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 𝑝 Inferred Solved 𝑝 duck eggs) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='7 × 10−4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='0 × 10−6 𝑝 chicken eggs) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='1 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='020 𝑝 duck soup) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='1 × 10−4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='8 × 10−3 𝑝 chicken soup) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='17 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='2 × 10−3 𝑝 eggs duck) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='7 × 10−3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='11 𝑝 soup duck) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='012 𝑝 eggs chicken) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='8 × 10−4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='7 × 10−5 𝑝 soup chicken) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='31 𝑝(eggs|duck) 𝑝(soup|duck) × 𝑝(duck|soup) 𝑝(chicken|soup) = 𝑝(duck|eggs) 𝑝(chicken|eggs) × 𝑝(eggs|chicken) 𝑝(soup|chicken) at lunch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' I had soup at lunch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' duck I had at lunch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' chicken Figure 2: An example of inconsistencies in the BERT-style MLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Each “inferred” value refers to the probability given by the MLM (RoBERTa-large in this figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Each “solved” value is obtained by passing the other 7 “inferred” values to the equation in the red square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' We see that the difference between each inferred and solved value is significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' And the solved value may even be larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Metric T5-base T5-large T5-3b T5-11b Relative difference (dr, median, %) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='5 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='8 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='7 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='0 Disagreement on comparison (%) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='64 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='85 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='53 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='54 Table 1: Inconsistencies in the T5 model tested on 19399 pairs of bigrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' We show the median value for relative difference as it is resilient to outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' a BERT-style MLM can be in serious inconsisten- cies with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' We build a testing dataset with 4 such plausible bigrams for each context and quantify consistencies in using difference of log probabilities: | log psolved − log pinferred| (6) Table 2 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 5 Summary This draft demonstrates and naively quantifies the inconsistencies that exist in large MLMs in the simple scenario of bigrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Such inconsistencies originate from the fact that the number of bidirec- tional conditionals MLMs can learn far exceeds what is needed for constructing the joint distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Given the recent evidence that MLM-based pretraining might be a powerful paradigm, we think that resolving the its consistency issue could be a necessary step for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Acknowledgements We would like to thank Fuzhao Xue for the useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' References Mohammad Bavarian, Heewoo Jun, Nikolas Tezak, John Schulman, Christine McLeavey, Jerry Tworek, and Mark Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Efficient training of lan- guage models to fill in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' arXiv preprint arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='14255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Tom B Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Language models are few-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' arXiv preprint arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='14165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Bert: Pre-training of deep bidirectional transformers for language understand- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='04805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Marjan Ghazvininejad, Omer Levy, Yinhan Liu, and Luke Zettlemoyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Mask-predict: Parallel Metric Roberta-base Roberta-large log-probability difference 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='836 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='792 Table 2: Difference of log-probabilities between inferred and solved conditionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' The difference would be 0 for self-consistent MLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Roughly a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='8 difference means that one is 120% larger than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' decoding of conditional masked language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' arXiv preprint arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='09324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Goyal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Characterizing and overcoming the limitations of neural autoregressive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' PhD thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Kartik Goyal, Chris Dyer, and Taylor Berg-Kirkpatrick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Exposing the implicit energy networks behind masked 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='13461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Jennifer Neville and David Jensen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Relational dependency networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Journal of Machine Learning Research, 8(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, Peter J Liu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Exploring the limits of transfer learning with a unified text-to-text trans- former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=', 21(140):1–67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Yi Tay, Jason Wei, Hyung Won Chung, Vinh Q Tran, David R So, Siamak Shakeri, Xavier Gar- cia, Huaixiu Steven Zheng, Jinfeng Rao, Aakanksha Chowdhery, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Transcending scaling laws with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='1% extra compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='11399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Alex Wang, Kyunghyun Cho, and CIFAR Azrieli Global Scholar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Bert has a mouth, and it must speak: Bert as a markov random field language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' NAACL HLT 2019, page 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Takateru Yamakoshi, Thomas L Griffiths, and Robert Hawkins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Probing bert’s priors with serial re- production chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' In Findings of the Association for Computational Linguistics: ACL 2022, pages 3977–3992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Aohan Zeng, Xiao Liu, Zhengxiao Du, Zihan Wang, Hanyu Lai, Ming Ding, Zhuoyi Yang, Yifan Xu, Wendi Zheng, Xiao Xia, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' Glm-130b: An open bilingual pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} +page_content='02414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_9AyT4oBgHgl3EQfRfam/content/2301.00068v1.pdf'} diff --git a/_dA0T4oBgHgl3EQfPf96/content/tmp_files/2301.02176v1.pdf.txt b/_dA0T4oBgHgl3EQfPf96/content/tmp_files/2301.02176v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..33dfd177bbd8a3b84ecc1aa12bee55d0aaa0f599 --- /dev/null +++ b/_dA0T4oBgHgl3EQfPf96/content/tmp_files/2301.02176v1.pdf.txt @@ -0,0 +1,1309 @@ +Draft version January 6, 2023 +Preprint typeset using LATEX style AASTeX6 v. 1.0 +AN EQUATION OF STATE OF CO FOR USE IN PLANETARY MODELING +M. Podolak +Dept. of Geosciences, Tel Aviv University, Tel Aviv, 69978 Israel +A. Levi +Braude College of Engineering, Karmiel, 2161002 Israel +A. Vazan +Astrophysics Research Center (ARCO), Dept. of Natural Sciences, Open University of Israel, Raanana, 43107 Israel +U. Malamud +Dept. of Geosciences, Tel Aviv University, Tel Aviv, 69978 Israel +Department of Physics, Technion – Israel Institute of Technology, Technion City, 3200003 Haifa, Israel +ABSTRACT +Although carbon monoxide (CO) is an abundant molecule and may have great importance for planetary interiors, +measurements of its properties are difficult due to its extreme volatility. We calculate the equation of state for CO over +a range of temperature and density that is applicable to the conditions in planetary interiors. Previous experimental +and theoretical studies cover only a limited temperature-density range. Our calculations match these early results +well, but now cover the full range of relevance. The method of calculation is based on the general-purpose quotidian +equation of state described by More et al. (1988), which is here used in order to generate a freely downloadable look-up +table to be used by the community. +1. INTRODUCTION +When modeling planetary interiors, it is necessary to have adequate descriptions for the behavior of the constituent +materials. +Thus equation of state (EOS) tables have been produced for the two most abundant elements in the +universe, hydrogen and helium (see, e.g. Chabrier et al. 2019), as well as other materials expected to be of importance +for planet models, such as water (see, e.g. Haldemann et al. 2020), various silicates such as dunite (Benz et al. 1989), +granite (Pierazzo et al. 1997), basalt (Pierazzo et al. 2005), quartz (Melosh 2007) and important metals such as iron +(e.g. Emsenhuber et al. 2018). +Since both carbon and oxygen have relatively high cosmic abundances, and since CO is a very stable molecule, CO +could be an important constituent in planetary interiors (see, e.g. Lisse et al. 2022). Yet this possibility cannot be +properly addressed because only limited regions of the CO EOS have been studied, and there are no complete equation +of state tables available in the literature. Empirical measurements of the density of solid (α-cubic, β-hexagonal) and +liquid CO have been made (Boon et al. 1967; Bierhals 2001), in addition to various other physical properties such as +viscosity, heat capacity (Rudesko & Schubnikow 1934; Tancredi et al. 1994), and elastic constants (Gammon 1978). All +of these studies are applicable to extremely low temperature and pressure conditions, and are ill-suited for planetary +interior applications. The behavior of CO at higher pressures and temperatures has been studied, to a limited extent +by Nellis et al. (1981) who reported the results of shock experiments. +More recent work by Zhang et al. (2011) +gives a more refined hugoniot for CO. In addition, theoretical calculations by Goodwin (1985) have investigated the +region of pressures below 100 MPa. Individual pressure-temperature-density points have been computed from quantum +molecular dynamics calculations by Massacrier et al. (2011), Wang & Zhang (2010), and Leonhardi & Militzer (2017). +However, all of this data is insufficient for planetary modeling, where a much larger range of pressures and temperatures +are encountered. +The fact that shock-derived carbon condensates have diameters of the order of a few nanometers (Titov et al. 1989; +Viecelli et al. 2001; Kr¨uger et al. 2005), and growth timescales of 100’s of picoseconds (Armstrong et al. 2020)) make +amitlevi.planetphys@gmail.com +arXiv:2301.02176v1 [astro-ph.EP] 5 Jan 2023 + +2 +direct DFT based molecular dynamics simulations of this system particularly challenging. Overcoming such immense +difficulties often requires some synthesis between a DFT based approach and more classical force field models using +various training models often referred to as machine learning approaches (see, e.g. Lindsey et al. 2020; Singraber et al. +2019). These techniques are very demanding computationally. Therefore, our model which is in good agreement with +experimental data and covers a very wide pressure-temperature domain is of merit. +To this end we have generated an equation of state table for CO which we describe below. +Our calculation is +admittedly more crude, but it should be sufficiently close to reality so as to be useful in establishing model trends such +as was done in the models of Podolak et al. (2022), for example. This paper is structured as follows: Section 2 gives a +brief description of the method for computing the quotidian EOS (QEOS). This computation requires the knowledge +of the density and bulk modulus at low energy. The DFT calculation of these parameters is described in section 3, +and the results are given in section 4. The resulting EOS table and its comparison to experimental and theoretical +work described above is given in section 5. It is hoped that this work will encourage more detailed EOS modeling for +CO in the future. +2. QUOTIDIAN EQUATION OF STATE +More et al. (1988) present a general-purpose method for computing equations of state at high pressure, called the +Quotidian Equation of State (QEOS). The QEOS is a statistical-mechanics-based method, in which thermodynamic +quantities are derived from the Helmholtz free energy. The Helmholtz free energy term is composed of three parts: an +ionic contribution, an electronic contribution, and a bonding correction. The ionic part is calculated by the Cowan +model, a semi-empirical model which interpolates between known limiting physical cases (ideal gas law, Lindemann +melting law, Dulong-Petit law, Gr¨uniesen EOS, Debye lattice). The electronic part is calculated using a modified +Thomas-Fermi (TF) model. The TF model neglects attractive (bonding) forces between neutral atoms and therefore +overestimates the critical point and the pressure near normal conditions. The bonding correction is used here to correct +for the electronic part failure by calibration of the EOS with density and bulk modulus at reference conditions of zero +(low) energy. +This method has been used to develop EOS tables for Fe, SiO2 and H2O for use in planetary modeling which compare +well with other EOS tables such as SESAME and ANEOS for these substances (Vazan et al. 2013, 2018, 2022). The +QEOS input variables are: atomic number, atomic weight, and reference conditions density and bulk modulus. The +calculated quantities are: pressure, specific internal energy, and specific entropy. The temperature-density range of +the calculation is 11.6 < T < 1.16 × 106 K, and 2.5 × 10−13 < ρ < 100 g cm−3. The liquid-vapor phase transition is +determined with regard to the Maxwell construction, based on finding equal Gibbs free energy on the liquid and the +vapor sides of each isotherm (up to the critical temperature). As a result, there is no coexistence of vapor and liquid +phases in the resulting smooth QEOS. +In order to calculate a QEOS for CO, the method requires prior knowledge of the density and bulk modulus of the +material at very low temperature and pressure. Unfortunately there have been no measurements of these quantities for +the α-phase of CO. We therefore performed a first-principles calculation for this state using density-functional theory +(DFT). This calculation described in the next section. +3. COMPUTATIONAL METHODS +Here we study the equation of state of α-CO at 0 K. The structure is taken from Hall & James (1976). We performed +static total energy relaxations with the CP2K code (K¨uhne et al. 2020). We use the quickstep framework within CP2K +with the Gaussian and plane waves mixed bases (GPW). We adopt the Gaussian basis sets from VandeVondele et al. +(2005); VandeVondele & Hutter (2007), in conjunction with the pseudopotentials (GTH-PBE) of Goedecker, Teter, +and Hutter (Goedecker et al. 1996; Hartwigsen et al. 1998; Krack 2005). +Our system is converged for a planewave cutoff energy of 600 Ry and a REL CUTOFF of 40 Ry. We use the revised +PBE exchange functional GGA X PBE R from Zhang & Yang (1998) and a PBE correlation functional, GGA C PBE +(Perdew et al. 1996, 1997). These are found to be adequate choices when describing an aqueous system in conjunction +with the non-local van der Waals correlation using the Grimme D3 method (Grimme et al. 2010), achieving convergence +for R CUTOFF of 14. The calculations were done done on a 2x2x2 supercell consisting of 32 CO molecules. The +derived data at 0 K is obtained using CELL OPT within CP2K and reported below. +4. THE EQUATION OF STATE +In table 1 and fig. 1 we give the volumes and energies derived for different pressures at 0 K. This data is fitted to a +third order Birch-Murnaghan equation of state with a bulk modulus B= 6.556 ± 0.074 GPa, a pressure derivative for + +3 +Table 1. +The volume, internal energy, +and derived enthalpy as a function of pres- +sure for the α-CO solid. Data is for a cubic +supercell consisting of 32 CO molecules. +P +V +U +H +[bar] +[˚A3] +[Ha] +[Ha] +30,000 +1012.664 +-694.3516 +-693.6548 +20,000 +1063.250 +-694.3819 +-693.8941 +10,000 +1134.213 +-694.4058 +-694.1456 +5000 +1186.408 +-694.4146 +-694.2785 +1000 +1243.957 +-694.4184 +-694.3899 +500 +1252.903 +-694.4185 +-694.4041 +250 +1257.361 +-694.4186 +-694.4114 +100 +1260.298 +-694.4186 +-694.4157 +50 +1261.144 +-694.4186 +-694.4172 +25 +1261.663 +-694.4186 +-694.4179 +10 +1261.931 +-694.4186 +-694.4183 +1 +1262.103 +-694.4186 +-694.4186 +the bulk modulus of B′ = 6.846 ± 0.120, and a zero pressure volume of V0 = 157.80 ± 0.05 ˚A3. The error bars are at +the 2σ level. As mentioned above, the QEOS requires a knowledge of ρ and B at reference conditions of zero energy, +and, based on this calculation, and a fit to the four lowest pressure points we take ρ = 1.179 g cm−3 and B= 2.676 GPa +as the input parameters. Note that this value of B falls between the best fit value of 6.556 GPa given above and the +value of 1.3 GPa measured by Gammon (1978) for β-CO. +125 +130 +135 +140 +145 +150 +155 +160 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +P [ GPa ] +Fitted data to BM3 +Figure 1. Pressure versus unit cell volume for α-CO. The blue circles are unit cell volumes from our optimization data at 0 K, and the +solid red curve is the fitted third order Birch-Murnaghan equation of state (BM3). +Using the results of the DFT calculation described above in the quotidian code, we produced an equation of state + +4 +table giving the pressure, energy and entropy of CO for a large range of temperatures and densities. +5. COMPARISON TO OTHER RESULTS +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +1.0E+08 +1.0E+09 +1.0E+10 +1.0E+11 +1.0E+12 +1.0E+13 +1.0E+14 +Density (g/cc) +Pressure (Pa) +Figure 2. Density as a function of pressure at zero temperature for the quotidian equation of state (black curve), and for the S-Z equation +of state (blue curve). The red dots are the results of the DFT calculation. +Salpeter & Zapolsky (1967) (S-Z) describe a semi-empirical formula for predicting the zero temperature pressure- +density relation for materials with any average atomic number. In principle, the S-Z EOS is similar to the More et al. +(1988) approach, since it relies on a Thomas-Fermi-Dirac model of the atom. However it does not include the effect +of temperature, so it is not always suitable for planet modeling. Fig. 2 shows the comparison between our quotidian +equation of state (QEOS) at zero temperature, and the S-Z EOS. As can be seen, the agreement is excellent, and +improves at higher pressures, as expected. The red dots in the figure are the DFT calculations given in table 1 and +fig. 1. These fall right on the QEOS curve. +The QEOS can be compared to experimental data at higher temperatures as well. +Goodwin (1985) gives the +thermophysical properties of CO up to a pressure of 100 MPa. Fig. 3 shows that data for an isotherm at 1000 K (red +dots) compared to the QEOS isotherm at that temperature (black curve). The discontinuity in the QEOS is due to +the fact that the QEOS finds two phases in present in this pressure-temperature range and traverses this region using +a Maxwell construction. As a result, the computed pressure remains constant over the relevant density range. The +actual pressure, as shown by the red dots, increases along the extrapolation of the lower part of the curve, as expected. +The exact position of the phase transition is sensitive to the choice of input parameters (zero energy density and bulk +modulus), and the actual value may be shifted somewhat. + +5 +1.0E-04 +1.0E-03 +1.0E-02 +1.0E-01 +1.0E+00 +1.0E+01 +1.0E+02 +1.0E+06 +1.0E+07 +1.0E+08 +1.0E+09 +1.0E+10 +1.0E+11 +1.0E+12 +1.0E+13 +1.0E+14 +1.0E+15 +Density (g/cc) +Pressure (Pa) +Figure 3. Density as a function of pressure for an isotherm at T = 1000 K (black curve), compared to the data in Goodwin (1985) (red +dots). See text for details. +At still higher pressures and temperatures, there are the shock wave experiments of Nellis et al. (1981). In this case +the temperatures are only inferred from the Hugoniot relations, and are different for the different pressures. More +recently, Zhang et al. (2011) have used quantum molecular dynamics calculations to compute points along a hugoniot. +These are shown (blue dots) together with the hugoniot calculated from our QEOS in Fig. 4. The black dots are the +experimental points of Nellis et al. (1981). As can be seen, the agreement is quite good and is in the range of these +works. At the highest temperatures (T ≳ 105 K) dissociation and ionization become important, and these effects are +not directly included in our calculation. Nonetheless, the energies we compute for CO at T = 5 × 105 K for densities +of 0.1, 1, 10, and 100 g cm−3 all fall within a factor of 1.5 or less from the values shown in fig. 9 of Massacrier et al. +(2011). +The full QEOS is summarized in Fig. 5. A short version for a range of pressures and temperatures that are expected +to be important for planetary interior modeling given in table 2, while the complete table is available at the following +site: CO EOS download. + +6 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +0 +50 +100 +150 +200 +250 +300 +350 +Density (g/cc) +Pressure (GPa) +Figure 4. Density as a function of pressure for a hugoniot (blue curve) corresponding to the conditions of the shock experiments of Nellis +et al. (1981) (black dots) and the quantum molecular dynamics calculations of Zhang et al. (2011) (blue dots). + +7 +Figure 5. Thermodynamic properties of CO as a function of density and temperature as computed from the quotidian equation of state. +Upper left: total pressure. Upper right: pressure divided by ideal gas pressure. This shows the region where an ideal gas approximation +may be used. Lower left: specific internal energy. Lower right: specific entropy. + +co: 1 +CO: log p (Pa) +0 +0 +log p (g/cm3) +10 +log p (g/cm3) +5 +5 +0og p (/pideal +5 +4 +3 +2 +1-10 +-10 +-5 +2 +4 +6 +2 +logT(K) +Co: log u (erg/g) +CO: Io +14 +0 +0 +13 +-5 +-5 +12 +11 +.10 +-10 +2 +4 +6 +2 +log T (K)0 +4 +6 +log T (K) +gs(erg/g/K) +9 +8 +7 +6 +5 +4 +3 +4 +6 +log T (K)8 +h +Table 2. Equation of state for CO. +log T +log ρ +log P +log u +log s +[K] +[g/cc] +[Pa] +[erg/g] +[erg/g − K] +1.06465 +0.10 +8.25060 +10.33294 +4.12858 +1.06465 +0.20 +9.33277 +10.36235 +3.94748 +1.06465 +0.30 +9.90369 +10.46041 +3.83167 +1.06465 +0.40 +10.35107 +10.63593 +3.74763 +1.06465 +0.50 +10.73560 +10.86075 +3.67754 +1.06465 +0.60 +11.08026 +11.10070 +3.61334 +1.06465 +0.70 +11.39668 +11.33587 +3.55157 +1.06465 +0.80 +11.69173 +11.55846 +3.49070 +1.06465 +0.90 +11.96991 +11.76663 +3.43006 +1.06465 +1.00 +12.23437 +11.96089 +3.36933 +1.06465 +1.10 +12.48741 +12.14250 +3.30834 +1.06465 +1.20 +12.73083 +12.31285 +3.24704 +1.06465 +1.30 +12.96602 +12.47327 +3.18540 +1.56465 +0.10 +8.25291 +10.33305 +5.29718 +1.56465 +0.20 +9.33286 +10.36239 +4.92635 +1.56465 +0.30 +9.90370 +10.46043 +4.63269 +1.56465 +0.40 +10.35107 +10.63594 +4.42170 +1.56465 +0.50 +10.73560 +10.86075 +4.27449 +1.56465 +0.60 +11.08026 +11.10070 +4.16723 +1.56465 +0.70 +11.39668 +11.33587 +4.08205 +1.56465 +0.80 +11.69173 +11.55846 +4.00839 +1.56465 +0.90 +11.96991 +11.76663 +3.94060 +1.56465 +1.00 +12.23437 +11.96089 +3.87576 +1.56465 +1.10 +12.48741 +12.14250 +3.81236 +1.56465 +1.20 +12.73083 +12.31285 +3.74960 +1.56465 +1.30 +12.96602 +12.47327 +3.68706 +2.06465 +0.10 +8.32026 +10.33617 +6.33748 +2.06465 +0.20 +9.33736 +10.36427 +6.10876 +2.06465 +0.30 +9.90446 +10.46124 +5.82885 +2.06465 +0.40 +10.35123 +10.63621 +5.53226 +2.06465 +0.50 +10.73564 +10.86083 +5.24767 +2.06465 +0.60 +11.08027 +11.10073 +5.00266 +2.06465 +0.70 +11.39669 +11.33588 +4.80674 +2.06465 +0.80 +11.69174 +11.55846 +4.65419 +2.06465 +0.90 +11.96991 +11.76663 +4.53392 +2.06465 +1.00 +12.23437 +11.96089 +4.43538 +2.06465 +1.10 +12.48741 +12.14250 +4.35065 +2.06465 +1.20 +12.73083 +12.31285 +4.27440 +2.06465 +1.30 +12.96602 +12.47327 +4.20327 +2.56465 +0.10 +8.59818 +10.35536 +6.81650 +2.56465 +0.20 +9.37536 +10.38058 +6.71214 +2.56465 +0.30 +9.91516 +10.47251 +6.59233 +2.56465 +0.40 +10.35484 +10.64227 +6.45354 +2.56465 +0.50 +10.73692 +10.86349 +6.29013 + +9 +Table 2. Equation of state for CO continued +log T +log ρ +log P +log u +log s +[K] +[g/cc] +[Pa] +[erg/g] +[erg/g − K] +2.56465 +0.60 +11.08076 +11.10182 +6.11803 +2.56465 +0.70 +11.39686 +11.33629 +5.91384 +2.56465 +0.80 +11.69180 +11.55861 +5.70071 +2.56465 +0.90 +11.96994 +11.76669 +5.48884 +2.56465 +1.00 +12.23438 +11.96091 +5.29094 +2.56465 +1.10 +12.48742 +12.14251 +5.11592 +2.56465 +1.20 +12.73083 +12.31286 +4.96608 +2.56465 +1.30 +12.96602 +12.47328 +4.83886 +3.06465 +0.10 +8.96463 +10.41261 +7.05238 +3.06465 +0.20 +9.49348 +10.43930 +7.00870 +3.06465 +0.30 +9.95968 +10.52059 +6.94781 +3.06465 +0.40 +10.37374 +10.67408 +6.87899 +3.06465 +0.50 +10.74579 +10.88182 +6.80456 +3.06465 +0.60 +11.08515 +11.11162 +6.72332 +3.06465 +0.70 +11.39912 +11.34141 +6.63379 +3.06465 +0.80 +11.69298 +11.56126 +6.53419 +3.06465 +0.90 +11.97055 +11.76803 +6.42218 +3.06465 +1.00 +12.23471 +11.96163 +6.31293 +3.06465 +1.10 +12.48758 +12.14285 +6.16560 +3.06465 +1.20 +12.73092 +12.31302 +6.00969 +3.06465 +1.30 +12.96606 +12.47335 +5.84475 +3.56465 +0.10 +9.36962 +10.57254 +7.21587 +3.56465 +0.20 +9.69405 +10.59071 +7.18659 +3.56465 +0.30 +10.06369 +10.65281 +7.15510 +3.56465 +0.40 +10.43076 +10.77475 +7.12114 +3.56465 +0.50 +10.77783 +10.94870 +7.08214 +3.56465 +0.60 +11.10251 +11.15104 +7.03402 +3.56465 +0.70 +11.40902 +11.36430 +6.98398 +3.56465 +0.80 +11.69886 +11.57468 +6.93149 +3.56465 +0.90 +11.97416 +11.77606 +6.87593 +3.56465 +1.00 +12.23696 +11.96647 +6.81651 +3.56465 +1.10 +12.48903 +12.14584 +6.75230 +3.56465 +1.20 +12.73184 +12.31488 +6.68219 +3.56465 +1.30 +12.96665 +12.47450 +6.60487 +4.06465 +-1.00 +8.72672 +11.12752 +7.61469 +4.06465 +-0.90 +8.84338 +11.11787 +7.60037 +4.06465 +-0.80 +8.96101 +11.10873 +7.58527 +4.06465 +-0.70 +9.07819 +11.09997 +7.56926 +4.06465 +-0.60 +9.19305 +11.09161 +7.55219 +4.06465 +-0.50 +9.30353 +11.08214 +7.53390 +4.06465 +-0.40 +9.40769 +11.07207 +7.51424 +4.06465 +-0.30 +9.50433 +11.06010 +7.49310 +4.06465 +-0.20 +9.59411 +11.04556 +7.47045 +4.06465 +-0.10 +9.68161 +11.02824 +7.44633 +4.06465 +0.00 +9.77804 +11.00927 +7.42091 +4.06465 +0.10 +9.90243 +10.99228 +7.39448 +4.06465 +0.20 +10.07443 +10.98353 +7.36684 + +10 +Table 2. Equation of state for CO continued +log T +log ρ +log P +log u +log s +[K] +[g/cc] +[Pa] +[erg/g] +[erg/g − K] +4.06465 +0.30 +10.30242 +10.99646 +7.33856 +4.06465 +0.40 +10.57225 +11.04588 +7.30967 +4.06465 +0.50 +10.86205 +11.14146 +7.28014 +4.06465 +0.60 +11.15517 +11.27981 +7.24992 +4.06465 +0.70 +11.44272 +11.44692 +7.21887 +4.06465 +0.80 +11.72104 +11.62736 +7.18684 +4.06465 +0.90 +11.98825 +11.80866 +7.14860 +4.06465 +1.00 +12.24611 +11.98690 +7.10900 +4.06465 +1.10 +12.49512 +12.15893 +7.06828 +4.06465 +1.20 +12.73600 +12.32343 +7.02608 +4.06465 +1.30 +12.96954 +12.48020 +6.98198 +4.56465 +-1.00 +9.39391 +11.85300 +7.82194 +4.56465 +-0.90 +9.49858 +11.83884 +7.80733 +4.56465 +-0.80 +9.60479 +11.82481 +7.79226 +4.56465 +-0.70 +9.71228 +11.81091 +7.77665 +4.56465 +-0.60 +9.82054 +11.79708 +7.76038 +4.56465 +-0.50 +9.92889 +11.78321 +7.74336 +4.56465 +-0.40 +10.03655 +11.76908 +7.72545 +4.56465 +-0.30 +10.14277 +11.75438 +7.70651 +4.56465 +-0.20 +10.24707 +11.73869 +7.68639 +4.56465 +-0.10 +10.34964 +11.72211 +7.66492 +4.56465 +0.00 +10.45180 +11.70292 +7.64198 +4.56465 +0.10 +10.55668 +11.68314 +7.61745 +4.56465 +0.20 +10.66971 +11.66302 +7.59129 +4.56465 +0.30 +10.79841 +11.64539 +7.56357 +4.56465 +0.40 +10.95060 +11.63526 +7.53444 +4.56465 +0.50 +11.13044 +11.64026 +7.50411 +4.56465 +0.60 +11.33652 +11.66954 +7.47311 +4.56465 +0.70 +11.56221 +11.72900 +7.44136 +4.56465 +0.80 +11.79910 +11.82012 +7.40932 +4.56465 +0.90 +12.04056 +11.93746 +7.37731 +4.56465 +1.00 +12.28172 +12.07201 +7.34534 +4.56465 +1.10 +12.51981 +12.21535 +7.31336 +4.56465 +1.20 +12.75346 +12.36132 +7.28123 +4.56465 +1.30 +12.98189 +12.50572 +7.24694 +5.06465 +-2.20 +9.01221 +12.79781 +8.22267 +5.06465 +-2.10 +9.10356 +12.78341 +8.20949 +5.06465 +-2.00 +9.19496 +12.76882 +8.19617 +5.06465 +-1.90 +9.28643 +12.75405 +8.18271 +5.06465 +-1.80 +9.37804 +12.73910 +8.16910 +5.06465 +-1.70 +9.46983 +12.72399 +8.15535 +5.06465 +-1.60 +9.56187 +12.70871 +8.14144 +5.06465 +-1.50 +9.65423 +12.69328 +8.12737 +5.06465 +-1.40 +9.74698 +12.67771 +8.11312 +5.06465 +-1.30 +9.84023 +12.66200 +8.09870 +5.06465 +-1.20 +9.93407 +12.64619 +8.08407 +5.06465 +-1.10 +10.02860 +12.63027 +8.06923 + +11 +Table 2. Equation of state for CO continued +log T +log ρ +log P +log u +log s +[K] +[g/cc] +[Pa] +[erg/g] +[erg/g − K] +5.06465 +-1.00 +10.12395 +12.61427 +8.05415 +5.06465 +-0.90 +10.22023 +12.59820 +8.03880 +5.06465 +-0.80 +10.31750 +12.58210 +8.02315 +5.06465 +-0.70 +10.41581 +12.56597 +8.00716 +5.06465 +-0.60 +10.51516 +12.54983 +7.99078 +5.06465 +-0.50 +10.61545 +12.53369 +7.97396 +5.06465 +-0.40 +10.71652 +12.51752 +7.95663 +5.06465 +-0.30 +10.81818 +12.50129 +7.93872 +5.06465 +-0.20 +10.92019 +12.48494 +7.92015 +5.06465 +-0.10 +11.02242 +12.46839 +7.90082 +5.06465 +0.00 +11.12490 +12.45155 +7.88063 +5.06465 +0.10 +11.22793 +12.43435 +7.85947 +5.06465 +0.20 +11.33226 +12.41683 +7.83721 +5.06465 +0.30 +11.43918 +12.39913 +7.81373 +5.06465 +0.40 +11.55056 +12.38222 +7.78890 +5.06465 +0.50 +11.66886 +12.36618 +7.76260 +5.06465 +0.60 +11.79684 +12.35360 +7.73472 +5.06465 +0.70 +11.93708 +12.34707 +7.70521 +5.06465 +0.80 +12.09132 +12.35034 +7.67410 +5.06465 +0.90 +12.26000 +12.36788 +7.64155 +5.06465 +1.00 +12.44186 +12.40353 +7.60770 +5.06465 +1.10 +12.63451 +12.45973 +7.57309 +5.06465 +1.20 +12.83475 +12.53549 +7.53795 +5.06465 +1.30 +13.03937 +12.62723 +7.50218 + +12 +6. ACKNOWLEDGEMENTS +The authors wish to thank Gilles Chabrier and an anonymous referee for many constructive comments. M.P. is +supported by a grant from the Pazy Fund of the Israel Atomic Energy Commission. A.L. is supported by a grant from +the Simons Foundation (SCOL #290360 to D.S.). The computations for this paper were run on the Odyssey cluster +supported by the FAS Division of Science, Research Computing Group at Harvard University. A.L. is grateful to the +administrative staff for their technical support. A.V. acknowledges support from ISF grants 770/21 and 773/21. +REFERENCES +Armstrong, M. R., Lindsey, R. K., Goldman, N., et al. 2020, +Nature Communications, 11, 353 1 +Benz, W., Cameron, A. G. W., & Melosh, H. J. 1989, Icarus, 81, +113, doi: 10.1016/0019-1035(89)90129-2 1 +Bierhals, J. 2001, in Ullmann’s Encyclopedia of Industrial +Chemistry, ed. F. Ullman (John Wiley & Sons, Ltd), +doi: https://doi.org/10.1002/14356007.a05_203 1 +Boon, J. P., Legros, J. 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Lett., 80, 890, +doi: 10.1103/PhysRevLett.80.890 3 + diff --git a/_dA0T4oBgHgl3EQfPf96/content/tmp_files/load_file.txt b/_dA0T4oBgHgl3EQfPf96/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a2ec859443c161883e6a5d35d6fa1044e27d8e44 --- /dev/null +++ b/_dA0T4oBgHgl3EQfPf96/content/tmp_files/load_file.txt @@ -0,0 +1,1388 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf,len=1387 +page_content='Draft version January 6, 2023 Preprint typeset using LATEX style AASTeX6 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='0 AN EQUATION OF STATE OF CO FOR USE IN PLANETARY MODELING M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Podolak Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' of Geosciences, Tel Aviv University, Tel Aviv, 69978 Israel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Levi Braude College of Engineering, Karmiel, 2161002 Israel A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Vazan Astrophysics Research Center (ARCO), Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' of Natural Sciences, Open University of Israel, Raanana, 43107 Israel U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Malamud Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' of Geosciences, Tel Aviv University, Tel Aviv, 69978 Israel Department of Physics, Technion – Israel Institute of Technology, Technion City, 3200003 Haifa, Israel ABSTRACT Although carbon monoxide (CO) is an abundant molecule and may have great importance for planetary interiors, measurements of its properties are difficult due to its extreme volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' We calculate the equation of state for CO over a range of temperature and density that is applicable to the conditions in planetary interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Previous experimental and theoretical studies cover only a limited temperature-density range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Our calculations match these early results well, but now cover the full range of relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The method of calculation is based on the general-purpose quotidian equation of state described by More et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' (1988), which is here used in order to generate a freely downloadable look-up table to be used by the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' INTRODUCTION When modeling planetary interiors, it is necessary to have adequate descriptions for the behavior of the constituent materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Thus equation of state (EOS) tables have been produced for the two most abundant elements in the universe, hydrogen and helium (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Chabrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 2019), as well as other materials expected to be of importance for planet models, such as water (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Haldemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 2020), various silicates such as dunite (Benz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 1989), granite (Pierazzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 1997), basalt (Pierazzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 2005), quartz (Melosh 2007) and important metals such as iron (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Emsenhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Since both carbon and oxygen have relatively high cosmic abundances, and since CO is a very stable molecule, CO could be an important constituent in planetary interiors (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Lisse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Yet this possibility cannot be properly addressed because only limited regions of the CO EOS have been studied, and there are no complete equation of state tables available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Empirical measurements of the density of solid (α-cubic, β-hexagonal) and liquid CO have been made (Boon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 1967;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Bierhals 2001), in addition to various other physical properties such as viscosity, heat capacity (Rudesko & Schubnikow 1934;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Tancredi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 1994), and elastic constants (Gammon 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' All of these studies are applicable to extremely low temperature and pressure conditions, and are ill-suited for planetary interior applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The behavior of CO at higher pressures and temperatures has been studied, to a limited extent by Nellis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' (1981) who reported the results of shock experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' More recent work by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' (2011) gives a more refined hugoniot for CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' In addition, theoretical calculations by Goodwin (1985) have investigated the region of pressures below 100 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Individual pressure-temperature-density points have been computed from quantum molecular dynamics calculations by Massacrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' (2011), Wang & Zhang (2010), and Leonhardi & Militzer (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' However, all of this data is insufficient for planetary modeling, where a much larger range of pressures and temperatures are encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The fact that shock-derived carbon condensates have diameters of the order of a few nanometers (Titov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Viecelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Kr¨uger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 2005), and growth timescales of 100’s of picoseconds (Armstrong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 2020)) make amitlevi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='planetphys@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='com arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='02176v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='EP] 5 Jan 2023 2 direct DFT based molecular dynamics simulations of this system particularly challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Overcoming such immense difficulties often requires some synthesis between a DFT based approach and more classical force field models using various training models often referred to as machine learning approaches (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Lindsey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Singraber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' These techniques are very demanding computationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Therefore, our model which is in good agreement with experimental data and covers a very wide pressure-temperature domain is of merit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' To this end we have generated an equation of state table for CO which we describe below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Our calculation is admittedly more crude, but it should be sufficiently close to reality so as to be useful in establishing model trends such as was done in the models of Podolak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' (2022), for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' This paper is structured as follows: Section 2 gives a brief description of the method for computing the quotidian EOS (QEOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' This computation requires the knowledge of the density and bulk modulus at low energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The DFT calculation of these parameters is described in section 3, and the results are given in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The resulting EOS table and its comparison to experimental and theoretical work described above is given in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' It is hoped that this work will encourage more detailed EOS modeling for CO in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' QUOTIDIAN EQUATION OF STATE More et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' (1988) present a general-purpose method for computing equations of state at high pressure, called the Quotidian Equation of State (QEOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The QEOS is a statistical-mechanics-based method, in which thermodynamic quantities are derived from the Helmholtz free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The Helmholtz free energy term is composed of three parts: an ionic contribution, an electronic contribution, and a bonding correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The ionic part is calculated by the Cowan model, a semi-empirical model which interpolates between known limiting physical cases (ideal gas law, Lindemann melting law, Dulong-Petit law, Gr¨uniesen EOS, Debye lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The electronic part is calculated using a modified Thomas-Fermi (TF) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The TF model neglects attractive (bonding) forces between neutral atoms and therefore overestimates the critical point and the pressure near normal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The bonding correction is used here to correct for the electronic part failure by calibration of the EOS with density and bulk modulus at reference conditions of zero (low) energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' This method has been used to develop EOS tables for Fe, SiO2 and H2O for use in planetary modeling which compare well with other EOS tables such as SESAME and ANEOS for these substances (Vazan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 2013, 2018, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The QEOS input variables are: atomic number, atomic weight, and reference conditions density and bulk modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The calculated quantities are: pressure, specific internal energy, and specific entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The temperature-density range of the calculation is 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='6 < T < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='16 × 106 K, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='5 × 10−13 < ρ < 100 g cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The liquid-vapor phase transition is determined with regard to the Maxwell construction, based on finding equal Gibbs free energy on the liquid and the vapor sides of each isotherm (up to the critical temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' As a result, there is no coexistence of vapor and liquid phases in the resulting smooth QEOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' In order to calculate a QEOS for CO, the method requires prior knowledge of the density and bulk modulus of the material at very low temperature and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Unfortunately there have been no measurements of these quantities for the α-phase of CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' We therefore performed a first-principles calculation for this state using density-functional theory (DFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' This calculation described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' COMPUTATIONAL METHODS Here we study the equation of state of α-CO at 0 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The structure is taken from Hall & James (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' We performed static total energy relaxations with the CP2K code (K¨uhne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' We use the quickstep framework within CP2K with the Gaussian and plane waves mixed bases (GPW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' We adopt the Gaussian basis sets from VandeVondele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' VandeVondele & Hutter (2007), in conjunction with the pseudopotentials (GTH-PBE) of Goedecker, Teter, and Hutter (Goedecker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Hartwigsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Krack 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Our system is converged for a planewave cutoff energy of 600 Ry and a REL CUTOFF of 40 Ry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' We use the revised PBE exchange functional GGA X PBE R from Zhang & Yang (1998) and a PBE correlation functional, GGA C PBE (Perdew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 1996, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' These are found to be adequate choices when describing an aqueous system in conjunction with the non-local van der Waals correlation using the Grimme D3 method (Grimme et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 2010), achieving convergence for R CUTOFF of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The calculations were done done on a 2x2x2 supercell consisting of 32 CO molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The derived data at 0 K is obtained using CELL OPT within CP2K and reported below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' THE EQUATION OF STATE In table 1 and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 1 we give the volumes and energies derived for different pressures at 0 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' This data is fitted to a third order Birch-Murnaghan equation of state with a bulk modulus B= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='556 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='074 GPa, a pressure derivative for 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The volume, internal energy, and derived enthalpy as a function of pres- sure for the α-CO solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Data is for a cubic supercell consisting of 32 CO molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' P V U H [bar] [˚A3] [Ha] [Ha] 30,000 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='664 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='3516 693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='6548 20,000 1063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='250 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='3819 693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='8941 10,000 1134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='213 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4058 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='1456 5000 1186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='408 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4146 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='2785 1000 1243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='957 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4184 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='3899 500 1252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='903 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4185 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4041 250 1257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='361 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4186 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4114 100 1260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='298 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4186 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4157 50 1261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='144 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4186 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4172 25 1261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='663 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4186 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4179 10 1261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='931 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4186 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4183 1 1262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='103 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4186 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='4186 the bulk modulus of B′ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='846 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='120, and a zero pressure volume of V0 = 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='05 ˚A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The error bars are at the 2σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' As mentioned above, the QEOS requires a knowledge of ρ and B at reference conditions of zero energy, and, based on this calculation, and a fit to the four lowest pressure points we take ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='179 g cm−3 and B= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='676 GPa as the input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Note that this value of B falls between the best fit value of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='556 GPa given above and the value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='3 GPa measured by Gammon (1978) for β-CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 125 130 135 140 145 150 155 160 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='5 P [ GPa ] Fitted data to BM3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Pressure versus unit cell volume for α-CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The blue circles are unit cell volumes from our optimization data at 0 K, and the solid red curve is the fitted third order Birch-Murnaghan equation of state (BM3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Using the results of the DFT calculation described above in the quotidian code, we produced an equation of state 4 table giving the pressure, energy and entropy of CO for a large range of temperatures and densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' COMPARISON TO OTHER RESULTS 0 2 4 6 8 10 12 14 16 18 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='0E+08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='0E+09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='0E+10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='0E+11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='0E+12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='0E+13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='0E+14 Density (g/cc) Pressure (Pa) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Density as a function of pressure at zero temperature for the quotidian equation of state (black curve), and for the S-Z equation of state (blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The red dots are the results of the DFT calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Salpeter & Zapolsky (1967) (S-Z) describe a semi-empirical formula for predicting the zero temperature pressure- density relation for materials with any average atomic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' In principle, the S-Z EOS is similar to the More et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' (1988) approach, since it relies on a Thomas-Fermi-Dirac model of the atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' However it does not include the effect of temperature, so it is not always suitable for planet modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 2 shows the comparison between our quotidian equation of state (QEOS) at zero temperature, and the S-Z EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' As can be seen, the agreement is excellent, and improves at higher pressures, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The red dots in the figure are the DFT calculations given in table 1 and fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' These fall right on the QEOS curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The QEOS can be compared to experimental data at higher temperatures as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Goodwin (1985) gives the thermophysical properties of CO up to a pressure of 100 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 3 shows that data for an isotherm at 1000 K (red dots) compared to the QEOS isotherm at that temperature (black curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The discontinuity in the QEOS is due to the fact that the QEOS finds two phases in present in this pressure-temperature range and traverses this region using a Maxwell construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' As a result, the computed pressure remains constant over the relevant density range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The actual pressure, as shown by the red dots, increases along the extrapolation of the lower part of the curve, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The exact position of the phase transition is sensitive to the choice of input parameters (zero energy density and bulk modulus), and the actual value may be shifted somewhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='0E-04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='0E-03 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='0E+15 Density (g/cc) Pressure (Pa) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Density as a function of pressure for an isotherm at T = 1000 K (black curve), compared to the data in Goodwin (1985) (red dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' At still higher pressures and temperatures, there are the shock wave experiments of Nellis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' In this case the temperatures are only inferred from the Hugoniot relations, and are different for the different pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' More recently, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' (2011) have used quantum molecular dynamics calculations to compute points along a hugoniot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' These are shown (blue dots) together with the hugoniot calculated from our QEOS in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The black dots are the experimental points of Nellis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' As can be seen, the agreement is quite good and is in the range of these works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' At the highest temperatures (T ≳ 105 K) dissociation and ionization become important, and these effects are not directly included in our calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Nonetheless, the energies we compute for CO at T = 5 × 105 K for densities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='1, 1, 10, and 100 g cm−3 all fall within a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='5 or less from the values shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 9 of Massacrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' The full QEOS is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' A short version for a range of pressures and temperatures that are expected to be important for planetary interior modeling given in table 2, while the complete table is available at the following site: CO EOS download.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='5 4 0 50 100 150 200 250 300 350 Density (g/cc) Pressure (GPa) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Density as a function of pressure for a hugoniot (blue curve) corresponding to the conditions of the shock experiments of Nellis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' (1981) (black dots) and the quantum molecular dynamics calculations of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' (2011) (blue dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' 7 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Thermodynamic properties of CO as a function of density and temperature as computed from the quotidian equation of state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Upper left: total pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Upper right: pressure divided by ideal gas pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' This shows the region where an ideal gas approximation may be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Lower left: specific internal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Lower right: specific entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' co: 1 CO: log p (Pa) 0 0 log p (g/cm3) 10 log p (g/cm3) 5 5 0og p (/pideal 5 4 3 2 1-10 10 5 2 4 6 2 logT(K) Co: log u (erg/g) CO: Io 14 0 0 13 5 5 12 11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='10 10 2 4 6 2 log T (K)0 4 6 log T (K) gs(erg/g/K) 9 8 7 6 5 4 3 4 6 log T (K)8 h Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Equation of state for CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' log T log ρ log P log u log s [K] [g/cc] [Pa] 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='30 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='03937 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='62723 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='50218 12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors wish to thank Gilles Chabrier and an anonymous referee for many constructive comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' is supported by a grant from the Pazy Fund of the Israel Atomic Energy Commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' is supported by a grant from the Simons Foundation (SCOL #290360 to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content=', 80, 890, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} +page_content='890 3' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dA0T4oBgHgl3EQfPf96/content/2301.02176v1.pdf'} diff --git a/_dAyT4oBgHgl3EQfdvdN/content/tmp_files/2301.00307v1.pdf.txt b/_dAyT4oBgHgl3EQfdvdN/content/tmp_files/2301.00307v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9d4ab471ca161fe67952a6205998fccf4248bab7 --- /dev/null +++ b/_dAyT4oBgHgl3EQfdvdN/content/tmp_files/2301.00307v1.pdf.txt @@ -0,0 +1,834 @@ +Bending Deformation Driven by Molecular Rotation +Pedro A. Santos-Florez,1 Shinnosuke Hattori,2 and Qiang Zhu1, ∗ +1Department of Physics and Astronomy, University of Nevada, Las Vegas, NV 89154, USA +2Advanced Research Laboratory, R&D Center, Sony Group Corporation, 4-14-1 Asahi-cho, Atsugi-shi 243-0014, Japan +(Dated: January 3, 2023) +Recently, some molecular crystals have been found to be surprisingly flexible by undergoing a large extent +of elastic or plastic deformation upon various mechanical loads. Despite the increasing experimental reports on +mechanically flexible crystals, this phenomenon has never been reproduced in numerical simulation and thus +there is no atomistic mechanism to explain its physical origin. Using three recently reported naphthalene diimide +derivatives as the examples, we perform the first direct molecular dynamics simulation to model their mechanical +behaviors from brittle fracture to elastic/plastic deformation upon mechanical bending. Our simulation reveals +that molecular rotational freedom is the key factor to determine the crystal’s mechanical response. Furthermore, +we propose the use of rotation-dependent potential energy surface to classify organic materials’ mechanical +response and screen new mechanically flexible candidates in future. +While most molecular crystals are known to be brittle, there +exists a class of compliant organic crystals that can easily bend +under a large mechanical stress loading1,2. Since early 2000, +there has been a growing number of experimental identifi- +cations of mechanically flexible crystals3–9. In general, the +mechanical response of an organic solid depends on both the +molecular substance and the corresponding crystal packing. +A remarkable example is shown in Fig. +1, three crystals, +made of similar molecules from naphthalene diimide deriva- +tives, were found to exhibit distinct responses from brittle +fracture to compliant deformation with either reversible (elas- +tic) or irreversible (plastic) characteristic10. The flexible na- +ture of such organic materials is vital for a variety of appli- +cations, e.g., high-performance modular organic solar cells11, +actuators12, photochemistry13, electronics14, optics15, as well +as drug tabulation16. +In the recent years, various computational techniques have +been introduced to characterize the observed mechanical +properties on different molecular systems10,17–19. +They in- +clude the topological analysis, elastic properties calculation17, +and the simulation of shear/tensile deformations10,18. These +techniques are partially successful in identifying the brittle +materials which usually exhibit a complex three dimensional +packing. Within such an interlocked environment, molecu- +lar motions are largely restricted, resulting a brittleness un- +der bending. On the other hand, the compliant class of ma- +terials are featured by a strong anisotropy with plausible slip +planes17,20. Therefore, these materials become compliant over +a broad range of applied stress along some specific crystallo- +graphic directions. However, all available techniques fail to +explain the difference between the elastic and plastic materi- +als. While there have been plenty of studies on the bending of +metals21–26, to our knowledge, no attempts have been made to +directly simulate the bending of organic materials at the atom- +istic level. +Among the compliant crystals, ductile materials are often +favored in engineering applications16. Hence, researchers at- +tempted to use the well established dislocation theory to ex- +plain the observed plasticity on organic materials2,3. Simi- +lar to the plastic deformation in ductile metals, it was found +that mechanical shearing can also occur via the slippage of +dislocated molecular layers on the molecular crystals with a +β +α +γ +α +β +γ +(a) +(b) +(c) + (degree) +α +x +z +x +y +z +γ +β +α +y +y +z +x +-30 +30 +0 +FIG. 1. +The simulated bending on three different materials based +on naphthalene diimide derivatives. (a) brittle Pr (50.3×7.0×6.8 +nm3), (b) elastic Et (50.7×6.4×6.6 nm3) and (c) elastic/plastic Me +(50.2×6.4×6.9 nm3). These three crystals consist of very similar +molecules that differ only in the side groups. In the left panel, the +initial and finally deformed configurations are colored by the molec- +ular alignment (α) along the x-axis. The corresponding molecules +and the definition of rotation angles are shown in the right panel. +layered packing27,28. Using these facile slip planes, a bend- +ing model was proposed accordingly to explain the underly- +ing mechanism3. Although the dislocation is not uncommon +in molecular crystals29,30, there has been no direct experimen- +tal evidence to support that the dislocation is present in the +organic crystals under bending. Furthermore, this mechanism +fails to explain the observed crystals that can also bend elas- +tically to a large extent. In fact, two crystals as shown in Fig. +arXiv:2301.00307v1 [cond-mat.mtrl-sci] 31 Dec 2022 + +X +z7X +z2 +1b-c possess very similar crystal packing. Give the apparent +similarity in both molecular structure and crystal packing, it +is expected that the elastic crystal (Fig. 1b) should undergo +similar molecular events like the plastic crystal (Fig. 1c) by +following the ending mechanism. But the actual deformation +was observed to be elastic. Clearly, our current understanding +on the elasticity and plasticity remains limited. +In this work, we present our efforts in questing the molec- +ular bending mechanism with the aid of atomistic simula- +tion. +To achieve this goal, we start by developing a ro- +bust simulation protocol that can directly model the bend- +ing of organic crystals at the atomic level. Specifically, we +employed a three-point bending model within a partial peri- +odic boundary condition31. In our calculation, we performed +non-equilibrium molecular dynamics simulation by applying +the indentation on the center of molecular slab under finite +temperature31. +We also carefully tested the choice of slab +models and thermal equilibration to ensure the robustness of +our simulation set up. In order to automate the simulation, +we developed a computational pipeline to automate the gen- +eration of molecular force fields from the AmberTools20 +software32. Force field parameters are assigned by the Gen- +eral Amber Force Field (GAFF) with atomic charges using +semi-empirical (AM1) with bond charge correction (BCC)33. +All simulations were performed on the LAMMPS package34 at +room temperature with the strain rate of 10 m/s. +In the following, we will focus on three naphthalenete- +tracarboxylic diimide crystals as discussed in Fig. 1. The +three molecules share the same backbone while differing only +in the side chains. The brittle crystal consists of the molecules +with the propyl group, featured by the orthorhombic space +group Pbca with one molecule in the asymmetric unit. On the +other hand, the elastic/plastic crystals have the ethyl/methyl +groups, both adopting the monoclinic space group P21/c with +half a molecule in the asymmetric unit. For convenience, we +follow the previous literature10 to name these systems accord- +ing to their molecular functional groups (i.e., Pr, Et, Me). +In all three cases, the weak interaction are formed by alkyl +groups at the (001) plane. However, the overall molecular +packing in the brittle-Pr crystal are more complex since there +exist eight different types of molecular alignments due to the +mmm symmetry operations. On the contrary, there are two +types of molecular alignments in the Et/Me crystals, and each +(001) layer contains only one type of molecular alignment (see +Fig. S1 and table S1). +Fig. +2 summarized the simulated evolution of average +molecular potential energy as a function of indentation depth +for all three materials. For a fair comparison, we set up the +model size close to ∼ 50.0 × 7.0 × 7.0 nm3. Encouragingly, +our calculations reproduced the experimentally observed brit- +tle fracture, elastic deformation and plastic bending, respec- +tively. First, Pr is clearly brittle as evidenced by the abrupt +drop of energy in Fig. 2a, which is also consistent to the +appearance of crack pattern in Fig. 1a when the indentation +depth reaches 3.5 nm. On the other hand, Et is more com- +plaint with a maximum indentation of 6.2 nm. Apply further +loading would lead to the formation of crack as well. If we re- +lease the indentation before Et reaches 6.2 nm, the simulation +FIG. 2. +The evolution of average molecular potential energy as +a function of indentation depth upon (a) loading and (b) unloading. +In (b), only two samples (Me-elastic and Me-plastic) are shown for +clarity. +will roughly return to the original state. Therefore, the defor- +mation is elastic. Interestingly, Me can survive under more +than 10 nm indentation without breaking with two different +setups. For the slab after a full isobaric-isothermal equilibra- +tion, it bends elastically, as evidenced by the reversible energy +versus indentation depth relation (denoted as Me-elastic in +Fig. 2b). When the slab has a small strain in the initial config- +uration (see Table S2), the corresponding energy curves upon +loading and unloading are no longer reversible. Compared to +the Me-elastic, this sample achieves lower energy stable when +it approaches the maximum indentation depth upon loading. +When the indentation is released, it does not return to the orig- +inal states, but maintains a relatively higher energy. Therefore, +the whole deformation process is irreversible and plastic. The +sample will be referred to Me-plastic from now on. It is also +important to note that the deformation is strongly anisotropic. +For the same Me sample, the deformations are brittle if the +indentation is applied on other directions. Such a direction- +dependence has also been observed in recent experiments16. +Although several recent computational studies attempted to +explain the observed mechanical properties, they were lim- +ited to indirect simulations such as pure tensile and shear +tests10,17–19. Here, our results provide the first direct evidence +from atomistic modeling and reproduce the experiment obser- +vations on their mechanical responses upon the bending de- +formation. Compared to the simulation results, the elastic and +plastic samples are found to bend under larger deformations in +real experiments10. This is because that the material’s length +on x-axis under the actual bending test can shrink to release +the tensile stress. However, our simulation model still obeys +the periodic boundary condition along the x-axis. Hence we +expect that the degree of bending from our simulation is un- + +(a) Loading +Pr: brittle +0.4 +Et: elastic +△E (kJ/mol) +Me: elastic +Me: plastic +0.2 +0.0 +0 +4 +6 +8 +10 +(b) Unloading +Me: elastic +△E (kJ/mol) +0.2 +Me: plastic +0.0 +0 +2 +4 +6 +8 +10 +Indentation Depth (nm)3 +derestimated as compared to the real situation. We also tried +to vary the strain rate. According to our attempts, it seems that +increasing the strain rate by 10 times does not qualitatively +change the results. However, an ultrafast strain rate (>200 +m/s) is likely to trigger some unrealistic phase transition thus +changes the nature of deformation significantly. Regardless +of these restrictions on parameter choices, our simulations are +robust in capturing the main physics. +0.0 +0.1 +Pr: brittle +Et: elastic +Me: plastic +0.0 +0.1 +40 +20 +0 +20 +40 +0.0 +0.1 +Distribution +Rotation (degree) +FIG. 3. The simulated distribution of accumulated rotational angles +(with respect to the initial configurations) for all materials upon the +bending loads. For clarity, the Me-elastic data was omitted. +While analyzing the dynamic trajectories, we observed that +molecules rotate strongly upon bending. Fig. 1 defines the +alignments (α, β, γ) for each molecule that can rotate along +the x, y, z axes in the Cartesian coordinates. Fig. 3 plots +the distribution of molecular rotations for all three directions. +Given that indentation direction acts on the z-axis and the +setup of three bending points aligns along the x-axis, we ex- +pect that the rotational mode along y axis (β) is the primary +motion under the loading. Indeed, Fig. 3 reveals that the rota- +tion in β is more pronounced that other directions for all three +molecules. According to the computed moments of rotational +inertia in Table I, the molecules with smaller size are easier to +rotate more. Therefore, Me has overall more rotational flexi- +bility than Et and Pr in all directions. +TABLE I. The computed moments of rotational inertia (Da· ˚A2) for +each system. +System +Number of atoms +Ixx +Iyy +Izz +Pr +44 +1707.95 4124.74 5606.78 +Et +38 +2332.63 2311.36 3610.28 +Me +32 +1911.78 1710.74 2854.21 +In Figs. S3-S531, we provided the detailed analysis on each +simulation trajectory. Among them, it is mostly interesting +to note that there is an obvious asymmetric distribution of β +for the plastic deformation as shown in Fig. 3. To quest its +origin, we plot a few representative structures from the cor- +responding trajectory in Fig. 4. Unlike the elastic deforma- +5.0 +7.5 +10.0 +Indentation depth (nm) + (degree) +β +-30 +30 +0 +-15 +15 +FIG. 4. +The list of representative snapshots from the simulation of +Me-plastic deformation. The molecules are colored by the β angle +values from red to blue. The domains of the secondary phase are +highlighted by the red dotted eclipses. The red dotted arrows indicate +the slip direction. The grey colored shapes represent the contacting +locations in the three-point bending test. +tion that all molecules are symmetrically aligned at the cen- +tered yz plane, we found that the region near the indenter +tip undergoes a phase transition through molecular rotation. +This region is also evident from non-zero rotations of α and +γ as shown in Fig. S5. This new domain, consisting of re- +aligned molecules (denoted as the red dotted eclipse), can +easily slip along its interface with the parent domain. Upon +indentation, the molecules in the secondary domain, located +on the upper surface of the slab, do not gain enough momen- +tum to go downward as compared to other molecules due to +the compressive stress from the bending forces. Therefore, +the relative slipping direction of the secondary domain is up- +ward and we observe the appearance of a bump near the in- +denter tip. As the tip continues to go down, the secondary +domain keeps climbing up until the bump reaches its maxi- +mum. Upon further compression, the molecules at the bottom +region are nearly flattened due to a large tensile stress, thus +creating much empty space along the z-axis. Thus, the sec- +ondary domain slips down to push the neighboring molecules +down to fill the empty space. Clearly, this secondary domain +serves as a buffer zone to help the system maintain a rela- +tively low energy state and postpone the formation of crack. +When the indentation is released, the process is supposed to +be irreversible at low temperature since triggering the back +transformation requires some energy barrier. Therefore, it is +a plastic deformation. However, it is driven by the molecu- +lar rotation, which is different from the plastic phenomenon +in the metals that requires the migration of dislocations. Due +to the phase transition driven by molecule rotation, the do- +main of new phase may appear near the indenter and coexist + +4 +20 +10 +0 +10 +20 +30 +40 +R1 (degree) +20 +10 +0 +10 +20 +30 +40 +R2 (degree) +GM +LM +(a) Brittle +20 +10 +0 +10 +20 +30 +40 +R1 (degree) +20 +10 +0 +10 +20 +30 +40 +GM +LM +(b) Elastic +20 +10 +0 +10 +20 +30 +40 +R1 (degree) +20 +10 +0 +10 +20 +30 +40 +GM +LM +(c) Plastic +10 +1 +100 +101 +102 +103 +104 +E (kJ/mol) +FIG. 5. The potential energy surface as a function of molecular rotation for three crystals with different mechanical response: (a) Pr-brittle, +(b) Et-elastic, and (c) Me-elastic/plastic deformations. The while region in (a) denotes the rotations leading to energy exceeding 104 kJ/mol. +with the parent phase via a low-energy interface. The newly +formed secondary phase can freely slide along the interface +due to the external stress conditions. In the early stage, the +upward movement of new phase results in a bump shape near +the indenter. We note that such a bump has actually been +found in the bending experiment10, but it was not discussed +in the literature. Our simulation here suggests that the forma- +tion of bump is a key characteristic of the plastic deformation +driven by molecular rotation. If the external temperature is +sufficiently high to cross the phase transition barrier, the pro- +cess may become reversible, similar to the previously reported +superelastic organic crystals4. +So far, we have established the relation between molecular +rotation and the observed mechanical responses. Clearly, the +degree of freedom of molecular rotation is the key factor that +determines the mechanical flexibility of organic crystals under +bending. However, we are still unclear why some materials +are more compliant than others and why we observed two dif- +ferent deformation behaviors on the Me crystal with slightly +different initial configurations. To quest their physical origins, +it is necessary to examine the potential energy surface (PES) +with respect to the molecular rotations. Therefore, we use the +relaxed crystal structure as the reference and then systemat- +ically rotate two groups of symmetrically-related molecules +(colored in red and blue in Fig. S1) along the y-axis in the +unit cell. For the Pr-crystal, each group has four molecules +with the same alignment in β. For both Et and Me crystals, +each group contains only one molecule. The computed poten- +tial energy maps as the function of the rotation angles (R1 and +R2) are summarized in Fig. 5. +As shown in Fig. +5a, Pr has a very stiff global mini- +mum (GM) at (0, 0). This indicates that even a slight rota- +tion can lead to a high energy penalty. The energy basin of +GM is aligned diagonally, suggesting that the low energy rota- +tion modes are synchronous due to the crystal symmetry con- +straint. In this energy basin, the total energy of the whole sys- +tem increase about 500 kJ/mol, when it reaches the (10, 10). +However, such high energy penalty would eventually lead to +the generation of crack. In addition, there is a local minimum +(LM) centered around (20, 20). But this state is nearly impos- +sible to reach due to a high energy barrier up to 104 kJ/mol. +Overall, Pr has a rather limited rotational freedom, which is +consistent with the fact that each molecule in Pr is surrounded +by multiple types of molecular alignments. +Compared to Pr, the Et sample (Fig. 5b) has more spreads +around the GM (0, 0). Therefore, the molecules can rotate +more under the mechanical load. As shown in Fig. 3, two +rotational peaks are symmetrically distributed at ±20 degrees +when the system reaches the elastic limit. According to Fig. +5b, the rotation around (20, 20) would lead to a penalty energy +of 500 kJ/mol. Therefore, the Et molecules can rotate more +than Pr before the crack event starts. Similarly, Et has another +LM around (30, 30), but it is unreachable due to a high energy +barrier. +On the other hand, the Me has a even flatter energy spread +around the GM basin (Fig. 3c). Using 500 kJ/mol as the +threshold, the computed area ratios are roughly 0.14 (Pr), 0.84 +(Et), 1.00 (Me). Hence, the Pr can sustain more elastic de- +formation than other materials. These values are qualitatively +consistent with our computed critical indentation depth val- +ues as shown in Fig. 2, and even fits the experimental values +better (given that Me is found to be significantly more elastic +than Pr). In addition, Me is remarkable because there exists +a low energy pathway that connects its LM at (30, 30). Under +the mechanical load, there exist two scenarios. One is to con- +tinue to expand in the GM basin and the system bends elasti- +cally, as we found in our simulation starting with the perfectly +equilibrated Me single crystal sample. Alternatively, it is also +possible to reach the neighboring LM basin. While the latter +case requires crossing a barrier on its PES map, it may be fa- +cilitated by the pre-existing structural defects or activated due +to kinetic reason. Indeed, we observed such a phase transition +when the initial configuration is strained. And this eventually +led to a plastic deformation as shown in Fig. 4. Correspond- +ingly, the existence of molecules at the LM (30, 30) region +resulted in a stronger peak around 30 degree as compared to +the peak at -30 degree for the distribution of β in Fig. 3, In +the real experiment, the latter scenario is more likely to occur +since the defects are unavoidable. Although the deformation +process is irreversible at low temperature upon the release of + +5 +indentation, it may become reversible at an elevated tempera- +ture when it is sufficient to cross the barrier between LM and +GM. +In summary, we perform the first molecular dynamics sim- +ulation to directly model the mechanical bending of organic +crystals. Using three recently reported naphthalene diimide +derivatives as the examples, our simulation successfully re- +produced the experimentally observed mechanical behaviors +from brittle fracture to elastic/plastic deformation upon me- +chanical bending. By analyzing their atomistic trajectories, +we found that molecular rotational freedom is the key factor +to determine whether or not the materials are bendable. This +phenomenon originates from the subtle interplay between ge- +ometry packing and intermolecular interaction. 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HE and B. MA, Trans. Nonferrous Met. Soc. China 32, 3687 +(2022). +27 C. M. Reddy, G. R. Krishna, and S. Ghosh, CrystEngComm 12, +2296 (2010). +28 C. Wang and C. C. Sun, Cryst. Growth Des. 18, 1909 (2018). +29 N. Mathew, C. R. Picu, and P. W. Chung, J. Phys. Chem. A 117, +5326 (2013). +30 I. A. Olson, A. G. Shtukenberg, B. Kahr, and M. D. Ward, Rep. +Prog. Phys. 81, 096501 (2018). +31 See Supplemental Material at http://link.aps.org/**** for a de- +tailed description of molecular packing, model setup and molec- +ular dynamics simulation results analysis for three naphthalenete- +tracarboxylic diimide crystals. +32 D. A. Case, K. Belfon, I. Y. Ben-Shalom, S. R. Brozell, D. S. +Cerutti, T. E. Cheatham, III, V. W. D. Cruzeiro, T. A. Darden, +R. E. Duke, G. Giambasu, M. K. Gilson, H. Gohlke, A. W. +Goetz, R. Harris, S. Izadi, S. A. Izmailov, K. Kasavajhala, A. Ko- +valenko, R. Krasny, T. Kurtzman, T. S. Lee, S. LeGrand, P. Li, +C. Lin, J. Liu, T. Luchko, R. Luo, V. Man, K. M. Merz, Y. Miao, +O. Mikhailovskii, G. Monard, H. Nguyen, A. Onufriev, F. Pan, +S. Pantano, R. Qi, D. R. Roe, A. Roitberg, C. Sagui, S. Schott- +Verdugo, J. Shen, C. L. Simmerling, N. R. Skrynnikov, J. Smith, +J. Swails, R. C. Walker, J. Wang, L. Wilson, R. M. Wolf, X. Wu, +Y. Xiong, Y. Xue, D. M. York, and P. A. Kollman, AMBER 2020 +(2020). +33 A. Jakalian, B. L. Bush, D. B. Jack, and C. I. Bayly, J. Comput. +Chem. 21, 132 (2000). +34 S. Plimpton, J. Comput. Phys. 117, 1 (1995). +35 Q. Zhu, W. Tang, and S. Hattori, Cryst. Growth Des. 22, 7308 +(2022). + +6 +Supplementary Online Materials: +Bending Deformation Driven by Molecular Rotation +A. Crystal structures +In this study, we focused on three systems consisting of naphthalene diimide derivatives as shown in Fig. S1. The three molecules +share the same backbone while differing only in the side chains. The brittle crystal consists of the molecules with the propyl +group (Pr), featured by the orthorhombic space group Pbca with one molecule in the asymmetric unit. On the other hand, the +elastic/plastic crystals have the ethyl/methyl groups, both adopting the monoclinic space group P21/c with half a molecule in +the asymmetric unit. In all three cases, the weak interaction plane formed by alkyl groups is (001). In Fig. S1, each molecule in +the unit cell is colored by the alignment along the y-axis. Clearly, the overall molecular packing in the brittle-Pr crystal are more +complex. Since there exist eight different types of molecular alignments due to the mmm symmetry operations, the Pr crystal +has molecules aligned in different ways within the same (001) layer. On the contrary, there are only two types of molecular +alignments in the Et/Me crystals. And the (001) layer in Et/Me crystals has all molecules aligned in the same direction. +FIG. S1. The crystal structures of (a) Pr, (b) Et (c) Me systems. +Table S1 summarizes the crystallographic information of three molecular crystals. Among them, Pr denotes the brittle crystal +with the CSD refcode of DAHLOQ; Et is the elastic crystal with the CSD refcode of BIYRIM01; and Me is the plastic crystal +with the CSD refcode of DAHMUX. In addition to the experimental cell parameters, the equilibrium cell parameters from our +Amber force field are also shown in the parentheses for a comparison. The excellent agreement between experiment and theory +warrants the use of Amber force field in our following simulations. +TABLE S1. The crystallographic information of three molecular crystals. +System +CSD Refcode +Space Group +Number of molecules +a ( ˚A) +b ( ˚A) +c ( ˚A) +β (◦) +Pr +DAHLOQ +Pbca +8 +6.96 (7.30) 17.24 (17.40) 27.58 (27.90) 90.0 (90.0) +Et +BIYRIM01 +P21/c +2 +4.84 (5.07) +7.74 (7.79) +18.32 (19.07) 90.1 (90.3) +Me +DAHMUX +P21/c +2 +4.62 (4.58) +8.02 (8.28) +17.02 (18.40) 94.0 (93.9) + +(a) +(b) +(c)7 +B. Simulation Setup +To enable the direct simulation of bending, we created the slab model as shown in Fig. S2. Both x and y-axes are under the +constraint of periodic boundary conditions, while the c-axis is not periodic. To reproduce the experimental results10, we rotated +the crystal structures with the matrix of [[0,0,1], [0,-1,0], [1,0,0]], and then built the super cell slab models according to Table +S2. In each case, we added the vacuum to allow the materials bend sufficiently. The slab correction was applied to remove the +slab-slab interactions from the periodic images. Due to the non-triclinic box restriction on the computation of slab correction, +the β angles for the slabs of Et and Me were to be set to 90◦, which are slightly different from the ideal values. However, this +compromise should not change the results largely. +For Me, two models were considered, including (i) the supercell after the isobaric-isothermal equilibration; and (ii) the +supercell with the experimental cell parameters. Although these two initial configurations only differ slightly, it has been found +they led to different elastic/plastic deformation processes in the subsequent bending simulation. +FIG. S2. The schematic setup of a bending simulation model. +TABLE S2. The details of models used in the bending simulation. +System +Deformation +Supercell +Number of molecules +a ( ˚A) +b ( ˚A) +c ( ˚A) +Vacuum ( ˚A) +Pr +brittle +18 × 4 × 5 +5760 +503.2 +69.9 +70.6 +120.0 +Et +elastic +27 × 4 × 5 +6480 +508.5 +63.6 +74.7 +120.0 +Me +elastic +29 × 8 × 15 +6960 +501.6 +65.2 +86.5 +120.0 +Me +plastic +30 × 8 × 15 +7200 +510.6 +64.2 +85.1 +120.0 +Along the non-periodic z-axis, a cylinderical indenter with the radius of 30 ˚A is applied on top of the slab center in the unit +cell. To mimic two other contacting points in the three-points bending simulation, the last one layer of molecules in the bottom +region were frozen in the entire simulation. In addition, the first columns of molecules on both left and right side of the unit +cell are defined as the border. The rest atoms not belonging the frozen and border groups are set to the moible group that can +move freely. To ensure a sufficient heat bath, we first perform Langevin thermostat on both mobile and border groups, followed +by a second thermal equilibration on only the border atoms. The fully equilibrated sample will be used to perform three-points +bending simulation with only the border atoms being under the Langevin thermostat to mimic the external temperature reservoir. +Upon bending, the indenter will be used to push into the simulation slab in a flow with the rate of 10 m/s. When the system +reaches the maximum indentation depth, the indenter will be kept for 300 ps to allow the system achieves thermal equilibrium. +Afterwards, the indenter will move upward with the previous rate to mimic the release of indenter process. + +Z +Border +Frozen +Mobile8 +C1. Deformation Analysis on Pr-Brittle +To quest the origin of Pr-Brittle, we plot a few representative structures from the corresponding trajectory in Fig. S3. Upon +deformation, we found that the sample continuously to bend from 0 to 3.0 nm (the first row of Fig. S3) and 4.0 nm (the second +row of Fig. S3). The molecules barely rotate around the x (α) and z (γ) axis. However, the rotation on y-axis is more pronounced +and it symmetrically distributed around the central indenter. When the indentation depth exceeds 4.2 nm (the last row of Fig. +S3), the lower surface cracks due to a large tensile stress. +2.5 +3.5 +4.5 +Indentation depth (nm) + (degree) +α +-30 +30 +0 +-15 +15 + (degree) +β +-30 +30 +0 +-15 +15 + (degree) +γ +-30 +30 +0 +-15 +15 +Pr-brittle +FIG. S3. The list of representative snapshots from the simulation of Pr-Brittle deformation. + +9 +C2. Deformation Analysis on Et-Elastic +To quest the origin of Et-Elastic, we plot a few representative structures from the corresponding trajectory in Fig. S4. At a +small indentation depth (4.0 nm as shown in the first row of Fig. S4), the molecules barely rotate around the x (α) and z (γ) axis, +while the rotation on y-axes (β) is more pronounced and it symmetrically distributed around the central indenter. However, it +is clear that the molecules around the center of y-axis do not rotate. Upon further indentation at 5.0 nm (the second row of Fig. +S4) and 6.2 nm (the last row of Fig. S4), the molecules at the center of lower surface undergo a large rotation around the x and +z due to a large compressive stress, but do not rotate around y. This suggests that molecules upon tension prefer a rotation on +α and γ, rather than the primary rotation mode at β due to the anisotropic behavior of its potential energy landscape. Since the +rotations are symmetrically distributed around the indenter, it is still an elastic deformation. When the indentation is released, +the process is supposed to be reversible. +4.0 +5.0 +6.0 +Indentation depth (nm) + (degree) +α +-30 +30 +0 +-15 +15 + (degree) +β +-30 +30 +0 +-15 +15 + (degree) +γ +-30 +30 +0 +-15 +15 +Et-elastic +FIG. S4. The list of representative snapshots from the simulation of Et-Elastic deformation. + +10 +C3. Deformation Analysis on Me-Plastic +To quest the origin of Me-Plastic, we plot a few representative structures from the corresponding trajectory in Fig. S5. At +the depth of 5.5 nm, we found that the molecules near the indenter tip (in the first row of Fig. S4) have alternative changes of +α and γ angles, which is similar to that in Fig. S4. However, these molecule has non-zero β angles. Therefore, it is no longer +symmetric and signals a phase transition trigger by the large compressive stress in the upper surface due to bending. This domain +of new phases, consisting of realigned molecules (denoted as the red dotted eclipse), can easily slip along its interface with the +parent domain. Upon indentation, the molecules in the secondary domain do not gain enough momentum to go downward as +compared to other molecules. Therefore, the relative slipping direction of the secondary domain is upward and we observe the +appearance of a bump near the indenter tip (in the second row of Fig. S3 at the indentation depth of 6.7 nm). As the tip continues +to go down, the secondary domain keeps climbing up until the bump reaches its maximum. In the mean time, the the molecules +at the center bottom region are nearly flattened, which can trigger another phase transition to form a new phase domain. Upon +further compression, the flattened molecules at the center bottom region create much empty space along the z-axis. Thus, the +secondary domain slips down to push the neighboring molecules down to fill the empty space (see the third row of Fig. S3 at +the indentation depth of 9.5 nm). When the indentation is released, the process is supposed to be irreversible at low temperature +since triggering the back transformation requires some energy barrier. Therefore, it is a plastic deformation. +5.0 +7.5 +10.0 +Indentation depth (nm) + (degree) +α +-30 +30 +0 +-15 +15 + (degree) +β +-30 +30 +0 +-15 +15 + (degree) +γ +-30 +30 +0 +-15 +15 +Me-plastic +FIG. S5. The list of representative snapshots from the simulation of Me-plastic deformation. + diff --git a/_dAyT4oBgHgl3EQfdvdN/content/tmp_files/load_file.txt b/_dAyT4oBgHgl3EQfdvdN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..45a52b810043027517d1b94eeb45c665159c1bed --- /dev/null +++ b/_dAyT4oBgHgl3EQfdvdN/content/tmp_files/load_file.txt @@ -0,0 +1,788 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf,len=787 +page_content='Bending Deformation Driven by Molecular Rotation Pedro A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Santos-Florez,1 Shinnosuke Hattori,2 and Qiang Zhu1, ∗ 1Department of Physics and Astronomy, University of Nevada, Las Vegas, NV 89154, USA 2Advanced Research Laboratory, R&D Center, Sony Group Corporation, 4-14-1 Asahi-cho, Atsugi-shi 243-0014, Japan (Dated: January 3, 2023) Recently, some molecular crystals have been found to be surprisingly flexible by undergoing a large extent of elastic or plastic deformation upon various mechanical loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Despite the increasing experimental reports on mechanically flexible crystals, this phenomenon has never been reproduced in numerical simulation and thus there is no atomistic mechanism to explain its physical origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Using three recently reported naphthalene diimide derivatives as the examples, we perform the first direct molecular dynamics simulation to model their mechanical behaviors from brittle fracture to elastic/plastic deformation upon mechanical bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Our simulation reveals that molecular rotational freedom is the key factor to determine the crystal’s mechanical response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Furthermore, we propose the use of rotation-dependent potential energy surface to classify organic materials’ mechanical response and screen new mechanically flexible candidates in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' While most molecular crystals are known to be brittle, there exists a class of compliant organic crystals that can easily bend under a large mechanical stress loading1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Since early 2000, there has been a growing number of experimental identifi- cations of mechanically flexible crystals3–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In general, the mechanical response of an organic solid depends on both the molecular substance and the corresponding crystal packing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' A remarkable example is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 1, three crystals, made of similar molecules from naphthalene diimide deriva- tives, were found to exhibit distinct responses from brittle fracture to compliant deformation with either reversible (elas- tic) or irreversible (plastic) characteristic10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The flexible na- ture of such organic materials is vital for a variety of appli- cations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=', high-performance modular organic solar cells11, actuators12, photochemistry13, electronics14, optics15, as well as drug tabulation16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In the recent years, various computational techniques have been introduced to characterize the observed mechanical properties on different molecular systems10,17–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' They in- clude the topological analysis, elastic properties calculation17, and the simulation of shear/tensile deformations10,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' These techniques are partially successful in identifying the brittle materials which usually exhibit a complex three dimensional packing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Within such an interlocked environment, molecu- lar motions are largely restricted, resulting a brittleness un- der bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' On the other hand, the compliant class of ma- terials are featured by a strong anisotropy with plausible slip planes17,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Therefore, these materials become compliant over a broad range of applied stress along some specific crystallo- graphic directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' However, all available techniques fail to explain the difference between the elastic and plastic materi- als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' While there have been plenty of studies on the bending of metals21–26, to our knowledge, no attempts have been made to directly simulate the bending of organic materials at the atom- istic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Among the compliant crystals, ductile materials are often favored in engineering applications16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Hence, researchers at- tempted to use the well established dislocation theory to ex- plain the observed plasticity on organic materials2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Simi- lar to the plastic deformation in ductile metals, it was found that mechanical shearing can also occur via the slippage of dislocated molecular layers on the molecular crystals with a β α γ α β γ (a) (b) (c) (degree) α x z x y z γ β α y y z x 30 30 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The simulated bending on three different materials based on naphthalene diimide derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' (a) brittle Pr (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='3×7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='8 nm3), (b) elastic Et (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='7×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='4×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='6 nm3) and (c) elastic/plastic Me (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='2×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='4×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='9 nm3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' These three crystals consist of very similar molecules that differ only in the side groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In the left panel, the initial and finally deformed configurations are colored by the molec- ular alignment (α) along the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The corresponding molecules and the definition of rotation angles are shown in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' layered packing27,28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Using these facile slip planes, a bend- ing model was proposed accordingly to explain the underly- ing mechanism3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Although the dislocation is not uncommon in molecular crystals29,30, there has been no direct experimen- tal evidence to support that the dislocation is present in the organic crystals under bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Furthermore, this mechanism fails to explain the observed crystals that can also bend elas- tically to a large extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In fact, two crystals as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='00307v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='mtrl-sci] 31 Dec 2022 X z7X z2 1b-c possess very similar crystal packing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Give the apparent similarity in both molecular structure and crystal packing, it is expected that the elastic crystal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 1b) should undergo similar molecular events like the plastic crystal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 1c) by following the ending mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' But the actual deformation was observed to be elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Clearly, our current understanding on the elasticity and plasticity remains limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In this work, we present our efforts in questing the molec- ular bending mechanism with the aid of atomistic simula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' To achieve this goal, we start by developing a ro- bust simulation protocol that can directly model the bend- ing of organic crystals at the atomic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Specifically, we employed a three-point bending model within a partial peri- odic boundary condition31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In our calculation, we performed non-equilibrium molecular dynamics simulation by applying the indentation on the center of molecular slab under finite temperature31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' We also carefully tested the choice of slab models and thermal equilibration to ensure the robustness of our simulation set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In order to automate the simulation, we developed a computational pipeline to automate the gen- eration of molecular force fields from the AmberTools20 software32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Force field parameters are assigned by the Gen- eral Amber Force Field (GAFF) with atomic charges using semi-empirical (AM1) with bond charge correction (BCC)33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' All simulations were performed on the LAMMPS package34 at room temperature with the strain rate of 10 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In the following, we will focus on three naphthalenete- tracarboxylic diimide crystals as discussed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The three molecules share the same backbone while differing only in the side chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The brittle crystal consists of the molecules with the propyl group, featured by the orthorhombic space group Pbca with one molecule in the asymmetric unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' On the other hand, the elastic/plastic crystals have the ethyl/methyl groups, both adopting the monoclinic space group P21/c with half a molecule in the asymmetric unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' For convenience, we follow the previous literature10 to name these systems accord- ing to their molecular functional groups (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=', Pr, Et, Me).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In all three cases, the weak interaction are formed by alkyl groups at the (001) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' However, the overall molecular packing in the brittle-Pr crystal are more complex since there exist eight different types of molecular alignments due to the mmm symmetry operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' On the contrary, there are two types of molecular alignments in the Et/Me crystals, and each (001) layer contains only one type of molecular alignment (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S1 and table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 2 summarized the simulated evolution of average molecular potential energy as a function of indentation depth for all three materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' For a fair comparison, we set up the model size close to ∼ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 nm3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Encouragingly, our calculations reproduced the experimentally observed brit- tle fracture, elastic deformation and plastic bending, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' First, Pr is clearly brittle as evidenced by the abrupt drop of energy in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 2a, which is also consistent to the appearance of crack pattern in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 1a when the indentation depth reaches 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' On the other hand, Et is more com- plaint with a maximum indentation of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='2 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Apply further loading would lead to the formation of crack as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' If we re- lease the indentation before Et reaches 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='2 nm, the simulation FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The evolution of average molecular potential energy as a function of indentation depth upon (a) loading and (b) unloading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In (b), only two samples (Me-elastic and Me-plastic) are shown for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' will roughly return to the original state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Therefore, the defor- mation is elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Interestingly, Me can survive under more than 10 nm indentation without breaking with two different setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' For the slab after a full isobaric-isothermal equilibra- tion, it bends elastically, as evidenced by the reversible energy versus indentation depth relation (denoted as Me-elastic in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' When the slab has a small strain in the initial config- uration (see Table S2), the corresponding energy curves upon loading and unloading are no longer reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Compared to the Me-elastic, this sample achieves lower energy stable when it approaches the maximum indentation depth upon loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' When the indentation is released, it does not return to the orig- inal states, but maintains a relatively higher energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Therefore, the whole deformation process is irreversible and plastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The sample will be referred to Me-plastic from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' It is also important to note that the deformation is strongly anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' For the same Me sample, the deformations are brittle if the indentation is applied on other directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Such a direction- dependence has also been observed in recent experiments16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Although several recent computational studies attempted to explain the observed mechanical properties, they were lim- ited to indirect simulations such as pure tensile and shear tests10,17–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Here, our results provide the first direct evidence from atomistic modeling and reproduce the experiment obser- vations on their mechanical responses upon the bending de- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Compared to the simulation results, the elastic and plastic samples are found to bend under larger deformations in real experiments10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' This is because that the material’s length on x-axis under the actual bending test can shrink to release the tensile stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' However, our simulation model still obeys the periodic boundary condition along the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Hence we expect that the degree of bending from our simulation is un- (a) Loading Pr: brittle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='4 Et: elastic △E (kJ/mol) Me: elastic Me: plastic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 0 4 6 8 10 (b) Unloading Me: elastic △E (kJ/mol) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='2 Me: plastic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 0 2 4 6 8 10 Indentation Depth (nm)3 derestimated as compared to the real situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' We also tried to vary the strain rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' According to our attempts, it seems that increasing the strain rate by 10 times does not qualitatively change the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' However, an ultrafast strain rate (>200 m/s) is likely to trigger some unrealistic phase transition thus changes the nature of deformation significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Regardless of these restrictions on parameter choices, our simulations are robust in capturing the main physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='1 Pr: brittle Et: elastic Me: plastic 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='1 40 20 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='1 Distribution Rotation (degree) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The simulated distribution of accumulated rotational angles (with respect to the initial configurations) for all materials upon the bending loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' For clarity, the Me-elastic data was omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' While analyzing the dynamic trajectories, we observed that molecules rotate strongly upon bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 1 defines the alignments (α, β, γ) for each molecule that can rotate along the x, y, z axes in the Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 3 plots the distribution of molecular rotations for all three directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Given that indentation direction acts on the z-axis and the setup of three bending points aligns along the x-axis, we ex- pect that the rotational mode along y axis (β) is the primary motion under the loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Indeed, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 3 reveals that the rota- tion in β is more pronounced that other directions for all three molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' According to the computed moments of rotational inertia in Table I, the molecules with smaller size are easier to rotate more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Therefore, Me has overall more rotational flexi- bility than Et and Pr in all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The computed moments of rotational inertia (Da· ˚A2) for each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' System Number of atoms Ixx Iyy Izz Pr 44 1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='95 4124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='74 5606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='78 Et 38 2332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='63 2311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='36 3610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='28 Me 32 1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='78 1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='74 2854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='21 In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S3-S531, we provided the detailed analysis on each simulation trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Among them, it is mostly interesting to note that there is an obvious asymmetric distribution of β for the plastic deformation as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' To quest its origin, we plot a few representative structures from the cor- responding trajectory in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Unlike the elastic deforma- 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 Indentation depth (nm) (degree) β 30 30 0 15 15 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The list of representative snapshots from the simulation of Me-plastic deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The molecules are colored by the β angle values from red to blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The domains of the secondary phase are highlighted by the red dotted eclipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The red dotted arrows indicate the slip direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The grey colored shapes represent the contacting locations in the three-point bending test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' tion that all molecules are symmetrically aligned at the cen- tered yz plane, we found that the region near the indenter tip undergoes a phase transition through molecular rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' This region is also evident from non-zero rotations of α and γ as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' This new domain, consisting of re- aligned molecules (denoted as the red dotted eclipse), can easily slip along its interface with the parent domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Upon indentation, the molecules in the secondary domain, located on the upper surface of the slab, do not gain enough momen- tum to go downward as compared to other molecules due to the compressive stress from the bending forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Therefore, the relative slipping direction of the secondary domain is up- ward and we observe the appearance of a bump near the in- denter tip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' As the tip continues to go down, the secondary domain keeps climbing up until the bump reaches its maxi- mum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Upon further compression, the molecules at the bottom region are nearly flattened due to a large tensile stress, thus creating much empty space along the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Thus, the sec- ondary domain slips down to push the neighboring molecules down to fill the empty space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Clearly, this secondary domain serves as a buffer zone to help the system maintain a rela- tively low energy state and postpone the formation of crack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' When the indentation is released, the process is supposed to be irreversible at low temperature since triggering the back transformation requires some energy barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Therefore, it is a plastic deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' However, it is driven by the molecu- lar rotation, which is different from the plastic phenomenon in the metals that requires the migration of dislocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Due to the phase transition driven by molecule rotation, the do- main of new phase may appear near the indenter and coexist 4 20 10 0 10 20 30 40 R1 (degree) 20 10 0 10 20 30 40 R2 (degree) GM LM (a) Brittle 20 10 0 10 20 30 40 R1 (degree) 20 10 0 10 20 30 40 GM LM (b) Elastic 20 10 0 10 20 30 40 R1 (degree) 20 10 0 10 20 30 40 GM LM (c) Plastic 10 1 100 101 102 103 104 E (kJ/mol) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The potential energy surface as a function of molecular rotation for three crystals with different mechanical response: (a) Pr-brittle, (b) Et-elastic, and (c) Me-elastic/plastic deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The while region in (a) denotes the rotations leading to energy exceeding 104 kJ/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' with the parent phase via a low-energy interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The newly formed secondary phase can freely slide along the interface due to the external stress conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In the early stage, the upward movement of new phase results in a bump shape near the indenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' We note that such a bump has actually been found in the bending experiment10, but it was not discussed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Our simulation here suggests that the forma- tion of bump is a key characteristic of the plastic deformation driven by molecular rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' If the external temperature is sufficiently high to cross the phase transition barrier, the pro- cess may become reversible, similar to the previously reported superelastic organic crystals4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' So far, we have established the relation between molecular rotation and the observed mechanical responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Clearly, the degree of freedom of molecular rotation is the key factor that determines the mechanical flexibility of organic crystals under bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' However, we are still unclear why some materials are more compliant than others and why we observed two dif- ferent deformation behaviors on the Me crystal with slightly different initial configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' To quest their physical origins, it is necessary to examine the potential energy surface (PES) with respect to the molecular rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Therefore, we use the relaxed crystal structure as the reference and then systemat- ically rotate two groups of symmetrically-related molecules (colored in red and blue in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S1) along the y-axis in the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' For the Pr-crystal, each group has four molecules with the same alignment in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' For both Et and Me crystals, each group contains only one molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The computed poten- tial energy maps as the function of the rotation angles (R1 and R2) are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 5a, Pr has a very stiff global mini- mum (GM) at (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' This indicates that even a slight rota- tion can lead to a high energy penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The energy basin of GM is aligned diagonally, suggesting that the low energy rota- tion modes are synchronous due to the crystal symmetry con- straint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In this energy basin, the total energy of the whole sys- tem increase about 500 kJ/mol, when it reaches the (10, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' However, such high energy penalty would eventually lead to the generation of crack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In addition, there is a local minimum (LM) centered around (20, 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' But this state is nearly impos- sible to reach due to a high energy barrier up to 104 kJ/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Overall, Pr has a rather limited rotational freedom, which is consistent with the fact that each molecule in Pr is surrounded by multiple types of molecular alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Compared to Pr, the Et sample (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 5b) has more spreads around the GM (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Therefore, the molecules can rotate more under the mechanical load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 3, two rotational peaks are symmetrically distributed at ±20 degrees when the system reaches the elastic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 5b, the rotation around (20, 20) would lead to a penalty energy of 500 kJ/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Therefore, the Et molecules can rotate more than Pr before the crack event starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Similarly, Et has another LM around (30, 30), but it is unreachable due to a high energy barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' On the other hand, the Me has a even flatter energy spread around the GM basin (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Using 500 kJ/mol as the threshold, the computed area ratios are roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='14 (Pr), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='84 (Et), 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='00 (Me).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Hence, the Pr can sustain more elastic de- formation than other materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' These values are qualitatively consistent with our computed critical indentation depth val- ues as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 2, and even fits the experimental values better (given that Me is found to be significantly more elastic than Pr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In addition, Me is remarkable because there exists a low energy pathway that connects its LM at (30, 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Under the mechanical load, there exist two scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' One is to con- tinue to expand in the GM basin and the system bends elasti- cally, as we found in our simulation starting with the perfectly equilibrated Me single crystal sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Alternatively, it is also possible to reach the neighboring LM basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' While the latter case requires crossing a barrier on its PES map, it may be fa- cilitated by the pre-existing structural defects or activated due to kinetic reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Indeed, we observed such a phase transition when the initial configuration is strained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' And this eventually led to a plastic deformation as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Correspond- ingly, the existence of molecules at the LM (30, 30) region resulted in a stronger peak around 30 degree as compared to the peak at -30 degree for the distribution of β in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 3, In the real experiment, the latter scenario is more likely to occur since the defects are unavoidable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Although the deformation process is irreversible at low temperature upon the release of 5 indentation, it may become reversible at an elevated tempera- ture when it is sufficient to cross the barrier between LM and GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In summary, we perform the first molecular dynamics sim- ulation to directly model the mechanical bending of organic crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Using three recently reported naphthalene diimide derivatives as the examples, our simulation successfully re- produced the experimentally observed mechanical behaviors from brittle fracture to elastic/plastic deformation upon me- chanical bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' By analyzing their atomistic trajectories, we found that molecular rotational freedom is the key factor to determine whether or not the materials are bendable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' This phenomenon originates from the subtle interplay between ge- ometry packing and intermolecular interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Furthermore, we found the use of rotation-dependent potential energy sur- face map can be used clearly explain the origin of different mechanical responses for organic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Together with the recently proposed crystal packing screening model35, our re- sults can be used to guide the search for new mechanically flexible candidates with improved functionality for future de- vice applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' This research is sponsored by the NSF (DMR-2142570) and Sony Group Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The computing resources are provided by ACCESS (TG-DMR180040).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' REFERENCES ∗ qiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='zhu@unlv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='edu 1 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Naumov, S.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Kadambi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Krishna, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Kammari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Buehler, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Ramamurty, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Reddy, Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 31, 1391 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 11 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Root, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Savagatrup, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Printz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Rodriquez, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Lipomi, Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 117, 6467 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 12 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Li, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Commins, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Al-Handawi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Karothu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Ha- labi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Schramm, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Weston, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Rezgui, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Naumov, Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 10, 7327 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 13 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Mutai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Sasaki, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Takamizawa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Photochem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Photobiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' C Photochem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' , 100479 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 14 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Sun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Ren, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Dong, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Hu, Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 48, 1492 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 15 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Lu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Wang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Zhang, Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 57, 8448 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 16 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Sun, Pharm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 34, 918 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 17 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Wang and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Sun, Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Pharmaceutics 16, 1732 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 18 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Ootani and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Kubo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' C 126, 10554 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 19 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Matveychuk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Yurchenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Masunov, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Bartashevich, Cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Growth Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 22, 6472 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 20 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Bryant, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Maloney, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Sykes, CrystEngComm 20, 2698 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 21 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Zhu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Wang, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Yang, Acta Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 60, 7112 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 22 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Yan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Liu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Sun, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Liang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' D: Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 47, 195301 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 23 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' N¨ohring, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' M¨oller, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Xie, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Bitzek, Extreme Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 8, 140 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 24 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Zhuo and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Beom, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 152, 331 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 25 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Katakam and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Yedla, Superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Microst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 146, 106674 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 26 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' HE and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' MA, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Nonferrous Met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' China 32, 3687 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 27 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Reddy, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Krishna, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Ghosh, CrystEngComm 12, 2296 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 28 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Wang and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Sun, Cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Growth Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 18, 1909 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 29 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Mathew, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Picu, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Chung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' A 117, 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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Ward, Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 81, 096501 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 31 See Supplemental Material at http://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='org/**** for a de- tailed description of molecular packing, model setup and molec- ular dynamics simulation results analysis for three naphthalenete- tracarboxylic diimide crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 32 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Case, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Belfon, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Ben-Shalom, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Brozell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Cerutti, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Cheatham, III, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Cruzeiro, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Darden, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Duke, G.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Monard, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Nguyen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Onufriev, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Pan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Pantano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Qi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Roe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Roitberg, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Sagui, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Schott- Verdugo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Shen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Simmerling, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Skrynnikov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Smith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Swails, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Walker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Wilson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Wolf, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Xiong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Xue, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' York, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Kollman, AMBER 2020 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 33 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Jakalian, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Bush, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Jack, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Bayly, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 21, 132 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 34 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Plimpton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 117, 1 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 35 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Zhu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Tang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Hattori, Cryst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Growth Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 22, 7308 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 6 Supplementary Online Materials: Bending Deformation Driven by Molecular Rotation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Crystal structures In this study, we focused on three systems consisting of naphthalene diimide derivatives as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The three molecules share the same backbone while differing only in the side chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The brittle crystal consists of the molecules with the propyl group (Pr), featured by the orthorhombic space group Pbca with one molecule in the asymmetric unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' On the other hand, the elastic/plastic crystals have the ethyl/methyl groups, both adopting the monoclinic space group P21/c with half a molecule in the asymmetric unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In all three cases, the weak interaction plane formed by alkyl groups is (001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S1, each molecule in the unit cell is colored by the alignment along the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Clearly, the overall molecular packing in the brittle-Pr crystal are more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Since there exist eight different types of molecular alignments due to the mmm symmetry operations, the Pr crystal has molecules aligned in different ways within the same (001) layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' On the contrary, there are only two types of molecular alignments in the Et/Me crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' And the (001) layer in Et/Me crystals has all molecules aligned in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The crystal structures of (a) Pr, (b) Et (c) Me systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Table S1 summarizes the crystallographic information of three molecular crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Among them, Pr denotes the brittle crystal with the CSD refcode of DAHLOQ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Et is the elastic crystal with the CSD refcode of BIYRIM01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' and Me is the plastic crystal with the CSD refcode of DAHMUX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In addition to the experimental cell parameters, the equilibrium cell parameters from our Amber force field are also shown in the parentheses for a comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The excellent agreement between experiment and theory warrants the use of Amber force field in our following simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' TABLE S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The crystallographic information of three molecular crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' System CSD Refcode Space Group Number of molecules a ( ˚A) b ( ˚A) c ( ˚A) β (◦) Pr DAHLOQ Pbca 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='96 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='30) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='24 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='40) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='58 (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='90) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0) Et BIYRIM01 P21/c 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='84 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='07) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='74 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='79) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='32 (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='07) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='1 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='3) Me DAHMUX P21/c 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='62 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='58) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='02 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='28) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='02 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='40) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='9) (a) (b) (c)7 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Simulation Setup To enable the direct simulation of bending, we created the slab model as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Both x and y-axes are under the constraint of periodic boundary conditions, while the c-axis is not periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' To reproduce the experimental results10, we rotated the crystal structures with the matrix of [[0,0,1], [0,-1,0], [1,0,0]], and then built the super cell slab models according to Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In each case, we added the vacuum to allow the materials bend sufficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The slab correction was applied to remove the slab-slab interactions from the periodic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Due to the non-triclinic box restriction on the computation of slab correction, the β angles for the slabs of Et and Me were to be set to 90◦, which are slightly different from the ideal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' However, this compromise should not change the results largely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' For Me, two models were considered, including (i) the supercell after the isobaric-isothermal equilibration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' and (ii) the supercell with the experimental cell parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Although these two initial configurations only differ slightly, it has been found they led to different elastic/plastic deformation processes in the subsequent bending simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The schematic setup of a bending simulation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' TABLE S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The details of models used in the bending simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' System Deformation Supercell Number of molecules a ( ˚A) b ( ˚A) c ( ˚A) Vacuum ( ˚A) Pr brittle 18 × 4 × 5 5760 503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='2 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='9 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='6 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 Et elastic 27 × 4 × 5 6480 508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='5 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='6 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='7 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 Me elastic 29 × 8 × 15 6960 501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='6 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='5 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 Me plastic 30 × 8 × 15 7200 510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='6 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='1 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 Along the non-periodic z-axis, a cylinderical indenter with the radius of 30 ˚A is applied on top of the slab center in the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' To mimic two other contacting points in the three-points bending simulation, the last one layer of molecules in the bottom region were frozen in the entire simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In addition, the first columns of molecules on both left and right side of the unit cell are defined as the border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The rest atoms not belonging the frozen and border groups are set to the moible group that can move freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' To ensure a sufficient heat bath, we first perform Langevin thermostat on both mobile and border groups, followed by a second thermal equilibration on only the border atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The fully equilibrated sample will be used to perform three-points bending simulation with only the border atoms being under the Langevin thermostat to mimic the external temperature reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Upon bending, the indenter will be used to push into the simulation slab in a flow with the rate of 10 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' When the system reaches the maximum indentation depth, the indenter will be kept for 300 ps to allow the system achieves thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Afterwards, the indenter will move upward with the previous rate to mimic the release of indenter process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Z Border Frozen Mobile8 C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Deformation Analysis on Pr-Brittle To quest the origin of Pr-Brittle, we plot a few representative structures from the corresponding trajectory in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Upon deformation, we found that the sample continuously to bend from 0 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 nm (the first row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S3) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 nm (the second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The molecules barely rotate around the x (α) and z (γ) axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' However, the rotation on y-axis is more pronounced and it symmetrically distributed around the central indenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' When the indentation depth exceeds 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='2 nm (the last row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S3), the lower surface cracks due to a large tensile stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='5 Indentation depth (nm) (degree) α 30 30 0 15 15 (degree) β 30 30 0 15 15 (degree) γ 30 30 0 15 15 Pr-brittle FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The list of representative snapshots from the simulation of Pr-Brittle deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 9 C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Deformation Analysis on Et-Elastic To quest the origin of Et-Elastic, we plot a few representative structures from the corresponding trajectory in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' At a small indentation depth (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 nm as shown in the first row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S4), the molecules barely rotate around the x (α) and z (γ) axis, while the rotation on y-axes (β) is more pronounced and it symmetrically distributed around the central indenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' However, it is clear that the molecules around the center of y-axis do not rotate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Upon further indentation at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 nm (the second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S4) and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='2 nm (the last row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S4), the molecules at the center of lower surface undergo a large rotation around the x and z due to a large compressive stress, but do not rotate around y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' This suggests that molecules upon tension prefer a rotation on α and γ, rather than the primary rotation mode at β due to the anisotropic behavior of its potential energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Since the rotations are symmetrically distributed around the indenter, it is still an elastic deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' When the indentation is released, the process is supposed to be reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 Indentation depth (nm) (degree) α 30 30 0 15 15 (degree) β 30 30 0 15 15 (degree) γ 30 30 0 15 15 Et-elastic FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The list of representative snapshots from the simulation of Et-Elastic deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 10 C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Deformation Analysis on Me-Plastic To quest the origin of Me-Plastic, we plot a few representative structures from the corresponding trajectory in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' At the depth of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='5 nm, we found that the molecules near the indenter tip (in the first row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S4) have alternative changes of α and γ angles, which is similar to that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' However, these molecule has non-zero β angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Therefore, it is no longer symmetric and signals a phase transition trigger by the large compressive stress in the upper surface due to bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' This domain of new phases, consisting of realigned molecules (denoted as the red dotted eclipse), can easily slip along its interface with the parent domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Upon indentation, the molecules in the secondary domain do not gain enough momentum to go downward as compared to other molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Therefore, the relative slipping direction of the secondary domain is upward and we observe the appearance of a bump near the indenter tip (in the second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S3 at the indentation depth of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='7 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' As the tip continues to go down, the secondary domain keeps climbing up until the bump reaches its maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' In the mean time, the the molecules at the center bottom region are nearly flattened, which can trigger another phase transition to form a new phase domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Upon further compression, the flattened molecules at the center bottom region create much empty space along the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Thus, the secondary domain slips down to push the neighboring molecules down to fill the empty space (see the third row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S3 at the indentation depth of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='5 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' When the indentation is released, the process is supposed to be irreversible at low temperature since triggering the back transformation requires some energy barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' Therefore, it is a plastic deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content='0 Indentation depth (nm) (degree) α 30 30 0 15 15 (degree) β 30 30 0 15 15 (degree) γ 30 30 0 15 15 Me-plastic FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} +page_content=' The list of representative snapshots from the simulation of Me-plastic deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAyT4oBgHgl3EQfdvdN/content/2301.00307v1.pdf'} diff --git a/atAyT4oBgHgl3EQf9_py/content/tmp_files/2301.00884v1.pdf.txt b/atAyT4oBgHgl3EQf9_py/content/tmp_files/2301.00884v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4b250fee32c238e89934e5346cb344d407817a01 --- /dev/null +++ b/atAyT4oBgHgl3EQf9_py/content/tmp_files/2301.00884v1.pdf.txt @@ -0,0 +1,889 @@ +Safety Filtering for Reinforcement +Learning-based Adaptive Cruise Control +Habtamu Hailemichael ∗ Beshah Ayalew ∗ Lindsey Kerbel ∗ +Andrej Ivanco ∗∗ Keith Loiselle ∗∗ +∗ Automotive Engineering, Clemson University, Greenville, SC 29607, +USA (hhailem, beshah, lsutto2)@clemson.edu. +∗∗ Allison Transmission Inc., One Allison Way, Indianapolis, IN, +46222, USA (andrej.ivanco, keith.loiselle)@allisontransmission.com +Abstract: Reinforcement learning (RL)-based adaptive cruise control systems (ACC) that +learn and adapt to road, traffic and vehicle conditions are attractive for enhancing vehicle +energy efficiency and traffic flow. However, the application of RL in safety critical systems such +as ACC requires strong safety guarantees which are difficult to achieve with learning agents +that have a fundamental need to explore. In this paper, we derive control barrier functions as +safety filters that allow an RL-based ACC controller to explore freely within a collision safe +set. Specifically, we derive control barrier functions for high relative degree nonlinear systems to +take into account inertia effects relevant to commercial vehicles. We also outline an algorithm +for accommodating actuation saturation with these barrier functions. While any RL algorithm +can be used as the performance ACC controller together with these filters, we implement the +Maximum A Posteriori Policy Optimization (MPO) algorithm with a hybrid action space that +learns fuel optimal gear selection and torque control policies. The safety filtering RL approach +is contrasted with a reward shaping RL approach that only learns to avoid collisions after +sufficient training. Evaluations on different drive cycles demonstrate significant improvements +in fuel economy with the proposed approach compared to baseline ACC algorithms. +Keywords: Adaptive cruise control, Safe reinforcement learning, Safety filtering, Control +barrier functions +1. INTRODUCTION +Adaptive cruise control (ACC) systems are one of the in- +creasingly prevalent driver assistance systems for modern +vehicles. An ACC system uses radar, computer vision, or +laser to understand the vehicle’s surrounding and make +control decisions. When another vehicle or object is not in +the sensing range, ACC compensates for the road grade, +friction, and aerodynamic resistances to maintain a speed +set by the driver. When another car or object is in front, +the ACC makes decisions to prevent collision and follow +the preceding vehicle as close as possible to avoid cut-ins. +ACC has been shown to decrease a driver’s workload, and +make traffic flows efficient and safer (Marsden et al., 2001; +Lang et al., 2014). +An effective ACC system should balance the traffic condi- +tion of the road, the vehicle performance, and the driver’s +demanded velocity. Currently available PID-based ACC +systems (Canale and Malan, 2003; Chamraz and Balogh, +2018) and proposed MPC-based approaches (Naus et al., +2008; Yang et al., 2021) are often tuned to balance this +trade-off for various operating environments. Although +’adaptive’ or gain-scheduled versions (Radke and Iser- +mann, 1987) can be sought, the fixed structure of these +approaches limits full adaptation throughout the lifetime +of the vehicle. Furthermore, MPC-based ACC also has +to find a reliable way of predicting the motion of the +leading vehicle for the future horizon. On the other hand, +data-driven reinforcement learning (RL) approaches offer +a mechanism to continuously customize to traffic, road +and vehicle conditions without a predefined control archi- +tecture (Li and G¨orges, 2020). In this work, we consider +applications of RL-based ACC to commercial vehicles. In +addition, while traditional ACC is primarily about the two +tasks of speed tracking and maintaining a safe gap, we +consider RL-based ACC (RL ACC for short) to explicitly +optimize fuel economy via gear selection and torque control +policies. +Despite the potential benefits of adaptability and im- +proved performance, RL ACC faces critical safety chal- +lenges. These derive from the needs of RL algorithms to +explore in order to learn the optimal policies. RL learns +how good the given state-action pair is after experiencing +it, but for applications like vehicle control, exploration in +an unsafe domain is unacceptable even during (on-road) +training of the RL algorithms. However, thanks to recent +progress in safe RL, different approaches are suggested +to encourage or limit the exploration only in the safe +domain. We briefly mention a few of them. Reward shaping +approaches put large penalties into the performance objec- +tive function if constraints are violated. On the other hand, +constrained Markov decision process (CMDP) approaches +assign safety constraint costs to each state-action pair and +limit the total safety constraint cost of a trajectory to +be lower than a certain threshold (Altman, 1999). The +arXiv:2301.00884v1 [eess.SY] 2 Jan 2023 + +reward shaping and CMDP approaches are implemented +on the performance controller itself to encourage respect- +ing safety constraints but they do not guarantee safety. +Another set of approaches involve the use of safety filters +that impose hard constraints. Such approaches separate +the performance-oriented RL controller, whose only aim is +to optimize the system’s performance objective function, +from the safety filters, which project the unsafe actions +proposed by the performance controller into the safe set. +The safety filters determine the safety condition of the +given state-action pair using the dynamical model of the +system, or they use offline data to learn constraints (Dalal +et al., 2018) and safety indexes (Thananjeyan et al., 2021; +Srinivasan et al., 2020). In this paper, we pursue dynamical +model-based safety guarantees to construct the safe set in +such a way that gives the RL performance controller the +freedom to explore within the safe boundaries. As its train- +ing progresses, the RL performance controller eventually +learns the safety boundaries and ceases to demand unsafe +actions (Thananjeyan et al., 2021). Note that even though +it does not interfere with the inner workings, the safety +filter affects control performance by dictating where the +performance controller can operate. +Of the model-based approaches to designing safety filters, +control barrier functions (CBFs) offer light computation +and scalability (Li, 2021). A CBF guarantees safety by +making the controller work in the invariant safe-set defined +by a superlevel set of a continuously differentiable function +h(x) : Rn → R. The actions selected by the performance +controllers are projected into the safe set in such a manner +that the proposed actions are modified minimally (Ames +et al., 2019), and no unsafe actions are passed to the +controlled system. Different approaches could be pursued +to specify CBFs with their pros and cons. The intuitive +one is to come up with a handcrafted CBF considering the +dynamics of the system and the action bounds associated +with it (Xu et al., 2018; Ames et al., 2014; Cheng et al., +2019). In collision avoidance problems, for instance, the +CBF can be derived by considering the maximum decel- +eration that the system could exert to close a distance +gap. When possible, it is also desirable to progressively +widen the safe set to get the maximal safe domain, a task +currently possible with polynomial plant dynamics and +polynomial CBFs via sum-of-squares (SOS) programming +(Chamraz and Balogh, 2018). Another approach that is +tailored to high relative degree nonlinear dynamical sys- +tems such as those involving inertia effects is the use of +exponential CBF (ECBF) (Nguyen and Sreenath, 2016). +In this work, we derive ECBFs to work as safety filters with +our RL-ACC controllers, thereby taking explicit consider- +ations of inertia effects that are important for commercial +vehicles that experience large changes in loading. +The main contributions of this paper are then the deriva- +tion and demonstration of CBF-based safe RL-ACC ap- +proach for commercial vehicles that optimizes fuel econ- +omy. While we derive ECBFs for safety certification, we +note that straight ECBFs (or CBFs in general) assume +unbounded actions, and in their natural form, they might +request actions that are not feasible for the vehicle’s pow- +ertrain to meet. We therefore put forward a method to +provide a safety guarantee for a given parameters of ECBF +within the vehicle action limits. Our performance RL-ACC +coordinates traction torque control and gear decisions +considering fuel consumption optimization objectives. The +RL ACC augmented with the safety certificate is trained +and evaluated on different driving cycles, and the vehicle +performance is compared with an RL ACC with reward- +shaping approach to safe RL, as well as with a conventional +PID-based ACC. +The rest of the paper is organized as follows. Section +2 describes our derivation of the ECBF as safety filters +for ACC and detail how we address actuation constraints +within them. Section 3 describes the algorithmic details of +our performance RL-ACC. Section 4 discusses results and +discussions, and Section 5 concludes the paper. +2. SAFETY FILTER FOR ACC +We briefly review the definition of CBFs as follows. Details +are given in Hsu et al. (2015). Consider a nonlinear control +affine system: +˙x = f (x) + g (x) u,. +(1) +where f and g are locally Lipschitz, x ∈ Rn is the system +state, u ∈ Rm is the control inputs. Assume a safe set +defined by C = {x ∈ Rn|h (x) ≥ 0}, where h : Rn → +R is a continuously differentiable function. Then h is a +control barrier function (CBF) if there exists an extended +class κ∞ function α such that for all x ∈ Int (C) = +{x ∈ Rn : h (x) > 0} : +sup +u∈U +[Lfh (x) + Lgh (x) u] ≥ −α (h (x)). +(2) +For high relative degree nonlinear affine systems, feedback +linearization could be used to develop exponential CBFs +(ECBF) as detailed in Nguyen and Sreenath (2016). This +is accomplished by transforming (input-output linearizing) +the high relative degree nonlinear systems into a virtual +linear system with new state variable ηb := [h(x), ˙h(x), · · +·, hr(x)]T , input µ and output h (x): +˙ηb = Fηb (x) + Gµ, +h (x) = Cηb +(3) +where F and G are matrices representing an integrator +chain, and C = [1, 0, · · · , 0]. A state feedback controller +can be designed for the transformed system as: µ = −Kαηb +with a suitable gain vector Kα that makes F − GKα +Hurwitz. For a system with relative degree r, µ is also rth +derivative of the output h(x), µ = Lr +fh(x)+Lg�Lr−1 +f +h(x)u. +If there exists a state feedback gain Kα that makes µ ≥ +−Kαηb (x) for all states, then one can show that h(x) is +an exponential control barrier function (see Nguyen and +Sreenath (2016)). +The ACC part of the present problem is modelled with the +state variables of separation distance z, the velocity of the +host vehicle vh and the velocity of the leading vehicle vl. +The corresponding state equations are: +˙z = vl − vh +(4a) +˙vl = al +(4b) +˙vh = +Tt +rwmv +− Fr (vh, mv, θ) +mv +(4c) +Fr = ρAcdv2 +h +2 ++ mvgf cos θ + mvg sin θ +(5) + +where Fr is the total resistance force including gravita- +tional, rolling and aerodynamic resistances, and Tt is the +traction torque at the wheels. The parameters cd, f, θ, mv , +ρ, Av, rw, al are aerodynamic coefficient, rolling resistance +coefficient, road grade, mass of the vehicle, density of +air, frontal area of the vehicle, radius of the wheels, and +acceleration of the leading vehicle, respectively. +We observe that the above model can be readily put in the +control affine form (1). Given a collision safety objective, +we seek the separation distance z to always be above a +specified minimum inter-vehicle distance z0. To this end, +we define the control barrier function (CBF) as the output +h (x) = z − z0. Considering that the control actuation is +the traction torque Tt, we have a control affine system of +relative degree two. In physical terms, the safety objective +is on position while traction torque directly manipulates +acceleration. Inertia effects come into play and must be +accounted for. The input-output linearization into the +form (3) then gives: +˙h(x) = vl − vh, +(6) +µ = ¨h (x) = Fr (vh, mv, θ) +mv ++ al − +Tt +mvrw +, +(7) +−Kαηb (x) = −kα1 (z − z0) − kα2 (vl−vh) +(8) +We now compute some bounds for the given control input +µ considering actuation limits on the traction torque (Tmin +and Tmax). For a given acceleration of the preceding +vehicle (al) and velocity of the host (vh), the feasible +bounds of µ are given as +µTmin/max = al + Fr (vh, θ, mv) +mv +− Tmin/max +mvrw +(9) +For a given gain vector Kα = [kα1, kα2], ECBF guar- +antees safety if the proposed state feedback control, +−kα1 (z − z0) − kα2 (vl − vh), is within the virtual linear +system action bound [µT max, µT min]. In general appli- +cation cases, however, this bound may not be respected. +Nevertheless, if Kα is chosen so that the poles are placed +sufficiently to the left in s-plane, the above ECBF could +still bound the safe set. Safety assurance for such pole +selections could be achieved by investigating the evolution +of the CBF control term h (x) in worst-case situation where +the linear virtual model is initialized with extreme possible +η0,xrm, and then the possible limiting torque actions are +applied. For a given minimum separation distance target +and maximum downhill road grade, this is equivalent to +applying the maximum possible traction torque output +of the performance RL-ACC agent, with the host vehicle +model (of largest loading) initialized in with the maximum +possible velocity while the preceding vehicle is under its +maximum deceleration. This extreme conditions gives the +feasible µ bounds as µT min−xrm and µT max−xrm using +equations (9). +To capture the evolution of h (x) under these extreme +conditions, a simulation rollout is discretized into timestep +∆t, and the action µ (saturated with µT min−xrm and +µT max−xrm) held piecewise constant. Algorithm 1 shows +how this is implemented by integrating the virtual system +(3). If the h (x) from this simulation is positive at infinity +(or after some finite time), the selected Kα guarantees +safety. Otherwise, the Kα needs to be changed until this +is satisfied. +Algorithm 1 An algorithm to enforce system bounds on +a virtual linear system +η ← η0 +µ ← µ0 +while t ≤ t∞ do +t ← t + ∆t +if µ < µT max−xrm then +µ ← µT max−xrm +else if µ > µT min−xrm then +µ ← µT min−xrm +end if +h (x(t)) ← C(eF ∆tη0 + eF ∆t � ∆t +0 +e−F τGµd(τ)) +µ ← −kα1h (x) − kα2 ˙h (x) +η0 ← +�h (x) +˙h (x) +� +end while +Once the suitable gain vector Kα is selected, the ECBF +safety constraint enforces safety by projecting the action +proposed by the outputs of the RL controller’s actor +network Ta (s) (see next section) to the control traction +torque Tt in a way that introduces minimal changes to it. +This is done by posing and solving the quadratic program: +T ∗ +t = arg min +Tt +1 +2 ∥Tt − Ta(s)∥2 +s.t. +al + Fr (vh, mv, θ) +mv +− +Tt +mvrw +≥ −kα1 (z − z0) +− kα2 (vl − vh) +(10) +3. VEHICLE ENVIRONMENT AND RL ACC +The powertrain controller is modeled as Markov decision +process (MDP) consisting of states s, actions a, a reward +function r (s, a), and discounting factor γ. The probability +of action choices is policy π(a|s, θ) where θ denotes the +parameters of the deep neural network used to approxi- +mate the policy. The host vehicle velocity vl, the relative +velocity between the preceding and host vehicles vrel, +the separation distance between the vehicles z, the gear +ng, the mass of the vehicle mv, the road grade θ, the +driver demanded velocity vset and a flag to show if the +vehicle is in ACC sensor range f constitute the states +of the RL agent, s = {vl, vrel, z, ng,mv, θ, vset, f}. The +RL performance controller is designed to perform both +traction torque Ta control and gear change selection ∆ng, +i.e. a = {Ta, ∆ng}. As shown in Fig.1, the proposed Ta is +filtered by the ECBF safety layer to safe traction torque +demand Tt (10). The engine torque and engine speed that +brings about this wheel traction torque are then calculated +utilizing transmission ratios of the selected gear and the +final drive, and the associated fuel rate is read from the +fuel map. Notice that while the RL controller’s actions are +Ta and ∆ng, the ECBF safety filter does not use ∆ng in +the safety constraint. However, taking into account that +gear selection is crucial for fuel economy and driver ac- +commodation, it is an integral part of the RL performance +controller. +The filtered traction torque Tt and the gear change ∆ng +actions are implemented in the vehicle environment, and + +Fig. 1. Training RL agent for ACC +the suitability of the actions is measured by the reward +function. The reward is designed to accomplish the in +range and out of range tasks, and different performance +objectives within each task are tuned by reward weights +(w). When there is not a vehicle present in the sensing +range (z > zsr), as shown in (11), the reward structure +requires the vehicle to maintain the driver-set velocity and +concurrently balances the fuel consumption and smooth +torque change considerations. When there is a vehicle in +the sensing range, on the other hand, the reward aims to +maintain a close distance from the preceding vehicle, as +shown in (12). In such proximity, in addition to smooth +torque change and fuel consumption considerations, the +reward ros discourages the host vehicle from overspeeding +beyond the driver demanded velocity (vset). Gear hunting +and the associated rough vehicle operation are mitigated +by including a gear reward term weighted by wg. +r = wv0.1 +|vh−vset| +Vrel,max + wf0.1 +˙ +mf +mf,max + wT 0.1 +|∆Te| +Te,max + +wg0.1 +|∆ng| +ng,max +(11) +r = wz0.1 +Z +Zsr + wf0.1 +˙ +mf +mf,max + wT 0.1 +|∆Te| +Te,max + +wg0.1 +|∆ng| +ng,max + ros +(12) +where ros = wos if vh ≤ vset, else : ros = wos0.1 +vh−vset +vrel,max , +˙mf is the fuel rate and Te is the engine torque. +To accommodate the continuous traction torque and the +discrete gear selection, Hybrid Maximum A Posteriori +Policy Optimization (HMPO) is found to be a good fit +for the RL training algorithm (Kerbel et al., 2022; Neunert +et al., 2020; Abdolmaleki et al., 2018). In addition to being +scalable and robust like state of the art Proximal Policy +Optimization (PPO) (Schulman et al., 2017) and Trust- +Region Policy Optimization (TRPO) (Schulman et al., +2015) algorithms, the fact that it is off-policy makes it +data efficient to apply it to the real world RL ACC +trainings. The RL agent comprises of an actor (parame- +terized by θ) and a critic (parameterized by φ) networks, +in which the former determines the control policy for +a given state π (s|θ) and the latter evaluates these ac- +tions by providing the associated action values Q (s, a|φ). +The actor network outputs the mean and variance of +a Gaussian distribution, from which traction torque is +sampled (13). In addition to that, it uses softmax ac- +tivation at the output layer with three choices for the +gear change decision, analogous to the available gear +changes ∆n = {1, 0, −1}(upshift, nochange, downshift). +Categorical sampling is then used to obtain the gear +change policy (14). Assuming independence between the +continuous πT +θ (Ta|s) and discrete πg +θ(∆ng|s) policies, the +total policy could be factorized as (15) for combine action +a = {Ta, ∆ng}. +πT +θ(Ta|s) = N +� +µθ (s) , σ2 +θ (s) +� +(13) +πg +θ(∆ng|s) = Cat(αθ(s)), ∀s +3 +� +k=1 +αk,θ (s) = 1 +(14) +πθ (a|s) = πT +θ (Ta|s) πg +θ(∆ng|s)) +(15) +In the policy improvement phase, MPO samples from the +Q-function for different actions and update the actor- +network parameters to output actions that maximize the +action values Q(s, a). This is accomplished by optimiz- +ing the likelihood function of acting optimally using the +expectation-maximization algorithm ( see Neunert et al. +(2020); Abdolmaleki et al. (2018)). The policy evaluation +phase of the training fits the Q-function Qθ (s, a, φ) of +the critic network, with parameters φ, by minimizing the +square loss of the current Qθ (s, a, φ) and a target defined +by retrace sampling Qret +t +(Munos et al., 2016). +min +φ L (φ) = min +φ E(s,a)∼R +� +Qθ (s, a|φ) − Qret +t +�2 +(16) +4. RESULTS AND DISCUSSIONS +The above RL ACC with the ECBF safety filter is applied +to a model of medium duty truck in urban and highway +driving conditions. The actor and critic networks are con- +structed with three hidden layers, and each layer consists +of 256 nodes. The simulation uses a 10-speed automated +manual transmission (AMT) truck that has a 5 to 10 +tons weight range. The preceding vehicle follows Federal +Test Procedure (FTP-75) drive cycle for the urban driv- +ing training, while for highway driving, a combination of +Highway Fuel Economy (HWFET) and ArtMw130 cycles +are used in succession (Barlow et al., 2009). Once trained, +we will use different drive cycles for evaluation as will be +described below. +In each simulation step, as shown in Fig.1, the actor +network proposes the torque and the gear actions for a +given state which will be filtered by the ECBF safety +layer. The vehicle environment then executes the safe +actions, and the associated rewards are calculated. To +accommodate the different objectives of each task, the +reward is structured with weights of [wv = 0.675, wf = +0.175, wT = 0.075, wg = 0.075] for in range, and [wz = +0.325, wf = 0.175, wos = 0.35, wT = 0.075, wg = 0.075] +for out of range conditions. The state, action and rewards +are stored in the memory buffer, and afterward, batches +of these data are used to train the networks using the +HMPO algorithm. In order to prevent RL from learning +the specific drive cycles, the vehicles are initialized in +random separation distance along with the addition of +noise to the velocity profile of the preceding vehicle. The +weight fluctuations are considered by varying the truck +weight within and between training episodes. + +m +ECBF filter +Memory buffer +Ta +Load actor +parameters +RL +Training +△ng +π(s) ={Ta,△ng} +Load critic +Q(s,a) +parameters +Actor +network +π(s) +Critic +networkDuring training, because of the careful choice of the gain +vector Kα = [0.2, 5] as per section 2, the vehicle never +crashes nor comes within safe distance z0. As the training +progresses, the RL learns to operate near the driver set +velocity when it is out of range and follows the preceding +vehicle more and more closely when it is in range. Even +though it is not provided with the engine efficiency map, +as exhibited by the improvement of MPG with training, +the RL network eventually learns the fuel optimal gear and +torque actions. +Table 1. Vehicle environment and RL hyperpa- +rameter setting +Vehicle Parameters +MPO Hyperparameters +Mass +5 - 10 tons +Actor, critic learning rate +10−4, 10−5 +Au +7.71m2 +Dual constraint +0.1 +Cd +0.08 +Retrace steps +15 +rw +0.498 +KL constraints ϵµ, ϵσ, ϵd +0.1, 0.001, 0.1 +f +0.015 +αd, αc +10 +zsr +350 +γ +0.99 +Even if it is not practical for safety critical systems, a +reward shaping approach of safeguarding safety is consid- +ered to compare against the ECBF-based safety filtering. +A penalty of rs = −1 is added to the reward function +when the host approaches closer than the minimum safe +distance limit z0 and, in the situation of a crash, the +penalty is enlarged to rc = −10. Due to these safety +violation penalties, unsafe actions reduce with training, +and eventually, the agent learns to maximize the reward +safely. In addition to the reward shaping approach, the +conventional PID ACC is used as a baseline which, like +in the case of RL, is designed by dividing the control into +phases for the in range and out of range conditions (Canale +and Malan, 2003). The traction torque Tt request is given +by PID controller and an optimal gear is chosen based on +the gear with the lowest fuel rate given the desired traction +torque and vehicle velocity (Yoon et al., 2020; Kerbel et al., +2022). +After the RL ACC with ECBF is trained, its performance +is evaluated and compared with PID ACC and RL ACC +with reward shaping counterparts on a 9-ton truck in +urban and highway driving conditions. For the urban case, +the preceding vehicle follows the ArtUrban drive cycle, +and the driver demanded velocity vset is set to be 15 m/s. +Similarly, a vset of 25 m/s is used for highway driving, and +to better capture different velocity profiles in the highway +situation, the preceding vehicle follows a combination of +ArtRoad and ARTMw150. The initial separation distance +between the vehicles is 1500 m in both cases. +In both driving conditions, the RL ACC successfully +meets the in range as well as out of range objectives +and, most importantly, safety constraints are respected. +Fig.2 shows the RL ACC has a similar velocity profile +to its PID ACC counterpart for the most part of the +simulation. However, when it comes to gear selection, the +RL ACC tends to operate at higher gears. As summarised +in Table 2, for highway driving, the RL ACC exhibited +an MPG improvement of 8.3%, whereas, in the case of +urban driving, it has 7.9% higher MPG than the PID +ACC baseline. When the preceding vehicle is in range, +the RL ACC is less susceptible to cut-in as it follows the +preceding vehicle closer, shown by the lower mean in range +Fig. 2. Simulation of separation distance, velocity, and gear +profiles of RL and PID ACC controllers in a highway +driving. +separation distance zir. Moreover, it is possible to see that +the RL ACC with ECBF filter and the RL ACC with +reward shaping arrangements achieve equivalent levels of +fuel economy and in range car following performances. +Table 3 shows the performance comparison with weight +fluctuation in which the vehicle’s weight ranges from 5 to +10-tons. The RL ACC maintains higher MPG than the +PID ACC throughout the given weight range, and the +separation distance is not significantly influenced. +Table 2. Performance comparison between PID +ACC, RL ACC with ECBF and RL ACC with +reward shaping +Highway driving +Urban driving +ACC +PID +RL +RL +PID +RL +RL +Safety +layer +- +ECBF +Reward +shaping +- +ECBF +Reward +shaping +MPG +8.6 +(-) +9.3 +(8.31%) +9.31 +(8.37%) +6.8 +(-) +7.35 +(7.9%) +7.38 +(8.4%) +Zir(m) +95 +74 +73 +42 +39 +38 +5. CONCLUSION +In this paper, an exponential control barrier function- +based safety filter is employed to instill safety into RL +based ACC system by projecting the learning exploration +to a safe set. Since practical systems operate with bounded +actions, we proposed an approach to verify the safety of a +given ECBF design by forward simulating in consideration +of worst case scenarios. After being filtered by this ECBF, +the traction torque and gear change actions proposed by +the RL-based ACC are implemented on a simulated vehi- +cle environment and the associated rewards are observed. +The RL networks are trained using Hybrid Maximum A +Table 3. Perandomizedof PID ACC and RL +ACC with vehicle mass fluctuation +Weight +(tons) +5 +6 +7 +8 +9 +10 +RL +with +ECBF +MPG +10.58 +(10.9%) +10.38 +(11.6%) +9.99 +(9.6%) +9.61 +(8.3%) +9.3 +(8.31%) +8.95 +(7.6%) +Zir(m) 67 +69 +73 +75 +74 +77 +PID +MPG +9.54 +9.3 +9.11 +8.87 +8.6 +8.32 +Zir(m) 95 +95 +94 +95 +95 +96 + +RL with Reward-shaping +PID +Sensing range +RL with CBF +Distance (m) +5000 +0 +25 +0 +10 +ear +5 +G +11 +0 +250 +500 +750 +1000 +1250 +1500 +1750 +2000 +Time (s)Posteriori Policy Optimization (HMPO) algorithm that +accommodates the continuous traction torque and discrete +gear change actions. Evaluation on a medium-duty truck +shows that the RL ACC fulfilled the velocity objectives +and, most importantly, respected the safety constraints. +Compared to PID ACC, the RL ACC augments MPG by +8.3% in highway driving conditions when the preceding +vehicle follows a combination of ArtRoad and ARTMw150 +drive cycles, and by 7.9% in urban driving conditions when +the preceding vehicle follows ArtUrban drive cycle. More- +over, the RL ACC learns to handle weight fluctuations +and maintains high performance throughout the vehicle’s +weight range. +The current algorithm training and evaluations are per- +formed on standard driving cycles. Future work will focus +on using randomized traffic data and measurement noise to +assess the performance and robustness of RL ACC in even +more realistic driving conditions. 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Actuators 2021, Vol. 10, Page 110, +10(6), 110. +Yoon, D.D., Ayalew, B., Ivanco, A., and Loiselle, K. +(2020). +Predictive kinetic energy management for an + +add-on driver assistance eco-driving of heavy vehicles. +IET Intelligent Transport Systems, 14(13), 1824–1834. + diff --git a/atAyT4oBgHgl3EQf9_py/content/tmp_files/load_file.txt b/atAyT4oBgHgl3EQf9_py/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..71e028fd2d9ea3fade6fa463ef32ea14e8bdcf15 --- /dev/null +++ b/atAyT4oBgHgl3EQf9_py/content/tmp_files/load_file.txt @@ -0,0 +1,489 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf,len=488 +page_content='Safety Filtering for Reinforcement Learning-based Adaptive Cruise Control Habtamu Hailemichael ∗ Beshah Ayalew ∗ Lindsey Kerbel ∗ Andrej Ivanco ∗∗ Keith Loiselle ∗∗ ∗ Automotive Engineering, Clemson University, Greenville, SC 29607, USA (hhailem, beshah, lsutto2)@clemson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' ∗∗ Allison Transmission Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', One Allison Way, Indianapolis, IN, 46222, USA (andrej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='ivanco, keith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='loiselle)@allisontransmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='com Abstract: Reinforcement learning (RL)-based adaptive cruise control systems (ACC) that learn and adapt to road, traffic and vehicle conditions are attractive for enhancing vehicle energy efficiency and traffic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' However, the application of RL in safety critical systems such as ACC requires strong safety guarantees which are difficult to achieve with learning agents that have a fundamental need to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' In this paper, we derive control barrier functions as safety filters that allow an RL-based ACC controller to explore freely within a collision safe set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Specifically, we derive control barrier functions for high relative degree nonlinear systems to take into account inertia effects relevant to commercial vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' We also outline an algorithm for accommodating actuation saturation with these barrier functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' While any RL algorithm can be used as the performance ACC controller together with these filters, we implement the Maximum A Posteriori Policy Optimization (MPO) algorithm with a hybrid action space that learns fuel optimal gear selection and torque control policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The safety filtering RL approach is contrasted with a reward shaping RL approach that only learns to avoid collisions after sufficient training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Evaluations on different drive cycles demonstrate significant improvements in fuel economy with the proposed approach compared to baseline ACC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Keywords: Adaptive cruise control, Safe reinforcement learning, Safety filtering, Control barrier functions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' INTRODUCTION Adaptive cruise control (ACC) systems are one of the in- creasingly prevalent driver assistance systems for modern vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' An ACC system uses radar, computer vision, or laser to understand the vehicle’s surrounding and make control decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' When another vehicle or object is not in the sensing range, ACC compensates for the road grade, friction, and aerodynamic resistances to maintain a speed set by the driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' When another car or object is in front, the ACC makes decisions to prevent collision and follow the preceding vehicle as close as possible to avoid cut-ins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' ACC has been shown to decrease a driver’s workload, and make traffic flows efficient and safer (Marsden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' An effective ACC system should balance the traffic condi- tion of the road, the vehicle performance, and the driver’s demanded velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Currently available PID-based ACC systems (Canale and Malan, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Chamraz and Balogh, 2018) and proposed MPC-based approaches (Naus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2021) are often tuned to balance this trade-off for various operating environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Although ’adaptive’ or gain-scheduled versions (Radke and Iser- mann, 1987) can be sought, the fixed structure of these approaches limits full adaptation throughout the lifetime of the vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Furthermore, MPC-based ACC also has to find a reliable way of predicting the motion of the leading vehicle for the future horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' On the other hand, data-driven reinforcement learning (RL) approaches offer a mechanism to continuously customize to traffic, road and vehicle conditions without a predefined control archi- tecture (Li and G¨orges, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' In this work, we consider applications of RL-based ACC to commercial vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' In addition, while traditional ACC is primarily about the two tasks of speed tracking and maintaining a safe gap, we consider RL-based ACC (RL ACC for short) to explicitly optimize fuel economy via gear selection and torque control policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Despite the potential benefits of adaptability and im- proved performance, RL ACC faces critical safety chal- lenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' These derive from the needs of RL algorithms to explore in order to learn the optimal policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' RL learns how good the given state-action pair is after experiencing it, but for applications like vehicle control, exploration in an unsafe domain is unacceptable even during (on-road) training of the RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' However, thanks to recent progress in safe RL, different approaches are suggested to encourage or limit the exploration only in the safe domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' We briefly mention a few of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Reward shaping approaches put large penalties into the performance objec- tive function if constraints are violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' On the other hand, constrained Markov decision process (CMDP) approaches assign safety constraint costs to each state-action pair and limit the total safety constraint cost of a trajectory to be lower than a certain threshold (Altman, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='00884v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='SY] 2 Jan 2023 reward shaping and CMDP approaches are implemented on the performance controller itself to encourage respect- ing safety constraints but they do not guarantee safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Another set of approaches involve the use of safety filters that impose hard constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Such approaches separate the performance-oriented RL controller, whose only aim is to optimize the system’s performance objective function, from the safety filters, which project the unsafe actions proposed by the performance controller into the safe set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The safety filters determine the safety condition of the given state-action pair using the dynamical model of the system, or they use offline data to learn constraints (Dalal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2018) and safety indexes (Thananjeyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Srinivasan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' In this paper, we pursue dynamical model-based safety guarantees to construct the safe set in such a way that gives the RL performance controller the freedom to explore within the safe boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' As its train- ing progresses, the RL performance controller eventually learns the safety boundaries and ceases to demand unsafe actions (Thananjeyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Note that even though it does not interfere with the inner workings, the safety filter affects control performance by dictating where the performance controller can operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Of the model-based approaches to designing safety filters, control barrier functions (CBFs) offer light computation and scalability (Li, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' A CBF guarantees safety by making the controller work in the invariant safe-set defined by a superlevel set of a continuously differentiable function h(x) : Rn → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The actions selected by the performance controllers are projected into the safe set in such a manner that the proposed actions are modified minimally (Ames et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2019), and no unsafe actions are passed to the controlled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Different approaches could be pursued to specify CBFs with their pros and cons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The intuitive one is to come up with a handcrafted CBF considering the dynamics of the system and the action bounds associated with it (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Ames et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' In collision avoidance problems, for instance, the CBF can be derived by considering the maximum decel- eration that the system could exert to close a distance gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' When possible, it is also desirable to progressively widen the safe set to get the maximal safe domain, a task currently possible with polynomial plant dynamics and polynomial CBFs via sum-of-squares (SOS) programming (Chamraz and Balogh, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Another approach that is tailored to high relative degree nonlinear dynamical sys- tems such as those involving inertia effects is the use of exponential CBF (ECBF) (Nguyen and Sreenath, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' In this work, we derive ECBFs to work as safety filters with our RL-ACC controllers, thereby taking explicit consider- ations of inertia effects that are important for commercial vehicles that experience large changes in loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The main contributions of this paper are then the deriva- tion and demonstration of CBF-based safe RL-ACC ap- proach for commercial vehicles that optimizes fuel econ- omy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' While we derive ECBFs for safety certification, we note that straight ECBFs (or CBFs in general) assume unbounded actions, and in their natural form, they might request actions that are not feasible for the vehicle’s pow- ertrain to meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' We therefore put forward a method to provide a safety guarantee for a given parameters of ECBF within the vehicle action limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Our performance RL-ACC coordinates traction torque control and gear decisions considering fuel consumption optimization objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The RL ACC augmented with the safety certificate is trained and evaluated on different driving cycles, and the vehicle performance is compared with an RL ACC with reward- shaping approach to safe RL, as well as with a conventional PID-based ACC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Section 2 describes our derivation of the ECBF as safety filters for ACC and detail how we address actuation constraints within them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Section 3 describes the algorithmic details of our performance RL-ACC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Section 4 discusses results and discussions, and Section 5 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' SAFETY FILTER FOR ACC We briefly review the definition of CBFs as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Details are given in Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Consider a nonlinear control affine system: ˙x = f (x) + g (x) u,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' (1) where f and g are locally Lipschitz, x ∈ Rn is the system state, u ∈ Rm is the control inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Assume a safe set defined by C = {x ∈ Rn|h (x) ≥ 0}, where h : Rn → R is a continuously differentiable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Then h is a control barrier function (CBF) if there exists an extended class κ∞ function α such that for all x ∈ Int (C) = {x ∈ Rn : h (x) > 0} : sup u∈U [Lfh (x) + Lgh (x) u] ≥ −α (h (x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' (2) For high relative degree nonlinear affine systems, feedback linearization could be used to develop exponential CBFs (ECBF) as detailed in Nguyen and Sreenath (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' This is accomplished by transforming (input-output linearizing) the high relative degree nonlinear systems into a virtual linear system with new state variable ηb := [h(x), ˙h(x), · · , hr(x)]T , input µ and output h (x): ˙ηb = Fηb (x) + Gµ, h (x) = Cηb (3) where F and G are matrices representing an integrator chain, and C = [1, 0, · · · , 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' A state feedback controller can be designed for the transformed system as: µ = −Kαηb with a suitable gain vector Kα that makes F − GKα Hurwitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' For a system with relative degree r, µ is also rth derivative of the output h(x), µ = Lr fh(x)+Lg�Lr−1 f h(x)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' If there exists a state feedback gain Kα that makes µ ≥ −Kαηb (x) for all states, then one can show that h(x) is an exponential control barrier function (see Nguyen and Sreenath (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The ACC part of the present problem is modelled with the state variables of separation distance z, the velocity of the host vehicle vh and the velocity of the leading vehicle vl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The corresponding state equations are: ˙z = vl − vh (4a) ˙vl = al (4b) ˙vh = Tt rwmv − Fr (vh, mv, θ) mv (4c) Fr = ρAcdv2 h 2 + mvgf cos θ + mvg sin θ (5) where Fr is the total resistance force including gravita- tional, rolling and aerodynamic resistances, and Tt is the traction torque at the wheels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The parameters cd, f, θ, mv , ρ, Av, rw, al are aerodynamic coefficient, rolling resistance coefficient, road grade, mass of the vehicle, density of air, frontal area of the vehicle, radius of the wheels, and acceleration of the leading vehicle, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' We observe that the above model can be readily put in the control affine form (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Given a collision safety objective, we seek the separation distance z to always be above a specified minimum inter-vehicle distance z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' To this end, we define the control barrier function (CBF) as the output h (x) = z − z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Considering that the control actuation is the traction torque Tt, we have a control affine system of relative degree two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' In physical terms, the safety objective is on position while traction torque directly manipulates acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Inertia effects come into play and must be accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The input-output linearization into the form (3) then gives: ˙h(x) = vl − vh, (6) µ = ¨h (x) = Fr (vh, mv, θ) mv + al − Tt mvrw , (7) −Kαηb (x) = −kα1 (z − z0) − kα2 (vl−vh) (8) We now compute some bounds for the given control input µ considering actuation limits on the traction torque (Tmin and Tmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' For a given acceleration of the preceding vehicle (al) and velocity of the host (vh), the feasible bounds of µ are given as µTmin/max = al + Fr (vh, θ, mv) mv − Tmin/max mvrw (9) For a given gain vector Kα = [kα1, kα2], ECBF guar- antees safety if the proposed state feedback control, −kα1 (z − z0) − kα2 (vl − vh), is within the virtual linear system action bound [µT max, µT min].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' In general appli- cation cases, however, this bound may not be respected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Nevertheless, if Kα is chosen so that the poles are placed sufficiently to the left in s-plane, the above ECBF could still bound the safe set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Safety assurance for such pole selections could be achieved by investigating the evolution of the CBF control term h (x) in worst-case situation where the linear virtual model is initialized with extreme possible η0,xrm, and then the possible limiting torque actions are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' For a given minimum separation distance target and maximum downhill road grade, this is equivalent to applying the maximum possible traction torque output of the performance RL-ACC agent, with the host vehicle model (of largest loading) initialized in with the maximum possible velocity while the preceding vehicle is under its maximum deceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' This extreme conditions gives the feasible µ bounds as µT min−xrm and µT max−xrm using equations (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' To capture the evolution of h (x) under these extreme conditions, a simulation rollout is discretized into timestep ∆t, and the action µ (saturated with µT min−xrm and µT max−xrm) held piecewise constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Algorithm 1 shows how this is implemented by integrating the virtual system (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' If the h (x) from this simulation is positive at infinity (or after some finite time), the selected Kα guarantees safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Otherwise, the Kα needs to be changed until this is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Algorithm 1 An algorithm to enforce system bounds on a virtual linear system η ← η0 µ ← µ0 while t ≤ t∞ do t ← t + ∆t if µ < µT max−xrm then µ ← µT max−xrm else if µ > µT min−xrm then µ ← µT min−xrm end if h (x(t)) ← C(eF ∆tη0 + eF ∆t � ∆t 0 e−F τGµd(τ)) µ ← −kα1h (x) − kα2 ˙h (x) η0 ← �h (x) ˙h (x) � end while Once the suitable gain vector Kα is selected,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' the ECBF safety constraint enforces safety by projecting the action proposed by the outputs of the RL controller’s actor network Ta (s) (see next section) to the control traction torque Tt in a way that introduces minimal changes to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' This is done by posing and solving the quadratic program: T ∗ t = arg min Tt 1 2 ∥Tt − Ta(s)∥2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' al + Fr (vh, mv, θ) mv − Tt mvrw ≥ −kα1 (z − z0) − kα2 (vl − vh) (10) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' VEHICLE ENVIRONMENT AND RL ACC The powertrain controller is modeled as Markov decision process (MDP) consisting of states s, actions a, a reward function r (s, a), and discounting factor γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The probability of action choices is policy π(a|s, θ) where θ denotes the parameters of the deep neural network used to approxi- mate the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The host vehicle velocity vl, the relative velocity between the preceding and host vehicles vrel, the separation distance between the vehicles z, the gear ng, the mass of the vehicle mv, the road grade θ, the driver demanded velocity vset and a flag to show if the vehicle is in ACC sensor range f constitute the states of the RL agent, s = {vl, vrel, z, ng,mv, θ, vset, f}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The RL performance controller is designed to perform both traction torque Ta control and gear change selection ∆ng, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' a = {Ta, ∆ng}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='1, the proposed Ta is filtered by the ECBF safety layer to safe traction torque demand Tt (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The engine torque and engine speed that brings about this wheel traction torque are then calculated utilizing transmission ratios of the selected gear and the final drive, and the associated fuel rate is read from the fuel map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Notice that while the RL controller’s actions are Ta and ∆ng, the ECBF safety filter does not use ∆ng in the safety constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' However, taking into account that gear selection is crucial for fuel economy and driver ac- commodation, it is an integral part of the RL performance controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The filtered traction torque Tt and the gear change ∆ng actions are implemented in the vehicle environment, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Training RL agent for ACC the suitability of the actions is measured by the reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The reward is designed to accomplish the in range and out of range tasks, and different performance objectives within each task are tuned by reward weights (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' When there is not a vehicle present in the sensing range (z > zsr), as shown in (11), the reward structure requires the vehicle to maintain the driver-set velocity and concurrently balances the fuel consumption and smooth torque change considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' When there is a vehicle in the sensing range, on the other hand, the reward aims to maintain a close distance from the preceding vehicle, as shown in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' In such proximity, in addition to smooth torque change and fuel consumption considerations, the reward ros discourages the host vehicle from overspeeding beyond the driver demanded velocity (vset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Gear hunting and the associated rough vehicle operation are mitigated by including a gear reward term weighted by wg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' r = wv0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='1 |vh−vset| Vrel,max + wf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='1 ˙ mf mf,max + wT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='1 |∆Te| Te,max + wg0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='1 |∆ng| ng,max (11) r = wz0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='1 Z Zsr + wf0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='1 ˙ mf mf,max + wT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='1 |∆Te| Te,max + wg0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='1 |∆ng| ng,max + ros (12) where ros = wos if vh ≤ vset, else : ros = wos0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='1 vh−vset vrel,max , ˙mf is the fuel rate and Te is the engine torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' To accommodate the continuous traction torque and the discrete gear selection, Hybrid Maximum A Posteriori Policy Optimization (HMPO) is found to be a good fit for the RL training algorithm (Kerbel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Neunert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Abdolmaleki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' In addition to being scalable and robust like state of the art Proximal Policy Optimization (PPO) (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2017) and Trust- Region Policy Optimization (TRPO) (Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2015) algorithms, the fact that it is off-policy makes it data efficient to apply it to the real world RL ACC trainings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The RL agent comprises of an actor (parame- terized by θ) and a critic (parameterized by φ) networks, in which the former determines the control policy for a given state π (s|θ) and the latter evaluates these ac- tions by providing the associated action values Q (s, a|φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The actor network outputs the mean and variance of a Gaussian distribution, from which traction torque is sampled (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' In addition to that, it uses softmax ac- tivation at the output layer with three choices for the gear change decision, analogous to the available gear changes ∆n = {1, 0, −1}(upshift, nochange, downshift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Categorical sampling is then used to obtain the gear change policy (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Assuming independence between the continuous πT θ (Ta|s) and discrete πg θ(∆ng|s) policies, the total policy could be factorized as (15) for combine action a = {Ta, ∆ng}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' πT θ(Ta|s) = N � µθ (s) , σ2 θ (s) � (13) πg θ(∆ng|s) = Cat(αθ(s)), ∀s 3 � k=1 αk,θ (s) = 1 (14) πθ (a|s) = πT θ (Ta|s) πg θ(∆ng|s)) (15) In the policy improvement phase, MPO samples from the Q-function for different actions and update the actor- network parameters to output actions that maximize the action values Q(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' This is accomplished by optimiz- ing the likelihood function of acting optimally using the expectation-maximization algorithm ( see Neunert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Abdolmaleki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' (2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The policy evaluation phase of the training fits the Q-function Qθ (s, a, φ) of the critic network, with parameters φ, by minimizing the square loss of the current Qθ (s, a, φ) and a target defined by retrace sampling Qret t (Munos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' min φ L (φ) = min φ E(s,a)∼R � Qθ (s, a|φ) − Qret t �2 (16) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS The above RL ACC with the ECBF safety filter is applied to a model of medium duty truck in urban and highway driving conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The actor and critic networks are con- structed with three hidden layers, and each layer consists of 256 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The simulation uses a 10-speed automated manual transmission (AMT) truck that has a 5 to 10 tons weight range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The preceding vehicle follows Federal Test Procedure (FTP-75) drive cycle for the urban driv- ing training, while for highway driving, a combination of Highway Fuel Economy (HWFET) and ArtMw130 cycles are used in succession (Barlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Once trained, we will use different drive cycles for evaluation as will be described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' In each simulation step, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='1, the actor network proposes the torque and the gear actions for a given state which will be filtered by the ECBF safety layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The vehicle environment then executes the safe actions, and the associated rewards are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' To accommodate the different objectives of each task, the reward is structured with weights of [wv = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='675, wf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='175, wT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='075, wg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='075] for in range, and [wz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='325, wf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='175, wos = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='35, wT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='075, wg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='075] for out of range conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The state, action and rewards are stored in the memory buffer, and afterward, batches of these data are used to train the networks using the HMPO algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' In order to prevent RL from learning the specific drive cycles, the vehicles are initialized in random separation distance along with the addition of noise to the velocity profile of the preceding vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The weight fluctuations are considered by varying the truck weight within and between training episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' m ECBF filter Memory buffer Ta Load actor parameters RL Training △ng π(s) ={Ta,△ng} Load critic Q(s,a) parameters Actor network π(s) Critic networkDuring training, because of the careful choice of the gain vector Kα = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='2, 5] as per section 2, the vehicle never crashes nor comes within safe distance z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' As the training progresses, the RL learns to operate near the driver set velocity when it is out of range and follows the preceding vehicle more and more closely when it is in range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Even though it is not provided with the engine efficiency map, as exhibited by the improvement of MPG with training, the RL network eventually learns the fuel optimal gear and torque actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Vehicle environment and RL hyperpa- rameter setting Vehicle Parameters MPO Hyperparameters Mass 5 - 10 tons Actor, critic learning rate 10−4, 10−5 Au 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='71m2 Dual constraint 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='1 Cd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='08 Retrace steps 15 rw 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='498 KL constraints ϵµ, ϵσ, ϵd 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='1 f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='015 αd, αc 10 zsr 350 γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='99 Even if it is not practical for safety critical systems, a reward shaping approach of safeguarding safety is consid- ered to compare against the ECBF-based safety filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' A penalty of rs = −1 is added to the reward function when the host approaches closer than the minimum safe distance limit z0 and, in the situation of a crash, the penalty is enlarged to rc = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Due to these safety violation penalties, unsafe actions reduce with training, and eventually, the agent learns to maximize the reward safely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' In addition to the reward shaping approach, the conventional PID ACC is used as a baseline which, like in the case of RL, is designed by dividing the control into phases for the in range and out of range conditions (Canale and Malan, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The traction torque Tt request is given by PID controller and an optimal gear is chosen based on the gear with the lowest fuel rate given the desired traction torque and vehicle velocity (Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Kerbel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' After the RL ACC with ECBF is trained, its performance is evaluated and compared with PID ACC and RL ACC with reward shaping counterparts on a 9-ton truck in urban and highway driving conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' For the urban case, the preceding vehicle follows the ArtUrban drive cycle, and the driver demanded velocity vset is set to be 15 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Similarly, a vset of 25 m/s is used for highway driving, and to better capture different velocity profiles in the highway situation, the preceding vehicle follows a combination of ArtRoad and ARTMw150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The initial separation distance between the vehicles is 1500 m in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' In both driving conditions, the RL ACC successfully meets the in range as well as out of range objectives and, most importantly, safety constraints are respected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='2 shows the RL ACC has a similar velocity profile to its PID ACC counterpart for the most part of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' However, when it comes to gear selection, the RL ACC tends to operate at higher gears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' As summarised in Table 2, for highway driving, the RL ACC exhibited an MPG improvement of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='3%, whereas, in the case of urban driving, it has 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='9% higher MPG than the PID ACC baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' When the preceding vehicle is in range, the RL ACC is less susceptible to cut-in as it follows the preceding vehicle closer, shown by the lower mean in range Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Simulation of separation distance, velocity, and gear profiles of RL and PID ACC controllers in a highway driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' separation distance zir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Moreover, it is possible to see that the RL ACC with ECBF filter and the RL ACC with reward shaping arrangements achieve equivalent levels of fuel economy and in range car following performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Table 3 shows the performance comparison with weight fluctuation in which the vehicle’s weight ranges from 5 to 10-tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The RL ACC maintains higher MPG than the PID ACC throughout the given weight range, and the separation distance is not significantly influenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Performance comparison between PID ACC, RL ACC with ECBF and RL ACC with reward shaping Highway driving Urban driving ACC PID RL RL PID RL RL Safety layer ECBF Reward shaping ECBF Reward shaping MPG 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='6 (-) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='3 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='31%) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='31 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='37%) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='8 (-) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='35 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='9%) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='38 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='4%) Zir(m) 95 74 73 42 39 38 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' CONCLUSION In this paper, an exponential control barrier function- based safety filter is employed to instill safety into RL based ACC system by projecting the learning exploration to a safe set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Since practical systems operate with bounded actions, we proposed an approach to verify the safety of a given ECBF design by forward simulating in consideration of worst case scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' After being filtered by this ECBF, the traction torque and gear change actions proposed by the RL-based ACC are implemented on a simulated vehi- cle environment and the associated rewards are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The RL networks are trained using Hybrid Maximum A Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Perandomizedof PID ACC and RL ACC with vehicle mass fluctuation Weight (tons) 5 6 7 8 9 10 RL with ECBF MPG 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='58 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='9%) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='38 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='6%) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='99 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='6%) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='61 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='3%) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='3 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='31%) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='95 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='6%) Zir(m) 67 69 73 75 74 77 PID MPG 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='54 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='11 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='87 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='32 Zir(m) 95 95 94 95 95 96 RL with Reward-shaping PID Sensing range RL with CBF Distance (m) 5000 0 25 0 10 ear 5 G 11 0 250 500 750 1000 1250 1500 1750 2000 Time (s)Posteriori Policy Optimization (HMPO) algorithm that accommodates the continuous traction torque and discrete gear change actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Evaluation on a medium-duty truck shows that the RL ACC fulfilled the velocity objectives and, most importantly, respected the safety constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Compared to PID ACC, the RL ACC augments MPG by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='3% in highway driving conditions when the preceding vehicle follows a combination of ArtRoad and ARTMw150 drive cycles, and by 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='9% in urban driving conditions when the preceding vehicle follows ArtUrban drive cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' More- over, the RL ACC learns to handle weight fluctuations and maintains high performance throughout the vehicle’s weight range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' The current algorithm training and evaluations are per- formed on standard driving cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Future work will focus on using randomized traffic data and measurement noise to assess the performance and robustness of RL ACC in 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', Ayalew, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', Ivanco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=', and Loiselle, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' Predictive kinetic energy management for an add-on driver assistance eco-driving of heavy vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} +page_content=' IET Intelligent Transport Systems, 14(13), 1824–1834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atAyT4oBgHgl3EQf9_py/content/2301.00884v1.pdf'} diff --git a/d9FKT4oBgHgl3EQfrC7t/content/tmp_files/2301.11877v1.pdf.txt b/d9FKT4oBgHgl3EQfrC7t/content/tmp_files/2301.11877v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..abb1d2a32397407a5ae435e182e978b085934149 --- /dev/null +++ b/d9FKT4oBgHgl3EQfrC7t/content/tmp_files/2301.11877v1.pdf.txt @@ -0,0 +1,1536 @@ +arXiv:2301.11877v1 [hep-th] 27 Jan 2023 +On KP-integrable skew Hurwitz τ-functions +and their β-deformations +A. Mironovb,c,d,∗, V. Mishnyakova,b,†, A. Morozova,c,d,‡, A. Popolitova,c,d,§, Wei-Zhong Zhaoe,¶ +FIAN/TD-02/23 +IITP/TH-02/23 +ITEP/TH-02/23 +MIPT/TH-02/23 +a MIPT, Dolgoprudny, 141701, Russia +b Lebedev Physics Institute, Moscow 119991, Russia +c ITEP, Moscow 117218, Russia +d Institute for Information Transmission Problems, Moscow 127994, Russia +eSchool of Mathematical Sciences, Capital Normal University, Beijing 100048, China +Abstract +We extend the old formalism of cut-and-join operators in the theory of Hurwitz τ-functions to description +of a wide family of KP-integrable skew Hurwitz τ-functions, which include, in particular, the newly discovered +interpolating WLZZ models. Recently, the simplest of them was related to a superintegrable two-matrix +model with two potentials and one external matrix field. Now we provide detailed proofs, and a generalization +to a multi-matrix representation, and propose the β-deformation of the matrix model as well. The general +interpolating WLZZ model is generated by a W -representation given by a sum of operators from a one- +parametric commutative sub-family (a commutative subalgebra of w∞). Different commutative families are +related by cut-and-join rotations. Two of these sub-families (‘vertical’ and ‘45-degree’) turn out to be nothing +but the trigonometric and rational Calogero-Sutherland Hamiltonians, the ‘horizontal’ family is represented +by simple derivatives. Other families require an additional analysis. +1 +Introduction +Matrix models are the simplest representatives of the universality classes of non-perturbative partition functions, +which exhibit their peculiar properties, well shadowed in perturbative Feynman diagram technique and in other +approaches based on the standard multi-dimensional quantum field theory. These properties include high hidden +symmetries, integrability, superintegability and W-representations, which express partition functions through +the action of generalized cut-and-join operators on vacuum states. By now, they are well-studied and understood +in particular models, and the new challenge is to classify and unite different models in a common entity, as a +step to a formulation of non-perturbative string theory. +Recently a big step was made in this direction by the suggestion of WLZZ models [1], which appeared to be +building blocks of a two-matrix model in the external matrix field [2]. In this paper, in particular, we provide +technical details behind this general claim made in [2]. They are based on using peculiar cut-and-join rotation +operators O and on an extension of the old formalism of [3] to skew Hurwitz partition functions, naturally +expanded in skew rather than ordinary Schur functions, and exploit the Littlewood-Richardson decompositions +of ones into the others. These decompositions get a little non-trivial after the β-deformation from the Schur +functions to the Jack polynomials, which we also explain. +∗mironov@lpi.ru,mironov@itep.ru +†mishnyakovvv@gmail.com +‡morozov@itep.ru +§popolit@gmail.com +¶zhaowz@cnu.edu.cn +1 + +Skew Hurwitz partition functions. +Generic Hurwitz partition functions depending on two sets of time +variables have the form [5], +ZH(u∆; g, p) = +� +R +SR{gk}SR{pk} exp +�� +∆ +u∆ψR(∆) +� +(1) +SR{pk} is the Schur function labelled by the partition R, which is a symmetric function of xi considered as a +graded polynomial of the power sums pk = � +i xk +i , ∆ is a partition, and ψR(∆) is the character of symmetric +group S|R|, |R| being the size of partition [4,6]. This partition function is generated by generalized cut-and-join +operators [5], and is not integrable in the KP/Toda sense [7,8], it becomes integrable only with a special choice +of the coefficients u∆: +exp +�� +∆ +u∆ψR(∆) +� +−→ exp +�� +k +ukC(k) +R +� +(2) +where C(k) +R +are eigenvalues of specially chosen kth Casimir operators [5, 9–11]1. Equivalently, one can choose +[3,13] +exp +�� +∆ +u∆ψR(∆) +� +−→ +� +i,j∈R +f(j − i) +(3) +with an arbitrary function f(x). KP-integrable Hurwitz partition functions of this kind are called hypergeometric +τ-functions of the KP hierarchy [13]. +In this paper, we propose a generalization of the Hurwitz partition functions (1) to the skew Hurwitz partition +functions, +ZH(u∆, v∆′; g, p) = +� +R +SR/Q{¯pk}SR{gk}SQ{pk} exp +�� +∆ +u∆ψR(∆) − +� +∆′ +v∆′ψQ(∆′) +� +(4) +where SR/Q is the skew Schur function [4]. They are also integrable only with choices analogous to (2)-(3) and +are called skew hypergeometric τ-functions [2]. +In [3], it was suggested to choose the function f(x) in (3) +f(x) = +n +� +i +(ui + x) +(5) +This choice has many applications, from generating numbers of isomorphism classes of the Belyi pairs (Grothendieck’s +dessins d’enfant) [14] to the Itzykson-Zuber integral [15]. +In this paper, as an extension of results of [3], we consider an extension of this choice to the skew Hurwitz +partition functions. Thus, we consider a partition functions of the form +Z(n,m)({ui}, {vj}; ¯p, g, p) = +� +R,Q +�n +i=1 ξR(ui) +�m +j=1 ξQ(vj)SR/Q{¯pk}SR{gk}SQ{pk} +(6) +where +ξR(u) = SR{pk = u} +SR{δk,1} += +� +i,j∈R +(u + j − i) +(7) +The partition function Z(n,m) depends on n parameters ui, m parameters vi and three infinite sets of time +variables {gk}, {pk}, and {¯pk}. The interpolating matrix model of [2] is just the Z(1,1) member of this two- +parametric family, while other models of [2] are the one-parametric family Z(n,n). +The case considered in [3] corresponds to the restriction of this partition function to all pk = 0. Then, it +becomes the sum +� +R +� n +� +i=1 +ξR(ui) +� +SR{¯pk}SR{gk} +(8) +1They can be also associated with values of characters on the completed cycles [10,12]. +2 + +which celebrates a series of properties: it is aτ-function of the Toda lattice, it is generated by the cut-and-join +rotation operators O, it has multi-matrix model representation, etc. Our goal in this paper is to demonstrate +that all these properties keep intact for the skew Hurwitz partition functions (6). Moreover, all the properties +but integrability (which is known not to survive any β-deformations) survive the β-deformation. +w∞-algebra pattern. +Another important issue, which we shortly touch in this paper, is that, from the point +of view of the W-representations, the original technique used in [1], the partition functions Z(n,n) (at some +special points) are generated by combinations of operators from w∞ algebra. The relevant for the construction +are the one-parametric commutative subalgebras depicted by the blue lines in the picture. The commutativity +of operators of each line follows from the w∞-algebra relations [16] +[Vm1,n1, Vm2,n2] = +� +(m1 − 1)n2 − (m2 − 1)n2 +� +Vm1+m2−2,n1+n2 +=⇒ +� +adm +ˆ +Fs ˆFs−1, adn +ˆ +Fs ˆFs−1 +� += 0 +(9) +for ˆFs = (−1)sVs+1,1, which is nothing but the commutation of generators +� +ˆH(s) +m , ˆH(s) +n +� += 0 +for +ˆH(s) +m := adm +ˆ +Fs+1 ˆFs = (−1)m+1Vms+1,m +(10) +After the Miwa transform from time-variables to the eigenvalues of λi of N × N matrices, H(s) +m +become +systems of commuting differential operators. Not surprisingly, some of them are familiar to us: that is, the +series at s = 2: adm +F2F1 (the 45-degree blue line) turns out to be Hamiltonians of the rational Calogero-Sutherland +system. This adds to the old knowledge that the vertical line is the trigonometric Calogero-Sutherland system, +while on the horizontal line one has just commuting first-order operators +∂ +∂pn . +In fact, these Calogero-Sutherland models are at the free fermion point. +In order to get the Calogero- +Sutherland system with a non-trivial coupling, one needs the β-deformation. Fortunately, this whole picture +survives the β-deformation, as we explain in sec.5. +∂p3 ∼ [ ˆF1, ∂p2] +∂p2 ∼ [ ˆF1, ∂p1] +∂p1 +ˆL0 +ˆW0 +ˆ +M0 +ˆF1 = [ ˆW0, ∂p1] +. . . +. . . +. . . +. . . +ad ˆ +F2 ˆF1 +ad2 +ˆ +F2 ˆF1 +ˆF2 = ad2 +ˆ +W0 ∂p1 +ˆF3 = ad3 +ˆ +W0 ∂p1 +ad ˆ +F3 ˆF2 +ˆHm +1 +The cut-and-join rotation operators O, one of the central personages of this paper have a spectacular +interpretation in these terms: they rotate blue lines, one into another. These operators are constructed from +the generalized cut-and-join operators [5]. Hence, the name. In this sense, the rational Calogero-Sutherland +system is just a simple rotation of the system {∂pn} while the trigonometric Calogero-Sutherland model is +obtained from the rational one through an infinite system of rotations trough a sequence of new integrable +systems intertwined by the operators O. +The paper is organized as follows. In section 2, we describe integrable properties of the generic skew Hurwitz +partition function (6) and its representation via generalized cut-and-join operators in parallel with [3, secs.2-4]. +In section 3, we concentrate on the particular case of Z(1,1) and derive its description as a two-matrix model +depending on the external matrix and two potentials. The matrix model description can be also provided for +Z(n,1), as we demonstrate in section 4. In section 5, we discuss the β-deformation of Z(1,1): the matrix model +description and the W-representation, which turns out to be associated with the rational Calogero-Sutherland +Hamiltonians. The crucial difference with the β = 1 case (which corresponds to the free fermion point of the +Calogero-Sutherland model) is that a description of the Hamiltonians in terms of matrix derivatives is no longer +available, instead one has to use the eigenvalue variables. Section 6 contains some concluding remarks. +3 + +2 +Properties of skew Hurwitz partition functions +2.1 +Representation via cut-and-join operators +The partition function (6) can be realized by action of operators constructed from the commutative set of +generalized cut-and-join operators ˆW∆ [5]. We call these operators the cut-and-join rotation operators, they +play one of the central roles in the present paper. +The generalized cut-and-join operators ˆW∆ form a commutative set of operators, the Schur functions being +their eigenfunctions [5]: +ˆW∆ SR = φR(∆) SR +(11) +where, for the diagram ∆ containing r unit cycles: ∆ = [ ˜∆, 1r], +φR,∆ = + + + + + + + +0 +|∆| > |R| +(|R| − |∆| + r)! +r!(|R| − |∆|)! +φR, ˆ∆ = (|R| − |∆| + r)! +r!(|R| − |∆|)! +ψR( ˆ∆) +z ˆ∆dR +|∆| ≤ |R| +(12) +where ˆ∆ := [∆, 1|R|−|∆|]. Now note that [17, Eq.(61)] +� +∆ +φR(∆)p∆ = SR(pk + δk1) +dR +(13) +Now we construct the cut-and-join rotation operator as follows: +ˆO(u) := +� +∆ +p∆ · ˆ +W∆, +with pk = u − δk,1 +(14) +Here we use the notation p∆ = �l∆ +i=1 pδi, where l∆ is the length of the partition ∆, and δi’s are its parts. +Then, +ˆO(u) · SR{pk} = +SR(u) +SR(δk,1)SR{pk} = ξR(u) SR{pk} +(15) +This operator was constructed earlier in [3, Eq.(21)] in order to insert additional factors SR(N) +SR(δk,1) into character +expansion of the partition function, and was written there in a different form. In particular, one can rewrite it +via Casimir operators [3, sec.3]. +Now we can straightforwardly obtain the skew Hurwitz partition function (6) by the action of these operators +ˆO(u): +Z(n,m)({ui}, {vj}v; ¯p, g, p) = +m +� +j=1 +ˆO−1 +g (vj) +n +� +i=1 +ˆOp(ui) exp +�� +k +(pk + ¯pk)gk +k +� +(16) +2.2 +Z(n,m) as a τ-function of Toda lattice +The skew Hurwitz partition function is proportional to a τ-function of the Toda lattice hierarchy [8]: +τN(g, p) = +N +� +k=1 +�n +i=1 Γ(1 − k + ui + N) +�m +j=1 Γ(1 − k + vi + N) · Z(n,m)({ui + N}, {vj + N}; ¯p, g, p) +(17) +Here N is the zeroth time of the hierarchy, and {kpk}, {kgk} are the two infinite sets of times of the hierarchy. +The third infinite set, {¯pk} describes the concrete solution to the hierarchy (the point of the infinite-dimensional +Grassmannian. +Let us note that, in accordance with [18], the sum +τn(g, p|ξ) = +� +R,Q +ζR,Q(n)SR{g}SQ{p} +(18) +4 + +is a τ-function of the Toda lattice hierarchy iff +ζR,Q(N) = det +i,j≤N F(Ri − i + N + 1, Qj − j + N + 1) +(19) +with some function F(x, y), and N playing the role of the zeroth time. +Hence, in order to prove that (17) is a Toda lattice τ-function, one has to prove that +� N +� +k=1 +�n +i=1 Γ(1 − k + ui + N) +�m +j=1 Γ(1 − k + vi + N) +� �n +i=1 ξR(ui + N) +�m +j=1 ξQ(vj + N) · SR/Q{¯pk} +(20) +has representation (19). To this end, we use the Jacobi-Trudi determinant representation for the skew Schur +functions, +SR/Q{¯p} = det +i,j≤lR hRi−i−Qj+j{¯p} +(21) +where hk are the complete homogeneous symmetric polynomials, that is, hk = S[k], and we put hk = 0 at k < 0. +Then, one obtains representation (18) for (17) with +ζR,Q(N) = det +i,j≤N +�n +k=1 Γ(Ri − i + 1 + uk + N) +�m +l=1 Γ(Qj − j + 1 + vl + N) hRi−i−Qj+j{¯p} +(22) +i.e. ζR,Q(N) is just of the form (19) with +F(x, y) = +�n +k=1 Γ(x + uk) +�m +l=1 Γ(y + vl) hx−y{¯p} +(23) +3 +Two-matrix model representation of Z(1,1) +In this section, we provide the matrix model representation for the skew Hurwitz partition function Z(1,1). We +explain that it is given by the two-matrix model +Z(N; ¯p, p, g) = +� � +N×N +dXdY exp +� +−Tr XY + Tr Y Λ + +� +k +gk +k Tr Xk + +� +k +¯pk +k Tr Y k +� +(24) +with pk = Tr Λk. Here the integral is understood as integration of a power series in gk, ¯pk and Tr Λk, and X +are Hermitian matrices, while Y are anti-Hermitian ones. +3.1 +From matrix to time derivatives +In order to deal with this matrix model, we need to know the action of invariant matrix derivatives, Tr +∂k +∂Y k +on polynomials of pk = Tr Y k. Note that, by comparing the actions on the Schur functions using [19, Eq.(37)] +and [1, Eq.(24)], one obtains +Tr +� ∂ +∂Y +�k += ˆO−1(N) +� +k ∂ +∂pk +� +ˆO(N) +(25) +Now let us use the identity2 +SR +� +k ∂ +∂pk +� +SQ{pk} = SQ/R{pk} +(27) +in order to calculate +SR +� +Tr +� ∂ +∂Y +�k� +SQ{Tr Y k +� �� � +pk +} = ξQ(N) ˆO−1(N)SR +� +k ∂ +∂pk +� +SQ{pk} = ξQ(N) ˆO−1(N)SQ/R{pk} = += ξQ(N) ˆO−1(N) +� +P +N Q +RP SP {pk} = +� +P +N Q +RP ξQ(N)ξ−1 +P (N)SP {pk} +(28) +2The simplest way to prove this formula is to use the Cauchy identity: +� +R +SR{p′ +k}SR +� +k ∂ +∂pk +� +SQ{pk} = exp +�� +k +p′ +k +∂ +∂pk +� +SQ{pk} = SQ{pk + p′ +k} = +� +R +SQ/R{pk}SR{p′ +k} +(26) +and compare the coefficients in front of SR{p′ +k}. +5 + +where N Q +RP are the Littlewood-Richardson coefficients. In particular, +SR +� +Tr +� ∂ +∂Y +�k� +SQ{Tr Y k +� �� � +pk +} +��� +Y =0 = ξR(N)δR,Q +(29) +3.2 +Evaluating the matrix integral +Now the two-matrix integral (24) can be rewritten in the form [3, Eq.(47)] +Z(N; ¯p, p, g) = +� +R,Q +SR{gk}SQ{¯pk} +� � +N×N +dXdY SR{Tr Xk}SQ{Tr Y k} exp (−Tr XY + Tr Y Λ) = += +� +R,Q +SR{gk}SQ{¯pk}SR +� +Tr +� ∂ +∂Y +�k� +SQ{Tr Y k} exp (Tr Y Λ) +��� +Y =0 +(30) +which follows from the formula of Fourier theory: +� +dxdyf(x)g(y)e−xy = f +� ∂ +∂y +� +g(y) +��� +y=0 +(31) +Let us note that the combination +AR,Q(Λ) := SR +� +Tr +� ∂ +∂Y +�k� +SQ{Tr Y k} exp (Tr Y Λ) +��� +Y =0 +(32) +is an invariant polynomial of Λ, i.e. +AR,Q(Λ) = +� +P +α(R,Q) +P +SP {Tr Λk} +(33) +where αP are yet unknown coefficients to be defined. Now let us apply to AR,Q(Λ) the derivative SP +� +Tr +� ∂ +∂Λ +�k� +, +then put Λ = 0, and use (51): +SR +� +Tr +� ∂ +∂Y +�k� +SQ{Tr Y k}SP {Tr Y k} +��� +Y =0 = ξP (N)α(R,Q) +P +(34) +It remains to note that the l.h.s. of this equality is +SR +� +Tr +� ∂ +∂Y +�k� +SQ{Tr Y k}SP {Tr Y k} +��� +Y =0 = +� +T +N T +QP SR +� +Tr +� ∂ +∂Y +�k� +ST {Tr Y k} +��� +Y =0 = N R +QP ξR(N) (35) +Hence, +α(R,Q) +P += ξR(N) +ξP (N)N R +QP +(36) +and +Z(N; ¯p, p, g) = � +R,Q,P SR{gk}SQ{¯pk}α(R,Q) +P +SP {Tr Λk} = � +R,Q,P +ξR(N) +ξP (N)N R +QP SR{gk}SQ{¯pk}SP {Tr Λk} = += � +R,P +ξR(N) +ξP (N)SR{gk}SR/P {¯pk}SP {Tr Λk} = Z(1,1)(N, N; ¯p, g, p) +(37) +Another derivation of this formula, which will be of use in the β-deformed case, is as follows: since the matrix +integral is an invariant polynomial of Λ, one can make a replace Λ → U −1ΛU with a unitary matrix U and then +perform an additional integration over U (normalized to the volume of the unitary group U(N)): +Z(N; ¯p, p, g) = +� +R,Q +SR{gk}SQ{¯pk}SR +� +Tr +� ∂ +∂Y +�k� +SQ{Tr Y k} exp (Tr Y Λ) +��� +Y =0 = += +� +R,Q +SR{gk}SQ{¯pk}SR +� +Tr +� ∂ +∂Y +�k� +SQ{Tr Y k} +� +N×N +dU exp +� +Tr Y U −1ΛU +� ��� +Y =0 +(38) +6 + +On the other hand, this integral can be evaluated using the Itzykson-Zuber formula [15, Eq.(4.5)], +� +N×N +dU exp +� +Tr Y U −1ΛU +� += +� +P +1 +ξP +SP {Tr Y k}SP {Λk} +(39) +immediately giving rise to (37). +4 +Multi-matrix model representation of Z(n,1) +4.1 +Evaluating 4-matrix model Z(2,1) +As a natural generalization of the two-matrix model, we now consider a similar four-matrix model: +Z(2)(N; ¯p, p(1), p(2), g) = +� +N1×N1 +dX1dY1 +� +N2×N2 +dX2dY2 × +× +exp +� +−Tr X1Y1 − Tr X2Y2 + Tr Y1Λ1 + Tr Y2Λ2 + +� +k +gk +k Tr Xk +1 + +� +k +¯pk +k Tr Y k +2 + +� +k +Tr Y k +1 Tr Xk +2 +k +� += += +� +R,Q,P +SR{gk}SQ{¯pk}SR +� +Tr +� ∂ +∂Y1 +�k� +SP {Tr Y k +1 }SP +� +Tr +� ∂ +∂Y2 +�k� +SQ{Tr Y k +2 } exp (Tr Y1Λ1 + Tr Y2Λ2) +��� +Y1=Y2=0 = += +� +R,Q,P +SR{gk}SQ{¯pk}AR,P (Λ1)AP,Q(Λ2) = +� +R,P,Q1,Q2 +ξRξP +ξQ1ξQ2 +SR{gk}SP/Q2{¯pk}N R +P Q1SQ1{Tr Λk +1}SQ2{Tr Λk +2} +where p(1) +k += Tr Λk +1, p(2) +k += Tr Λk +2. +If Λ1 = 0, +Z(2)(N1, N2; ¯p, p(2), g) = +� +R,Q +ξ2 +R +ξQ +SR{gk}SR/Q{¯pk}SQ{Tr Λk +2} = Z(2,1)(N1, N2, N2; ¯p, g, p) +(40) +4.2 +2n-matrix model +Similarly, for the 2n-matrix model, one obtains +Z(n)(Ni; ¯p, {p(i)}, g) = +n +� +i=1 +� +Ni×Ni +dXidYi × +× +exp +� +− +n +� +i=1 +Tr XiYi + +n +� +i=1 +Tr YiΛi + +� +k +gk +k Tr Xk +1 + +� +k +¯pk +k Tr Y k +n + +n−1 +� +i=1 +� +k +Tr Y k +i Tr Xk +i+1 +k +� += += +� +{Ri,Qi} +SR1{gk}SRn/Qn{¯pk} +n−1 +� +i=1 +N Ri +Ri+1Qi +n +� +i=1 +ξRi +ξQi +SQi{Tr Λk +i } +(41) +and, at all Λi = 0 but Λn, +Z(n)(N; ¯p, p(n), g) = +� +R,Q +ξn +R +ξQ +SR{gk}SR/Q{¯pk}SQ{Tr Λk +n} = Z(n,1)({Ni}, Nn; ¯p, g, p) +(42) +5 +β-deformation +5.1 +The Jack polynomials +In order to deal with the β-deformation of the matrix model (24), we need to replace the Schur functions of +sec.3 with the Jack polynomials, and to replace correspondingly a few properties. +The orthogonality relation in this case follows from +JQ +� k +β +∂ +∂pk +� +JR{pk} = ||JQ|| JR/Q{pk} +(43) +7 + +where ||JQ|| is the norm square of the Jack polynomial, +||JR|| := G +β +R∨R(0) +Gβ +RR∨(0) +β|R| +Gβ +R′R′′(x) := +� +(i,j)∈R′ +� +x + R′ +i − j + β(R′′ +j − i + 1) +� +(44) +with the bar over the functions denoting the substitution β → β−1. The ratio +JR{N} +JR{δk,1} = +� +i,j∈R +(N + (j − 1)β−1 − i + 1) +(45) +Now we again use the operator ˆOβ +N with the property +ˆOβ(N) · JR{pk} = ξβ +R(N) JR{pk} +ξβ +R(N) := +� +i,j∈N +(N + β−1(j − 1) − i + 1) +(46) +We do not need the manifest form of this operator. +Now comparing Eqs.(59) and (60) from [1], one obtains that there exists a set of commuting differential +operators ˆHk such that +ˆHk = +� +ˆOβ(N) +�−1 � k +β +∂ +∂pk +� +ˆOβ(N) +(47) +These operators ˆHn defined in [1] do no longer have a meaning of matrix derivatives. We discuss them in detail +in secs.5.3-5.4. +Now let us use the identity (43) in order to obtain +JR +� +ˆHk +� +JQ{pk} = ξβ +Q(N) +� +ˆOβ(N) +�−1 +JR +� k +β +∂ +∂pk +� +JQ{pk} = ξβ +Q(N)||JQ|| +� +ˆOβ(N) +�−1 +JQ/R{pk} = += ξβ +Q(N)||JQ|| +� +ˆOβ(N) +�−1 � +P +β−1N Q∨ +R∨P ∨JP {pk} = +� +P +β−1N Q∨ +R∨P ∨||JQ|| +ξβ +Q(N) +ξβ +P (N) +JP {pk} +(48) +where +βN Q +RP are the Littlewood-Richardson coefficients, and we used that +JR/P = +� +Q +β−1N R∨ +Q∨P ∨ JQ +(49) +Note that +β−1N R∨ +Q∨P ∨ = βN R +QP +||JR|| +||JQ|| ||JP || +(50) +In particular, +JR +� +ˆHk +� +JQ{pk} +��� +pk=0 = ξβ +R(N)||JQ||δR,Q +(51) +5.2 +β-deformed matrix model +We introduce the β-deformed matrix model: +Zβ(N; ¯p, p, g) = +� � +N×N +[dXdY ]β exp +� +−Tr XY + Tr Y Λ + β +� +k +gk +k Tr Xk + β +� +k +¯pk +k Tr Y k +� +(52) +where the β-deformed integration measure is defined as +� � +N×N +[dXdY ]βJR{Tr Xk}JQ{Tr Y k} exp (−Tr XY + Tr Y Λ) = JR +� +ˆHk +� +JQ{Tr Y k} exp (Tr Y Λ) +��� +Y =0 +(53) +8 + +In order to evaluate the matrix integral (52), we use the same trick as before and insert an additional unitary +matrix integration using the β-deformed Itzykson-Zuber formula [20, sec.2] +� +N×N +[dU]β exp +� +Tr Y U −1ΛU +� += +� +P +1 +ξβ +P +JP {Tr Y k}JP{Λk} +(54) +Then, using the Cauchy identity for the Jack polynomials +� +R +JR{pk}JR{p′ +k} +||JR|| += exp +� +β +� +k +pkp′ +k +k +� +(55) +one repeats the calculation of the non-deformed case and, using (51), immediately gets that +Zβ(N; ¯p, p, g) = +� +R,Q,P +ξβ +R(N) +ξβ +P (N) +βN R +QP +JR{gk}JQ{¯pk}JP {Tr Λk} +||JP || ||JQ|| +(50) += +� +R,Q +ξβ +R(N) +ξβ +P (N) +JR/Q{¯pk}JR{gk}JP {Tr Λk} +||JR|| +(56) +This is exactly the formula that is generated by the W-representation of [1]. +The β-deformation of the multi-matrix model (41) is absolutely immediate. +5.3 +W-representation +The β-deformed matrix model has the W-representation similar to that in the non-deformed case [2]. In order +to construct it, we need a set of operators ˆHk discussed above. These operators are manifestly constructed in +the following way [1]. First of all, we define an auxiliary operator +ˆW0 += +1 +2 +� +a,b=0 +� +β(a + b)papb +∂ +∂pa+b ++ abpa+b +∂2 +∂pa∂pb +� ++ 1 − β +2 +� +k +(k − 1)kpk +∂ +∂pk +(57) +which is a deformation of the cut-and-join operator [5, 21]. Using this operator, we can construct a pair of +another auxiliary operators +ˆF1 += +[β−1 ∂ +∂p1 +, ˆW0] = +� +b=0 +(b + 1)pb +∂ +∂pb+1 +ˆF2 += +[ ˆF1, ˆ +W0] = +� +a,b=0 +� +βpapb(a + b + 1) +∂ +∂pa+b+1 ++ abpa+b−1 +∂2 +∂pa∂pb +� ++ (1 − β) +� +b +b(b + 1)pb +∂ +∂pb+1 +(58) +Now we construct the whole series of operators ˆHk by the recursion relation +ˆHk+1 = 1 +k [ ˆHk, ˆF2] +(59) +with the initial condition ˆH1 = ˆF1. The first few operators ˆHk are +ˆH1 += +� +b=0 +(b + 1)pb +∂ +∂pb+1 +ˆH2 += +[ ˆH1, ˆF2] = +� +a,b=0 +� +βpapb(a + b + 2) +∂ +∂pa+b+2 ++ abpa+b−2 +∂2 +∂pa∂pb +� ++ (1 − β) +� +b=0 +(b + 1)(b + 2)pb +∂ +∂pb+1 +ˆH3 += +1 +2[ ˆH2, ˆF2] = +� +a,b,c=0 +� +β2(a + b + c + 3)papbpc +∂ +∂pa+b+c ++ abcpa+b+c−3 +∂3 +∂pa∂pb∂pc +� ++ ++ +3(1 − β) +2 +� +a,b=0 +� +ab(a + b − 2)pa+b−3 +∂2 +∂pa∂pb ++ β(a + b + 2)(a + b + 3)papb +∂ +∂pa+b+3 +� ++ ++ +3β +2 +� +a,b,c,d=0 +δ3+a+b,c+dpapb +∂2 +∂pc∂pd ++ 2β2 − 3β + 2 +2 +� +a=0 +a(a − 1)(a − 2)pa−3 +∂ +∂pa +(60) +and we put p0 = N. +9 + +With these operators one obtains for ¯pk = δk,2: +Zβ(N; ¯p = δk,2, p, g) = e +ˆ +H2 +2 · eβ � +k +pkgk +k += +� +R,Q,P +β +|Q| +2 ξβ +R(N) +ξβ +P (N) +JR/Q{δk,2}JR{gk}JP {Tr Λk} +||JR|| +(61) +and, for ¯pk = δk,3, +Zβ(N; ¯p = δk,3, p, g) = e +ˆ +H3 +3 · eβ � +k +pkgk +k += +� +R,Q,P +β +2|Q| +3 ξβ +R(N) +ξβ +P (N) +JR/Q{δk,3}JR{gk}JP {Tr Λk} +||JR|| +(62) +The case of generic ¯pk looks like +Zβ(N; ¯p, p, g) = exp +�� +k +β +1−k +k ¯pk ˆHk +k +� +· eβ � +k +pkgk +k += +� +R,Q,P +ξβ +R(N) +ξβ +P (N) +JR/Q{¯pk}JR{gk}JP {Tr Λk} +||JR|| +(63) +which can be proved along the line of [2]. +5.4 +Operators ˆHk as Calogero-Sutherland Hamiltonians +Note that, for pk = �N +i=1 λk +i , and when acting on symmetric functions of λi,3 the operators ˆHm are realized as +ˆH1 += +� +i +∂ +∂λi +ˆH2 += +2β +� +i̸=j +1 +λi − λj +∂ +∂λi ++ +� +i +∂2 +∂λ2 +i +ˆH3 += +3β2 � +i̸=j̸=k +1 +(λi − λj)(λi − λk) +∂ +∂λi ++ 3β +� +i̸=j +1 +λi − λj +∂2 +∂λ2 +i ++ +� +i +∂3 +∂λ3 +i +. . . +ˆHn += +n +� +k=1 +Cn +k βn−k � +i +� +I⊂[1,...,N]\i +|I|=k +� +j∈I +1 +λi − λj +∂k +∂λk +i +(64) +where Cn +k are the binomial coefficients. This is a set of the (mutually commuting) rational Calogero-Sutherland +Hamiltonians. At β = 1, it reduces to [23, Eq.(21)] +ˆHn = Tr ∂n +∂Λn = N +� +i +� +I⊂[1,...,N] +|I|=n−1 +� +j∈I +1 +λi − λj +∂ +∂λi +(65) +where Λ is an N × N matrix with eigenvalues λi, and the operator ˆHn is understood as acting on invariant +functions of Λ. Note that the sum includes the terms with poles at i = j, which are resolved by the L’Hˆospital’s +rule. +The commutativity of operators ˆHk immediately follows from their realization (47). One can also consider, +instead of (47), the rotation [2] +ˆH(m) +k += +m +� +i=1 +� +ˆOβ(ui) +�−1 � k +β +∂ +∂pk +� m +� +i=1 +ˆOβ(ui) +(66) +which gives rise to a commutative family of Hamiltonians for each m. We will return to this issue elsewhere. +One can also realize ˆHn in terms of the Dunkl operators ˆDi: +ˆDi = +∂ +∂λi ++ β +� +j̸=i +1 +λi − λj +(1 − Pij) +(67) +3Formal subtleties of the procedure can be found in [22]. +10 + +where Pij is the operator permuting i and j. When acting on symmetric functions of λi, +ˆHk = +� +i +ˆDk +i +(68) +Note that the standard Calogero-Sutherland Hamiltonians are obtained by the rotation: +HCal +k += ∆(λ)β � +i +ˆDk +i ∆(λ)−β +(69) +where ∆(λ) = � +i̸=j(λi − λj). +Thus, we observe a surprising way of constructing the rational Calogero-Sutherland Hamiltonians: by suc- +cessive commutators with ˆF2 starting from ˆH1, ˆF2 being constructed by commutating of ˆH1 with ˆW0. The form +of all these auxiliary and operators and Hamiltonians in terms of the matrix derivative at β = 1 is +ˆW0 = 1 +2 : Tr +� +Λ ∂ +∂Λ +�2 +: +ˆF1 = Tr ∂ +∂Λ +ˆF2 = Tr Λ ∂2 +∂Λ2 +ˆHn = Tr ∂n +∂Λn +(70) +where : . . . : denotes the normal ordering, i.e. all the derivatives moved to the right. At generic β, ˆF1 is still +given by the same formula, while the other operators can be rewritten only in terms of the eigenvalues: the +auxiliary operators are +ˆW0 += +β +� +i̸=j +λ2 +i +λi − λj +∂ +∂λi ++ 1 +2 +� +i +λ2 +i +∂2 +∂λ2 +i +ˆF2 += +2β +� +i̸=j +λi +λi − λj +∂ +∂λi ++ +� +i +λi +∂2 +∂λ2 +i +(71) +while the Hamiltonians are given by (64). +Note also that ˆW0 gives the trigonometric Calogero-Sutherland Hamiltonian [24, 25]. Generally, they are +given by the generalized cut-and-join operators ˆW[k] [5], and ˆ +W0 = ˆW[2]. In the w∞-algebra picture of the +Introduction, ˆ +W[k] are associated with the vertical line 1 +kVk+1,0. In particular, ˆL0 = ˆW[1]. At the same picture, +the horizontal line is made from k +β +∂ +∂pk = V1,k. +6 +Conclusion +In this paper, we developed the theory of skew Hurwitz partition functions. They are τ-functions of the Toda +lattice hierarchy of the skew hypergeometric type. +Specifically, we discussed the formalism of cut-and-join +operators and of the rotation operator ˆO made from them, and explained how to apply them to the skew +Hurwitz partition functions. This formalism allows one to substitute an explicit representation of w∞ action on +the Young diagrams [1,25–27] by nearly trivial manipulations with abstract operators. We applied it to prove +the equivalence of the simplest skew Hurwitz partition function to a two-matrix models in background field +(a kind of 2-matrix generalization of the generalized Kontsevich model in the character phase, [28]). We also +pointed out peculiarities of the β-deformation of this formalism. +We also explained the interpretation of operators ˆO as intertwiners of commuting 1-parametric sub-algebras +of w∞, which look like rotations in its pictorial representation. After reduction to the Miwa locus (from times +variables to matrix eigenvalues), these subalgebras (the blue lines in the picture of the Introduction) describe +integrable systems, which, in the two simplest cases, are just the rational and trigonometric Calogero-Sutherland +systems having arbitrary coupling constant only after the β-deformation. This is a pattern that deserves further +analysis and better understanding. +While the β-deformation of the skew Hurwitz partition functions is immediate and preserves all the structures +but integrability, the q, t-deformation of the picture is somewhat less straightforward, and exploits other technical +means. It deserves a separate discussion. +11 + +Acknowledgements +Our work is partly supported by the grant of the Foundation for the Advancement of Theoretical Physics +“BASIS” and by the joint grant 21-51-46010-ST-a, and by the National Natural Science Foundation of China +(Nos. 11875194). +References +[1] R. Wang, F. Liu, C. H. Zhang and W. Z. Zhao, arXiv:2206.13038 +[2] A. Mironov, V. Mishnyakov, A. Morozov, A. Popolitov, R. Wang and W. Z. Zhao, arXiv:2301.04107 +[3] A. Alexandrov, A. Mironov, A. Morozov and S. Natanzon, JHEP 11 (2014) 080, arXiv:1405.1395 +[4] I.G. Macdonald, Symmetric functions and Hall polynomials, Second Edition, Oxford University Press, 1995 +[5] A. Mironov, A. Morozov and S. Natanzon, Theor.Math.Phys. 166 (2011) 1-22, arXiv:0904.4227; Journal +of Geometry and Physics 62 (2012) 148-155, arXiv:1012.0433 +[6] W. Fulton, Young tableaux: with applications to representation theory and geometry, LMS, 1997 +[7] E. Date, M. Jimbo, M .Kashiwara and T. 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A11 (1996) 5031, hep-th/9404005 +13 + diff --git a/d9FKT4oBgHgl3EQfrC7t/content/tmp_files/load_file.txt b/d9FKT4oBgHgl3EQfrC7t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..204499cf4923c91e35a57804e0a1f944233a4ced --- /dev/null +++ b/d9FKT4oBgHgl3EQfrC7t/content/tmp_files/load_file.txt @@ -0,0 +1,536 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf,len=535 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='11877v1 [hep-th] 27 Jan 2023 On KP-integrable skew Hurwitz τ-functions and their β-deformations A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Mironovb,c,d,∗, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Mishnyakova,b,†, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Morozova,c,d,‡, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Popolitova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='§,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Wei-Zhong Zhaoe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='¶ FIAN/TD-02/23 IITP/TH-02/23 ITEP/TH-02/23 MIPT/TH-02/23 a MIPT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Dolgoprudny,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' 141701,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Russia b Lebedev Physics Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Moscow 119991,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Russia c ITEP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Moscow 117218,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Russia d Institute for Information Transmission Problems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Moscow 127994,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Russia eSchool of Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Capital Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Beijing 100048,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' China Abstract We extend the old formalism of cut-and-join operators in the theory of Hurwitz τ-functions to description of a wide family of KP-integrable skew Hurwitz τ-functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' which include,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' in particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' the newly discovered interpolating WLZZ models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Recently, the simplest of them was related to a superintegrable two-matrix model with two potentials and one external matrix field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Now we provide detailed proofs, and a generalization to a multi-matrix representation, and propose the β-deformation of the matrix model as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The general interpolating WLZZ model is generated by a W -representation given by a sum of operators from a one- parametric commutative sub-family (a commutative subalgebra of w∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Different commutative families are related by cut-and-join rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Two of these sub-families (‘vertical’ and ‘45-degree’) turn out to be nothing but the trigonometric and rational Calogero-Sutherland Hamiltonians, the ‘horizontal’ family is represented by simple derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Other families require an additional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' 1 Introduction Matrix models are the simplest representatives of the universality classes of non-perturbative partition functions, which exhibit their peculiar properties, well shadowed in perturbative Feynman diagram technique and in other approaches based on the standard multi-dimensional quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' These properties include high hidden symmetries, integrability, superintegability and W-representations, which express partition functions through the action of generalized cut-and-join operators on vacuum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' By now, they are well-studied and understood in particular models, and the new challenge is to classify and unite different models in a common entity, as a step to a formulation of non-perturbative string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Recently a big step was made in this direction by the suggestion of WLZZ models [1], which appeared to be building blocks of a two-matrix model in the external matrix field [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' In this paper, in particular, we provide technical details behind this general claim made in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' They are based on using peculiar cut-and-join rotation operators O and on an extension of the old formalism of [3] to skew Hurwitz partition functions, naturally expanded in skew rather than ordinary Schur functions, and exploit the Littlewood-Richardson decompositions of ones into the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' These decompositions get a little non-trivial after the β-deformation from the Schur functions to the Jack polynomials, which we also explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ∗mironov@lpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='ru,mironov@itep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='ru †mishnyakovvv@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='com ‡morozov@itep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='ru §popolit@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='com ¶zhaowz@cnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='cn 1 Skew Hurwitz partition functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Generic Hurwitz partition functions depending on two sets of time variables have the form [5], ZH(u∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' g, p) = � R SR{gk}SR{pk} exp �� ∆ u∆ψR(∆) � (1) SR{pk} is the Schur function labelled by the partition R, which is a symmetric function of xi considered as a graded polynomial of the power sums pk = � i xk i , ∆ is a partition, and ψR(∆) is the character of symmetric group S|R|, |R| being the size of partition [4,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' This partition function is generated by generalized cut-and-join operators [5], and is not integrable in the KP/Toda sense [7,8], it becomes integrable only with a special choice of the coefficients u∆: exp �� ∆ u∆ψR(∆) � −→ exp �� k ukC(k) R � (2) where C(k) R are eigenvalues of specially chosen kth Casimir operators [5, 9–11]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Equivalently, one can choose [3,13] exp �� ∆ u∆ψR(∆) � −→ � i,j∈R f(j − i) (3) with an arbitrary function f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' KP-integrable Hurwitz partition functions of this kind are called hypergeometric τ-functions of the KP hierarchy [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' In this paper, we propose a generalization of the Hurwitz partition functions (1) to the skew Hurwitz partition functions, ZH(u∆, v∆′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' g, p) = � R SR/Q{¯pk}SR{gk}SQ{pk} exp �� ∆ u∆ψR(∆) − � ∆′ v∆′ψQ(∆′) � (4) where SR/Q is the skew Schur function [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' They are also integrable only with choices analogous to (2)-(3) and are called skew hypergeometric τ-functions [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' In [3], it was suggested to choose the function f(x) in (3) f(x) = n � i (ui + x) (5) This choice has many applications, from generating numbers of isomorphism classes of the Belyi pairs (Grothendieck’s dessins d’enfant) [14] to the Itzykson-Zuber integral [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' In this paper, as an extension of results of [3], we consider an extension of this choice to the skew Hurwitz partition functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Thus, we consider a partition functions of the form Z(n,m)({ui}, {vj};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, g, p) = � R,Q �n i=1 ξR(ui) �m j=1 ξQ(vj)SR/Q{¯pk}SR{gk}SQ{pk} (6) where ξR(u) = SR{pk = u} SR{δk,1} = � i,j∈R (u + j − i) (7) The partition function Z(n,m) depends on n parameters ui, m parameters vi and three infinite sets of time variables {gk}, {pk}, and {¯pk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The interpolating matrix model of [2] is just the Z(1,1) member of this two- parametric family, while other models of [2] are the one-parametric family Z(n,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The case considered in [3] corresponds to the restriction of this partition function to all pk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Then, it becomes the sum � R � n � i=1 ξR(ui) � SR{¯pk}SR{gk} (8) 1They can be also associated with values of characters on the completed cycles [10,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' 2 which celebrates a series of properties: it is aτ-function of the Toda lattice, it is generated by the cut-and-join rotation operators O, it has multi-matrix model representation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Our goal in this paper is to demonstrate that all these properties keep intact for the skew Hurwitz partition functions (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Moreover, all the properties but integrability (which is known not to survive any β-deformations) survive the β-deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' w∞-algebra pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Another important issue, which we shortly touch in this paper, is that, from the point of view of the W-representations, the original technique used in [1], the partition functions Z(n,n) (at some special points) are generated by combinations of operators from w∞ algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The relevant for the construction are the one-parametric commutative subalgebras depicted by the blue lines in the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The commutativity of operators of each line follows from the w∞-algebra relations [16] [Vm1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Vm2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='n2] = � (m1 − 1)n2 − (m2 − 1)n2 � Vm1+m2−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='n1+n2 =⇒ � adm ˆ Fs ˆFs−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' adn ˆ Fs ˆFs−1 � = 0 (9) for ˆFs = (−1)sVs+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' which is nothing but the commutation of generators � ˆH(s) m ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ˆH(s) n � = 0 for ˆH(s) m := adm ˆ Fs+1 ˆFs = (−1)m+1Vms+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='m (10) After the Miwa transform from time-variables to the eigenvalues of λi of N × N matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' H(s) m become systems of commuting differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Not surprisingly, some of them are familiar to us: that is, the series at s = 2: adm F2F1 (the 45-degree blue line) turns out to be Hamiltonians of the rational Calogero-Sutherland system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' This adds to the old knowledge that the vertical line is the trigonometric Calogero-Sutherland system, while on the horizontal line one has just commuting first-order operators ∂ ∂pn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' In fact, these Calogero-Sutherland models are at the free fermion point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' In order to get the Calogero- Sutherland system with a non-trivial coupling, one needs the β-deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Fortunately, this whole picture survives the β-deformation, as we explain in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ∂p3 ∼ [ ˆF1, ∂p2] ∂p2 ∼ [ ˆF1, ∂p1] ∂p1 ˆL0 ˆW0 ˆ M0 ˆF1 = [ ˆW0, ∂p1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ad ˆ F2 ˆF1 ad2 ˆ F2 ˆF1 ˆF2 = ad2 ˆ W0 ∂p1 ˆF3 = ad3 ˆ W0 ∂p1 ad ˆ F3 ˆF2 ˆHm 1 The cut-and-join rotation operators O, one of the central personages of this paper have a spectacular interpretation in these terms: they rotate blue lines, one into another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' These operators are constructed from the generalized cut-and-join operators [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Hence, the name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' In this sense, the rational Calogero-Sutherland system is just a simple rotation of the system {∂pn} while the trigonometric Calogero-Sutherland model is obtained from the rational one through an infinite system of rotations trough a sequence of new integrable systems intertwined by the operators O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' In section 2, we describe integrable properties of the generic skew Hurwitz partition function (6) and its representation via generalized cut-and-join operators in parallel with [3, secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='2-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' In section 3, we concentrate on the particular case of Z(1,1) and derive its description as a two-matrix model depending on the external matrix and two potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The matrix model description can be also provided for Z(n,1), as we demonstrate in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' In section 5, we discuss the β-deformation of Z(1,1): the matrix model description and the W-representation, which turns out to be associated with the rational Calogero-Sutherland Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The crucial difference with the β = 1 case (which corresponds to the free fermion point of the Calogero-Sutherland model) is that a description of the Hamiltonians in terms of matrix derivatives is no longer available, instead one has to use the eigenvalue variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Section 6 contains some concluding remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' 3 2 Properties of skew Hurwitz partition functions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='1 Representation via cut-and-join operators The partition function (6) can be realized by action of operators constructed from the commutative set of generalized cut-and-join operators ˆW∆ [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' We call these operators the cut-and-join rotation operators, they play one of the central roles in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The generalized cut-and-join operators ˆW∆ form a commutative set of operators, the Schur functions being their eigenfunctions [5]: ˆW∆ SR = φR(∆) SR (11) where, for the diagram ∆ containing r unit cycles: ∆ = [ ˜∆, 1r], φR,∆ = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 0 |∆| > |R| (|R| − |∆| + r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' (|R| − |∆|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' φR, ˆ∆ = (|R| − |∆| + r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' (|R| − |∆|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ψR( ˆ∆) z ˆ∆dR |∆| ≤ |R| (12) where ˆ∆ := [∆, 1|R|−|∆|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Now note that [17, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' (61)] � ∆ φR(∆)p∆ = SR(pk + δk1) dR (13) Now we construct the cut-and-join rotation operator as follows: ˆO(u) := � ∆ p∆ · ˆ W∆, with pk = u − δk,1 (14) Here we use the notation p∆ = �l∆ i=1 pδi, where l∆ is the length of the partition ∆, and δi’s are its parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Then, ˆO(u) · SR{pk} = SR(u) SR(δk,1)SR{pk} = ξR(u) SR{pk} (15) This operator was constructed earlier in [3, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' (21)] in order to insert additional factors SR(N) SR(δk,1) into character expansion of the partition function, and was written there in a different form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' In particular, one can rewrite it via Casimir operators [3, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Now we can straightforwardly obtain the skew Hurwitz partition function (6) by the action of these operators ˆO(u): Z(n,m)({ui}, {vj}v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, g, p) = m � j=1 ˆO−1 g (vj) n � i=1 ˆOp(ui) exp �� k (pk + ¯pk)gk k � (16) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='2 Z(n,m) as a τ-function of Toda lattice The skew Hurwitz partition function is proportional to a τ-function of the Toda lattice hierarchy [8]: τN(g, p) = N � k=1 �n i=1 Γ(1 − k + ui + N) �m j=1 Γ(1 − k + vi + N) · Z(n,m)({ui + N}, {vj + N};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, g, p) (17) Here N is the zeroth time of the hierarchy, and {kpk}, {kgk} are the two infinite sets of times of the hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The third infinite set, {¯pk} describes the concrete solution to the hierarchy (the point of the infinite-dimensional Grassmannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Let us note that, in accordance with [18], the sum τn(g, p|ξ) = � R,Q ζR,Q(n)SR{g}SQ{p} (18) 4 is a τ-function of the Toda lattice hierarchy iff ζR,Q(N) = det i,j≤N F(Ri − i + N + 1, Qj − j + N + 1) (19) with some function F(x, y), and N playing the role of the zeroth time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Hence, in order to prove that (17) is a Toda lattice τ-function, one has to prove that � N � k=1 �n i=1 Γ(1 − k + ui + N) �m j=1 Γ(1 − k + vi + N) � �n i=1 ξR(ui + N) �m j=1 ξQ(vj + N) · SR/Q{¯pk} (20) has representation (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' To this end, we use the Jacobi-Trudi determinant representation for the skew Schur functions, SR/Q{¯p} = det i,j≤lR hRi−i−Qj+j{¯p} (21) where hk are the complete homogeneous symmetric polynomials, that is, hk = S[k], and we put hk = 0 at k < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Then, one obtains representation (18) for (17) with ζR,Q(N) = det i,j≤N �n k=1 Γ(Ri − i + 1 + uk + N) �m l=1 Γ(Qj − j + 1 + vl + N) hRi−i−Qj+j{¯p} (22) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ζR,Q(N) is just of the form (19) with F(x, y) = �n k=1 Γ(x + uk) �m l=1 Γ(y + vl) hx−y{¯p} (23) 3 Two-matrix model representation of Z(1,1) In this section, we provide the matrix model representation for the skew Hurwitz partition function Z(1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' We explain that it is given by the two-matrix model Z(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, p, g) = � � N×N dXdY exp � −Tr XY + Tr Y Λ + � k gk k Tr Xk + � k ¯pk k Tr Y k � (24) with pk = Tr Λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Here the integral is understood as integration of a power series in gk, ¯pk and Tr Λk, and X are Hermitian matrices, while Y are anti-Hermitian ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='1 From matrix to time derivatives In order to deal with this matrix model, we need to know the action of invariant matrix derivatives, Tr ∂k ∂Y k on polynomials of pk = Tr Y k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Note that, by comparing the actions on the Schur functions using [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' (37)] and [1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' (24)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' one obtains ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='Tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='∂Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='�k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='= ˆO−1(N) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='k ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='∂pk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='ˆO(N) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='(25) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='Now let us use the identity2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='SR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='k ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='∂pk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='SQ{pk} = SQ/R{pk} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='(27) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='in order to calculate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='SR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='Tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='∂Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='�k� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='SQ{Tr Y k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� �� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='pk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='} = ξQ(N) ˆO−1(N)SR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='k ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='∂pk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='SQ{pk} = ξQ(N) ˆO−1(N)SQ/R{pk} = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='= ξQ(N) ˆO−1(N) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='N Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='RP SP {pk} = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='N Q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='RP ξQ(N)ξ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='P (N)SP {pk} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='(28) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='2The simplest way to prove this formula is to use the Cauchy identity: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='SR{p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='k}SR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='k ∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='∂pk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='SQ{pk} = exp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='∂pk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='SQ{pk} = SQ{pk + p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='k} = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='SQ/R{pk}SR{p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='k} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='(26) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='and compare the coefficients in front of SR{p′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' 5 where N Q RP are the Littlewood-Richardson coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' In particular, SR � Tr � ∂ ∂Y �k� SQ{Tr Y k � �� � pk } ��� Y =0 = ξR(N)δR,Q (29) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='2 Evaluating the matrix integral Now the two-matrix integral (24) can be rewritten in the form [3, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' (47)] Z(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, p, g) = � R,Q SR{gk}SQ{¯pk} � � N×N dXdY SR{Tr Xk}SQ{Tr Y k} exp (−Tr XY + Tr Y Λ) = = � R,Q SR{gk}SQ{¯pk}SR � Tr � ∂ ∂Y �k� SQ{Tr Y k} exp (Tr Y Λ) ��� Y =0 (30) which follows from the formula of Fourier theory: � dxdyf(x)g(y)e−xy = f � ∂ ∂y � g(y) ��� y=0 (31) Let us note that the combination AR,Q(Λ) := SR � Tr � ∂ ∂Y �k� SQ{Tr Y k} exp (Tr Y Λ) ��� Y =0 (32) is an invariant polynomial of Λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' AR,Q(Λ) = � P α(R,Q) P SP {Tr Λk} (33) where αP are yet unknown coefficients to be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Now let us apply to AR,Q(Λ) the derivative SP � Tr � ∂ ∂Λ �k� , then put Λ = 0, and use (51): SR � Tr � ∂ ∂Y �k� SQ{Tr Y k}SP {Tr Y k} ��� Y =0 = ξP (N)α(R,Q) P (34) It remains to note that the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' of this equality is SR � Tr � ∂ ∂Y �k� SQ{Tr Y k}SP {Tr Y k} ��� Y =0 = � T N T QP SR � Tr � ∂ ∂Y �k� ST {Tr Y k} ��� Y =0 = N R QP ξR(N) (35) Hence, α(R,Q) P = ξR(N) ξP (N)N R QP (36) and Z(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, p, g) = � R,Q,P SR{gk}SQ{¯pk}α(R,Q) P SP {Tr Λk} = � R,Q,P ξR(N) ξP (N)N R QP SR{gk}SQ{¯pk}SP {Tr Λk} = = � R,P ξR(N) ξP (N)SR{gk}SR/P {¯pk}SP {Tr Λk} = Z(1,1)(N, N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, g, p) (37) Another derivation of this formula, which will be of use in the β-deformed case, is as follows: since the matrix integral is an invariant polynomial of Λ, one can make a replace Λ → U −1ΛU with a unitary matrix U and then perform an additional integration over U (normalized to the volume of the unitary group U(N)): Z(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, p, g) = � R,Q SR{gk}SQ{¯pk}SR � Tr � ∂ ∂Y �k� SQ{Tr Y k} exp (Tr Y Λ) ��� Y =0 = = � R,Q SR{gk}SQ{¯pk}SR � Tr � ∂ ∂Y �k� SQ{Tr Y k} � N×N dU exp � Tr Y U −1ΛU � ��� Y =0 (38) 6 On the other hand, this integral can be evaluated using the Itzykson-Zuber formula [15, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='5)], � N×N dU exp � Tr Y U −1ΛU � = � P 1 ξP SP {Tr Y k}SP {Λk} (39) immediately giving rise to (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' 4 Multi-matrix model representation of Z(n,1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='1 Evaluating 4-matrix model Z(2,1) As a natural generalization of the two-matrix model, we now consider a similar four-matrix model: Z(2)(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' p(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' p(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' g) = � N1×N1 dX1dY1 � N2×N2 dX2dY2 × × exp � −Tr X1Y1 − Tr X2Y2 + Tr Y1Λ1 + Tr Y2Λ2 + � k gk k Tr Xk 1 + � k ¯pk k Tr Y k 2 + � k Tr Y k 1 Tr Xk 2 k � = = � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='P SR{gk}SQ{¯pk}SR � Tr � ∂ ∂Y1 �k� SP {Tr Y k 1 }SP � Tr � ∂ ∂Y2 �k� SQ{Tr Y k 2 } exp (Tr Y1Λ1 + Tr Y2Λ2) ��� Y1=Y2=0 = = � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='P SR{gk}SQ{¯pk}AR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='P (Λ1)AP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='Q(Λ2) = � R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='Q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='Q2 ξRξP ξQ1ξQ2 SR{gk}SP/Q2{¯pk}N R P Q1SQ1{Tr Λk 1}SQ2{Tr Λk 2} where p(1) k = Tr Λk 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' p(2) k = Tr Λk 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' If Λ1 = 0, Z(2)(N1, N2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, p(2), g) = � R,Q ξ2 R ξQ SR{gk}SR/Q{¯pk}SQ{Tr Λk 2} = Z(2,1)(N1, N2, N2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, g, p) (40) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='2 2n-matrix model Similarly, for the 2n-matrix model, one obtains Z(n)(Ni;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, {p(i)}, g) = n � i=1 � Ni×Ni dXidYi × × exp � − n � i=1 Tr XiYi + n � i=1 Tr YiΛi + � k gk k Tr Xk 1 + � k ¯pk k Tr Y k n + n−1 � i=1 � k Tr Y k i Tr Xk i+1 k � = = � {Ri,Qi} SR1{gk}SRn/Qn{¯pk} n−1 � i=1 N Ri Ri+1Qi n � i=1 ξRi ξQi SQi{Tr Λk i } (41) and, at all Λi = 0 but Λn, Z(n)(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, p(n), g) = � R,Q ξn R ξQ SR{gk}SR/Q{¯pk}SQ{Tr Λk n} = Z(n,1)({Ni}, Nn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, g, p) (42) 5 β-deformation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='1 The Jack polynomials In order to deal with the β-deformation of the matrix model (24), we need to replace the Schur functions of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='3 with the Jack polynomials, and to replace correspondingly a few properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The orthogonality relation in this case follows from JQ � k β ∂ ∂pk � JR{pk} = ||JQ|| JR/Q{pk} (43) 7 where ||JQ|| is the norm square of the Jack polynomial, ||JR|| := G β R∨R(0) Gβ RR∨(0) β|R| Gβ R′R′′(x) := � (i,j)∈R′ � x + R′ i − j + β(R′′ j − i + 1) � (44) with the bar over the functions denoting the substitution β → β−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The ratio JR{N} JR{δk,1} = � i,j∈R (N + (j − 1)β−1 − i + 1) (45) Now we again use the operator ˆOβ N with the property ˆOβ(N) · JR{pk} = ξβ R(N) JR{pk} ξβ R(N) := � i,j∈N (N + β−1(j − 1) − i + 1) (46) We do not need the manifest form of this operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Now comparing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' (59) and (60) from [1], one obtains that there exists a set of commuting differential operators ˆHk such that ˆHk = � ˆOβ(N) �−1 � k β ∂ ∂pk � ˆOβ(N) (47) These operators ˆHn defined in [1] do no longer have a meaning of matrix derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' We discuss them in detail in secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='3-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Now let us use the identity (43) in order to obtain JR � ˆHk � JQ{pk} = ξβ Q(N) � ˆOβ(N) �−1 JR � k β ∂ ∂pk � JQ{pk} = ξβ Q(N)||JQ|| � ˆOβ(N) �−1 JQ/R{pk} = = ξβ Q(N)||JQ|| � ˆOβ(N) �−1 � P β−1N Q∨ R∨P ∨JP {pk} = � P β−1N Q∨ R∨P ∨||JQ|| ξβ Q(N) ξβ P (N) JP {pk} (48) where βN Q RP are the Littlewood-Richardson coefficients,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' and we used that JR/P = � Q β−1N R∨ Q∨P ∨ JQ (49) Note that β−1N R∨ Q∨P ∨ = βN R QP ||JR|| ||JQ|| ||JP || (50) In particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' JR � ˆHk � JQ{pk} ��� pk=0 = ξβ R(N)||JQ||δR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='Q (51) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='2 β-deformed matrix model We introduce the β-deformed matrix model: Zβ(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, p, g) = � � N×N [dXdY ]β exp � −Tr XY + Tr Y Λ + β � k gk k Tr Xk + β � k ¯pk k Tr Y k � (52) where the β-deformed integration measure is defined as � � N×N [dXdY ]βJR{Tr Xk}JQ{Tr Y k} exp (−Tr XY + Tr Y Λ) = JR � ˆHk � JQ{Tr Y k} exp (Tr Y Λ) ��� Y =0 (53) 8 In order to evaluate the matrix integral (52), we use the same trick as before and insert an additional unitary matrix integration using the β-deformed Itzykson-Zuber formula [20, sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='2] � N×N [dU]β exp � Tr Y U −1ΛU � = � P 1 ξβ P JP {Tr Y k}JP{Λk} (54) Then, using the Cauchy identity for the Jack polynomials � R JR{pk}JR{p′ k} ||JR|| = exp � β � k pkp′ k k � (55) one repeats the calculation of the non-deformed case and, using (51), immediately gets that Zβ(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, p, g) = � R,Q,P ξβ R(N) ξβ P (N) βN R QP JR{gk}JQ{¯pk}JP {Tr Λk} ||JP || ||JQ|| (50) = � R,Q ξβ R(N) ξβ P (N) JR/Q{¯pk}JR{gk}JP {Tr Λk} ||JR|| (56) This is exactly the formula that is generated by the W-representation of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The β-deformation of the multi-matrix model (41) is absolutely immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='3 W-representation The β-deformed matrix model has the W-representation similar to that in the non-deformed case [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' In order to construct it, we need a set of operators ˆHk discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' These operators are manifestly constructed in the following way [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' First of all, we define an auxiliary operator ˆW0 = 1 2 � a,b=0 � β(a + b)papb ∂ ∂pa+b + abpa+b ∂2 ∂pa∂pb � + 1 − β 2 � k (k − 1)kpk ∂ ∂pk (57) which is a deformation of the cut-and-join operator [5, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Using this operator, we can construct a pair of another auxiliary operators ˆF1 = [β−1 ∂ ∂p1 , ˆW0] = � b=0 (b + 1)pb ∂ ∂pb+1 ˆF2 = [ ˆF1, ˆ W0] = � a,b=0 � βpapb(a + b + 1) ∂ ∂pa+b+1 + abpa+b−1 ∂2 ∂pa∂pb � + (1 − β) � b b(b + 1)pb ∂ ∂pb+1 (58) Now we construct the whole series of operators ˆHk by the recursion relation ˆHk+1 = 1 k [ ˆHk, ˆF2] (59) with the initial condition ˆH1 = ˆF1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The first few operators ˆHk are ˆH1 = � b=0 (b + 1)pb ∂ ∂pb+1 ˆH2 = [ ˆH1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ˆF2] = � a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='b=0 � βpapb(a + b + 2) ∂ ∂pa+b+2 + abpa+b−2 ∂2 ∂pa∂pb � + (1 − β) � b=0 (b + 1)(b + 2)pb ∂ ∂pb+1 ˆH3 = 1 2[ ˆH2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ˆF2] = � a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='c=0 � β2(a + b + c + 3)papbpc ∂ ∂pa+b+c + abcpa+b+c−3 ∂3 ∂pa∂pb∂pc � + + 3(1 − β) 2 � a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='b=0 � ab(a + b − 2)pa+b−3 ∂2 ∂pa∂pb + β(a + b + 2)(a + b + 3)papb ∂ ∂pa+b+3 � + + 3β 2 � a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='d=0 δ3+a+b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='c+dpapb ∂2 ∂pc∂pd + 2β2 − 3β + 2 2 � a=0 a(a − 1)(a − 2)pa−3 ∂ ∂pa (60) and we put p0 = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' 9 With these operators one obtains for ¯pk = δk,2: Zβ(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p = δk,2, p, g) = e ˆ H2 2 · eβ � k pkgk k = � R,Q,P β |Q| 2 ξβ R(N) ξβ P (N) JR/Q{δk,2}JR{gk}JP {Tr Λk} ||JR|| (61) and, for ¯pk = δk,3, Zβ(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p = δk,3, p, g) = e ˆ H3 3 · eβ � k pkgk k = � R,Q,P β 2|Q| 3 ξβ R(N) ξβ P (N) JR/Q{δk,3}JR{gk}JP {Tr Λk} ||JR|| (62) The case of generic ¯pk looks like Zβ(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ¯p, p, g) = exp �� k β 1−k k ¯pk ˆHk k � eβ � k pkgk k = � R,Q,P ξβ R(N) ξβ P (N) JR/Q{¯pk}JR{gk}JP {Tr Λk} ||JR|| (63) which can be proved along the line of [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='4 Operators ˆHk as Calogero-Sutherland Hamiltonians Note that, for pk = �N i=1 λk i , and when acting on symmetric functions of λi,3 the operators ˆHm are realized as ˆH1 = � i ∂ ∂λi ˆH2 = 2β � i̸=j 1 λi − λj ∂ ∂λi + � i ∂2 ∂λ2 i ˆH3 = 3β2 � i̸=j̸=k 1 (λi − λj)(λi − λk) ∂ ∂λi + 3β � i̸=j 1 λi − λj ∂2 ∂λ2 i + � i ∂3 ∂λ3 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' ˆHn = n � k=1 Cn k βn−k � i � I⊂[1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=',N]\\i |I|=k � j∈I 1 λi − λj ∂k ∂λk i (64) where Cn k are the binomial coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' This is a set of the (mutually commuting) rational Calogero-Sutherland Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' At β = 1, it reduces to [23, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' (21)] ˆHn = Tr ∂n ∂Λn = N � i � I⊂[1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=',N] |I|=n−1 � j∈I 1 λi − λj ∂ ∂λi (65) where Λ is an N × N matrix with eigenvalues λi, and the operator ˆHn is understood as acting on invariant functions of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Note that the sum includes the terms with poles at i = j, which are resolved by the L’Hˆospital’s rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The commutativity of operators ˆHk immediately follows from their realization (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' One can also consider, instead of (47), the rotation [2] ˆH(m) k = m � i=1 � ˆOβ(ui) �−1 � k β ∂ ∂pk � m � i=1 ˆOβ(ui) (66) which gives rise to a commutative family of Hamiltonians for each m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' We will return to this issue elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' One can also realize ˆHn in terms of the Dunkl operators ˆDi: ˆDi = ∂ ∂λi + β � j̸=i 1 λi − λj (1 − Pij) (67) 3Formal subtleties of the procedure can be found in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' 10 where Pij is the operator permuting i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' When acting on symmetric functions of λi, ˆHk = � i ˆDk i (68) Note that the standard Calogero-Sutherland Hamiltonians are obtained by the rotation: HCal k = ∆(λ)β � i ˆDk i ∆(λ)−β (69) where ∆(λ) = � i̸=j(λi − λj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Thus, we observe a surprising way of constructing the rational Calogero-Sutherland Hamiltonians: by suc- cessive commutators with ˆF2 starting from ˆH1, ˆF2 being constructed by commutating of ˆH1 with ˆW0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' The form of all these auxiliary and operators and Hamiltonians in terms of the matrix derivative at β = 1 is ˆW0 = 1 2 : Tr � Λ ∂ ∂Λ �2 : ˆF1 = Tr ∂ ∂Λ ˆF2 = Tr Λ ∂2 ∂Λ2 ˆHn = Tr ∂n ∂Λn (70) where : .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' : denotes the normal ordering, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' all the derivatives moved to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' At generic β, ˆF1 is still given by the same formula, while the other operators can be rewritten only in terms of the eigenvalues: the auxiliary operators are ˆW0 = β � i̸=j λ2 i λi − λj ∂ ∂λi + 1 2 � i λ2 i ∂2 ∂λ2 i ˆF2 = 2β � i̸=j λi λi − λj ∂ ∂λi + � i λi ∂2 ∂λ2 i (71) while the Hamiltonians are given by (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Note also that ˆW0 gives the trigonometric Calogero-Sutherland Hamiltonian [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Generally, they are given by the generalized cut-and-join operators ˆW[k] [5], and ˆ W0 = ˆW[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' In the w∞-algebra picture of the Introduction, ˆ W[k] are associated with the vertical line 1 kVk+1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' In particular, ˆL0 = ˆW[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' At the same picture, the horizontal line is made from k β ∂ ∂pk = V1,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' 6 Conclusion In this paper, we developed the theory of skew Hurwitz partition functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' They are τ-functions of the Toda lattice hierarchy of the skew hypergeometric type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Specifically, we discussed the formalism of cut-and-join operators and of the rotation operator ˆO made from them, and explained how to apply them to the skew Hurwitz partition functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' This formalism allows one to substitute an explicit representation of w∞ action on the Young diagrams [1,25–27] by nearly trivial manipulations with abstract operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' We applied it to prove the equivalence of the simplest skew Hurwitz partition function to a two-matrix models in background field (a kind of 2-matrix generalization of the generalized Kontsevich model in the character phase, [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' We also pointed out peculiarities of the β-deformation of this formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' We also explained the interpretation of operators ˆO as intertwiners of commuting 1-parametric sub-algebras of w∞, which look like rotations in its pictorial representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' After reduction to the Miwa locus (from times variables to matrix eigenvalues), these subalgebras (the blue lines in the picture of the Introduction) describe integrable systems, which, in the two simplest cases, are just the rational and trigonometric Calogero-Sutherland systems having arbitrary coupling constant only after the β-deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' This is a pattern that deserves further analysis and better understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' While the β-deformation of the skew Hurwitz partition functions is immediate and preserves all the structures but integrability, the q, t-deformation of the picture is somewhat less straightforward, and exploits other technical means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' It deserves a separate discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' 11 Acknowledgements Our work is partly supported by the grant of the Foundation for the Advancement of Theoretical Physics “BASIS” and by the joint grant 21-51-46010-ST-a, and by the National Natural Science Foundation of China (Nos.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} +page_content=' A11 (1996) 5031, hep-th/9404005 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/d9FKT4oBgHgl3EQfrC7t/content/2301.11877v1.pdf'} diff --git a/gNFLT4oBgHgl3EQfYy_o/content/tmp_files/2301.12067v1.pdf.txt b/gNFLT4oBgHgl3EQfYy_o/content/tmp_files/2301.12067v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a517c1e95014b7737b5f76f8d05ad492e3dad168 --- /dev/null +++ b/gNFLT4oBgHgl3EQfYy_o/content/tmp_files/2301.12067v1.pdf.txt @@ -0,0 +1,2449 @@ +Learning Optimal Features via Partial Invariance +Moulik Choraria1*, Ibtihal Ferwana1, Ankur Mani2, Lav R. Varshney1 +1University of Illinois at Urbana-Champaign, 2 University of Minnesota, Twin Cities +Abstract +Learning models that are robust to test-time distribution shifts +is a key concern in domain generalization, and in the wider +context of their real-life applicability. Invariant Risk Mini- +mization (IRM) is one particular framework that aims to learn +deep invariant features from multiple domains, and has sub- +sequently led to further variants. A key assumption for the +success of these methods requires that the underlying causal +mechanisms/features remain invariant across domains and the +true invariant features be sufficient to learn the optimal pre- +dictor. In practical problem settings, these assumptions are +often not satisfied, which leads to IRM learning a sub-optimal +predictor for that task. In this work, we propose the notion of +partial invariance as a relaxation of the IRM framework. Under +our problem setting, we first highlight the sub-optimality of +the IRM solution. We then demonstrate how partitioning the +training domains, assuming access to some meta-information +about the domains, can help improve the performance of invari- +ant models via partial invariance. Finally, we conduct several +experiments, both in linear settings as well as with classifica- +tion tasks in language and images with deep models, which +verify our conclusions. +Introduction +Standard machine learning models trained using classical +Empirical Risk Minimization (ERM) can be expected to gen- +eralize well to unseen data drawn from the same distribution +as the training data (Vapnik 2013). However, distribution +shifts during test time (when data is from different sources +or under different conditions) can severely degrade model +performance (Lake et al. 2017; Marcus 2018). For instance, +in a vision task to classify camels and cows, (Beery, V. Horn, +and Perona 2018) showed that during testing, a model with +perfect training loss misclassified cows as camels at test time +when the image background was a desert. The error can be +attributed to the model picking up a strong but spurious cor- +relation: training data for most cow images included green +pastures, whereas camel images were mostly taken in deserts. +Such statistically informative but spurious correlations can +hamper performance in Out-of-Distribution (OoD) tasks and +limit applicability in real-life settings, wherein the data distri- +bution pertaining to the actual use-case almost always differs +*Corresponding author: moulikc2@illinois.edu. +from training. Thus, several lines of research explore alter- +nate learning objectives for training robust models. +One particular line of research stems from the Invariant +Causal Prediction framework (Peters, Bühlmann, and Mein- +shausen 2015), where the goal is to learn causal mechanisms +that work well under interventions; our work focuses on the +similarly inspired Invariant Risk Minimization (IRM) frame- +work, which aims to learn a predictor that relies only on fea- +tures that are invariant across all training environments. The +underlying motivation for invariance is rooted in its strong +links with causality (Pearl 2009), with the intuition being +that by invariance can help the model distinguish the causal +features from domain-specific spurious features, which it can +then discard for better generalization. +A standard assumption in such invariance-based objectives is +that of sufficiency (Ahuja et al. 2020b), in that there exists a +predictor, relying solely on invariant features, which achieves +optimal risk in all environments. In a concept drift setting +i.e. wherein the weights w.r.t. the causal features changes +across environments, the set of invariant features are clearly +insufficient and it is unclear if IRM (or similar objectives) +can achieve optimality. However, such situations often arise +in practice, for instance in language tasks spanning different +communities, in which linguistic features might have dif- +ferent connotations within different communities (Gallacher +2021; Mani, Varshney, and Pentland 2021) or in tasks with +distribution shifts across time (Luu et al. 2021). In practice +however, IRM (or a related objective) is often directly ap- +plied to the entire set of available data/training environments +(Peyrard et al. 2021; Adragna et al. 2020), without account- +ing for these factors. Thus, imposing invariance constraints +across all environments can over-constrain the predictor and +cause performance to degrade, since it is directly incentivized +to discard such non-invariant yet informative features via the +IRM learning objective. +To address this, we propose a relaxation for IRM via the +Partial invariance (P-IRM) framework, that imposes invari- +ance constraints only within partitions/sub-groups of training +environments. This increases model flexibility by allowing +learning of features that are locally invariant within the parti- +tion, without concerning about training environments outside +the partition. Naturally, the cost of finding the optimal parti- +tion in an information agnostic setting grows combinatorially +with the number of environments. However, access to meta- +arXiv:2301.12067v1 [cs.LG] 28 Jan 2023 + +information about environments can often allow us to easily +infer the ‘optimal’ training partition for a given use-case. In +doing so however, we move away from the OoD minimax +regime, and instead focus on optimality in a Bayesian sense +i.e. conditioned on this meta-information. In this work, we +first formally quantify this notion of meta-information and +then assuming access to it, we theoretically and empirically +demonstrate how the partially invariant solution can improve +performance under distribution shifts. The rest paper is orga- +nized as follows: we begin with a literature review in Related +Work, and motivate P-IRM and present our main results in +Theory. We report our empirical evaluations in Experiments +and wrap up with some concluding remarks in Discussion. +Related Work +Many approaches aim to learn deep invariant feature repre- +sentations: some focus on domain adaptation by finding a rep- +resentation whose distribution is invariant across source and +target distributions (Ben-David et al. 2010; Zhang, Gong, and +Schoelkopf 2015), while others focus on conditional domain- +invariance (Gong et al. 2016; Li et al. 2018). However, there +is evidence that domain adaption approaches are insufficient +when the test distribution may lie outside the convex hull of +training distributions (Lee and Raginsky 2018; Duchi, Glynn, +and Namkoong 2021; Mohri, Sivek, and Suresh 2019). Other +approaches include Bayesian Deep Learning (Neal 1996), +which tries to account for model uncertainty during test- +time, and Robust Optimization (Ben-Tal, El Ghaoui, and +Nemirovski 2009), which aims to generalize well to distribu- +tions close to training. +Our work focuses particularly on the IRM framework (Ar- +jovsky et al. 2019), which relates to domain generalization +wherein access to the test distribution is not assumed. IRM is +rooted in the theory of causality (Schölkopf et al. 2012) and +proposes invariance for achieving OoD generalization (Peters, +Bühlmann, and Meinshausen 2016; Heinze-Deml, Peters, and +Meinshausen 2018). In (Ahuja et al. 2020a), the authors re- +formulate IRM via a game-theoretic approach, wherein the +invariant representation corresponds to the Nash equilibrium +of a game. While the IRM framework assumes only the in- +variance of the conditional expectation of the label given the +representation, some follow-ups rely on stronger invariance +assumptions (Xie et al. 2021; Mahajan, Tople, and Sharma +2021). As mentioned before, this line of work assumes suffi- +ciency of invariant features whereas we specifically focus on +distribution shifts when sufficiency is violated. +Several follow-up works attempt to characterize IRM’s per- +formance under different settings and model assumptions. +It has been noted that carefully tuned ERM can often out- +perform state-of-the-art domain generalization approaches, +including IRM, across multiple benchmarks (Gulrajani and +Lopez-Paz 2020). The failure of IRM may stem from the +gap between the proposed framework and its practical “lin- +ear” version (IRMv1), which fails to capture natural invari- +ances (Kamath Pritish and Srebro 2021). Indeed, the authors +of (Rosenfeld, Ravikumar, and Risteski 2020) demonstrate +that a near-optimal solution to the IRMv1 objective, which +matches IRM on training environments, does no better than +ERM on environments that differ significantly from training. +Following these deficiencies, several works propose alter- +nate objectives for achieving invariance (Krueger et al. 2021; +Bellot and van der Schaar 2020; Jin, Barzilay, and Jaakkola +2020; Ahuja et al. 2021; Shui, Wang, and Gagné 2021). +However, unlike previous works that aim to improve the in- +variance learning objective, we question whether invariance +as a constraint can be improved upon for better performance. +To that end, our notion of partial invariance generalizes not +only IRM, but all similar invariance learning objectives. The +use of meta-information for invariant learning has been pro- +posed in (Lin, Zhu, and Cui 2022). However, unlike partition- +ing, the focus therein is to artificially generate environment +membership for samples when not available a priori. Finally, +a related idea appears in (Yu et al. 2022), which proposes +applying different invariance penalty weights for different +domains, but with the goal of addressing data quality variance +across domains. +Theory +In this section, we present the notion of partial invariance. +Notation: We use upper-case boldface U to denote ma- +trix/tensor/vector valued random variables, and lowercase +boldface u to denote scalar valued random variables. We use +upper-case U to denote matrices/vectors/tensors and lower- +case u to denote scalars. +Invariant Risk Minimization +The IRM setup assumes access to datasets of the form +De := {Xe +i , ye +i }ne +i=1 collected from multiple training environ- +ments e ∈ Etr. The samples in dataset De are i.i.d. from the +environment’s joint distribution, P(Xe, ye). The task is to +estimate a map f : X → Y or alternatively, the conditional +distribution P(Y |X), so that it performs well across unseen +environments Eall ⊃ Etr. Formally, the IRM framework aims +to minimize the Out-of-Distribution (OoD) risk: ROoD(f) = +maxe∈Eall Re(f), where Re(f) := EXe,ye[ℓ(f(Xe), ye)] +is the expected risk in environment e. The predictor f is +parametrized as w ◦ Φ, wherein Φ : X → Z represents the +learned representation and w : Z → Y is a linear predictor +over said representation. The IRM learning objective is posed +as a constrained optimization problem: +min +Φ,w +� +e∈Eobs +Re(w ◦ Φ) +(IRM) +s.t. w ∈ arg min +˜ +w Re′( ˜w ◦ Φ) ∀ e′ ∈ Etr. +(1) +To avoid the inner optimization, the minimization constraint +is replaced by a more tractable gradient penalty: +min +Φ,w +� +e∈Eobs +Re(w ◦ Φ) +(IRMgc) +s.t. w ∈ { ˜w : ∥∇wRe′(w ◦ Φ)∥ = 0 ∀ e′ ∈ Etr}, +(2) +where IRMgc is shorthand for the gradient constrained IRM. +In practice, this constraint is enforced via a regularizer λ: +min +Φ +� +e∈Eobs +Re(Φ) + λ∥∇w,w=1.0Re(Φ)∥, +(IRMv1) + +where the implicit overparametrization in having a separate +classifier and representation map is removed by fixing a +dummy classifier w = 1.0. Thus, Φ becomes the entire invari- +ant predictor and the strictness of the gradient norm penalty, +which enforces invariance, is via λ. Note that when λ = ∞, +IRMv1 is equivalent to IRMgc, which in turn is the first order +approximation for the true IRM objective. +An intrinsic assumption in the IRM learning setup for prov- +ing minimax optimality is the ideal scenario of sufficiency i.e. +there exists a Φ that is invariant across all e ∈ Etr and is suf- +ficient i.e ye ⊥ Xe| Φ(Xe) ∀e ∈ Eall (Ahuja et al. 2020b). +However if sufficiency is violated for an environment, one +would expect the IRM model, which relies solely on invariant +features, to be sub-optimal for that environment (compared +to a model that utilizes non-invariant features along with in- +variant ones). Such a situation may arise under concept drift, +wherein the the conditional expectation of the label ye given +“causal” features may change across environments. Thus, in +practice, if an invariant Φ that is also sufficient for environ- +ments does not exist for the desired use-case, we expect the +performance of IRM (or related frameworks) to degrade. We +illustrate this with a simple example. +Example 1 We adapt the generative model from (Arjovsky +et al. 2019): the goal is to predict target y using X = +[x1, x2, x3], in environment e such that e ∈ Eall can affect +the distribution of X as well as the conditional distribution +of y given X via a deterministic map c(e) : Eall → {−1, 1}: +x1 ← N(0, σ(e)2), x2 ← N(0, σ(e)2), +c(e) ∈ {1, −1}, ϵ ∼ N(0, σ(e)2) +y ← x1 + c(e)x2 + ϵ, ϵ ⊥ x1, ϵ ⊥ x2 +x3 ← y + N(0, 1), σ(e)2 ∈ [0, σ2 +MAX]. +We estimate y as ˆy = α1x1 + α2x2 + α3x3. Within +the IRM framework, the only feasible representation Φ +(upto scaling) that yields invariant predictors across all e is +Φ([x1, x2, x3]) = [x1, 0, 0], with corresponding regression +coefficients [1, 0, 0]. Although this minimizes the OoD error +for arbitrary e, it does so by discarding the non-invariant +but informative x2. However, if our predictor is privy to +some knowledge of c(e), we could first partition the set of +training environments Etr into two partitions, such that envi- +ronments within a partition have the same c(e) value. Then, +applying IRM within each partition yields models with bet- +ter performance that can exploit x2 as an invariant feature +in the partition. Note that this partial notion of invariance +still retains the ability to discard spurious/non-causal x3. Ad- +ditionally with partitioning, we can improve generalization +if information about c(eunseen) is available, by choosing +the right model/partition for prediction. Next, we study the +conditions under which partitioning can improve upon IRM +performance and we refer to this method as P-IRM. +Model +For our analysis, we consider a simple regression task to +succinctly capture our intuition about the conditions under +which partitioning is feasible. To begin with, we assume +access to the underlying causal features and instead, focus +on understanding the nature of the IRM solution set under +distribution shifts. In the next part, we extend this analysis to +study learning under partial invariance. +We consider the following generative model: we observe +samples (Xe +i , ye +i )in environment e and the goal is to predict +ye +i from Xe +i . Xe +i ’s are samples corresponding to the random +variable Xe ∼ P(Xe), as described below: +Xe = [xe +1, xe +2, . . . , xe +c]⊤ ∼ P(Xe), +where each xe +i denotes an individual feature. To simplify our +initial analysis, we assume that the individual features are +independent of each other and are normalized i.e. E[Xe] = 0 +and E[XeXe⊤] = I ∀ e. The target ye for given Xe can +be characterized as: +ye = ⟨W e, Xe⟩ + ϵy, +W e = [we +1, we +2, . . . , we +c] ∈ Rc, ϵy ∼ N(0, σ2 +y(e)). +(3) +where weights W e encode the conditional distribution of +observing label ye given Xe in environment e and are fixed +for that environment, and ⟨·, ·⟩ denotes the standard inner +product in Rc. For a given feature xe +i in environment e, the +corresponding feature weight we +i is independently and uni- +formly sampled from set Ai for each environment e. Once +sampled however, these weights remain fixed for that envi- +ronment. Additionally |A1| = 1, so that feature weight we +1 is +fixed and thus xe +1 is invariant for all e: +W e = [we +1, we +2, . . . , we +c], +where wi +e ∼ Unif({Ai}) ∀ i ∈ {1, 2, . . . , c}, +|A1| = 1, |Ai| > 1 ∀ i > 1. +(4) +We make note of some important aspects. As per our model, +xe +1 is the only truly invariant feature since E[ye|xe +1] = +winv.xe +1, is fixed for all e, where A1 = {winv} is a singleton, +and winv denotes the invariant feature weight. Additionally, +the cardinality of set Ai, |Ai| defines an implicit notion of +the variance of feature xi, with a higher cardinality indicat- +ing that the feature weight is more likely to change across +environments and is thus, less invariant. +With our generative model in place, we next consider the +task of predicting ye given Xe, under the mean squared +loss. Recall that the IRM framework considers predictors +of the form w ◦ Φ, where the transformation Φ extracts a +suitable representation and w is the linear predictor acting on +that representation. Due to the implicit overparametrization, +we fix w = 1.0 to a scalar value as proposed in (Arjovsky +et al. 2019) and analyze the corresponding IRM solutions +with Φ ∈ Rc. For simplicity, we ignore finite sample effects +and consider the objective in (IRMv1) when λ = ∞, or +equivalently, the gradient penalty constraint in (2) which ide- +ally approximates the true IRM objective. Additionally, we +assume the following for training environments Etr. +Assumption 1 (Sufficiency for IRM). Assume ∃ an envi- +ronment e ∈ Etr for which the truly invariant predictor +is sufficient, i.e. the corresponding feature weights satisfy +we +1 = winv and wei = 0 ∀ i ∈ {2, . . . , c}. +In other words, we assume existence of a training environ- +ment in which the invariant predictor that only recovers the + +invariant feature xe +1 achieves optimal MSE risk, which is a +standard assumption in related literature (Ahuja et al. 2020b). +Lemma 1. As per above parametrization (with w = 1.0), +under Assumption 1, the values for Φ that satisfies the IRM +solution constraints in (2) is a singleton and the value of the +corresponding predictor equates to Φ = [winv, 0, 0 . . . , 0], +the predictor only recovers the invariant feature. +The proof of the lemma is included in the Appendix and +relies on showing that any predictor which assigns non-zero +weights to any of non-invariant features would violate the +gradient penalty constraints. More importantly, the previous +lemma roughly says that any non-invariant feature will be +discarded by the IRM predictor. Note that while this is a +desirable property for minimax optimality, we ask whether +we can do better given additional contextual information. +We formalize the notion of contextual information explicitly +by defining an oracle ω(e) = 1[∥W eref − W e∥0 ≤ δ], that +provides us a notion of distance between environments, from +a fixed reference environment eref. Alternatively, it identifies +whether environment e is close to eref. +Remark 1: The choice of the ℓ0 metric for the oracle suits +our combinatorial setting, since we do make any assumptions +on the individual elements in the feature weight sets (i.e. +Ai’s). +Next we characterize our objective to utilize this information. +Suppose we know that our test environment shares the feature +weight with the reference environment for a given feature xe +i. +Then we can define the goal of minimizing the risk w.r.t. to +the predictor f, conditioned on this information: +Rcond(f) = Ee s.t. w +eref +i +=we +i Re(f), +where the expectation is over the draw of environments as per +the uniform sampling. We note that a predictor that accounts +for the prior condition (reference feature) will improve per- +formance (i.e. with a lower MSE risk Rcond), as compared to +the truly invariant predictor in the previous lemma. However, +to obtain the required feature as a feasible solution via IRM +constraints, we need to first isolate a subset of training en- +vironments Epartition ⊆ Etr such that within this set, we +i is +invariant and secondly, that we avoid learning the rest of the +non-invariant features to avoid feature weight mismatches in +unseen environments. It turns out that with access to the ora- +cle and under certain mild conditions, we can ensure exactly +that in our uniform distribution shift model. Before stating +the result, we require a similar sufficiency assumption for the +partially invariant predictor. +Assumption 2 (Sufficiency for P-IRM). Assume ∃ an envi- +ronment e ∈ Etr for which the partially invariant predictor +is sufficient, i.e. the corresponding feature weights satisfy +we +1 = winv, we +i = weref +i +and wej = 0 ∀ j ∈ {2, . . . , c}\{i}. +Theorem 1. Under the model (4), under Assumption 2, with +access to oracle ω(e) = 1[∥W eref − W e∥0 ≤ δ] and +δ < (c − 2)/2, isolate Epartition := {e ∈ Etr|ω(e) = +1} ∪ {eref} ⊆ Etr. Next, let |Ai| = k, where Ai is the set +corresponding to the feature weight weref +i +of interest. Then, +if the sets {Aj}∀ j ∈ {2, . . . , c} \ {i} satisfy |Aj| > αk +for some α > 1, we have with probability greater or equal +to ( +p +p+1)|Epartition|, where p ≥ (c−1−δ)α +δ +, the IRM solution +over set Epartition will recover the feature of interest weref +i +. +The proof, available in the Appendix, utilizes the generative +model by showing that within the partition that satisfies the +oracle condition, the probability of successfully isolating the +required feature is high. Then the result follows as a conse- +quence of Lemma 1. +In words, the theorem says that if we can identify a parti- +tion in which the environments are not too different, then +with high probability, the IRM solution will recover features +which do not vary too much (i.e. non-invariant but still close +to invariant). Note that in case of erroneous partitioning, the +solution set allowed by the non-convex penalty becomes +harder to characterize due to the presence of other feature +weights besides the reference. Nevertheless, if the conditions +are such that probability of that happening is sufficiently low, +we can safely assume that partitioning will achieve a better +expected risk. Additionally, our result suggests that P-IRM +becomes more feasible as the oracle becomes more precise +and if feature of interest is the closest to invariance. +Remark 2: While P-IRM does improve upon the IRM so- +lution, both variants are likely to be outperformed by ERM +in this setting. However, we point out that this is a simpli- +fied setting wherein access to causal features is assumed. In +more general settings when the causal features need to be +inferred from complex data, ERM may be susceptible in- +variance to confounders/anti-causal variables and thus, we +require invariance as a means to make the solution robust. +Partitioning and Partial Invariance +Next, we study P-IRM in a general setup, using previous +results to characterize the required number of training en- +vironments as in IRM. As before, we assume access to the +oracle, ω to identify the partition, i.e. Epartition ⊆ Etr. +Learning Setup: We consider the same causal mechanism +for regression task (xe, ye) from before. The goal is to find a +partition using the oracle such that a feature of interest cor- +responding to the reference environment, weref +i +is retained. +Note that since we want to retain only the invariant features +denoted as Xinv +e = [xe +1, xe +i], and discard the non-invariant +(or non-partially invariant) features, we encapsulate them into +the noise term as ˜ϵy = ϵy + (Xe +{1···c}\{1,i})⊤W e +{1···c}\{1,i}. +Then, notice that we still have ˜ϵy ⊥ Xinv +e and that E[˜ϵy] = +0, due to feature independence and centering assumptions. +Next, we consider a realistic learning setup where we observe +a scrambled version ˜ +Xe of the true causal features Xe: +ye = (Xinv +e)⊤Winv +e + ˜ϵy, ˜ϵy ⊥ Xinv +e, E[˜ϵy] = 0 +˜ +Xe = S(Xe, X′e). +(5) +Here, Xe = [Xinv +e, Xe +{1···c}\{1,i}] ∈ Rc denote the causal +features with respect to the label, X′e ∈ Rq, +˜ +Xe = +S(xe, de) ∈ Rd with S ∈ Rd×(c+q). The variable X′e +may be arbitrarily correlated with Xinv +e, ˜ϵy or the label +ye and is intended to represent the spurious correlations +in data. However, we require S to be such that ∃ ˜S s.t. +˜S(S(Xe, X′e)) = Xinv +e i.e. an inverse map such that the + +recovery of the desired features is feasible. +Next, we define γ = +1 +k +√ +2n exp(−nD(δ/n∥1/αk)), where +as before, δ is the oracle distance parameter, k is the cardinal- +ity of the set Ai, |Ai|, α is as defined in Theorem 1, n = c−2 +and D(m∥n) denotes KL divergence between Bern(m) and +Bern(n). Intuitively, γ estimates the lower bound on the +probability of sampling an environment under the genera- +tive model that satisfies the oracle condition of close distance +to the reference environment. Then we have the following +sample complexity on the number of required environments. +Theorem 2 (Informal). Assume we observe ( ˜ +Xe, ye) as +per (5), with environments e ∈ Etr sampled as per (4) +and let Epartition := {e ∈ Etr|ω(e) = 1} ∪ eref ⊆. Let +Φ ∈ Rd×d have rank r > 0. Then sampling |Etr| > 1 +γ (d − +r+d/r) log(1/ϵ) ensures partition cardinality |Epartition| > +d − r + d/r with probability > 1 − ϵ. Furthermore, if +e ∈ Epartition lie in linear general position of degree r +(Assumption 3 in Appendix), then with probability greater +than or equal to ( +p +p+1)|Epartition|, where p ≥ (c−1−δ)α +δ +, the +oracle identifies Epartition such that the predictor w ◦ Φ +learnt via IRM within that partition recovers the desired fea- +tures/weights and corresponding prediction (Xe +inv)⊤W e +inv, +∀e ∈ Eall which satisfy we +i = weref +i +. +The proof along with the formal statement is included in +the Appendix and follows from our previous results by ap- +plying concentration bounds on the draw of environments, +and subsequently using prior generalization results for IRM. +In words, Theorem 2 states that if the obtained partition is +accurate, is of sufficient cardinality and is sufficiently diverse, +then Φ recovers the partially invariant features. However, no- +tice that the required number of environments grows inversely +with γ, meaning that we need stronger priors (i.e. sample en- +vironments close to the reference) to obtain feasible sample +complexities in the number of required environments. +Partial Invariance in Practice +Next, we state the P-IRM objective more formally. We first +assume a distance metric d between environments (known +directly or via contextual information). Then, our goal is to +identify a subset of training environments Epartition ⊆ Etr +such that its average distance w.r.t. a reference environment +eref roughly satisfies: +1 +|Epartition| +� +e∈Epartition +d(e, eref) < +1 +|Etr| +� +e∈Etr +d(e, eref). +Thus, the predictor is trained on a subset of observed environ- +ments. However, discarding environments is not data-efficient +and can lead to lower fidelity and worse generalization, espe- +cially in high-complexity models. To avoid this, we introduce +the notion of conditional invariance as an alternative. For- +mally, consider the set of observed training environments Etr +and a subset corresponding to the partition Epartition (chosen +suitably via d), satisfying Epartition ⊆ Etr. We propose the +following two variants of P-IRM: +min +Φ,w +� +e∈E1 +Re(w ◦ Φ) s.t. w ∈ arg min +˜ +w Re′( ˜w ◦ Φ) ∀ e′ ∈ E2, +if E1 = E2 = Epartition, +(P-IRM (Partitioning)) +if E1 = Etr & E2 = Epartition +(P-IRM (Conditioning)) +where the empirical risk minimization objective is over envi- +ronments in E1 and the IRM invariance constraint is applied +on environments in E2. For P-IRM (Conditioning), note that +while the model uses data from all environments, the in- +variance penalty is applied only to environments within the +chosen partition, which mitigates the issue of having fewer +data samples. Intuitively, it serves as a relaxation of the IRM +objective to allow for partially invariant features. Next, we +qualitatively discuss some potential issues in the application +of P-IRM. Firstly, fulfilling the requirements as per Theorem +2, for the required worst case number of environments is +infeasible. Fortunately, in practice, IRM can pick up the re- +quired invariances from just two environments and we expect +P-IRM to overcome that issue as well. +Next, we revisit the oracle which provides the distance be- +tween environments. Note that the extraction of the set of +causal features is in itself the holy grail of machine learning, +and therefore in practice, we do not have access to this met- +ric. Additionally, note that assuming access to a prior on the +distance to the unseen environment implies that we no longer +solve for minimax optimality. However, in certain situations, +the nature of the distribution shift can be inferred via avail- +able contextual information which, while often discarded by +practictioners, can serve as an effective pseudo-metric for +the same. For instance, authors of (Luu et al. 2021) pointed +out that temporal mis-alignments of distributions in language +tasks leads to performance degradation, noting that degrada- +tion increases with an increase in the time duration between +test and train environments. Thus, learning from only the +recent past could yield a larger and more relevant set of in- +variant features for a use-case on future data. +Experiments +In this section, we first perform a basic sanity check via a +synthetic experiment, which in essence is an extension of the +example presented in the previous section. This synthetic set- +ting serves as a simple visualization of how IRM can end up +suppressing non-invariant features causal features, leading to +performance degradation. With a better understanding of the +pitfalls associated with full invariance, we then evaluate the +efficacy of the P-IRM framework (both partitioning or condi- +tioning) on four tasks: a regression task for housing price pre- +diction, an image classification task on the MetaShift dataset +(Liang and Zou 2022), an entity recognition task for scientific +texts on the SciERC dataset (Luan et al. 2018) dataset, and a +text classification task for prediction of venues of scientific +papers. Within image classification, we consider two sub- +tasks: Domain Generalization and Sub-population shifts. Due +to space constraints, we defer the synthetic experiment on +IRM, along with the text classification and Sub-population +shift tasks to the Appendix. +For comparison against other learning algorithms aside from + +IRM, we evaluate the results for standard ERM as well In- +formation Bottleneck IRM (IB_IRM) (Ahuja et al. 2021). +Apart from these benchmarks, we also include additional +experiments in the image and language classification tasks +to empirically characterize the effect of partitioning on ERM +and IB_IRM, which we dub as P-ERM and P-IB_IRM re- +spectively. +A common underlying thread for our choice of experiments +is that for each of the selected tasks, we have access to meta- +information that allow us to estimate a notion of distance +or similarity between environments, which P-IRM can then +exploit to construct the required partitions. Specifically, in +both housing price prediction and entity recognition task, +our environments are partitioned across time and due to dis- +tribution shifts, we expect environments closer in time to +have higher similarity. Similarly in MetaShift, meta-labels +for each image is made available within the data-set, that +allows an explicit notion of the distance between training and +testing environments. In all our experiments we employ the +train-domain validation strategy(Gulrajani and Lopez-Paz +2020) for hyper-parameter tuning. The code is available at +https://github.com/IbtihalFerwana/pirm and other implemen- +tation details are deferred to Appendix. +Linear Regression +We consider a regression task to predict house prices based +on house features 1, built across years [1910-2010]. Each data +point consists of 79 predictive features (for instance, num- +ber of bedrooms or house area) and a corresponding target, +which is the house price. As pre-processing step, we drop all +non-numerical features, dropping samples with missing val- +ues and normalizing each feature and the price label to zero +mean, unit variance and the samples, {Xi, yi}i ∈ (R32 ×R). +Experiment Setup To adapt this task to OoD prediction, +following (Lin, Zhu, and Cui 2022), we manually split the +training data-set into 10-year segments and use the house +year built as a meta-data for partitioning, with the intuition +being that factors affecting house prices change over time +with societal perceptions. +For prediction, we consider a linear regression model for +the task. Since the IRM framework to learn w ◦ Φ is inher- +ently overparametrized, we fix w = 1.0 ∈ R and we con- +sider Φ ∈ R32 (prediction (Φ⊤X)) with the Adam optimizer +(Kingma and Ba 2015). We consider 6 training environments +corresponding to years [1910-1970], while the test samples +draw from 4 OoD environments [1970-2010]. We expect par- +titions closer to our test set to yield better predictors. +Results We report the test MSE error (both average and worst +group) over the set of testing OoD environments, averaged +over 5 random seeds in Table 1. We find that P-IRM sig- +nificantly improves the average and worst group OoD error +over IRM. Partitioning also benefits ERM, showing more +evidence of a distribution shift over time, as in Fig. 3 in +Appendix. Finally, note that for the two variants for P-IRM, +partitioning performs much better in this regime, where we +have more samples than parameters. In the Appendix, we +1House Prices Dataset: https://www.kaggle.com/c/house-prices- +advanced-regression-techniques +include a comparison over different partition sizes. +Model +Training Years +Avg. MSE +Worst Group MSE +ERM +1910-1970 +0.475 (0.000) +1.037 (0.000) +ERM +1930-1970 +0.431 (0.000) +0.963 (0.000) +IRM +1910-1970 +0.522 (0.015) +1.129 (0.038) +P-IRM (partitioned) +1930-1970 +0.427 (0.009) +0.873 (0.024) +P-IRM (conditioned) +1930-1970 +0.490 (0.014) +1.035 (0.034) +Table 1: Prices Shifts in Housing: Training on partitioned +data shows an improvement in model performance for both +ERM and IRM models. Testing comprises of 4 OoD environ- +ments, on houses built from years between 1970-2010. +Image Classification +We evaluate P-IRM on a binary image classification task on +the MetaShift dataset (Liang and Zou 2022). +Dataset In MetaShift dataset, each image is associated with +a set of tags that describe the image context (e.g., cat on a +rug, cat beside a chair). Thus, for each given tag (e.g. rug, +chair), there is an associated set of images and these sets can +overlap if an image has multiple tags. This structure naturally +induces a graph, with each image context Ci denotes a node +(or community) in the graph. This graph is weighted and the +weights between nodes is determined by the number of im- +ages that are shared between the communities. The weights +between each pair of communities, Ci and Cj, estimate the +similarity between two communities and are calculated us- +ing the Szymkiewicz-Simpson coefficient, which yields the +corresponding adjacency matrix G: +G(i, j) = +|Ci∩Cj| +min(|Ci|,|Cj|) +(6) +Having access to such an undirected weighted graph over +sets of images thus allows us to derive an implicit notion of +distance between the corresponding communities. +Notion of Distance To introduce partitioning, we develop +a notion of distance, which then allows us to quantify +the relatedness between training and testing environments. +These environments are assumed to be sets of communi- +ties. To estimate the distance d between any two given +nodes/communities, given that our data is structured as a +weighted graph, we can make use of the spectral embeddings +(Belkin and Niyogi 2001). Spectral embeddings are based on +graph Laplacian connectivity (Ng, Jordan, and Weiss 2001). +The graph Laplacian L is calculated by L = Ddiag − G, +where Ddiag is a diagonal degree matrix of the graph G. +The corresponding eigenvectors of L, u1, . . . , uk, computed +and normalized to form the matrix U, are the corresponding +embeddings for the graph. Once we calculate the spectral +embeddings, we measure d between communities as the eu- +clidean distance between the corresponding spectral embed- +dings of each community node. With our notion of distance, +we can partition the graph based on distances between sets of +communities. This in turn, allows us to identify a subset of +training communities which is closer to the test environment. +Experiment Setup For all our experiments, we consider the +same set of training communities as in (Liang and Zou 2022). + +The set of communities are split into two environments in +the IRM setting, and we proceed with the same split as in +(Liang and Zou 2022). To introduce partitioning, we assume +distances d between the training environments and the test +communities is known/can be estimated via the meta-labels. +For learning the P-IRM model, we consider the training envi- +ronment for IRM which is closer to the test set on average, +and split it into two sub-environments. Note that under this +split, P-IRM has access to roughly only half the training +samples compared to IRM. To remedy this, we consider addi- +tional data splits wherein we add samples from communities +in the other IRM training environment, that are close to the +test set. These additional samples amount to a percentage p +of samples in that environment, allowing P-IRM access to a +slightly larger portion of the training set. Following (Liang +and Zou 2022), we consider multiple settings by fixing the +test community to be dog(shelf) and observing performance +as the distance between dog train vs test communities, d, is +varied. The cat training set remains unchanged. +Results For all experiments, we report the binary classifica- +Experiment 1 Experiment 2 Experiment 3 Experiment 4 +d = 0.17 +d = 0.54 +d = 0.81 +d = 0.92 +Avg. Performance +ERM +0.777(0.078) +0.560(0.179) +0.493(0.119) +0.667(0.114) +0.62425 +P-ERM (p = 0) +0.823(0.045) +0.790(0.086) +0.387(0.074) +0.663(0.192) +0.66575 +P-ERM (p = 10) +0.820(0.098) +0.770(0.057) +0.493(0.141) +0.663(0.128) +0.6865 +P-ERM (p = 25) +0.867(0.050) +0.740(0.079) +0.557(0.056) +0.430(0.079) +0.6485 +IRM +0.757(0.231) +0.477(0.172) +0.757(0.110) +0.687(0.309) +0.6695 +P-IRM (p = 0) +0.960(0.050) +0.817(0.045) +0.487(0.083) +0.650(0.142) +0.7285 +P-IRM (p = 10) +0.710(0.107) +0.813(0.147) +0.727(0.087) +0.690(0.184) +0.735 +P-IRM (p = 25) +0.820(0.148) +0.742(0.138) +0.597(0.243) +0.753(0.209) +0.728 +IB_IRM +0.647(0.197) +0.740(0.171) +0.750(0.155) +0.303(0.241) +0.61 +P-IB_IRM (p = 0) +0.663(0.242) +0.643(0.137) +0.437(0.289) +0.617(0.059) +0.59 +P-IB_IRM (p = 10) 0.690(0.340) +0.790(0.070) +0.377(0.214) +0.837(0.160) +0.6735 +P-IB_IRM (p = 25) 0.613(0.386) +0.740(0.171) +0.203(0.029) +0.343(0.464) +0.47475 +Table 2: Domaing Generalization in Metashift. For each +experiment the training environments are d away from the test +community dog(shelf). The partitioned models are applied +with additional samples up to percentage p ∈ {0, 10, 25}. +Communities in training are not observed during testing +tion accuracy averaged over 3 seeds, with the randomness +solely arising from the learning algorithm. We compare the +performance of P-IRM against IRM, as well other bench- +marks and their corresponding partitioned versions in table 2. +We highlight the best performing model between each model +and its corresponding model with partitioning. In most of the +experiments, especially with higher deviation between the +training and testing data, models with partitioning tend to +perform better even with p = 0 of additional samples. +Named Entity Recognition (NER) +Distributional shifts are common in language tasks, given +that societal changes are known to influence language usage +over time. These changes are also reflected in word embed- +dings (words vectors to represent language) (Garg et al. 2018). +Within this context, we explore possible benefits arising out +of partitioning (Lazaridou et al. 2021; Luu et al. 2021). +Experiment Setup We consider the SciERC (Luan et al. +2018) dataset, which consists of CS publications from 1980 to +2016. The specific task is Named Entity Recognition, a multi- +class classification task, that labels each scientific mention in +a sentence into six possible categories (Task, Method, Evalua- +tion Metric, Material, Other-Scientific-Term, or Generic).The +training set comprises of years from 1980-2009 and we test +the model on data obtained between 2010-2016, with an in- +tention to study distribution shift over time. For creating the +training environments, we split training years into smaller +intervals, 1990-2009, 2000-2009 and 2005-2009, such that +each interval has roughly the same number of samples. For +partitioning, we consider contiguous partitions of time in- +tervals, based on the intuition that vocabularies in text have +higher overlap when closer in time (Gururangan et al. 2020). +For building the model, we train a classifier over the BERT +pretrained language model (Devlin et al. 2019). Due to high +sample complexity, we also consider the conditioned P-IRM +method that makes use of all training environments. +Results We report the classification accuracy, averaged over +3 seeds in table 3. We find that both variants of P-IRM in- +deed improve performance over IRM. Additionally, we find +that leveraging more training data using conditioned P-IRM +leads to marginally better predictors, when compared against +standard partitioning. Comparisons against IB_IRM as well +as ERM demonstrate that partitioning can improve efficacy +of other learning algorithms as well. +Model +Number of envs Training Years Testing accuracy (2010-2016) +ERM +4 +1980-2009 +0.800 (0.012) +P-ERM +3 +1990-2009 +0.804 (0.020) +P-ERM +2 +2000-2009 +0.804 (0.016) +IRM +4 +1980-2009 +0.795 (0.005) +P-IRM (partitioned) +3 +1990-2009 +0.795 (0.017) +P-IRM (partitioned) +2 +2000-2009 +0.807 (0.005) +P-IRM (conditioned) +3 +1990-2009 +0.812 (0.008) +P-IRM (conditioned) +2 +2000-2009 +0.807 (0.015) +IB_IRM +4 +1980-2009 +0.800 (0.010) +P-IB_IRM (partitioned) +3 +1990-2009 +0.800 (0.015) +P-IB_IRM (partitioned) +2 +2000-2009 +0.794 (0.015) +P-IB_IRM (conditioned) +3 +1990-2009 +0.807 (0.008) +P-IB_IRM (conditioned) +2 +2000-2009 +0.805 (0.020) +Table 3: Language Shifts in SciERC dataset. The partition- +ing improves performance for not only IRM but also other +learning objectives. Additionally, we find that the choice of +optimal partition (1990-2009) is consistent across training +algorithms. +Discussion +In this work, we propose P-IRM: a relaxation of the IRM +objective via partial invariance. Through our analysis, we +determine conditions under which P-IRM becomes feasible. +We then experimentally verify, in both linear regression and +deep learning settings across multiple domains, that when +contextual information allows to interpret a distance metric, +we indeed improve upon the IRM predictor as well as other +learning frameworks. +We note that the application of partitioning/P-IRM framework +is naturally limited by the informativeness of the available +information about training/deployment domains, which often +may not be readily available. Additionally, while distribu- +tion shifts across time allows for partitions to be contiguous +time intervals, in general, finding the appropriate partition +is non-trivial under more complex shifts. 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In Proceedings of the +29th Association for the Advancement of Artificial Intelli- +gence Conference (AAAI’15). + +Proofs +Proof of Lemma 1 +The proof follows as a consequence of the parametrization. +Under the MSE loss, note that the expected risk takes the +following form: +Re(w ◦ Φ) = Eye,Xe(ye − w.Φ⊤Xe)2, +(7) +wherein the scalar w is fixed at 1.0. The gradient penalty w.r.t. +w in environment e can then be obtained as: +∥∇w,w=1.0Re′(w ◦ Φ)∥ = |∇w,w=1.0Eye,Xe(ye − w.Φ⊤Xe)2| += |Eye,Xe[∇w,w=1.0(ye − w.Φ⊤Xe)]2| += |Eye,Xe[2(w.Φ⊤Xe − ye)Φ⊤Xe]| += |Eϵye,Xe[2(Φ − W e)⊤Xe − ϵy +e)Xe⊤Φ]| += |EXe[2(Φ − W e)⊤(XeXe⊤)Φ] + 0| += |2(Φ − W e)⊤EXe[(XeXe⊤)]Φ| += |2(Φ − W e)⊤Φ| += 2| +c +� +i=1 +(Φ2 +i − we +i .Φi)|, +(8) +wherein the simplifications follow through due to feature +independence, normalization and zero mean noise assump- +tion. Notice that as per the constraints in (2), we need the +risk penalty term above to be equal to zero for all training +environments: +| +c +� +i=1 +(Φ2 +i − we +i .Φi)| = 0 ∀ e ∈ Etr. +Then note from a risk minimization incentive, we naturally +have Φ1 = winv without incurring any penalty. Thus the +penalty boils to: +| +c +� +i=2 +(Φ2 +i − we +i .Φi)| = 0 ∀ e ∈ Etr. +But from Assumption 1, we have an environment e in the +training set in which IRM is sufficient, i.e. we +i = 0 ∀ i ̸= 1. +Thus, the constraint in that environment equates to: +| +c +� +i=2 +(Φ2 +i − 0.Φi)| = | +c +� +i=2 +Φ2 +i | = 0 ∀ e ∈ Etr. +Thus to satisfy this constraint, Φi = 0 ∀ i ̸= 1, which means +the set of feasible solutions is comprised solely of the per- +fectly invariant predictor, as claimed. +Proof of Lemma 2 +Under the uniform feature model, we sample Etr such that +|Etr| = t. Notice that the cardinality m of the required parti- +tion Epartition := {e ∈ Etr|ω(e) = 1} is random variable. +We note that for an environment sample e ∈ Etr, for a +given reference environment eref, we have from our anal- +ysis in Proof of Theorem 1 that P(∥W e − W eref ∥ ≤ +δ) = P(E1) + P(E2) and that P(E1) > pP(E2), wherein +p ≥ (c−1−δ)α +δ +>> 1. Let n = c − 2 for brevity. Then, we +have the following approximation: +P(∥W e − W eref ∥ ≤ δ) ≈ P(E1) +≥ γ = +1 +k +√ +2n exp(−nD(δ/n∥1/αk)), +where +the +final +result +follows +from +standard +anti- +concentration bounds on a Binomial distribution. Then it is +easy to see that given |Etr| = t, P(|Epartition| ≥ m) = +P(| � +e∈Etr 1[ω(e) = 1] ≥ m)] ≥ P(�t +j=1 Dj ≥ m] +where Dj is Bernoulli random variable distributed as Dj ∼ +Bern(γ). +On the RHS, we get a sum of i.i.d. Bernoulli variables for +and is thus, �t +j=1 Dj ∼ Bin(n, γ). Let M = �t +j=1 Dj and +notice that we need that M > m with high probability. To +achieve this, we first upper bound the probability of event +M < m using Chernoff’s tail bound and derive conditions +under which this upper bound is small. Specifically: +P(M < m) < exp(−tD +�m +t ∥γ +� +) < ϵ, +(9) +where D(a∥b) = a log( a +b )+(1−a) log( 1−a +1−b ). Next, assume +that t = Cm. Then we can simplify the inequality: +−Cm +� 1 +C log +1 +γ +C + C − 1 +C +log +(C − 1) 1 +γ +C( 1 +γ − 1) +� +< log(ϵ) +(10a) +=> +� +log +1 +γ +C + (C − 1) log +(C − 1) 1 +γ +C( 1 +γ − 1) +� +> 1/m log(1/ϵ) +(10b) +=> +� +C log +(C − 1) 1 +γ +C( 1 +γ − 1) − log C − 1 +1 +γ − 1 +� +> 1/m log(1/ϵ) +(10c) +≍ +� +C log +(C − 1) 1 +γ +C( 1 +γ − 1) +� +> 1/m log(1/ϵ). +(10d) +Our inequality will be satisfied if a) C > 1 +γ log(1/ϵ) and b) +log( +(C−1) 1 +γ +C( 1 +γ −1)) > γm. Then we can show that for sufficiently +large 1 +γ m, condition b) roughly amounts C > ci(1 + +1 +m−1). +So if C > max{ 1 +γ log(1/ϵ), 1 +γ (1 + +1 +m−1} ∼ 1 +γ log(1/ϵ) (for +small ϵ), then we have P(M < m) < ϵ. Hence, for t = +Cm ∼ 1 +γ m log(1/ϵ), we get that M = �t +j=1 Dj with high +probability. But note that we already have P(|Epartition| ≥ +m) ≥ P(�t +j=1 Dj ≥ m]. Thus, P(|Epartition| ≥ m) > +1 − ϵ. + +Proof of Theorem 1 +We begin by restating the theorem. +Theorem. 1 Under the model (4), under Assumption 2, with +access to oracle ω(e) = 1[∥W eref − W e∥0 ≤ δ] and +δ < (c − 2)/2, isolate Epartition := {e ∈ Etr|ω(e) = +1} ∪ {eref} ⊆ Etr. Next, let |Ai| = k, where Ai is the set +corresponding to the feature weight weref +i +of interest. Then, +if the sets {Aj}∀ j ∈ {2, . . . , c} \ {i} satisfy |Aj| > αk +for some α > 1, we have with probability greater or equal +to ( +p +p+1)|Epartition|, where p ≥ (c−1−δ)α +δ +, the IRM solution +over set Epartition will recover the feature of interest weref +i +. +The proof sketch is as follows. We first characterize the prob- +ability of error in using the oracle as an indicator for the +partition membership. Assuming the partition is identified, +we can then directly apply Lemma 1 to obtain the desired +result. We begin by characterizing the two possible cases that +arise when oracle ω(e) = 1. +• E1: If feature at level i remains unchanged, this means +that out of c − 2 features (discarding the feature at level i +and the invariant feature at level 1), a maximum of up-to +δ features changed in the worst case. +• E2: If feature at level i changed, this means that out of +c − 2 features (discarding the feature at level i and the +invariant feature at level 1), a maximum of up-to δ − 1 +features changed in the worst case. +Note that both of the events can be modelled as a sum of +Bernoulli random variables with different probabilities of +success. Consider Bj ∼ Bern(1 − 1/|Aj|), which indicates +whether the feature value at level j changed under the uniform +model. In the first case, we model the conditional probability +as: +P(E1) = P(we +i = weref +i +|ω(e) = 1) += (1/k)P(1 ≤ +c +� +j=2,j̸=i +Bj ≤ δ), +(11) +while in the alternate case, +P(E2) = P(we +i ̸= weref +i +|ω(e) = 1) += (1 − 1/k)P(0 ≤ +c +� +j=2,j̸=i +Bj ≤ δ − 1). +(12) +Next, we make use of the assumption on cardinalities and +analyze the probabilities corresponding to each term. We note +that: +P(E1) = +δ +� +m=1 +(1/k)P( +c +� +j=2,j̸=i +Bj = m). +Similarly: +P(E2) = +δ−1 +� +m=0 +(1 − 1/k)P( +c +� +j=2,j̸=i +Bj = m) +For brevity, we represent each individual term on the +right hand side of the summations as Pa,m (a += +1 or 2 depending on the event), corresponding to that value +of m. Next, we analyze the ratio for the final two terms in +either sequence. Let c − 2 = n and note that using the as- +sumption on cardinalities of the feature sets, we can bound +the ratio as follows: +P1,δ +P2,δ−1 +≥ +(1/k)( +�n +δ +� +(1 − 1/αk)δ(1/αk)n−δ) +(1 − 1/k)( +� n +δ−1 +� +(1 − 1/αk)δ−1(1/αk)n+1−δ), +wherein the greater than equal to sign holds since cardinality +of each feature set is greater than or equal to αk. Simplifying +the terms, we get: +P1,δ +P2,δ−1 +≥ (n + 1 − δ)(αk − 1) +(k − 1)(δ) +≥ (c − 1 − δ)α +δ += p. +Note that this ratio increases as δ reduces and therefore, we +conclude that: +P(E1)/P(E2) = +�δ +m=1 P1,m +�δ−1 +m=1 P2,m +≥ p. +Thus, we obtain that Perror = P(E2) ≤ 1/(p + 1). Thus, +for our training partition Epartition = {e ∈ Etr|ω(e) = 1}, +with probability greater than ( +p +p+1)|Epartition|, we will learn +an accurate partition. Since within this partition the feature +weight corresponding to xi is same as the reference for all +environments, we obtain the required result from Lemma 1. + +Proof of Theorem 2 +First, we restate the theorem more formally. +Theorem. 2 (Formal) Assume we observe samples ( ˜ +Xe, ye) +as per equation (5), with environments e ∈ Etr sampled as +per equation (4) and let Epartition := {e ∈ Etr|ω(e) = +1}. Let Φ ∈ Rd×d have rank r. Then sampling |Etr| > +1 +γ (d − r + d/r) log(1/ϵ) ensures that partition cardinality +|Epartition| > d − r + d/r with probability > 1 − ϵ. Further- +more, if e ∈ Etr lie in linear general position of degree r +(Assumption 3), then with probability greater than or equal to +( +p +p+1)|Epartition|, where p ≥ (c−1−δ)α +δ +, the oracle identifies +Epartition such that we have: +ΦE ˜ +Xe[ ˜ +Xe( ˜ +Xe)⊤]Φ⊤W = ΦE ˜ +Xe,ye[ ˜ +Xeye], +holds for all e ∈ Epartition iff Φ elicits an invariant predictor +Φ⊤W ∀ e ∈ Eall whose feature weights satisfy we +i = weref +i +. +We begin by recollecting the requisite tools from (Arjovsky +et al. 2019). +Assumption 3. With observation model as in equation (5), +a set of training environments Epartition ⊆ Etr lie in linear +general position of degree r if |Epartition| > d − r + d/r for +some r ∈ N, r < d, and for all non-zero X ∈ Rd: +dim +� +span +�� +E ˜ +Xe[ ˜ +Xe( ˜ +Xe)⊤]X − E ˜ +Xe,˜ϵy[ ˜ +Xe˜ϵy] +� +e∈Epartition +�� +> d − r. +Intuitively, this assumption states that we require the training +environments in our partition partition Etr to be sufficiently +diverse, with limited co-linearity. +Next, recall that in our setup, instead of directly observing the +training partition for a given level i and feature weight v, we +need to identify the a subset Epartition from the available set +of training environments Etr. Thus, for the training partition +to be at least of size m, we need certain conditions on |Etr|. +We characterize this in the following result. +Lemma 2. Under environment sampling as per equation +(4), if the cardinality of observed environments |Etr| = n ∼ +1 +γ m log(1/ϵ), for the subset that satisfies the oracle distance +condition i.e. Epartition := {e ∈ Etr|ω(e) = 1}, we have +that P(|Epartition| ≥ m) > 1 − ϵ. +Having obtained the correct partitioning with high probability, +we call upon the out of domain generalization result from +(Arjovsky et al. 2019). +Proposition 1 (Theorem 9 in (Arjovsky et al. 2019)). Assume +that +Y e = (Ze +1)⊤β + ϵe, ϵe ⊥ Ze +1, E[ϵe] = 0 +Xe = S(Ze +1, Ze +2). +(13) +Here, β ∈ Rc, Ze +1 takes values in Rc, Ze +2 takes values +in R1 and S ∈ Rd×(c+q). Assume that the Z1 component +of S is invertible: that there exists ˜S ∈ Rc×d such that +˜S(S(z1, z2)) = z1, for all z1 ∈ Rc, z2 ∈ Rq. Let Φ ∈ Rd×d +have rank r. Then, if atleast d − r + d/r training environ- +ments Etr ⊆ Eall lie in linear general position of degree r, +then we have: +ΦEXe[Xe(Xe)⊤]Φ⊤w = ΦEXe,Y e[XeY e], +holds for ∀e ∈ Etr iff Φ elicits an invariant predictor Φ⊤w +for all e ∈ Eall. +We now provide the proof of Theorem 2. First, let m = +d − r + d/r. Then from Lemma 2, we know that sampling +n ∼ 1 +γ m log(1/δ) environments gives us |Epartition| ≥ m +with high probability. From Assumption 3, we also have +that environments in Epartition lie in linear general position +of degree r. Finally, note that our scheme is contingent on +the oracle correctly identifying the required partition, which +happens with probability greater than ( +p +p+1)|Epartition|, as +noted in Proof of Theorem 1. Armed with these, we can +apply Proposition 1 to our learning setup in equation (5) to +learn an predictor Φ, W that can recover features Xe +inv and +corresponding desired weights W e +inv ∀e ∈ Eall which satisfy +we +i = weref +i +. + +Additional Experiments +In all our experiments when implementing IRM/P-IRM, we +keep the penalty parameter sufficiently high λ = 102/103. +The rationale for this is to have λ high enough so that the +invariance penalty term dominates the fidelity loss term and +features are close to invariant. +Synthetic Experiment +The experimental setting is adapted from (Arjovsky et al. +2019). We assume the following generative model: +X1 ← N(0, e2I), X2 ← N(0, e2I), +c(e) ∈ {0, 1}, ϵ ∼ N(0, e2I) +y ← XT +1 W1 + XT +2 (c(e)W2) + ϵ, ϵ ⊥ X1, X2 +The task is to predict target y ∈ R based on observed +X ∈ R20, where X = (X1 ∈ R10, X2 ∈ R10). The predic- +tor w ◦ Φ is learnt via the IRM objective and as noted previ- +ously, the overparametrization in the objective is handled by +parametrizing Φ ∈ R20 to be rank 1 and fixing w = 1.0 as a +scalar. The true weights W1, W2 ∈ R10 are fixed Gaussian +entries, but the sampling of c(e) for different environments +controls whether X2 is a causal feature for y. +The goal is to visualize the intuition in Lemma 1 i.e. how +IRM can discard causal features which are non-invariant. To +that end, we sample 1000 data points from four environments +characterized by e ∈ {0.2, 1, 2, 5}. The c(e) for each envi- +ronment is assigned uniformly, such that the final training set +has two environments for each of {0, 1}, and feature X2 is +non-invariant. The learning procedure for IRM is consistent +with the rest of the paper, with λ = 103 and the initial 4000 +epochs for annealing the IRM loss. +To study learning of each feature, first denote Φ = (Φ1, Φ2), +Φ1, Φ2 ∈ R10. Then note that Φi captures the contribution of +feature Xi in the prediction. We then look at ∥Φi∥ +∥Wi∥ (averaged +over the random draw of environment weights). Intuitively, +this ratio indicates the information captured by the IRM pre- +dictor for that feature. We visualize the results in Fig. 1, which +demonstrates the tendency for IRM to suppress learning of +non-invariant features. +Figure 1: The plot demonstrates that IRM is incentivized to +suppress non-invariant features, as is the case for feature_2. +Linear Regression +For our choice of learning rate, number of iterations and +optimizer and annealing iterations, we refer to ((Lin, Zhu, +and Cui 2022)). While the reported results were for λ = 102, +we verified similar trends for λ = 103. +Hyperparameter +Values +Number of Iterations +4000 +Learning rate +10−3 +Optimizer +Adam +IRM Penalty +102 +Annealing Iterations +2000 +Table 4: Hyperparameters for experiments on the housing +dataset, following (Lin, Zhu, and Cui 2022) +. +Figure 2: Partitioning vs Conditioning: In the under- +parametrized linear regression setting where the number of +data points is much greater than learnable parameters, condi- +tioning is not helpful in terms of improving P-IRM accuracy +and partitioning consistently performs better. +Figure 3: For the regression experiment in the main paper, +we find that for both ERM and IRM, there exists an optimal +partition. Note that while ERM consistently finds the unique +optima, the IRM solution has some variance due to the non- +convex objective. + +Demonstration of iRM supressing non-invariant features +Ratio of Feature Weight Norm (Learnt over Actual) +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +feature 1 (invariant) +feature 2 (non-invariant)AverageTestError([1970-2010oj)foriRMvariantsforLinearRegression +0.54 +IRM(Conditioned) +IRM(Partitioned) +0.52 +0.50 +Error +0.48 +MSE +0.46 +0.44 +0.42 +20 +30 +40 +50 +60 +TrainingPartitionsSize(inyears))0.54 +IRM +ERM +0.52 +0.50 +rror +E +0.48 +MSE +0.46 +0.44 +0.42 +20 +30 +40 +50 +60 +TrainingPartitions Size (inyears))Image Classification +For the image classification experiments on MetaShift (Liang +and Zou 2022) dataset, we follow a similar training pipeline +as in (Gulrajani and Lopez-Paz 2020; Liang and Zou 2022). +Following (Gulrajani and Lopez-Paz 2020), we consider +ResNet-50 (He et al. 2016), since larger ResNets are known +to generalize better. ResNet-50 was pre-trained on ImageNet +(He et al. 2016), and for domain generalization, the batch nor- +malization and final softmax layers of ResNet are chopped +off (Gulrajani and Lopez-Paz 2020). Then ResNet-50 layers +are followed by non-linear functions, e.g. ReLU, and a final +dropout layer (Gulrajani and Lopez-Paz 2020). +Domain Generalization +Herein, we report the relevant +choice of hyperparameters in Table 5, to reproduce our re- +sults pertaining to the domain generalization experiment in +the main paper. +Model +IRM Penalty +weight +IRM +An- +nealing +Iterations +IB_IRM +Penalty +weight +IB_IRM an- +nealing itera- +tions +Experiment 1 +IRM +10 +20 +P-IRM (p=0) +10 +20 +P-IRM (p=10) +10 +40 +P-IRM (p=25) +10 +20 +IB_IRM +10 +40 +10 +20 +P-IB_IRM (p=0) +10 +20 +10 +40 +P-IB_IRM (p=10) +100 +20 +10 +20 +P-IB_IRM (p=25) +10 +40 +10 +20 +Experiment 2 +IRM +10 +20 +P-IRM (p=0) +10 +20 +P-IRM (p=10) +10 +20 +P-IRM (p=25) +10 +20 +IB_IRM +10 +20 +10 +20 +P-IB_IRM (p=0) +10 +20 +10 +20 +P-IB_IRM (p=10) +10 +20 +10 +20 +P-IB_IRM (p=25) +10 +20 +10 +20 +Experiment 3 +IRM +10 +40 +P-IRM (p=0) +10 +40 +P-IRM (p=10) +10 +40 +P-IRM (p=25) +10 +40 +IB_IRM +10 +20 +10 +20 +P-IB_IRM (p=0) +10 +20 +10 +40 +P-IB_IRM (p=10) +10 +20 +10 +20 +P-IB_IRM (p=25) +10 +20 +10 +20 +Experiment 4 +IRM +10 +20 +P-IRM (p=0) +10 +40 +P-IRM (p=10) +10 +20 +P-IRM (p=25) +10 +20 +IB_IRM +10 +20 +10 +20 +P-IB_IRM (p=0) +10 +20 +10 +20 +P-IB_IRM (p=10) +10 +40 +10 +20 +P-IB_IRM (p=25) +10 +40 +10 +20 +Table 5: Domain Generalization in Metashift: IRM and +IB_IRM hyperparameters obtained via model selection +Subpopulation Shift +Following our setup described in Im- +age Classification setting, we conduct further experiments to +study performance under subpopulation shifts for the binary +classification task. In subpopulation shifts, communities used +for training and testing are the same, but their relative propor- +tions differ between training and testing environments, with +certain groups often subject to under-representation. The goal +is to obtain a model to do well even for minority groups in +the training data (Koh et al. 2021). +Following (Liang and Zou 2022), the communities are +grouped into two environments: indoor and outdoor. In train- +ing, cat(outdoor) and dog(indoor) subsets are the minority +groups, while cat(indoor) and dog(outdoor) are majority +groups. We vary the percentage of minority groups within +the training set to be m ∈ {0.12, 0.01} of the total training +set, and we keep the size of the training set fixed with 1700 +samples. We use a balanced set of testing by equally sampling +from each environment with balanced labels. Tables 6 and 7 +show that the results under subpopulation shift settings. We +report the average accuracy over the four groups, and worst +group accuracy (the group with the worst performace), and +the average minority accuracy which is the average of mi- +nority groups in training i.e. cat(outdoor) and dog(indoor). +On the more challenging setting of m = 0.01 where minority +groups are observed minimally, P-IRM models achieve better +worst group and average minority performance. However as +expected, it becomes harder to improve performance for par- +titioned models as we lower the amount of available training +data, and it is best rely on IRM/ERM. +For reproducibility of results, Table 8 shows the selected +hyperparameters. +m = 0.12 +Avg. Acc. +Worst Group Acc. +Avg. Minority Acc. +ERM +0.816(0.209) +0.722(0.103) +0.737(0.021) +P-ERM (p = 0) +0.78(0.129) +0.623(0.157) +0.675(0.074) +P-ERM (p = 10) +0.76(0.189) +0.526(0.089) +0.616(0.127) +P-ERM (p = 25) +0.779(0.154) +0.590(0.043) +0.736(0.206) +IRM +0.638(0.250) +0.336(0.160) +0.558(0.314) +P-IRM (p = 0) +0.703(0.158) +0.560(0.230) +0.5705(0.015) +P-IRM (p = 10) +0.681(0.190) +0.518(0.143) +0.663(0.205) +P-IRM (p = 25) +0.737(0.147) +0.604(0.227) +0.7335(0.183) +IB_IRM +0.639(0.209) +0.380(0.082) +0.47(0.127) +P-IB_IRM (p = 0) +0.587(0.136) +0.451(0.234) +0.4695(0.026) +P-IB_IRM (p = 10) +0.613(0.301) +0.264(0.137) +0.5635(0.424) +P-IB_IRM (p = 25) +0.596(0.178) +0.426(0.115) +0.448(0.031) +Table 6: Subpopulation shift on Metashift The value m +represents the portion of minority groups within a training +environment. The partitioned models are applied with addi- +tional samples up to percentage p ∈ {0, 10, 25} + +m = 0.01 +Avg. Acc. +Worst Group Acc. +Avg. Minority Acc. +ERM +0.744(0.209) +0.514(0.113) +0.559(0.064) +P-ERM (p = 0) +0.747(0.186) +0.574(0.112) +0.587(0.018) +P-ERM (p = 10) +0.729(0.254) +0.481(0.133) +0.51(0.041) +P-ERM (p = 25) +0.745(0.248) +0.488(0.013) +0.532(0.062) +IRM +0.725(0.185) +0.509(0.216) +0.5775(0.097) +P-IRM (p = 0) +0.677(0.232) +0.426(0.116) +0.484(0.082) +P-IRM (p = 10) +0.629(0.277) +0.341(0.260) +0.6355(0.269) +P-IRM (p = 25) +0.687(0.181) +0.525(0.189) +0.531(0.008) +IB_IRM +0.269(0.179) +0.470(0.379) +0.5555(0.121) +P-IB_IRM (p = 0) +0.56(0.27) +0.208(0.102) +0.3955(0.265) +P-IB_IRM (p = 10) +0.556(0.206) +0.366(0.108) +0.4805(0.162) +P-IB_IRM (p = 25) +0.568(0.138) +0.398(0.412) +0.583(0.092) +Table 7: Subpopulation shift on Metashift The value m +represents the portion of minority groups within a training +environment. The partitioned models are applied with addi- +tional samples up to percentage p ∈ 0, 10, 25 +Model +IRM Penalty +weight +IRM +An- +nealing +Iterations +IB_IRM +Penalty +weight +IB_IRM an- +nealing itera- +tions +Experiment 1 (m = 0.12 ) +IRM +10 +40 +P-IRM (p = 0) +10 +40 | 20 +P-IRM (p = 10) +10 +40 | 20 +P-IRM (p = 25) +10 +20 | 40 +IB_IRM +10 +40 +10 +20 +P-IB_IRM (p = 0) +10 +20 +10 +20 +P-IB_IRM (p = 10) +10 +20 | 40 +10 +40 +P-IB_IRM (p = 25) +10 +20 +10 +20 | 40 +Experiment 2 (m = 0.01 ) +IRM +10 +40 +P-IRM (p = 0) +1000 | 10 +40 +P-IRM (p = 10) +100 | 10 +40 | 20 +P-IRM (p = 25) +100 | 10 +20 | 40 +IB_IRM +1000 +40 +10 +40 +P-IB_IRM (p = 0) +10 +40 | 20 +100 | 10 +20 +P-IB_IRM (p = 10) +100 | 10 +20 +100 | 10 +40 | 20 +P-IB_IRM (p = 25) +1000 | 10 +20 +100 | 10 +40 | 20 +Table 8: Subpopulation shift in Metashift: IRM and +IB_IRM hyperparameters based on model selection. (For +partitioned models since we are using two models, we are +reporting the parameters for both, if they are the same we +report one value only) + +Language Experiments +For language experiments, NER and TC, we build a classifier +based on the pre-trained language model BERT (Devlin et al. +2019), followed by a dropout and a linear layer. We also con- +sidered DistillBERT (Sanh et al. 2019) and GPT-2 (Radford +et al. 2019),but found that BERT-based models outperformed +other networks. We train the models for the maximum num- +ber of iterations, (details in tables 9, and 11) for one seed. +Then we select the best number of iterations to apply for +other seeds. The results are the average of three seeds. +Named Entity Recognition (NER) +For the language NER +experiments, the best hyperparameter values are reported in +Table 9, and Table 10 which have been selected based on the +best model performance on the validation set. The training +was done for 80 epochs, around which both training and in- +domain validation losses stabilize and remain the same. For +annealing epochs, we considered [10, 20, 30, 35, 40] epochs +and found that for all variants of P-IRM/IRM, 40 epochs +yielded best performance. The optimizer and learning rate +was based on standard choice for using pre-trained BERT +models. +Hyperparameter +Values +Maximum Number of epochs +80 +Batch size +8 +Learning rate +10−6 +Optimizer +Adam +Number of GPUs +4 +Table 9: Hyperparameters choices for experiments on the +NER dataset +Model +# envs +IRM +Penalty +weight +IRM +an- +nealing +iterations +IB_IRM +Penalty +weight +IB_IRM an- +nealing itera- +tions +#epochs +ERM +4 +44 +P-ERM +3 +54 +P-ERM +4 +58 +IRM +4 +1000 +30 +53 +P-IRM (partitioned) +3 +1000 +40 +70 +P-IRM (partitioned) +2 +1000 +40 +76 +P-IRM (conditioned) +3 +100 +40 +64 +P-IRM (conditioned) +2 +100 +30 +66 +IB_IRM +4 +100 +40 +1 +40 +57 +P-IB_IRM (partitioned) +3 +100 +40 +1 +40 +77 +P-IB_IRM (partitioned) +2 +100 +40 +1 +40 +76 +P-IB_IRM (conditioned) 3 +100 +40 +1 +40 +76 +P-IB_IRM (conditioned) 2 +100 +40 +1 +40 +59 +Table 10: NER dataset: Best IRM hyperparameters values +selected based on early stopping on validation data +Hyperparameter +Values +Maximum Number of epochs +40 +Batch size +8 +Learning rate +10−6 +Optimizer +Adam +Number of GPUs +4 +Table 11: Hyperparameters choices for the Text Classification +task +Model +# envs +IRM Penalty +weight +IRM +An- +nealing +Iterations +IB_IRM +Penalty +weight +IB_IRM an- +nealing itera- +tions +# epochs +ERM +4 +22 +P-ERM +3 +39 +P-ERM +4 +38 +IRM +4 +1000 +20 +37 +P-IRM (parti- +tioned) +3 +1000 +20 +36 +P-IRM (parti- +tioned) +2 +1000 +20 +37 +P-IRM (condi- +tioned) +3 +1000 +20 +33 +P-IRM (condi- +tioned) +2 +1000 +20 +33 +IB_IRM +4 +1000 +20 +0.1 +20 +37 +P-IB_IRM +(partitioned) +3 +1000 +20 +0.1 +20 +36 +P-IB_IRM +(partitioned) +2 +1000 +20 +0.1 +20 +39 +P-IB_IRM +(conditioned) +3 +1000 +20 +0.1 +20 +33 +P-IB_IRM +(conditioned) +2 +1000 +20 +0.1 +20 +33 +Table 12: TC dataset: Best IRM hyperparameters values se- +lected based on early stopping on validation data +Text Classification (TC) +We consider another language +classification task, which identifies the venue of a published +paper 2, selecting AAAI and ICML conferences for classifi- +cation. This task represents a topic classification task. Our +temporal partitions are: 2006-2008, 2009-2011, 2012-2014, +and 2015-2017 and we test on papers published between +2018-2020. The model selection follows as before, training +for a maximum of 40 epochs. The final hyperparameters are +reported in Table 12 and 11. Table 13 shows how partition- +ing outperforms their baseline methods. P-IRM with two +environments performed the best among all other models. +Model +Number of +envs +Training +Years +Testing accuracy (2018-2020) +ERM +4 +2006-2017 +0.862 (0.008) +P-ERM +3 +2009-2017 +0.862 (0.004) +P-ERM +2 +2012-2017 +0.875 (0.014) +IRM +4 +2006-2017 +0.846 (0.013) +P-IRM (paritioned) +3 +2009-2017 +0.862 (0.008) +P-IRM (paritioned) +2 +2012-2017 +0.882 (0.016) +P-IRM (conditioned) +3 +2009-2017 +0.869 (0.007) +P-IRM (conditioned) +2 +2012-2017 +0.853 (0.010) +IB_IRM +4 +2006-2017 +0.846 (0.014) +P-IB_IRM (partitioned) +3 +2009-2017 +0.868 (0.001) +P-IB_IRM (partitioned) +2 +2012-2017 +0.874 (0.016) +P-IB_IRM (conditioned) +3 +2009-2017 +0.860 (0.016) +P-IB_IRM (conditioned) +2 +2012-2017 +0.862 (0.011) +Table 13: Results on text classification, comparison between +ERM, IRM, IB_IRM and their partitioned variants. +2https://www.semanticscholar.org/product/api + diff --git a/gNFLT4oBgHgl3EQfYy_o/content/tmp_files/load_file.txt b/gNFLT4oBgHgl3EQfYy_o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..52210d354b27aed82a1928ddfcf742ce9d5bab2b --- /dev/null +++ b/gNFLT4oBgHgl3EQfYy_o/content/tmp_files/load_file.txt @@ -0,0 +1,1879 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf,len=1878 +page_content='Learning Optimal Features via Partial Invariance Moulik Choraria1*, Ibtihal Ferwana1, Ankur Mani2, Lav R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Varshney1 1University of Illinois at Urbana-Champaign, 2 University of Minnesota, Twin Cities Abstract Learning models that are robust to test-time distribution shifts is a key concern in domain generalization, and in the wider context of their real-life applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Invariant Risk Mini- mization (IRM) is one particular framework that aims to learn deep invariant features from multiple domains, and has sub- sequently led to further variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' A key assumption for the success of these methods requires that the underlying causal mechanisms/features remain invariant across domains and the true invariant features be sufficient to learn the optimal pre- dictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In practical problem settings, these assumptions are often not satisfied, which leads to IRM learning a sub-optimal predictor for that task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In this work, we propose the notion of partial invariance as a relaxation of the IRM framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Under our problem setting, we first highlight the sub-optimality of the IRM solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We then demonstrate how partitioning the training domains, assuming access to some meta-information about the domains, can help improve the performance of invari- ant models via partial invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Finally, we conduct several experiments, both in linear settings as well as with classifica- tion tasks in language and images with deep models, which verify our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Introduction Standard machine learning models trained using classical Empirical Risk Minimization (ERM) can be expected to gen- eralize well to unseen data drawn from the same distribution as the training data (Vapnik 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' However, distribution shifts during test time (when data is from different sources or under different conditions) can severely degrade model performance (Lake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Marcus 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' For instance, in a vision task to classify camels and cows, (Beery, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Horn, and Perona 2018) showed that during testing, a model with perfect training loss misclassified cows as camels at test time when the image background was a desert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The error can be attributed to the model picking up a strong but spurious cor- relation: training data for most cow images included green pastures, whereas camel images were mostly taken in deserts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Such statistically informative but spurious correlations can hamper performance in Out-of-Distribution (OoD) tasks and limit applicability in real-life settings, wherein the data distri- bution pertaining to the actual use-case almost always differs Corresponding author: moulikc2@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' from training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Thus, several lines of research explore alter- nate learning objectives for training robust models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' One particular line of research stems from the Invariant Causal Prediction framework (Peters, Bühlmann, and Mein- shausen 2015), where the goal is to learn causal mechanisms that work well under interventions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' our work focuses on the similarly inspired Invariant Risk Minimization (IRM) frame- work, which aims to learn a predictor that relies only on fea- tures that are invariant across all training environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The underlying motivation for invariance is rooted in its strong links with causality (Pearl 2009), with the intuition being that by invariance can help the model distinguish the causal features from domain-specific spurious features, which it can then discard for better generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' A standard assumption in such invariance-based objectives is that of sufficiency (Ahuja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2020b), in that there exists a predictor, relying solely on invariant features, which achieves optimal risk in all environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In a concept drift setting i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' wherein the weights w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' the causal features changes across environments, the set of invariant features are clearly insufficient and it is unclear if IRM (or similar objectives) can achieve optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' However, such situations often arise in practice, for instance in language tasks spanning different communities, in which linguistic features might have dif- ferent connotations within different communities (Gallacher 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Mani, Varshney, and Pentland 2021) or in tasks with distribution shifts across time (Luu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In practice however, IRM (or a related objective) is often directly ap- plied to the entire set of available data/training environments (Peyrard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Adragna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2020), without account- ing for these factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Thus, imposing invariance constraints across all environments can over-constrain the predictor and cause performance to degrade, since it is directly incentivized to discard such non-invariant yet informative features via the IRM learning objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' To address this, we propose a relaxation for IRM via the Partial invariance (P-IRM) framework, that imposes invari- ance constraints only within partitions/sub-groups of training environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' This increases model flexibility by allowing learning of features that are locally invariant within the parti- tion, without concerning about training environments outside the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Naturally, the cost of finding the optimal parti- tion in an information agnostic setting grows combinatorially with the number of environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' However, access to meta- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='12067v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='LG] 28 Jan 2023 information about environments can often allow us to easily infer the ‘optimal’ training partition for a given use-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In doing so however, we move away from the OoD minimax regime, and instead focus on optimality in a Bayesian sense i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' conditioned on this meta-information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In this work, we first formally quantify this notion of meta-information and then assuming access to it, we theoretically and empirically demonstrate how the partially invariant solution can improve performance under distribution shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The rest paper is orga- nized as follows: we begin with a literature review in Related Work, and motivate P-IRM and present our main results in Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We report our empirical evaluations in Experiments and wrap up with some concluding remarks in Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Related Work Many approaches aim to learn deep invariant feature repre- sentations: some focus on domain adaptation by finding a rep- resentation whose distribution is invariant across source and target distributions (Ben-David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Zhang, Gong, and Schoelkopf 2015), while others focus on conditional domain- invariance (Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' However, there is evidence that domain adaption approaches are insufficient when the test distribution may lie outside the convex hull of training distributions (Lee and Raginsky 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Duchi, Glynn, and Namkoong 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Mohri, Sivek, and Suresh 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Other approaches include Bayesian Deep Learning (Neal 1996), which tries to account for model uncertainty during test- time, and Robust Optimization (Ben-Tal, El Ghaoui, and Nemirovski 2009), which aims to generalize well to distribu- tions close to training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Our work focuses particularly on the IRM framework (Ar- jovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2019), which relates to domain generalization wherein access to the test distribution is not assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' IRM is rooted in the theory of causality (Schölkopf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2012) and proposes invariance for achieving OoD generalization (Peters, Bühlmann, and Meinshausen 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Heinze-Deml, Peters, and Meinshausen 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In (Ahuja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2020a), the authors re- formulate IRM via a game-theoretic approach, wherein the invariant representation corresponds to the Nash equilibrium of a game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' While the IRM framework assumes only the in- variance of the conditional expectation of the label given the representation, some follow-ups rely on stronger invariance assumptions (Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Mahajan, Tople, and Sharma 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' As mentioned before, this line of work assumes suffi- ciency of invariant features whereas we specifically focus on distribution shifts when sufficiency is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Several follow-up works attempt to characterize IRM’s per- formance under different settings and model assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' It has been noted that carefully tuned ERM can often out- perform state-of-the-art domain generalization approaches, including IRM, across multiple benchmarks (Gulrajani and Lopez-Paz 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The failure of IRM may stem from the gap between the proposed framework and its practical “lin- ear” version (IRMv1), which fails to capture natural invari- ances (Kamath Pritish and Srebro 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Indeed, the authors of (Rosenfeld, Ravikumar, and Risteski 2020) demonstrate that a near-optimal solution to the IRMv1 objective, which matches IRM on training environments, does no better than ERM on environments that differ significantly from training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Following these deficiencies, several works propose alter- nate objectives for achieving invariance (Krueger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Bellot and van der Schaar 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Jin, Barzilay, and Jaakkola 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Ahuja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Shui, Wang, and Gagné 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' However, unlike previous works that aim to improve the in- variance learning objective, we question whether invariance as a constraint can be improved upon for better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' To that end, our notion of partial invariance generalizes not only IRM, but all similar invariance learning objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The use of meta-information for invariant learning has been pro- posed in (Lin, Zhu, and Cui 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' However, unlike partition- ing, the focus therein is to artificially generate environment membership for samples when not available a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Finally, a related idea appears in (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2022), which proposes applying different invariance penalty weights for different domains, but with the goal of addressing data quality variance across domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Theory In this section, we present the notion of partial invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Notation: We use upper-case boldface U to denote ma- trix/tensor/vector valued random variables, and lowercase boldface u to denote scalar valued random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We use upper-case U to denote matrices/vectors/tensors and lower- case u to denote scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Invariant Risk Minimization The IRM setup assumes access to datasets of the form De := {Xe i , ye i }ne i=1 collected from multiple training environ- ments e ∈ Etr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The samples in dataset De are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' from the environment’s joint distribution, P(Xe, ye).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The task is to estimate a map f : X → Y or alternatively, the conditional distribution P(Y |X), so that it performs well across unseen environments Eall ⊃ Etr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Formally, the IRM framework aims to minimize the Out-of-Distribution (OoD) risk: ROoD(f) = maxe∈Eall Re(f), where Re(f) := EXe,ye[ℓ(f(Xe), ye)] is the expected risk in environment e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The predictor f is parametrized as w ◦ Φ, wherein Φ : X → Z represents the learned representation and w : Z → Y is a linear predictor over said representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The IRM learning objective is posed as a constrained optimization problem: min Φ,w � e∈Eobs Re(w ◦ Φ) (IRM) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' w ∈ arg min ˜ w Re′( ˜w ◦ Φ) ∀ e′ ∈ Etr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' (1) To avoid the inner optimization, the minimization constraint is replaced by a more tractable gradient penalty: min Φ,w � e∈Eobs Re(w ◦ Φ) (IRMgc) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' w ∈ { ˜w : ∥∇wRe′(w ◦ Φ)∥ = 0 ∀ e′ ∈ Etr}, (2) where IRMgc is shorthand for the gradient constrained IRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In practice, this constraint is enforced via a regularizer λ: min Φ � e∈Eobs Re(Φ) + λ∥∇w,w=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='0Re(Φ)∥, (IRMv1) where the implicit overparametrization in having a separate classifier and representation map is removed by fixing a dummy classifier w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Thus, Φ becomes the entire invari- ant predictor and the strictness of the gradient norm penalty, which enforces invariance, is via λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Note that when λ = ∞, IRMv1 is equivalent to IRMgc, which in turn is the first order approximation for the true IRM objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' An intrinsic assumption in the IRM learning setup for prov- ing minimax optimality is the ideal scenario of sufficiency i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' there exists a Φ that is invariant across all e ∈ Etr and is suf- ficient i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e ye ⊥ Xe| Φ(Xe) ∀e ∈ Eall (Ahuja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' However if sufficiency is violated for an environment, one would expect the IRM model, which relies solely on invariant features, to be sub-optimal for that environment (compared to a model that utilizes non-invariant features along with in- variant ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Such a situation may arise under concept drift, wherein the the conditional expectation of the label ye given “causal” features may change across environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Thus, in practice, if an invariant Φ that is also sufficient for environ- ments does not exist for the desired use-case, we expect the performance of IRM (or related frameworks) to degrade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We illustrate this with a simple example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Example 1 We adapt the generative model from (Arjovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2019): the goal is to predict target y using X = [x1, x2, x3], in environment e such that e ∈ Eall can affect the distribution of X as well as the conditional distribution of y given X via a deterministic map c(e) : Eall → {−1, 1}: x1 ← N(0, σ(e)2), x2 ← N(0, σ(e)2), c(e) ∈ {1, −1}, ϵ ∼ N(0, σ(e)2) y ← x1 + c(e)x2 + ϵ, ϵ ⊥ x1, ϵ ⊥ x2 x3 ← y + N(0, 1), σ(e)2 ∈ [0, σ2 MAX].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We estimate y as ˆy = α1x1 + α2x2 + α3x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Within the IRM framework, the only feasible representation Φ (upto scaling) that yields invariant predictors across all e is Φ([x1, x2, x3]) = [x1, 0, 0], with corresponding regression coefficients [1, 0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Although this minimizes the OoD error for arbitrary e, it does so by discarding the non-invariant but informative x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' However, if our predictor is privy to some knowledge of c(e), we could first partition the set of training environments Etr into two partitions, such that envi- ronments within a partition have the same c(e) value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then, applying IRM within each partition yields models with bet- ter performance that can exploit x2 as an invariant feature in the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Note that this partial notion of invariance still retains the ability to discard spurious/non-causal x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Ad- ditionally with partitioning, we can improve generalization if information about c(eunseen) is available, by choosing the right model/partition for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Next, we study the conditions under which partitioning can improve upon IRM performance and we refer to this method as P-IRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Model For our analysis, we consider a simple regression task to succinctly capture our intuition about the conditions under which partitioning is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' To begin with, we assume access to the underlying causal features and instead, focus on understanding the nature of the IRM solution set under distribution shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In the next part, we extend this analysis to study learning under partial invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We consider the following generative model: we observe samples (Xe i , ye i )in environment e and the goal is to predict ye i from Xe i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Xe i ’s are samples corresponding to the random variable Xe ∼ P(Xe), as described below: Xe = [xe 1, xe 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' , xe c]⊤ ∼ P(Xe), where each xe i denotes an individual feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' To simplify our initial analysis, we assume that the individual features are independent of each other and are normalized i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' E[Xe] = 0 and E[XeXe⊤] = I ∀ e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The target ye for given Xe can be characterized as: ye = ⟨W e, Xe⟩ + ϵy, W e = [we 1, we 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' , we c] ∈ Rc, ϵy ∼ N(0, σ2 y(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' (3) where weights W e encode the conditional distribution of observing label ye given Xe in environment e and are fixed for that environment, and ⟨·, ·⟩ denotes the standard inner product in Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' For a given feature xe i in environment e, the corresponding feature weight we i is independently and uni- formly sampled from set Ai for each environment e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Once sampled however, these weights remain fixed for that envi- ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Additionally |A1| = 1, so that feature weight we 1 is fixed and thus xe 1 is invariant for all e: W e = [we 1, we 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' , we c], where wi e ∼ Unif({Ai}) ∀ i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' , c}, |A1| = 1, |Ai| > 1 ∀ i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' (4) We make note of some important aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' As per our model, xe 1 is the only truly invariant feature since E[ye|xe 1] = winv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='xe 1, is fixed for all e, where A1 = {winv} is a singleton, and winv denotes the invariant feature weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Additionally, the cardinality of set Ai, |Ai| defines an implicit notion of the variance of feature xi, with a higher cardinality indicat- ing that the feature weight is more likely to change across environments and is thus, less invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' With our generative model in place, we next consider the task of predicting ye given Xe, under the mean squared loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Recall that the IRM framework considers predictors of the form w ◦ Φ, where the transformation Φ extracts a suitable representation and w is the linear predictor acting on that representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Due to the implicit overparametrization, we fix w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='0 to a scalar value as proposed in (Arjovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2019) and analyze the corresponding IRM solutions with Φ ∈ Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' For simplicity, we ignore finite sample effects and consider the objective in (IRMv1) when λ = ∞, or equivalently, the gradient penalty constraint in (2) which ide- ally approximates the true IRM objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Additionally, we assume the following for training environments Etr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Assumption 1 (Sufficiency for IRM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Assume ∃ an envi- ronment e ∈ Etr for which the truly invariant predictor is sufficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' the corresponding feature weights satisfy we 1 = winv and wei = 0 ∀ i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' , c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In other words, we assume existence of a training environ- ment in which the invariant predictor that only recovers the invariant feature xe 1 achieves optimal MSE risk, which is a standard assumption in related literature (Ahuja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' As per above parametrization (with w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='0), under Assumption 1, the values for Φ that satisfies the IRM solution constraints in (2) is a singleton and the value of the corresponding predictor equates to Φ = [winv, 0, 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' , 0], the predictor only recovers the invariant feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The proof of the lemma is included in the Appendix and relies on showing that any predictor which assigns non-zero weights to any of non-invariant features would violate the gradient penalty constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' More importantly, the previous lemma roughly says that any non-invariant feature will be discarded by the IRM predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Note that while this is a desirable property for minimax optimality, we ask whether we can do better given additional contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We formalize the notion of contextual information explicitly by defining an oracle ω(e) = 1[∥W eref − W e∥0 ≤ δ], that provides us a notion of distance between environments, from a fixed reference environment eref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Alternatively, it identifies whether environment e is close to eref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Remark 1: The choice of the ℓ0 metric for the oracle suits our combinatorial setting, since we do make any assumptions on the individual elements in the feature weight sets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Ai’s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Next we characterize our objective to utilize this information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Suppose we know that our test environment shares the feature weight with the reference environment for a given feature xe i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then we can define the goal of minimizing the risk w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' to the predictor f, conditioned on this information: Rcond(f) = Ee s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' w eref i =we i Re(f), where the expectation is over the draw of environments as per the uniform sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We note that a predictor that accounts for the prior condition (reference feature) will improve per- formance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' with a lower MSE risk Rcond), as compared to the truly invariant predictor in the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' However, to obtain the required feature as a feasible solution via IRM constraints, we need to first isolate a subset of training en- vironments Epartition ⊆ Etr such that within this set, we i is invariant and secondly, that we avoid learning the rest of the non-invariant features to avoid feature weight mismatches in unseen environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' It turns out that with access to the ora- cle and under certain mild conditions, we can ensure exactly that in our uniform distribution shift model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Before stating the result, we require a similar sufficiency assumption for the partially invariant predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Assumption 2 (Sufficiency for P-IRM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Assume ∃ an envi- ronment e ∈ Etr for which the partially invariant predictor is sufficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' the corresponding feature weights satisfy we 1 = winv, we i = weref i and wej = 0 ∀ j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' , c}\\{i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Under the model (4), under Assumption 2, with access to oracle ω(e) = 1[∥W eref − W e∥0 ≤ δ] and δ < (c − 2)/2, isolate Epartition := {e ∈ Etr|ω(e) = 1} ∪ {eref} ⊆ Etr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Next, let |Ai| = k, where Ai is the set corresponding to the feature weight weref i of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then, if the sets {Aj}∀ j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' , c} \\ {i} satisfy |Aj| > αk for some α > 1, we have with probability greater or equal to ( p p+1)|Epartition|, where p ≥ (c−1−δ)α δ , the IRM solution over set Epartition will recover the feature of interest weref i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The proof, available in the Appendix, utilizes the generative model by showing that within the partition that satisfies the oracle condition, the probability of successfully isolating the required feature is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then the result follows as a conse- quence of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In words, the theorem says that if we can identify a parti- tion in which the environments are not too different, then with high probability, the IRM solution will recover features which do not vary too much (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' non-invariant but still close to invariant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Note that in case of erroneous partitioning, the solution set allowed by the non-convex penalty becomes harder to characterize due to the presence of other feature weights besides the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Nevertheless, if the conditions are such that probability of that happening is sufficiently low, we can safely assume that partitioning will achieve a better expected risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Additionally, our result suggests that P-IRM becomes more feasible as the oracle becomes more precise and if feature of interest is the closest to invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Remark 2: While P-IRM does improve upon the IRM so- lution, both variants are likely to be outperformed by ERM in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' However, we point out that this is a simpli- fied setting wherein access to causal features is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In more general settings when the causal features need to be inferred from complex data, ERM may be susceptible in- variance to confounders/anti-causal variables and thus, we require invariance as a means to make the solution robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Partitioning and Partial Invariance Next, we study P-IRM in a general setup, using previous results to characterize the required number of training en- vironments as in IRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' As before, we assume access to the oracle, ω to identify the partition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Epartition ⊆ Etr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Learning Setup: We consider the same causal mechanism for regression task (xe, ye) from before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The goal is to find a partition using the oracle such that a feature of interest cor- responding to the reference environment, weref i is retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Note that since we want to retain only the invariant features denoted as Xinv e = [xe 1, xe i], and discard the non-invariant (or non-partially invariant) features, we encapsulate them into the noise term as ˜ϵy = ϵy + (Xe {1···c}\\{1,i})⊤W e {1···c}\\{1,i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then, notice that we still have ˜ϵy ⊥ Xinv e and that E[˜ϵy] = 0, due to feature independence and centering assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Next, we consider a realistic learning setup where we observe a scrambled version ˜ Xe of the true causal features Xe: ye = (Xinv e)⊤Winv e + ˜ϵy, ˜ϵy ⊥ Xinv e, E[˜ϵy] = 0 ˜ Xe = S(Xe, X′e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' (5) Here, Xe = [Xinv e, Xe {1···c}\\{1,i}] ∈ Rc denote the causal features with respect to the label, X′e ∈ Rq, ˜ Xe = S(xe, de) ∈ Rd with S ∈ Rd×(c+q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The variable X′e may be arbitrarily correlated with Xinv e, ˜ϵy or the label ye and is intended to represent the spurious correlations in data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' However, we require S to be such that ∃ ˜S s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' ˜S(S(Xe, X′e)) = Xinv e i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' an inverse map such that the recovery of the desired features is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Next, we define γ = 1 k √ 2n exp(−nD(δ/n∥1/αk)), where as before, δ is the oracle distance parameter, k is the cardinal- ity of the set Ai, |Ai|, α is as defined in Theorem 1, n = c−2 and D(m∥n) denotes KL divergence between Bern(m) and Bern(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Intuitively, γ estimates the lower bound on the probability of sampling an environment under the genera- tive model that satisfies the oracle condition of close distance to the reference environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then we have the following sample complexity on the number of required environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Theorem 2 (Informal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Assume we observe ( ˜ Xe, ye) as per (5), with environments e ∈ Etr sampled as per (4) and let Epartition := {e ∈ Etr|ω(e) = 1} ∪ eref ⊆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Let Φ ∈ Rd×d have rank r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then sampling |Etr| > 1 γ (d − r+d/r) log(1/ϵ) ensures partition cardinality |Epartition| > d − r + d/r with probability > 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Furthermore, if e ∈ Epartition lie in linear general position of degree r (Assumption 3 in Appendix), then with probability greater than or equal to ( p p+1)|Epartition|, where p ≥ (c−1−δ)α δ , the oracle identifies Epartition such that the predictor w ◦ Φ learnt via IRM within that partition recovers the desired fea- tures/weights and corresponding prediction (Xe inv)⊤W e inv, ∀e ∈ Eall which satisfy we i = weref i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The proof along with the formal statement is included in the Appendix and follows from our previous results by ap- plying concentration bounds on the draw of environments, and subsequently using prior generalization results for IRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In words, Theorem 2 states that if the obtained partition is accurate, is of sufficient cardinality and is sufficiently diverse, then Φ recovers the partially invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' However, no- tice that the required number of environments grows inversely with γ, meaning that we need stronger priors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' sample en- vironments close to the reference) to obtain feasible sample complexities in the number of required environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Partial Invariance in Practice Next, we state the P-IRM objective more formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We first assume a distance metric d between environments (known directly or via contextual information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then, our goal is to identify a subset of training environments Epartition ⊆ Etr such that its average distance w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' a reference environment eref roughly satisfies: 1 |Epartition| � e∈Epartition d(e, eref) < 1 |Etr| � e∈Etr d(e, eref).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Thus, the predictor is trained on a subset of observed environ- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' However, discarding environments is not data-efficient and can lead to lower fidelity and worse generalization, espe- cially in high-complexity models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' To avoid this, we introduce the notion of conditional invariance as an alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' For- mally, consider the set of observed training environments Etr and a subset corresponding to the partition Epartition (chosen suitably via d), satisfying Epartition ⊆ Etr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We propose the following two variants of P-IRM: min Φ,w � e∈E1 Re(w ◦ Φ) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' w ∈ arg min ˜ w Re′( ˜w ◦ Φ) ∀ e′ ∈ E2, if E1 = E2 = Epartition, (P-IRM (Partitioning)) if E1 = Etr & E2 = Epartition (P-IRM (Conditioning)) where the empirical risk minimization objective is over envi- ronments in E1 and the IRM invariance constraint is applied on environments in E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' For P-IRM (Conditioning), note that while the model uses data from all environments, the in- variance penalty is applied only to environments within the chosen partition, which mitigates the issue of having fewer data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Intuitively, it serves as a relaxation of the IRM objective to allow for partially invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Next, we qualitatively discuss some potential issues in the application of P-IRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Firstly, fulfilling the requirements as per Theorem 2, for the required worst case number of environments is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Fortunately, in practice, IRM can pick up the re- quired invariances from just two environments and we expect P-IRM to overcome that issue as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Next, we revisit the oracle which provides the distance be- tween environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Note that the extraction of the set of causal features is in itself the holy grail of machine learning, and therefore in practice, we do not have access to this met- ric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Additionally, note that assuming access to a prior on the distance to the unseen environment implies that we no longer solve for minimax optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' However, in certain situations, the nature of the distribution shift can be inferred via avail- able contextual information which, while often discarded by practictioners, can serve as an effective pseudo-metric for the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' For instance, authors of (Luu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2021) pointed out that temporal mis-alignments of distributions in language tasks leads to performance degradation, noting that degrada- tion increases with an increase in the time duration between test and train environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Thus, learning from only the recent past could yield a larger and more relevant set of in- variant features for a use-case on future data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Experiments In this section, we first perform a basic sanity check via a synthetic experiment, which in essence is an extension of the example presented in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' This synthetic set- ting serves as a simple visualization of how IRM can end up suppressing non-invariant features causal features, leading to performance degradation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' With a better understanding of the pitfalls associated with full invariance, we then evaluate the efficacy of the P-IRM framework (both partitioning or condi- tioning) on four tasks: a regression task for housing price pre- diction, an image classification task on the MetaShift dataset (Liang and Zou 2022), an entity recognition task for scientific texts on the SciERC dataset (Luan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2018) dataset, and a text classification task for prediction of venues of scientific papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Within image classification, we consider two sub- tasks: Domain Generalization and Sub-population shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Due to space constraints, we defer the synthetic experiment on IRM, along with the text classification and Sub-population shift tasks to the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' For comparison against other learning algorithms aside from IRM, we evaluate the results for standard ERM as well In- formation Bottleneck IRM (IB_IRM) (Ahuja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Apart from these benchmarks, we also include additional experiments in the image and language classification tasks to empirically characterize the effect of partitioning on ERM and IB_IRM, which we dub as P-ERM and P-IB_IRM re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' A common underlying thread for our choice of experiments is that for each of the selected tasks, we have access to meta- information that allow us to estimate a notion of distance or similarity between environments, which P-IRM can then exploit to construct the required partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Specifically, in both housing price prediction and entity recognition task, our environments are partitioned across time and due to dis- tribution shifts, we expect environments closer in time to have higher similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Similarly in MetaShift, meta-labels for each image is made available within the data-set, that allows an explicit notion of the distance between training and testing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In all our experiments we employ the train-domain validation strategy(Gulrajani and Lopez-Paz 2020) for hyper-parameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='com/IbtihalFerwana/pirm and other implemen- tation details are deferred to Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Linear Regression We consider a regression task to predict house prices based on house features 1, built across years [1910-2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Each data point consists of 79 predictive features (for instance, num- ber of bedrooms or house area) and a corresponding target, which is the house price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' As pre-processing step, we drop all non-numerical features, dropping samples with missing val- ues and normalizing each feature and the price label to zero mean, unit variance and the samples, {Xi, yi}i ∈ (R32 ×R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Experiment Setup To adapt this task to OoD prediction, following (Lin, Zhu, and Cui 2022), we manually split the training data-set into 10-year segments and use the house year built as a meta-data for partitioning, with the intuition being that factors affecting house prices change over time with societal perceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' For prediction, we consider a linear regression model for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Since the IRM framework to learn w ◦ Φ is inher- ently overparametrized, we fix w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='0 ∈ R and we con- sider Φ ∈ R32 (prediction (Φ⊤X)) with the Adam optimizer (Kingma and Ba 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We consider 6 training environments corresponding to years [1910-1970], while the test samples draw from 4 OoD environments [1970-2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We expect par- titions closer to our test set to yield better predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Results We report the test MSE error (both average and worst group) over the set of testing OoD environments, averaged over 5 random seeds in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We find that P-IRM sig- nificantly improves the average and worst group OoD error over IRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Partitioning also benefits ERM, showing more evidence of a distribution shift over time, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 3 in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Finally, note that for the two variants for P-IRM, partitioning performs much better in this regime, where we have more samples than parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In the Appendix, we 1House Prices Dataset: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='com/c/house-prices- advanced-regression-techniques include a comparison over different partition sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Model Training Years Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' MSE Worst Group MSE ERM 1910-1970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='475 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='000) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='037 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='000) ERM 1930-1970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='431 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='000) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='963 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='000) IRM 1910-1970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='522 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='015) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='129 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='038) P-IRM (partitioned) 1930-1970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='427 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='009) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='873 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='024) P-IRM (conditioned) 1930-1970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='490 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='014) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='035 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='034) Table 1: Prices Shifts in Housing: Training on partitioned data shows an improvement in model performance for both ERM and IRM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Testing comprises of 4 OoD environ- ments, on houses built from years between 1970-2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Image Classification We evaluate P-IRM on a binary image classification task on the MetaShift dataset (Liang and Zou 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Dataset In MetaShift dataset, each image is associated with a set of tags that describe the image context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=', cat on a rug, cat beside a chair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Thus, for each given tag (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' rug, chair), there is an associated set of images and these sets can overlap if an image has multiple tags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' This structure naturally induces a graph, with each image context Ci denotes a node (or community) in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' This graph is weighted and the weights between nodes is determined by the number of im- ages that are shared between the communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The weights between each pair of communities, Ci and Cj, estimate the similarity between two communities and are calculated us- ing the Szymkiewicz-Simpson coefficient, which yields the corresponding adjacency matrix G: G(i, j) = |Ci∩Cj| min(|Ci|,|Cj|) (6) Having access to such an undirected weighted graph over sets of images thus allows us to derive an implicit notion of distance between the corresponding communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Notion of Distance To introduce partitioning, we develop a notion of distance, which then allows us to quantify the relatedness between training and testing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' These environments are assumed to be sets of communi- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' To estimate the distance d between any two given nodes/communities, given that our data is structured as a weighted graph, we can make use of the spectral embeddings (Belkin and Niyogi 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Spectral embeddings are based on graph Laplacian connectivity (Ng, Jordan, and Weiss 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The graph Laplacian L is calculated by L = Ddiag − G, where Ddiag is a diagonal degree matrix of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The corresponding eigenvectors of L, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' , uk, computed and normalized to form the matrix U, are the corresponding embeddings for the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Once we calculate the spectral embeddings, we measure d between communities as the eu- clidean distance between the corresponding spectral embed- dings of each community node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' With our notion of distance, we can partition the graph based on distances between sets of communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' This in turn, allows us to identify a subset of training communities which is closer to the test environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Experiment Setup For all our experiments, we consider the same set of training communities as in (Liang and Zou 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The set of communities are split into two environments in the IRM setting, and we proceed with the same split as in (Liang and Zou 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' To introduce partitioning, we assume distances d between the training environments and the test communities is known/can be estimated via the meta-labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' For learning the P-IRM model, we consider the training envi- ronment for IRM which is closer to the test set on average, and split it into two sub-environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Note that under this split, P-IRM has access to roughly only half the training samples compared to IRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' To remedy this, we consider addi- tional data splits wherein we add samples from communities in the other IRM training environment, that are close to the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' These additional samples amount to a percentage p of samples in that environment, allowing P-IRM access to a slightly larger portion of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Following (Liang and Zou 2022), we consider multiple settings by fixing the test community to be dog(shelf) and observing performance as the distance between dog train vs test communities, d, is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The cat training set remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Results For all experiments, we report the binary classifica- Experiment 1 Experiment 2 Experiment 3 Experiment 4 d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='17 d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='54 d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='81 d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='92 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Performance ERM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='777(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='078) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='560(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='179) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='493(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='119) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='667(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='114) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='62425 P-ERM (p = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='823(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='045) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='790(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='086) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='387(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='074) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='663(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='192) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='66575 P-ERM (p = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='820(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='098) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='770(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='057) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='493(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='141) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='663(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='128) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='6865 P-ERM (p = 25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='867(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='050) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='740(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='079) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='557(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='056) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='430(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='079) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='6485 IRM 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='309) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='6695 P-IRM (p = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='960(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='050) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='817(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='045) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='487(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='083) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='650(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='142) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='7285 P-IRM (p = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='710(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='107) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='813(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='147) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='727(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='087) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='690(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='184) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='735 P-IRM (p = 25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='820(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='148) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='742(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='138) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='597(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='243) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='753(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='209) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='728 IB_IRM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='647(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='197) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='740(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='171) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='750(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='155) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='303(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='241) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='61 P-IB_IRM (p = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='663(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='242) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='643(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='137) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='437(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='289) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='617(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='059) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='59 P-IB_IRM (p = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='690(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='340) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='790(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='070) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='377(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='214) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='837(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='160) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='6735 P-IB_IRM (p = 25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='613(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='386) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='740(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='171) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='203(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='029) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='343(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='464) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='47475 Table 2: Domaing Generalization in Metashift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' For each experiment the training environments are d away from the test community dog(shelf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The partitioned models are applied with additional samples up to percentage p ∈ {0, 10, 25}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Communities in training are not observed during testing tion accuracy averaged over 3 seeds, with the randomness solely arising from the learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We compare the performance of P-IRM against IRM, as well other bench- marks and their corresponding partitioned versions in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We highlight the best performing model between each model and its corresponding model with partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In most of the experiments, especially with higher deviation between the training and testing data, models with partitioning tend to perform better even with p = 0 of additional samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Named Entity Recognition (NER) Distributional shifts are common in language tasks, given that societal changes are known to influence language usage over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' These changes are also reflected in word embed- dings (words vectors to represent language) (Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Within this context, we explore possible benefits arising out of partitioning (Lazaridou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Luu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Experiment Setup We consider the SciERC (Luan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2018) dataset, which consists of CS publications from 1980 to 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The specific task is Named Entity Recognition, a multi- class classification task, that labels each scientific mention in a sentence into six possible categories (Task, Method, Evalua- tion Metric, Material, Other-Scientific-Term, or Generic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='The training set comprises of years from 1980-2009 and we test the model on data obtained between 2010-2016, with an in- tention to study distribution shift over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' For creating the training environments, we split training years into smaller intervals, 1990-2009, 2000-2009 and 2005-2009, such that each interval has roughly the same number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' For partitioning, we consider contiguous partitions of time in- tervals, based on the intuition that vocabularies in text have higher overlap when closer in time (Gururangan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' For building the model, we train a classifier over the BERT pretrained language model (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Due to high sample complexity, we also consider the conditioned P-IRM method that makes use of all training environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Results We report the classification accuracy, averaged over 3 seeds in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We find that both variants of P-IRM in- deed improve performance over IRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Additionally, we find that leveraging more training data using conditioned P-IRM leads to marginally better predictors, when compared against standard partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Comparisons against IB_IRM as well as ERM demonstrate that partitioning can improve efficacy of other learning algorithms as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Model Number of envs Training Years Testing accuracy (2010-2016) ERM 4 1980-2009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='800 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='012) P-ERM 3 1990-2009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='804 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='020) P-ERM 2 2000-2009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='804 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='016) IRM 4 1980-2009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='795 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='005) P-IRM (partitioned) 3 1990-2009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='795 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='017) P-IRM (partitioned) 2 2000-2009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='807 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='005) P-IRM (conditioned) 3 1990-2009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='812 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='008) P-IRM (conditioned) 2 2000-2009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='807 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='015) IB_IRM 4 1980-2009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='800 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='010) P-IB_IRM (partitioned) 3 1990-2009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='800 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='015) P-IB_IRM (partitioned) 2 2000-2009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='794 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='015) P-IB_IRM (conditioned) 3 1990-2009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='807 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='008) P-IB_IRM (conditioned) 2 2000-2009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='805 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='020) Table 3: Language Shifts in SciERC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The partition- ing improves performance for not only IRM but also other learning objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Additionally, we find that the choice of optimal partition (1990-2009) is consistent across training algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Discussion In this work, we propose P-IRM: a relaxation of the IRM objective via partial invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Through our analysis, we determine conditions under which P-IRM becomes feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We then experimentally verify, in both linear regression and deep learning settings across multiple domains, that when contextual information allows to interpret a distance metric, we indeed improve upon the IRM predictor as well as other learning frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We note that the application of partitioning/P-IRM framework is naturally limited by the informativeness of the available information about training/deployment domains, which often may not be readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Additionally, while distribu- tion shifts across time allows for partitions to be contiguous time intervals, in general, finding the appropriate partition is non-trivial under more complex shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In that sense, our work provides the first step towards understanding the need for choosing the right set of training domains in invariant learning settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Developing more general heuristics for identifying the right partition is an important direction of fu- ture work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Finally, 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Causal View.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In Proceedings of the 29th Association for the Advancement of Artificial Intelli- gence Conference (AAAI’15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Proofs Proof of Lemma 1 The proof follows as a consequence of the parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Under the MSE loss, note that the expected risk takes the following form: Re(w ◦ Φ) = Eye,Xe(ye − w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Φ⊤Xe)2, (7) wherein the scalar w is fixed at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The gradient penalty w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' w in environment e can then be obtained as: ∥∇w,w=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='0Re′(w ◦ Φ)∥ = |∇w,w=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='0Eye,Xe(ye − w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Φ⊤Xe)2| = |Eye,Xe[∇w,w=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='0(ye − w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Φ⊤Xe)]2| = |Eye,Xe[2(w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Φ⊤Xe − ye)Φ⊤Xe]| = |Eϵye,Xe[2(Φ − W e)⊤Xe − ϵy e)Xe⊤Φ]| = |EXe[2(Φ − W e)⊤(XeXe⊤)Φ] + 0| = |2(Φ − W e)⊤EXe[(XeXe⊤)]Φ| = |2(Φ − W e)⊤Φ| = 2| c � i=1 (Φ2 i − we i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Φi)|, (8) wherein the simplifications follow through due to feature independence, normalization and zero mean noise assump- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Notice that as per the constraints in (2), we need the risk penalty term above to be equal to zero for all training environments: | c � i=1 (Φ2 i − we i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Φi)| = 0 ∀ e ∈ Etr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then note from a risk minimization incentive, we naturally have Φ1 = winv without incurring any penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Thus the penalty boils to: | c � i=2 (Φ2 i − we i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Φi)| = 0 ∀ e ∈ Etr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' But from Assumption 1, we have an environment e in the training set in which IRM is sufficient, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' we i = 0 ∀ i ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Thus, the constraint in that environment equates to: | c � i=2 (Φ2 i − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Φi)| = | c � i=2 Φ2 i | = 0 ∀ e ∈ Etr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Thus to satisfy this constraint, Φi = 0 ∀ i ̸= 1, which means the set of feasible solutions is comprised solely of the per- fectly invariant predictor, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Proof of Lemma 2 Under the uniform feature model, we sample Etr such that |Etr| = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Notice that the cardinality m of the required parti- tion Epartition := {e ∈ Etr|ω(e) = 1} is random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We note that for an environment sample e ∈ Etr, for a given reference environment eref, we have from our anal- ysis in Proof of Theorem 1 that P(∥W e − W eref ∥ ≤ δ) = P(E1) + P(E2) and that P(E1) > pP(E2), wherein p ≥ (c−1−δ)α δ >> 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Let n = c − 2 for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then, we have the following approximation: P(∥W e − W eref ∥ ≤ δ) ≈ P(E1) ≥ γ = 1 k √ 2n exp(−nD(δ/n∥1/αk)), where the final result follows from standard anti- concentration bounds on a Binomial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then it is easy to see that given |Etr| = t, P(|Epartition| ≥ m) = P(| � e∈Etr 1[ω(e) = 1] ≥ m)] ≥ P(�t j=1 Dj ≥ m] where Dj is Bernoulli random variable distributed as Dj ∼ Bern(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' On the RHS, we get a sum of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Bernoulli variables for and is thus, �t j=1 Dj ∼ Bin(n, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Let M = �t j=1 Dj and notice that we need that M > m with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' To achieve this, we first upper bound the probability of event M < m using Chernoff’s tail bound and derive conditions under which this upper bound is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Specifically: P(M < m) < exp(−tD �m t ∥γ � ) < ϵ, (9) where D(a∥b) = a log( a b )+(1−a) log( 1−a 1−b ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Next, assume that t = Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then we can simplify the inequality: −Cm � 1 C log 1 γ C + C − 1 C log (C − 1) 1 γ C( 1 γ − 1) � < log(ϵ) (10a) => � log 1 γ C + (C − 1) log (C − 1) 1 γ C( 1 γ − 1) � > 1/m log(1/ϵ) (10b) => � C log (C − 1) 1 γ C( 1 γ − 1) − log C − 1 1 γ − 1 � > 1/m log(1/ϵ) (10c) ≍ � C log (C − 1) 1 γ C( 1 γ − 1) � > 1/m log(1/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' (10d) Our inequality will be satisfied if a) C > 1 γ log(1/ϵ) and b) log( (C−1) 1 γ C( 1 γ −1)) > γm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then we can show that for sufficiently large 1 γ m, condition b) roughly amounts C > ci(1 + 1 m−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' So if C > max{ 1 γ log(1/ϵ), 1 γ (1 + 1 m−1} ∼ 1 γ log(1/ϵ) (for small ϵ), then we have P(M < m) < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Hence, for t = Cm ∼ 1 γ m log(1/ϵ), we get that M = �t j=1 Dj with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' But note that we already have P(|Epartition| ≥ m) ≥ P(�t j=1 Dj ≥ m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Thus, P(|Epartition| ≥ m) > 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Proof of Theorem 1 We begin by restating the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 1 Under the model (4), under Assumption 2, with access to oracle ω(e) = 1[∥W eref − W e∥0 ≤ δ] and δ < (c − 2)/2, isolate Epartition := {e ∈ Etr|ω(e) = 1} ∪ {eref} ⊆ Etr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Next, let |Ai| = k, where Ai is the set corresponding to the feature weight weref i of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then, if the sets {Aj}∀ j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' , c} \\ {i} satisfy |Aj| > αk for some α > 1, we have with probability greater or equal to ( p p+1)|Epartition|, where p ≥ (c−1−δ)α δ , the IRM solution over set Epartition will recover the feature of interest weref i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The proof sketch is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We first characterize the prob- ability of error in using the oracle as an indicator for the partition membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Assuming the partition is identified, we can then directly apply Lemma 1 to obtain the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We begin by characterizing the two possible cases that arise when oracle ω(e) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' E1: If feature at level i remains unchanged, this means that out of c − 2 features (discarding the feature at level i and the invariant feature at level 1), a maximum of up-to δ features changed in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' E2: If feature at level i changed, this means that out of c − 2 features (discarding the feature at level i and the invariant feature at level 1), a maximum of up-to δ − 1 features changed in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Note that both of the events can be modelled as a sum of Bernoulli random variables with different probabilities of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Consider Bj ∼ Bern(1 − 1/|Aj|), which indicates whether the feature value at level j changed under the uniform model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In the first case, we model the conditional probability as: P(E1) = P(we i = weref i |ω(e) = 1) = (1/k)P(1 ≤ c � j=2,j̸=i Bj ≤ δ), (11) while in the alternate case, P(E2) = P(we i ̸= weref i |ω(e) = 1) = (1 − 1/k)P(0 ≤ c � j=2,j̸=i Bj ≤ δ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' (12) Next, we make use of the assumption on cardinalities and analyze the probabilities corresponding to each term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We note that: P(E1) = δ � m=1 (1/k)P( c � j=2,j̸=i Bj = m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Similarly: P(E2) = δ−1 � m=0 (1 − 1/k)P( c � j=2,j̸=i Bj = m) For brevity, we represent each individual term on the right hand side of the summations as Pa,m (a = 1 or 2 depending on the event), corresponding to that value of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Next, we analyze the ratio for the final two terms in either sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Let c − 2 = n and note that using the as- sumption on cardinalities of the feature sets, we can bound the ratio as follows: P1,δ P2,δ−1 ≥ (1/k)( �n δ � (1 − 1/αk)δ(1/αk)n−δ) (1 − 1/k)( � n δ−1 � (1 − 1/αk)δ−1(1/αk)n+1−δ), wherein the greater than equal to sign holds since cardinality of each feature set is greater than or equal to αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Simplifying the terms, we get: P1,δ P2,δ−1 ≥ (n + 1 − δ)(αk − 1) (k − 1)(δ) ≥ (c − 1 − δ)α δ = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Note that this ratio increases as δ reduces and therefore, we conclude that: P(E1)/P(E2) = �δ m=1 P1,m �δ−1 m=1 P2,m ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Thus, we obtain that Perror = P(E2) ≤ 1/(p + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Thus, for our training partition Epartition = {e ∈ Etr|ω(e) = 1}, with probability greater than ( p p+1)|Epartition|, we will learn an accurate partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Since within this partition the feature weight corresponding to xi is same as the reference for all environments, we obtain the required result from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Proof of Theorem 2 First, we restate the theorem more formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2 (Formal) Assume we observe samples ( ˜ Xe, ye) as per equation (5), with environments e ∈ Etr sampled as per equation (4) and let Epartition := {e ∈ Etr|ω(e) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Let Φ ∈ Rd×d have rank r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then sampling |Etr| > 1 γ (d − r + d/r) log(1/ϵ) ensures that partition cardinality |Epartition| > d − r + d/r with probability > 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Further- more, if e ∈ Etr lie in linear general position of degree r (Assumption 3), then with probability greater than or equal to ( p p+1)|Epartition|, where p ≥ (c−1−δ)α δ , the oracle identifies Epartition such that we have: ΦE ˜ Xe[ ˜ Xe( ˜ Xe)⊤]Φ⊤W = ΦE ˜ Xe,ye[ ˜ Xeye], holds for all e ∈ Epartition iff Φ elicits an invariant predictor Φ⊤W ∀ e ∈ Eall whose feature weights satisfy we i = weref i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We begin by recollecting the requisite tools from (Arjovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' With observation model as in equation (5), a set of training environments Epartition ⊆ Etr lie in linear general position of degree r if |Epartition| > d − r + d/r for some r ∈ N, r < d, and for all non-zero X ∈ Rd: dim � span �� E ˜ Xe[ ˜ Xe( ˜ Xe)⊤]X − E ˜ Xe,˜ϵy[ ˜ Xe˜ϵy] � e∈Epartition �� > d − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Intuitively, this assumption states that we require the training environments in our partition partition Etr to be sufficiently diverse, with limited co-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Next, recall that in our setup, instead of directly observing the training partition for a given level i and feature weight v, we need to identify the a subset Epartition from the available set of training environments Etr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Thus, for the training partition to be at least of size m, we need certain conditions on |Etr|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We characterize this in the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Under environment sampling as per equation (4), if the cardinality of observed environments |Etr| = n ∼ 1 γ m log(1/ϵ), for the subset that satisfies the oracle distance condition i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Epartition := {e ∈ Etr|ω(e) = 1}, we have that P(|Epartition| ≥ m) > 1 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Having obtained the correct partitioning with high probability, we call upon the out of domain generalization result from (Arjovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Proposition 1 (Theorem 9 in (Arjovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2019)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Assume that Y e = (Ze 1)⊤β + ϵe, ϵe ⊥ Ze 1, E[ϵe] = 0 Xe = S(Ze 1, Ze 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' (13) Here, β ∈ Rc, Ze 1 takes values in Rc, Ze 2 takes values in R1 and S ∈ Rd×(c+q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Assume that the Z1 component of S is invertible: that there exists ˜S ∈ Rc×d such that ˜S(S(z1, z2)) = z1, for all z1 ∈ Rc, z2 ∈ Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Let Φ ∈ Rd×d have rank r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then, if atleast d − r + d/r training environ- ments Etr ⊆ Eall lie in linear general position of degree r, then we have: ΦEXe[Xe(Xe)⊤]Φ⊤w = ΦEXe,Y e[XeY e], holds for ∀e ∈ Etr iff Φ elicits an invariant predictor Φ⊤w for all e ∈ Eall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We now provide the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' First, let m = d − r + d/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then from Lemma 2, we know that sampling n ∼ 1 γ m log(1/δ) environments gives us |Epartition| ≥ m with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' From Assumption 3, we also have that environments in Epartition lie in linear general position of degree r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Finally, note that our scheme is contingent on the oracle correctly identifying the required partition, which happens with probability greater than ( p p+1)|Epartition|, as noted in Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Armed with these, we can apply Proposition 1 to our learning setup in equation (5) to learn an predictor Φ, W that can recover features Xe inv and corresponding desired weights W e inv ∀e ∈ Eall which satisfy we i = weref i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Additional Experiments In all our experiments when implementing IRM/P-IRM, we keep the penalty parameter sufficiently high λ = 102/103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The rationale for this is to have λ high enough so that the invariance penalty term dominates the fidelity loss term and features are close to invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Synthetic Experiment The experimental setting is adapted from (Arjovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We assume the following generative model: X1 ← N(0, e2I), X2 ← N(0, e2I), c(e) ∈ {0, 1}, ϵ ∼ N(0, e2I) y ← XT 1 W1 + XT 2 (c(e)W2) + ϵ, ϵ ⊥ X1, X2 The task is to predict target y ∈ R based on observed X ∈ R20, where X = (X1 ∈ R10, X2 ∈ R10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The predic- tor w ◦ Φ is learnt via the IRM objective and as noted previ- ously, the overparametrization in the objective is handled by parametrizing Φ ∈ R20 to be rank 1 and fixing w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='0 as a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The true weights W1, W2 ∈ R10 are fixed Gaussian entries, but the sampling of c(e) for different environments controls whether X2 is a causal feature for y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The goal is to visualize the intuition in Lemma 1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' how IRM can discard causal features which are non-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' To that end, we sample 1000 data points from four environments characterized by e ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='2, 1, 2, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The c(e) for each envi- ronment is assigned uniformly, such that the final training set has two environments for each of {0, 1}, and feature X2 is non-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The learning procedure for IRM is consistent with the rest of the paper, with λ = 103 and the initial 4000 epochs for annealing the IRM loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' To study learning of each feature, first denote Φ = (Φ1, Φ2), Φ1, Φ2 ∈ R10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then note that Φi captures the contribution of feature Xi in the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We then look at ∥Φi∥ ∥Wi∥ (averaged over the random draw of environment weights).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Intuitively, this ratio indicates the information captured by the IRM pre- dictor for that feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We visualize the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 1, which demonstrates the tendency for IRM to suppress learning of non-invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Figure 1: The plot demonstrates that IRM is incentivized to suppress non-invariant features, as is the case for feature_2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Linear Regression For our choice of learning rate, number of iterations and optimizer and annealing iterations, we refer to ((Lin, Zhu, and Cui 2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' While the reported results were for λ = 102, we verified similar trends for λ = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Hyperparameter Values Number of Iterations 4000 Learning rate 10−3 Optimizer Adam IRM Penalty 102 Annealing Iterations 2000 Table 4: Hyperparameters for experiments on the housing dataset, following (Lin, Zhu, and Cui 2022) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Figure 2: Partitioning vs Conditioning: In the under- parametrized linear regression setting where the number of data points is much greater than learnable parameters, condi- tioning is not helpful in terms of improving P-IRM accuracy and partitioning consistently performs better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Figure 3: For the regression experiment in the main paper, we find that for both ERM and IRM, there exists an optimal partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Note that while ERM consistently finds the unique optima, the IRM solution has some variance due to the non- convex objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Demonstration of iRM supressing non-invariant features Ratio of Feature Weight Norm (Learnt over Actual) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='0 feature 1 (invariant) feature 2 (non-invariant)AverageTestError([1970-2010oj)foriRMvariantsforLinearRegression 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='54 IRM(Conditioned) IRM(Partitioned) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='50 Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='48 MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='42 20 30 40 50 60 TrainingPartitionsSize(inyears))0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='54 IRM ERM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='50 rror E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='48 MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='42 20 30 40 50 60 TrainingPartitions Size (inyears))Image Classification For the image classification experiments on MetaShift (Liang and Zou 2022) dataset, we follow a similar training pipeline as in (Gulrajani and Lopez-Paz 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Liang and Zou 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Following (Gulrajani and Lopez-Paz 2020), we consider ResNet-50 (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2016), since larger ResNets are known to generalize better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' ResNet-50 was pre-trained on ImageNet (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2016), and for domain generalization, the batch nor- malization and final softmax layers of ResNet are chopped off (Gulrajani and Lopez-Paz 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then ResNet-50 layers are followed by non-linear functions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' ReLU, and a final dropout layer (Gulrajani and Lopez-Paz 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Domain Generalization Herein, we report the relevant choice of hyperparameters in Table 5, to reproduce our re- sults pertaining to the domain generalization experiment in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='IRM Penalty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='weight ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='IRM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='An- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='nealing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='IB_IRM ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='P-IB_IRM (p=10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='P-IB_IRM (p=25) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Table 5: Domain Generalization in Metashift: IRM and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='IB_IRM hyperparameters obtained via model selection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Subpopulation Shift ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Following our setup described in Im- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='age Classification setting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' we conduct further experiments to study performance under subpopulation shifts for the binary classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In subpopulation shifts, communities used for training and testing are the same, but their relative propor- tions differ between training and testing environments, with certain groups often subject to under-representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The goal is to obtain a model to do well even for minority groups in the training data (Koh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Following (Liang and Zou 2022), the communities are grouped into two environments: indoor and outdoor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' In train- ing, cat(outdoor) and dog(indoor) subsets are the minority groups, while cat(indoor) and dog(outdoor) are majority groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We vary the percentage of minority groups within the training set to be m ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='12, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='01} of the total training set, and we keep the size of the training set fixed with 1700 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We use a balanced set of testing by equally sampling from each environment with balanced labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Tables 6 and 7 show that the results under subpopulation shift settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We report the average accuracy over the four groups, and worst group accuracy (the group with the worst performace), and the average minority accuracy which is the average of mi- nority groups in training i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' cat(outdoor) and dog(indoor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' On the more challenging setting of m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='01 where minority groups are observed minimally, P-IRM models achieve better worst group and average minority performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' However as expected, it becomes harder to improve performance for par- titioned models as we lower the amount of available training data, and it is best rely on IRM/ERM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' For reproducibility of results, Table 8 shows the selected hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='12 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Worst Group Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Minority Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' ERM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='816(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='209) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='722(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='103) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='737(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='021) P-ERM (p = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='78(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='129) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='623(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='157) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='675(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='074) P-ERM (p = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='76(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='189) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='526(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='089) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='616(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='127) P-ERM (p = 25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='779(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='154) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='590(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='043) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='736(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='206) IRM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='638(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='250) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='336(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='160) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='558(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='314) P-IRM (p = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='703(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='158) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='560(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='230) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='5705(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='015) P-IRM (p = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='681(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='190) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='518(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='143) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='663(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='205) P-IRM (p = 25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='737(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='147) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='604(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='227) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='7335(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='183) IB_IRM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='639(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='209) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='380(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='082) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='47(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='127) P-IB_IRM (p = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='587(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='136) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='451(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='234) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='4695(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='026) P-IB_IRM (p = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='613(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='301) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='264(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='137) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='5635(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='424) P-IB_IRM (p = 25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='596(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='178) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='426(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='115) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='448(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='031) Table 6: Subpopulation shift on Metashift The value m represents the portion of minority groups within a training environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The partitioned models are applied with addi- tional samples up to percentage p ∈ {0, 10, 25} m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='01 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Worst Group Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Minority Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' ERM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='744(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='209) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='514(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='113) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='559(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='064) P-ERM (p = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='747(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='186) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='574(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='112) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='587(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='018) P-ERM (p = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='729(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='254) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='481(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='133) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='51(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='041) P-ERM (p = 25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='745(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='248) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='488(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='013) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='532(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='062) IRM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='725(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='185) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='509(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='216) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='5775(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='097) P-IRM (p = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='677(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='232) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='426(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='116) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='484(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='082) P-IRM (p = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='629(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='277) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='341(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='260) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='6355(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='269) P-IRM (p = 25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='687(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='181) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='525(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='189) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='531(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='008) IB_IRM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='269(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='179) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='470(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='379) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='5555(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='121) P-IB_IRM (p = 0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='56(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='27) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='208(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='102) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='3955(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='265) P-IB_IRM (p = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='556(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='206) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='366(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='108) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='4805(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='162) P-IB_IRM (p = 25) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='568(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='138) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='398(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='412) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='583(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='092) Table 7: Subpopulation shift on Metashift The value m represents the portion of minority groups within a training environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The partitioned models are applied with addi- tional samples up to percentage p ∈ 0, 10, 25 Model IRM Penalty weight IRM An- nealing Iterations IB_IRM Penalty weight IB_IRM an- nealing itera- tions Experiment 1 (m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='12 ) IRM 10 40 P-IRM (p = 0) 10 40 | 20 P-IRM (p = 10) 10 40 | 20 P-IRM (p = 25) 10 20 | 40 IB_IRM 10 40 10 20 P-IB_IRM (p = 0) 10 20 10 20 P-IB_IRM (p = 10) 10 20 | 40 10 40 P-IB_IRM (p = 25) 10 20 10 20 | 40 Experiment 2 (m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='01 ) IRM 10 40 P-IRM (p = 0) 1000 | 10 40 P-IRM (p = 10) 100 | 10 40 | 20 P-IRM (p = 25) 100 | 10 20 | 40 IB_IRM 1000 40 10 40 P-IB_IRM (p = 0) 10 40 | 20 100 | 10 20 P-IB_IRM (p = 10) 100 | 10 20 100 | 10 40 | 20 P-IB_IRM (p = 25) 1000 | 10 20 100 | 10 40 | 20 Table 8: Subpopulation shift in Metashift: IRM and IB_IRM hyperparameters based on model selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' (For partitioned models since we are using two models, we are reporting the parameters for both, if they are the same we report one value only) Language Experiments For language experiments, NER and TC, we build a classifier based on the pre-trained language model BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2019), followed by a dropout and a linear layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We also con- sidered DistillBERT (Sanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2019) and GPT-2 (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 2019),but found that BERT-based models outperformed other networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' We train the models for the maximum num- ber of iterations, (details in tables 9, and 11) for one seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Then we select the best number of iterations to apply for other seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The results are the average of three seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Named Entity Recognition (NER) For the language NER experiments, the best hyperparameter values are reported in Table 9, and Table 10 which have been selected based on the best model performance on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The training was done for 80 epochs, around which both training and in- domain validation losses stabilize and remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' For annealing epochs, we considered [10, 20, 30, 35, 40] epochs and found that for all variants of P-IRM/IRM, 40 epochs yielded best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The optimizer and learning rate was based on standard choice for using pre-trained BERT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Hyperparameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Values ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Maximum Number of epochs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Batch size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Learning rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='10−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Optimizer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Adam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Number of GPUs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Table 9: Hyperparameters choices for experiments on the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='NER dataset ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='# envs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='IRM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Penalty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='weight ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='IRM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='an- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='nealing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='IB_IRM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Penalty ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='weight ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='IB_IRM an- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='nealing itera- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='tions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='#epochs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='ERM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='44 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='P-ERM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='54 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='P-ERM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='IRM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='P-IRM (partitioned) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='P-IRM (partitioned) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='P-IRM (conditioned) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='P-IRM (conditioned) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='66 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='IB_IRM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='57 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='P-IB_IRM (partitioned) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='77 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='P-IB_IRM (partitioned) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='P-IB_IRM (conditioned) 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='76 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='P-IB_IRM (conditioned) 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='59 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Table 10: NER dataset: Best IRM hyperparameters values ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='selected based on early stopping on validation data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Hyperparameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Values ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Maximum Number of epochs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Batch size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Learning rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='10−6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Optimizer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Adam ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Number of GPUs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Table 11: Hyperparameters choices for the Text Classification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='task ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='# envs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='P-IRM (condi- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='tioned) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='33 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='IB_IRM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='1 20 37 P-IB_IRM (partitioned) 3 1000 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='1 20 36 P-IB_IRM (partitioned) 2 1000 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='1 20 39 P-IB_IRM (conditioned) 3 1000 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='1 20 33 P-IB_IRM (conditioned) 2 1000 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='1 20 33 Table 12: TC dataset: Best IRM hyperparameters values se- lected based on early stopping on validation data Text Classification (TC) We consider another language classification task, which identifies the venue of a published paper 2, selecting AAAI and ICML conferences for classifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' This task represents a topic classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Our temporal partitions are: 2006-2008, 2009-2011, 2012-2014, and 2015-2017 and we test on papers published between 2018-2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The model selection follows as before, training for a maximum of 40 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' The final hyperparameters are reported in Table 12 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Table 13 shows how partition- ing outperforms their baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' P-IRM with two environments performed the best among all other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' Model Number of envs Training Years Testing accuracy (2018-2020) ERM 4 2006-2017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='862 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='008) P-ERM 3 2009-2017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='862 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='004) P-ERM 2 2012-2017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='875 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='014) IRM 4 2006-2017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='846 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='013) P-IRM (paritioned) 3 2009-2017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='862 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='008) P-IRM (paritioned) 2 2012-2017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='882 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='016) P-IRM (conditioned) 3 2009-2017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='869 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='007) P-IRM (conditioned) 2 2012-2017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='853 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='010) IB_IRM 4 2006-2017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='846 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='014) P-IB_IRM (partitioned) 3 2009-2017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='868 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='001) P-IB_IRM (partitioned) 2 2012-2017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='874 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='016) P-IB_IRM (conditioned) 3 2009-2017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='860 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='016) P-IB_IRM (conditioned) 2 2012-2017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='862 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content='011) Table 13: Results on text classification, comparison between ERM, IRM, IB_IRM and their partitioned variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFLT4oBgHgl3EQfYy_o/content/2301.12067v1.pdf'} +page_content=' 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new file mode 100644 index 0000000000000000000000000000000000000000..724fba75933636c5163b470a9631d7ee926fd2f8 --- /dev/null +++ b/idE3T4oBgHgl3EQf4wsK/content/tmp_files/2301.04774v1.pdf.txt @@ -0,0 +1,2062 @@ +1 +A Decentralized Pilot Assignment Methodology +for Scalable O-RAN Cell-Free Massive MIMO +Myeung Suk Oh, Student Member, IEEE, Anindya Bijoy Das, Member, IEEE, +Seyyedali Hosseinalipour, Member, IEEE, Taejoon Kim, Senior Member, IEEE, +David J. Love, Fellow, IEEE, and Christopher G. Brinton, Senior Member, IEEE +Abstract +Radio access networks (RANs) in monolithic architectures have limited adaptability to supporting +different network scenarios. Recently, open-RAN (O-RAN) techniques have begun adding enormous +flexibility to RAN implementations. O-RAN is a natural architectural fit for cell-free massive multiple- +input multiple-output (CFmMIMO) systems, where many geographically-distributed access points (APs) +are employed to achieve ubiquitous coverage and enhanced user performance. In this paper, we address the +decentralized pilot assignment (PA) problem for scalable O-RAN-based CFmMIMO systems. We propose +a low-complexity PA scheme using a multi-agent deep reinforcement learning (MA-DRL) framework in +which multiple learning agents perform distributed learning over the O-RAN communication architecture +to suppress pilot contamination. Our approach does not require prior channel knowledge but instead +relies on real-time interactions made with the environment during the learning procedure. In addition, +we design a codebook search (CS) scheme that exploits the decentralization of our O-RAN CFmMIMO +architecture, where different codebook sets can be utilized to further improve PA performance without +any significant additional complexities. Numerical evaluations verify that our proposed scheme provides +substantial computational scalability advantages and improvements in channel estimation performance +compared to the state-of-the-art. +Index Terms +M. S. Oh, A. B. Das, D. J. Love, and C. G. Brinton are with the School of Electrical and Computer Engineering, Purdue +University, West Lafayette, IN, 47907 USA (e-mail: {oh223, das207, djlove, cgb}@purdue.edu). +S. Hosseinalipour is with the Department of Electrical Engineering, University at Buffalo, NY, 14260 USA (email: +alipour@buffalo.edu). +T. Kim is with the Department of Electrical Engineering and Computer Science, the University of Kansas, Lawrence, KS, +66045 USA (email: taejoonkim@ku.edu). +arXiv:2301.04774v1 [eess.SP] 12 Jan 2023 + +2 +Open-RAN (O-RAN), cell-free massive MIMO, deep reinforcement learning, pilot assignment. +I. INTRODUCTION +A. Open Radio Access Network (O-RAN) +Next generation wireless technologies will likely employ many dispersed radio access networks +(RANs) for ubiquitous coverage and enhanced user performance [1], [2]. However, interconnecting +different RANs to create one seamless network requires well-defined network functions and +interfaces which are flexible in their integration capability. Recently, the evolution of software- +defined open RAN (O-RAN) solutions have added enormous flexibility to the implementation +of current 5G networks [3]–[5] and development of emerging 6G networks. O-RAN offers +software-defined disaggregation on virtual network functions (VNFs) and necessary interfaces +to support their coordination, allowing system implementations that are adaptive to various +architectural settings. With this openness and flexibility, O-RAN promotes interoperability across +different RAN vendors and allows network operators to adapt to different wireless environments. +O-RAN adopts the functional split defined in 3GPP [6] and defines three distinct units [7]: the +open central unit (O-CU), open distributed unit (O-DU), and open radio unit (O-RU). Moreover, +O-RAN operation is divided into three different control loops [7]: the real-time (RT), near-RT, +and non-RT loops executing at different time-scales. The resulting O-RAN architecture, and +standard names of interfaces between these elements which enable practical implementations of +many RAN operations, are depicted in Fig. 1a. +O-RAN offers two types of RAN intelligent controllers (RICs) [7] as shown in Fig. 1a: near-RT +RIC and non-RT RIC. Each of these RICs handles tasks manageable in different time-scales. +O-RAN offers virtualization of both RICs, which promotes flexibility in implementing data-driven +intelligence tasks that will be key components of emerging wireless networks. Various operations +can be implemented via custom third-party applications called xApps/rApps [7], allowing RICs +to be much more accessible to the public. In this work, we will consider the implementation of +machine learning (ML) algorithms over these RICs to optimize pilot signal assignments. +Due to these aforementioned advantages offered by O-RAN, a number of opportunities to +utilize O-RAN on future wireless technologies seem promising, some of which are: +• Massive multiple-input multiple-output (MIMO) beamforming (BF): To implement ML-based +BF strategies that handle both latency-intensive (e.g., RT beam selection) and data-intensive + +3 +Near-RT Control Loop +10ms < +< 1s +Non-RT RIC +rApp +xApp +xApp +Near-RT RIC +A1 +O-RU +RAN +Database +E2 +O-FH +O-DU +O-DU +O-RU +O-RU +O-RU +O-RU +O-RU +Non-RT Control Loop +> 1s +RT Control Loop +< 10ms +rApp +O-CU +O-CU +O-DU +F1 +(a) O-RAN architecture with different types of control loops. +O-DU +O-RU +User +O-Cloud +Uplink Pilot +Transmission +User-centric +RU Clusters +Backhaul +O-FH +RIC +VNF +Database +Inter-DU Connection +(b) A decentralized CFmMIMO system realized in O-RAN. +Fig. 1: Illustrations of O-RAN architecture (left) and decentralized O-RAN CFmMIMO system (right). +(e.g., policy update via real-world dataset) tasks is challenging, and O-RAN provides a platform +for realizing their framework [8]–[10]. ML tasks are implemented in RICs, and BF operation +can be split over O-RU and O-DU (e.g., option 7.2x [11]) to maximize computational efficiency. +• Unmanned aerial vehicle (UAV) network: UAVs are typically deployed in dynamic environments +(e.g., emergency rescue and aerial surveillance [12]), where the network infrastructure is required +to be extremely flexible and adaptive. Flexibility and interoperability offered by O-RAN can +be exploited to meet this architectural need [13], [14]. +• Localization via channel charting: Channel charting is a data-driven localization technique [15] +that maps a user to radio geometry using channel information. For the practical implementation +of channel charting, O-RAN can offer a balanced distribution of heavy computational load +coming from the data that is consistently collected and updated for each user. +B. Cell-free Massive MIMO +One innovative idea to address the shortcomings of 5G cellular networks is to remove cell +boundaries using many dispersed transmission/reception points. This idea falls within the academic +definition of cell-free massive MIMO (CFmMIMO) [16]–[18]. By deploying many geo-distributed +access points (APs), CFmMIMO system alleviates the existing cell-edge problems by substantially +improving both the reliability [19] and energy efficiency [20] compared to cellular massive MIMO. +These enhancements are due to the user-centric paradigm offered by CFmMIMO, where a group +of APs are dynamically selected to form a cluster to serve each user. +In early CFmMIMO literature, a system with APs connected to a single processing unit (PU) +was considered for centralized operation. However, in a scalable system where the number of users + +0User4 +and APs grow large, the resulting complexity becomes prohibitive [21]. Thus, CFmMIMO with +multiple decentralized PUs (Fig. 1b), each of which is connected to a disjoint subset of APs, has +been introduced to consider scalability [21]–[24]. The decentralization allows the system to scale +but still be practical by reducing computational and fronthaul load on each PU [18]. Nevertheless, +implementing centralized CFmMIMO techniques (e.g., signal adaptation and resource allocation) +into a decentralized architecture is a challenging task. +C. CFmMIMO Pilot Assignment Problem +In CFmMIMO, reliable channel estimation at both transmitter and receiver is absolutely critical +to facilitate advanced diversity and signal processing techniques. For channel estimation, a set of +orthogonal pilots are used. However, when the number of users grows beyond the number of +available pilots, some users must share their pilots with others, leading to pilot contamination +(PC) that can significantly degrade the channel estimation performance [25]. To cope with PC, +various pilot assignment (PA) methods have been studied in the CFmMIMO literature [26]–[32]. +In [26], a greedy PA scheme with iterative pilot updates was proposed to mitigate PC. A dynamic +pilot reuse scheme to acquire a set of user-pairs for pilot sharing was proposed in [28]. In [29], a +user-group PA strategy, in which the same pilot is assigned to users with minimum overlapping +APs, was proposed. Other methods to solve the PA problem include k-means clustering [27], +graph coloring [31], tabu-search [30], and Hungarian [32] algorithms. +These prior works [26]–[32], however, require a centralized processing for PA and thus are +not scalable computationally. They also utilize closed-form expressions derived from Bayesian +estimation, requiring any relevant information (e.g., pathloss) to be known a priori. For large-scale +systems, especially under a dynamic environment, accurate prior information is often not available, +underscoring the need to develop a PA scheme that does not require prior knowledge. +D. Overview of Methodology and Contributions +Motivated by the aforementioned challenges, we focus on PA in scalable CFmMIMO systems. +As CFmMIMO deploys a large number of APs for ubiquitous coverage, it is crucial to maintain a +great level of implementation flexibility and interoperability across different RANs for scalability. +Hence, we propose to design our CFmMIMO system in O-RAN architecture. As O-RAN keeps +balance in operational complexities and computational loads via functional split along the network +(i.e., O-RU/DUs and RICs), O-RAN becomes a natural solution for scalable CFmMIMO systems. + +5 +Based on the O-RAN CFmMIMO system, we formulate a decentralized PA problem and +develop a learning-based PA scheme to solve it. In doing so, we resort to multi-agent deep +reinforcement learning (MA-DRL) framework, in which a group of agents individually perform +their learning that provides a low-complexity solution without an explicit training stage [33]–[35]. +Our PA scheme is designed to operate in the near-RT RIC of O-RAN. +We summarize the key contributions of our work as below. +• We design our CFmMIMO system based on the O-RAN architecture (Sec. II). We specifically +focus on channel estimation and pilot allocation models considering practical aspects (e.g., +fronthaul overhead and operational complexity by each functional unit), which can be adopted +to the O-RAN CFmMIMO systems. +• We design a Markov game model (Sec. III-C) for our MA-DRL which leads to an efficient +solution for our decentralized PA problem. In particular, we formulate our reward based on +observations that are directly measurable at the O-RUs. Thus, our scheme does not require +prior knowledge of channel statistics, which is different from previous PA algorithms [26]–[32]. +• Leverage the availability of RICs, we propose a novel learning-based PA scheme (Sec. III-D) +aiming to minimize the total mean squared error (MSE) across the users. By adopting the +distributed learning framework of MA-DRL, our scheme provides low-complexity PA solutions +and therefore offers scalability to support large-scale systems. +• Utilizing the decentralization of our system, we consider two effective ways to improve the +PA performance: (i) inter-DU message passing for observation sharing and (ii) low-complexity +codebook search (CS) algorithm (Sec. III-E) that jointly operates with our PA scheme. Numerical +results verify that these approaches can further improve the PA performance. +• We show that our PA scheme can maintain its performance over a mobile environment, which +is possible due to (i) the DRL framework that naturally performs adaptive learning and (ii) +the CS algorithm with iterative greedy search. Previous PA methods only consider a static +environment and do not address the user mobility. +• We numerically evaluate (Sec. IV) the performance of our PA scheme against the state-of- +the-art [30], [32] in both channel estimation performance and computational complexity. The +results show that our scheme outperforms the benchmarks in terms of sum-MSE and scalability. + +6 +II. SYSTEM MODEL AND PROBLEM FORMULATION +In this section, we first describe the CFmMIMO system realized in O-RAN architecture (Sec. II-A). +Then, after describing the channel model (Sec. II-B), we provide details on codebook-based +channel estimation (Sec. II-C) and formulate our decentralized PA problem (Sec. II-D). +A. CFmMIMO Configuration in O-RAN Architecture +We consider M single-antenna O-RUs and U O-DUs collected in sets M = {1, 2, . . . , M} +and U = {1, 2, . . . , U}, respectively. We assume each O-RU is connected to one of the O-DUs +in U via an open fronthaul (O-FH) connection. We define MDU +u +⊆ M as the set of O-RUs +connected to O-DU u ∈ U. We assume inter-DU connections [36] to form RU clusters that are +fully user-centric since the users can be served by RUs from different sets of MDU +u . We focus +our work on the PA task while making an assumption that O-FH and inter-DU connections are +error-free with no delay. Our O-RAN CFmMIMO system is illustrated in Fig. 1b. +Here, we have our O-DUs connected to O-Cloud [7] via backhaul network (Fig. 1b). O-Cloud +is the cloud computing platform that supports the virtualized network functions (VNFs) within +O-RAN, which include RICs. In designing our PA scheme, we specifically focus on the near-RT +RIC that communicates with O-DUs via E2 interface (Fig. 1a). Now, within the near-RT RIC, we +assume U independent learning agents, each of which has a one-to-one correspondence to one of +the O-DUs in the system. Note that we assume multiple agents to fully impose decentralization +on our system. Each agent in near-RT RIC conducts local learning through the O-DU and O-RUs +connected. In addition, we consider a single non-RT RIC interacting with the near-RT RIC via +A1 interface (Fig. 1a), which is responsible for learning model updates of near-RT RIC. +Next, we consider K single-antenna users in a set K = {1, 2, . . . , K}. For each user k, a +user-centric RU cluster is formed such that only M UE +k +≪ M O-RUs are engaged to serve the user, +where we define MUE +k +⊂ M to be the set of O-RUs serving user k ∈ K (i.e., M UE +k += |MUE +k | +where | · | denotes the set cardinality). Each MUE +k +is assumed to be selected and updated using a +procedure independent from our PA. (e.g., radio resource control (RRC) setup procedure [37]). +We also define KRU +m ⊂ K to be the set of users served by O-RU m ∈ M. +Since we have U multiple agents performing PA, each user k ∈ K must belong to one of these +agents. To develop user-to-agent pairings, we consider two different types of users: (i) user k +whose MUE +k +is connected to a single O-DU u, i.e., MUE +k +⊆ MDU +u , which we simply pair that user + +7 +O-RU 3 +O-DU 3 +O-RU 1 +User 1 +User 2 +User 3 +O-DU 2 +O-DU 1 +O-RU 2 +O-RU 4 +O-RU 5 +O-RU 6 +O-RU 7 +O-RU 8 +O-RU 9 +PA Control +Fig. 2: A list of our defined sets and their visual examples for the given decentralized cell-free O-RAN layout. +k to the corresponding agent u, and (ii) user k whose MUE +k +consists of O-RUs from different +O-DUs. For the second type, a serving O-DU [36], which can be defined by any reasonable +criterion (e.g., the O-DU with the most number of O-RUs serving the user), is determined and +paired with the user. We define KDU +u +to be the set of users whose PA is managed by O-DU u. +Example 1. Here we consider a scenario with U = 3, M = 9, and K = 3, and the sets that we +have defined are illustrated in Fig. 2. Each O-DU controls three O-RUs that are closest (e.g., +MDU +1 += {1, 2, 3}), and user-centric RU clusters with M UE +k += 4 are formed for each user (e.g., +MUE +1 += {1, 2, 4, 5}). Note that O-RU can serve multiple users (e.g., KRU +2 += {1, 2}). Since each +user needs an agent for PA, the user is paired to one of the three O-DUs (e.g., KDU +1 += {1, 2}). +B. Time-varying Channel Model +We assume a periodic channel estimation with time interval Te and indicate each estimation +instance using index i = 0, 1, . . . , N. The channel between user k ∈ K and O-RU m ∈ M during +channel estimation instance i is formally expressed as +g(i) +km = +� +β(i) +kmh(i) +km, +(1) +where h(i) +km = µkh(i−1) +km ++ +� +(1 − µ2 +k)n(i) +km is the small-scale fading factor following the first-order +time-varying Gauss-Markov process for i = 1, 2, . . . , N. The perturbation term n(i) +km is a zero- +mean, unit-variance complex Gaussian random variable independent and identically distributed +(i.i.d.) over k, m, and i, i.e., n(i) +km ∼ CN(0, 1). At i = 0, we assume h(0) +km ∼ CN(0, 1) to +be mutually independent from n(1) +km. The correlation coefficient µk for user k is defined as +µk = J0(2π vk +c fcTe) [38], where J0(·) is the Bessel function of the first kind of order zero, vk is +the velocity of user k, fc is the carrier frequency, and c = 3 × 108 m/s is the speed of light. The + +8 +Backhaul +O-DU u +User k +O-FH +O-RU m +RIC (Agent u) +Near-RT +RT +DU-based +Channel +Estimation +PA +Channel Estimation +(a) DU-based channel estimation. +Backhaul +O-RU m +O-DU u +RIC (Agent u) +User k +PA +Channel Estimation +RU-based +Channel +Estimation +O-FH +Near-RT +RT +(b) RU-based channel estimation. +Fig. 3: A block diagram of two different channel estimation structures. +term β(i) +km in (1) is the large-scale fading factor inversely proportional to the distance between +user k and O-RU m at the channel estimation instance i. +C. Codebook-based Channel Estimation +We consider uplink channel estimation with Tp channel uses dedicated for each estima- +tion instance. This allows Tp orthogonal pilots to be available for channel estimation. For +channel estimation, user k ∈ KDU +u +is assigned with one of the Tp pilots in a codebook +T (i) +u += {φ(i) +u,1, φ(i) +u,2, . . . , φ(i) +u,Tp}, where each φ(i) +u,t for t = 1, 2, . . . , Tp is a unit-norm complex +vector of length Tp. For each T (i) +u , we assume mutual orthogonality. Thus, for t, t′ = 1, 2, . . . , Tp, +(φ(i) +u,t)Hφ(i) +u,t′ = 1 if t = t′, and zero otherwise, where (·)H denotes the conjugate transpose. We +denote the pilot assigned to user k for the channel estimation instance i as x(i) +k . +To conduct channel estimation, each user k ∈ K transmits the assigned pilot x(i) +k . The signal +vector (of length Tp) received by O-RU m ∈ M is then expressed as +y(i) +m = X(i)g(i) +m + w(i) +m = +� +k∈K +g(i) +kmx(i) +k + w(i) +m , +(2) +where X(i) = [x(i) +1 x(i) +2 · · · x(i) +K ] is the Tp × K pilot matrix and g(i) +m = [g(i) +1m g(i) +2m · · · g(i) +Km]⊤ is the +channel vector (of length K) for O-RU m. Here, w(i) +m ∼ CN(0, σ2ITp) is the zero-mean complex +Gaussian noise vector of length Tp with covariance σ2ITp, where In is the n × n identity matrix. +We discuss two different channel estimation structures within O-RAN architecture, which +we illustrate in Fig. 3. One structure (Fig. 3a) performs channel estimation at O-DU whereas +the estimation occurs at O-RU in the other structure (Fig. 3b). For the DU-based channel +estimation, y(i) +m from each O-RU m ∈ MDU +u +must be collected by the O-DU in RT scale, +significantly increasing the scheduling and data transfer overhead on O-FH as the number of +O-RUs grows. Such an increasing overhead is critical for the scalability of CFmMIMO. On the +other hand, channel estimation at O-RU only requires the pilot information of the served users + +k +KEKDUmmk +KEKDUk9 +(i.e., {x(i) +k }k∈KRU +m ) to be informed to each individual O-RU in near-RT scale, which does not +involve as much O-FH overhead as the DU-based estimation. Hence, similar to the work in [26], +we assume our channel estimation to take place at O-RUs. +Next, in case of user-centric RU clustering, each RU m ∈ M only needs to estimate |KRU +m | +different channels (i.e., {g(i) +km}k∈KRU +m ) associated with users in KRU +m . For estimating the channel, +we consider two different techniques called pilot-matching [19] and least-square [39] estimations. +If we set �g(i) +m = [�g(i) +km]⊤ +k∈KRU +m as the |KRU +m |-length estimated channel vector from O-RU m during +the channel estimation instance i, pilot-matching and least-square estimations are expressed as +�g(i) +m = ( ¯X(i) +m )Hy(i) +m +(3) +and +�g(i) +m = ( ¯X(i) +m )H(X(i)(X(i))H)−1y(i) +m , +(4) +respectively, where ¯X(i) +m = [x(i) +k ]k∈KRU +m is the Tp × |KRU +m | pilot matrix of the users served by O-RU +m. Now, when some of |KRU +m | users share the pilot, ¯X(i) +m is not unitary (i.e., ( ¯X(i) +m )H ¯X(i) +m ̸= I|KRU +m |), +so the least-square estimation in (4), which utilizes the pseudo-inverse term (X(i)(X(i))H)−1 to +negate the PC, yields better estimation performance. However, in the least-square approach, +since X(i) needs to be known to every O-RU and the size of X(i) increases linearly with K, the +resulting overhead causes significant delay as the number of users grows. This motivates the +pilot-matching channel estimation scheme in (3) for scalability [19]. The estimated channel �g(i) +km +between O-RU m and user k ∈ KRU +m is then expressed as +�g(i) +km= (x(i) +k )Hy(i) +m = +� +k′∈K +g(i) +k′m(x(i) +k )Hx(i) +k′ +(x(i) +k )Hw(i) +m = g(i) +km+ +� +k′∈K +k′̸=k +g(i) +k′m(x(i) +k )Hx(i) +k′ +(x(i) +k )Hw(i) +m . (5) +Note that the summation term the in last equality captures the effect of PC. +D. Problem Formulation +We use MSE of the channel estimation described in Sec. II-C for our PA performance metric. +For user k served by the O-RUs in MUE +k , we define the MSE of the channel estimate in (5) as +MSE(i) +k = E +� � +m∈MUE +k +����g(i) +km − g(i) +km +��� +2 +� += +� +m∈MUE +k +E +�����g(i) +km − g(i) +km +��� +2� += +� +m∈MUE +k +E +���� +� +k′∈K +k′̸=k +g(i) +k′m(x(i) +k )Hx(i) +k′ + (x(i) +k )Hw(i) +m +��� +2 +� += +� +m∈MUE +k +� +k′∈K +k′̸=k +β(i) +k′m +���(x(i) +k )Hx(i) +k′ +��� +2 ++ σ2, +(6) + +10 +where the expectation is taken over the channel and noise. The third equality holds as we +substitute �g(i) +km with (5). Next, the last equality holds since g(i) +km and w(i) +m are i.i.d. across k and +m with E[|g(i) +km|2] = β(i) +km and E[∥w(i) +m ∥2 +2] = σ2, respectively. From (6), we see that the MSE is +directly proportional to the interference caused by PC, and thus can be used as an effective +metric to quantify the PA performance. +Since our system involves U agents, each of which handles the PA of user k ∈ KDU +u , we can +formulate the PA optimization problem for agent u as +(Pu) : +min +{x(i) +k }k∈KDU +u +� +k∈K +MSE(i) +k +(7) +s.t. x(i) +k ∈ T (i) +u , ∀k ∈ KDU +u , +(8) +∥φ(i) +u,t∥2 +2 = 1, +� +φ(i) +u,t +�H +φ(i) +u,t′ = 0 if t ̸= t′, ∀t, t′ = 1, 2, . . . , Tp. +(9) +If β(i) +km, ∀k, m is known, one can directly evaluate � +k∈K MSE(i) +k +using (6) and solve Pu using +PA algorithms (e.g., the previous works [26]–[32]). However, in large-scale systems, such prior +knowledge is often not available, and one can no longer evaluate the objective function in a +straightforward manner. Suppose the knowledge is somehow available for the MSE to be evaluated, +but some of these algorithms (e.g., PAs with Tabu-search [30] and Hungarian algorithm [32] +having the complexities of O(NtabuK2M) and O(KT 3 +p ), respectively) still cannot be considered +as the complexity becomes prohibitive for a large number of users. To address both issues, we +propose to solve Pu via a distributed learning framework, details of which are given in Sec. III. +The decentralization imposed in this work allows our PA scheme to be much more scalable. +III. SCALABLE LEARNING-BASED PILOT ASSIGNMENT SCHEME FOR O-RAN CFMMIMO +In this section, we first describe how our proposed PA scheme is framed in O-RAN (Sec. III-A). +Next, after providing preliminaries on MA-DRL (Sec. III-B), we design a Markov game model +perceiving our PA problem (Sec. III-C), and show that the action selection in our learning +framework corresponds to minimizing the PC (Theorem 1). Finally, we provide implementation +details for our DRL-based PA scheme (Sec. III-D) and iterative CS algorithm (Sec. III-E). +A. Pilot Assignment Framework in O-RAN Architecture +Our learning-based PA scheme for CFmMIMO is designed based on O-RAN architecture +defined in Sec. II-A. Its conceptual block diagram is illustrated in Fig. 4. Here the PA is conducted +under three different O-RAN control loops which have been described earlier in Fig. 1a. + +11 +Updated +Codebook +Information +Observation +Non-RT RIC +RT loop +Near-RT loop +Non-RT loop +Near-RT RIC +O-RU +User +Pilot Assignment +Pilot Sequence +Channel +Estimation +Pilot +Assignment +Information +Weight Update +Agent +Agent +O-DU +Codebook Search +Inter-DU Message +RT Loop +Near-RT Loop +Fig. 4: A block diagram of the proposed PA scheme. +1) RT loop: We assume that a single round of channel estimation steps described in Sec. II-C +takes place in each RT loop. Hence, we denote the index of each RT loop using the same notation +used for indexing the channel estimation instance. In each RT loop i, users transmit their assigned +pilots, and the O-RU m completes the channel estimation to obtain �g(i) +km for k ∈ KRU +m . +2) Near-RT loop: Near-RT loop occurs once in every Nn RT loops. During each near-RT loop, +O-DU u collects observation data, which we describe later in Sec. III-C, from the O-RUs in +MDU +u +and transfers it to the near-RT RIC to be used for learning. At the same time, each agent +u in the near-RT RIC conducts PA on the users in KDU +u . We use ℓ = 0, 1, . . . , ⌊ N +Nn⌋ to denote the +index of near-RT loop, thus, ℓ-th near-RT loop occurs during the Nnℓ-th RT loop (or the Nnℓ-th +channel estimation instance). The relationship between i and ℓ is visualized in Fig. 4. +To further improve our PA performance, two acceleration techniques are introduced: +• Inter-DU message passing: We consider inter-DU message passing which occurs at each +near-RT loop. The inter-DU connection is essential for fully realizing user-centric RU clusters +in decentralized CFmMIMO [36], and we exploit this feature to improve our PA performance. +With inter-DU messages, we aim to reinforce the data observed by the local group of O-RUs +(i.e., O-RUs of MDU +u ). The details on inter-DU message passing are provided in Sec. III-D. +• Codebook searching: We leverage the decentralization of our system and develop a CS algorithm +that operates jointly with our PA scheme. In doing so, we adopt the idea of quasi-orthogonal +codebooks [40], [41] to be used across the agents. In multi-cell systems, where each cell +conducts its own PA to the serving users, using non-identical orthogonal codebooks across the +cells has shown improved system performance. Inspired by this, we rotate the codebook of +each agent in an iterative manner to find the codebook orientation that yields the minimum + +12 +MSE of channel estimation. The detailed steps of our CS scheme is provided in Sec. III-E. +3) Non-RT loop: The non-RT loop is utilized to handle time-insensitive tasks. In our PA +scheme, the update of the learning parameters for near-RT RIC occurs over this loop. Here, the +non-RT loop occurs once in every Nnon RT loops, and we denote q = 0, 1, . . . , ⌊ N +Nnon⌋ as the +non-RT loop index. As described in Fig. 1a, a near-RT loop duration can be as short as 10 ms +while the shortest duration for non-RT loop is a second. Hence, we assume Nnon ≫ Nn. +B. Preliminaries on Multi-agent Deep Reinforcement Learning +MA-DRL addresses scenarios where multiple agents perform simultaneous decision-making +based on a Markov game model [42]. For our decentralized PA problem, we define MA-DRL +using a tuple ({S(ℓ) +u }u∈U, {a(ℓ) +u }u∈U, {r(ℓ) +u }u∈U), where S(ℓ) +u , a(ℓ) +u , and r(ℓ) +u +are respectively the +state, action, and reward of the agent u during the ℓ-th near-RT loop. For each loop ℓ, agent u +with a state S(ℓ) +u +makes an action a(ℓ) +u +to interact with the environment. Subsequently, the agent +makes an observation and computes a reward r(ℓ) +u +which helps to find the next state S(ℓ+1) +u +. +In the non-RT loop, once an agent has completed multiple interactions with the environment, its +policy on action selection for a given state is optimized by updating the weights of its respective +deep neural network (DNN). Here the action is determined based on the Q-value [43] denoted by +Q(S(ℓ) +u , a(ℓ) +u ). The Q-value quantifies the quality of an agent’s action for a given state. Thus, it is +important for the agent to obtain accurate Q-values to make correct decisions. In DRL, these +Q-values are computed via a DNN, the weights of which are trained with experiences so that a +correct (i.e., Q-value-maximizing) action can be selected upon each decision-making. +Now, in perceiving our PA task as a multi-agent learning problem, there are two conditions +we need to consider [44]. First, multiple agents making independent decisions simultaneously +implies the environment is never seen as stationary to an action of a single agent. Second, due +to the decentralized architecture, each agent only obtains a part of the observation available from +the entire environment. Due to these conditions, in multi-agent learning, careful design of the +Markov game model is crucial for achieving performance comparable to centralized learning. +C. Markov Game Model for Decentralized Pilot Assignment +In our O-RAN CFmMIMO setting, channel estimation is repeated for every RT loop i, forming +a periodic interaction with the environment. The near-RT PA corresponds to action selection +that affects the environment and resulting observation. Based on this, we formally define each +component of the tuple presented in Sec. III-B to perceive our PA task as a Markov game model. + +13 +1) States: To represent the PA status of agent u on users in KDU +u +at the start of near-RT loop ℓ, +we define the state as S(ℓ) +u += Φ(ℓ) +u +which is a |KDU +u | × Tp sized matrix where +[Φ(ℓ) +u ]k,t = +� +� +� +� +� +1 +if x(Nnℓ) +k += φ(Nnℓ) +u,t +, +0 +otherwise. +(10) +As discussed previously, PC occurs when users share a pilot, and this can be indicated by the ones +in each column of Φ(ℓ) +u . Hence, Φ(ℓ) +u +can become an effective means to represent the condition of +PA for each agent, and we aim to have the agents accurately perceive the relationship between +their PA (i.e., their actions) and the resulting PC. +2) Actions: We consider sequential updates on the pilots, where the pilot of only a single user +is changed with every action. If we consider actions that assign pilots to all |KDU +u | users at once, +this would lead our action space to take T |KDU +u | +p +possible combinations and suffer from the “curse +of dimensionality”. We hence define actions as 2-tuples indicating the user of interest and the +pilot to be assigned, respectively. The action of agent u at near-RT PA ℓ is formally defined as +a(ℓ) +u = (k, t), where k ∈ KDU +u +and t ∈ {1, 2, . . . , Tp}. With this setting, there are total |KDU +u |Tp +possible actions for agent u to take, resulting in a more computationally scalable action space. +3) Rewards: We propose to compute the reward of each agent u on the ℓ-th near-RT PA based +on the average sum-power of the channel estimates obtained by the O-RUs. Note that, for each +action (i.e., near-RT PA) taken by an agent, Nn channel estimations are conducted by O-RU m +to acquire a set of �g(i) +m for Nnℓ ≤ i < Nn(ℓ + 1). Using this information, O-RU m computes +p(ℓ) +km = 1 +Nn +Nn−1 +� +n=0 +����g(Nnℓ+n) +km +��� +2 +(11) +on user k ∈ KRU +m during the near-RT loop ℓ and sends it to the corresponding O-DU. At the +end of this transfer, O-DU u collects different sets of p(ℓ) +km from each O-RU m ∈ MDU +u +(i.e., +{{p(ℓ) +km}k∈KRU +m }m∈MDU +u ). In decentralized PA, each agent u ∈ U is responsible for a disjoint subset of +K users, and it is desirable for the agent to have access to p(ℓ) +km from all O-RUs associated with the +users (i.e., {{p(ℓ) +km}m∈MUE +k }k∈KDU +u ). However, as each O-DU u is only connected to O-RUs of MDU +u , +{{p(ℓ) +km}m∈MUE +k ∩MDU +u }k∈KDU +u +only gets collected by the agent. Hence, O-DU u ends up computing +the observation data to be transferred to the agent u as ¯p(ℓ) +u = � +k∈KDU +u +� +m∈MUE +k ∩MDU +u p(ℓ) +km. +Note that the rest of information required by agent u (i.e., {{p(ℓ) +km}m∈MUE +k \MDU +u }k∈KDU +u ) has +been collected by other O-DUs. As mentioned earlier in Sec. III-A, since we consider inter-DU + +14 +messages, this information can be transferred to each corresponding O-DU. Then, each O-DU u +can now compute the reinforced observation data which is expressed as +�p(ℓ) +u = ¯p(ℓ) +u + +� +k∈KDU +u +� +m∈MUE +k \MDU +u +p(ℓ) +km = +� +k∈KDU +u +� +m∈MUE +k +p(ℓ) +km. +(12) +The observation data computed by O-DU u in (12) is transferred to agent u via backhaul, and +the reward for agent u at near-RT loop ℓ is subsequently computed using the mapping function +r(ℓ) +u (p) = (pmax − p)/(pmax − pmin), +(13) +where p = �p(ℓ) +u +by the availability of inter-DU message. The mapping function (13) converts +the observation data into a reward range such that lower values of p are rewarded higher. Here +[pmin, pmax] is the range of observation data, which we assume is set by the non-RT RIC. +We now show that the learning via our Markov model leads to taking an action that minimizes +the degree of PC. The basic mechanism of learning we utilize is that, for each given state Su, +we want the agent u to select the action that maximizes its Q-value [43], i.e., +a⋆ +u = arg max +au∈Au +Q(Su, au), +(14) +where Au is the set of all possible actions for agent u. The training in DRL is done by updating +the network weights via regression toward the experiences obtained. The Q-value, which is +the numerical output of the trained network, is then expected to follow the average of these +experiences, i.e., the Q-value is updated through training to yield Q(Su, au) = E[ru(p)|(Su, au)]. +For each near-RT loop ℓ, the following theorem shows that, with inter-DU message passing, +the action selected via (14) is the best action in terms of minimizing the degree of local PC. +Theorem 1. With �p(ℓ) +u +available, for a given state S(ℓ) +u , taking the action a(ℓ) +u +which satisfies (14) +is equivalent to finding the action that minimizes the degree of pilot contamination occurring on +local users in KDU +u +during the near-RT loop ℓ, which is expressed as +� +k∈KDU +u +� +m∈MUE +k +Nn−1 +� +n=0 +� +k′∈K,k′̸=k +β(Nnℓ+n) +k′m +���(x(Nnℓ) +k +)Hx(Nnℓ) +k′ +��� +2 +. +(15) +Proof. First, in terms of the parameters defined in our model, we find the expected reward at +near-RT loop ℓ for a given state-action pair (S(ℓ) +u , a(ℓ) +u ), which is expressed as +E[r(ℓ) +u (�p(ℓ) +u )|(S(ℓ) +u , a(ℓ) +u )] = pmax − E[�p(ℓ) +u ] +pmax − pmin +, +(16) + +15 +where the equality holds from (13). Recalling (14), the learning conducted at each agent u aims +to find the action achieving the maximum Q-value Q(Su, au), which we discussed to yield +E[ru(p)|(Su, au)]. Thus, the action selection mechanism of agent u can be expressed as +a(ℓ) +u = arg max +au∈Au +E[r(ℓ) +u (�p(ℓ) +u )|(S(ℓ) +u , au)]. +(17) +Now combining (16) and (17), we can say that +a(ℓ) +u = arg min +au∈Au +� +k∈KDU +u +� +m∈MUE +k +E[p(ℓ) +km] = arg min +au∈Au +1 +Nn +� +k∈KDU +u +� +m∈MUE +k +Nn−1 +� +n=0 +E +�����g(Nnℓ+n) +km +��� +2� +, +(18) +where the first and second equalities are obtained using (12) and (11), respectively. Now, for +n = 0, 1, . . . , Nn − 1, using (5) we have +E +�����g(Nnℓ+n) +km +��� +2� += E +����g(Nnℓ+n) +km +��� +2� ++ +� +k′∈K +k′̸=k +E +����g(Nnℓ+n) +k′m +(x(Nnℓ+n) +k +)Hx(Nnℓ+n) +k′ +��� +2� ++ E +����(x(Nnℓ+n) +k +)Hw(Nnℓ+n) +m +��� +2� += β(Nnℓ+n) +km ++ ξ(ℓ,n) +km + σ2, +(19) +where ξ(ℓ,n) +km += � +k′∈K +k′̸=k +β(Nnℓ+n) +k′m +��(x(Nnℓ+n) +k +)Hx(Nnℓ+n) +k′ +��2 reflects the PC discussed in Sec. II-D. By +the definition of �g(i) +km in (5), taking the expectation of |�g(i) +km|2 leaves only the autocorrelation terms +for �g(i) +km and w(i) +m , corresponding to β(Nnℓ+n) +km += E[|g(Nnℓ+n) +km +|]2 and σ2 = E[|(x(Nnℓ+n) +k +)Hw(Nnℓ+n) +m +|]2 +in (19). This is because the channel and noise are assumed uncorrelated across k and m. +Now, since (i) ξ(ℓ,n) +km +is the only term that is impacted by action au, i.e., β(Nnℓ+n) +km +and σ2 in (19) +are independent from PA and (ii) x(i) +k +only changes once every Nn RT loops, i.e., x(Nnℓ+n) +k +is +fixed for n = 0, 1, . . . , Nn − 1, by ignoring +1 +Nn as a scaling factor, (18) is equivalent to +a(ℓ) +u = arg min +au∈Au +� +k∈KDU +u +� +m∈MUE +k +Nn−1 +� +n=0 +� +k′∈K,k′̸=k +β(Nnℓ+n) +k′m +���(x(Nnℓ) +k +)Hx(Nnℓ) +k′ +��� +2 +, +(20) +which represents the degree of PC at near-RT loop ℓ over the users in KDU +u . +■ +From Theorem 1, we conclude that learning based on our Markov games model is equivalent +to performing the pilot update which minimizes the interference due to PC at each near-RT PA. +According to (15), the PA made at each near-RT loop ℓ couples with the pathloss occurring over +the corresponding Nn RT loops. Since we do not assume prior knowledge on the pathloss β(i) +km, +we cannot evaluate the exact MSE. However, through the reward we define and the learning +mechanism of DRL, we can still design our PA scheme such that the MSE performance is + +16 +Replay +Memory +Action +State +Agent in near-RT RIC +Reward +Observation +Random +Action +Compute +Reward +Train DNN +Weight +Copy +Deep Q-Network +Target DNN +Experience +O-DU +User +Inter-DU Message +O-RU +Compute Reward +Experience +Weight Update +Evaluate +Codebook +Rotate +Codebook +Codebook +Decision +Codebook Search +Fig. 5: A block diagram overview of our PA scheme, consisting of non-RT DNN training and near-RT PA updates. +improved over time. For static scenarios, where β(i) +km is constant over i, all the actions taken +over near-RT loops (i.e., the entire series of successive pilot updates) are contributing to look +for a single optimal PA solution that minimizes the sum-MSE. On the other hand, for mobile +scenarios, each action is led to focus on minimizing the sum-MSE resulted from the current +channel statistics by leveraging the past information. Our PA scheme is designed to cope with +time varying small-scale and large-scale fading factors upon continuous training. +D. MA-DRL-based Pilot Assignment Algorithm +Given the setting in the previous subsections, we describe our PA algorithm in detail which uses +MA-DRL framework to find the solution to our decentralized PA problem. We incorporate MA- +DRL using the deep Q-network (DQN), which utilizes neural network layers for approximating +Q-values. An individual DQN is implemented at each agent in the near-RT RIC for distributed +learning. Fig. 5 provides an overview of our methodology which is also outlined in Alg. 1. We +detail each of the steps in the following: +Near-RT PA: At ℓ = 0, each agent u randomly assigns one of the Tp sequences in T (0) +u +to its +associated users in KDU +u , from which the state S(0) +u +is generated. For each subsequent near-RT +loop ℓ, the agent u takes an action a(ℓ) +u +via an ϵ-greedy method [43] and assigns a different pilot +sequence to one of its users, obtaining a new state S(ℓ+1) +u +. Since Nn RT channel estimations +occur during a single loop of near-RT PA, each O-DU u collects necessary information, i.e., +{p(ℓ) +km}k∈KRU +m , from the O-RUs in MDU +u +and computes �p(ℓ) +u +with the aid of inter-DU message +passing. The O-DU transfers �p(ℓ) +u +to its agent in the near-RT RIC, which computes the reward +r(ℓ) +u (p) and stores an experience tuple (S(ℓ) +u , a(ℓ) +u , r(ℓ) +u (p), S(ℓ+1) +u +) in a replay memory of size Dm. + +k E K!kmm EMDU2 +ku(l,l)rint,u17 +Algorithm 1: Proposed Pilot Assignment (PA) Scheme +1 Input: Pilot length Tp, number of RT loops N, number of RT loops per near-RT loop Nn, number of +internal loops L, set of users managed by O-DU u KDU +u , set of O-RUs managed by O-DU u MDU +u , set of +users served by O-RU m KRU +m , set of O-RUs serving the user k MUE +k , training period, update period +2 Initialize near-RT loop index ℓ = 0; randomize the parameter vectors θtr +u and θta +u +3 Generate codebook T (Nnℓ) +u +; randomly assign {φ(Nnℓ) +k +}k∈KDU +u +4 for ℓ = 0 to N do +5 +Compute S(ℓ) +u +using (10) +6 +if ℓ > 0 then +7 +Compute r(ℓ−1) +u +(�p(ℓ−1) +u +) using (13); store (S(ℓ−1) +u +, a(ℓ−1) +u +, r(ℓ−1) +u +(�p(ℓ−1) +u +), S(ℓ) +u ) in the memory +8 +for l = 0 to L − 1 do +9 +Select a(ℓ,l) +int,u randomly; compute S(ℓ,l) +int,u using (10); compute r(ℓ,l) +int,u using (22) +10 +Store (S(ℓ) +u , a(ℓ,l) +int,u, r(ℓ,l) +int,u, S(ℓ,l) +int,u) in the memory +11 +if ϵ-greedy then select a(ℓ) +u +randomly else a(ℓ) +u += arg maxau Qθtru(S(ℓ) +u , au) +12 +Update the PA according to a(ℓ) +u +13 +for i = 0 to Nn − 1 do +14 +User k ∈ KDU +u +transmits φ(Nnℓ+i) +k +; O-RU m ∈ MDU +u +estimates {�g(i) +km}k∈KRU +m using (5) +15 +if mod(ℓ, training period) = 0 then generate a batch from the memory and train θtr +u via SGD on (21) +16 +if mod(ℓ, update period) = 0 then set θta +u = θtr +u +17 Output: Updated pilot sequences {φ(N) +k +}k∈KDU +u +Non-RT DNN Training: The learning of each agent u is carried out by two DNNs called the +train and target networks [33], [45], where their network parameter vectors are denoted by θtr +u +and θta +u , respectively. Once enough experiences have been collected in the memory, a mini-batch +of size Db is randomly selected from the memory and used to update θtr +u minimizing the loss: +L(θtr +u ) = Eℓ +� +yℓ − Qθtru(S(ℓ) +u , a(ℓ) +u ) +� +, +(21) +where yℓ = r(ℓ) +u + γ maxa Qθta +u (S(ℓ+1) +u +, a) with γ being the discount factor. Here Qθ(S, a) +represents the Q-value for a given pair of state S and action a computed via a DNN of weight +vector θ. The update is done using stochastic gradient descent (SGD). Here, the weights of θtr +u +are periodically copied to target network θta +u , with the length of this period as a design parameter. +Experience generation: By the O-RAN capability, the value of Nn can vary and impact the + +18 +rate of experiences being collected to each agent, i.e., the number of experiences collected for +a given amount of time varies by Nn. If Nn is too large, a sufficient size of data required to +perform effective training may not be collected within a desired time period. To resolve the issue +and utilize time more efficiently, we exploit the architecture of O-RAN and introduce an internal +experience-generating loop inside the near-RT RIC. This internal loop is executed L times during +a single near-RT loop. In particular, once an experience is obtained via the ℓ-th near-RT loop, we +generate L extra experiences by taking a random action and evaluating the corresponding reward +for each internal loop. We define the reward by the l-th internal loop of the ℓ-th near-RT PA as +r(ℓ,l) +int,u(p) = +� +1 − κ(ℓ,l) +u +/κmax +� +r(ℓ) +u (p), +(22) +where κ(ℓ,l) +u += +����Tp +t=1 +�� +k∈KDU +u (φ(Nnℓ) +u,t +)Hx(ℓ,l) +k +− +� +|KDU +u | +Tp +����� is the penalty for having more than +necessary number of users sharing the same pilot sequence and κmax = 2|KDU +u |(Tp − 1)/Tp is +the maximum penalty obtainable. Integrating this internal loop alongside near-RT PA, we can +generate L more experiences to accelerate the convergence of our scheme and train our DNNs to +favor sequence combinations that have more evenly spread number of users across Tp sequences. +E. Iterative Codebook Search (CS) Algorithm +We describe our CS algorithm that is designed to work with the PA scheme in Sec. III-D. As +each agent assigns pilots to its local users using the codebook T (i) +u , CS is iteratively conducted +so that the final set of U codebook sets, when combined with our PA solution, suppresses the +PC to the minimum degree. We detail each of the steps in the following. +First, we assign each agent u ∈ U with an identical codebook, i.e., T (0) +1 += T (0) +2 += · · · = T (0) +U , +and initiate our PA scheme without CS to ensure that the agents first learn and improve their PA +only based on the interference resulted from pilot sharing. We design our algorithm to begin its +iterative CS only after the learning on PA is stabilized so that the PA and CS do not impair each +other from converging. We determine the PA of agent u to be stable when the state S(ℓ) +u +remains +unchanged over Ncs near-RT loops. Once the agent u has given the same PA for Ncs consecutive +times at the end of near-RT loop ℓ⋆ +u, the agent is perceived as stable and becomes subject for CS. +Note that ℓ⋆ +u is likely to vary for each agent due to our decentralized PA framework. +If we design our agents to conduct CS in parallel, it becomes difficult to accurately evaluate a +codebook as multiple actions simultaneously affect the environment. Hence, we propose to have +each agent take a turn and conduct CS while the rest of agents is paused from the search. To + +19 +implement a such design, we define an operation called the CS run in which an isolated CS is +conducted for each agent u ∈ U(v) +cs , where U (v) +cs +is the set of agents subject for CS during the +v-th CS run. For each isolated search, the following steps are performed. +Suppose it is the turn of the w-th element of U (v) +cs , denoted by uv,w, to perform the isolated CS, +where w = 1, 2, . . . , |U(v) +cs |. We first define ℓv,w to be the near-RT loop in which the agent uv,w +begins its search. We also let Ns define the number of near-RT loops to be spent for codebook +evaluation. During the first Ns near-RT loops (i.e., ℓv,w ≤ ℓ < ℓv,w + Ns), the quality of current +codebook matrix Told +v,w = [φ(Nnℓv,w) +uv,w,1 , φ(Nnℓv,w) +uv,w,2 , . . . , φ(Nnℓv,w) +uv,w,Tp ] is evaluated by computing +¯rold +v,w = 1 +Ns +Ns−1 +� +n=0 +r(ℓv,w+n) +uv,w +(p), +(23) +which is the average of the most Ns recent rewards collected at agent uv,w via our PA algorithm. +Note that (23) represents the quality of PA performed using the codebook T (Nnℓv,w) +uv,w +. +After obtaining (23), the agent generates a Tp × Tp column-normalized random perturbation +matrix Pv,w and computes the rotation matrix as Rv,w = +� +1 − η2 +uv,wITp + ηuv,wPv,w, where +ηuv,w = 1 − +ℓv,w−ℓ⋆ +uv,w +N/Nn−ℓ⋆uv,w is the perturbation degree designed to decrease with ℓv,w so that a +converged solution is obtained. Note that larger ηuv,w results in Rv,w with greater perturbation. +After acquiring Rv,w, the agent rotates the current codebook to obtain a new codebook matrix +Tnew +v,w = proj(Rv,wTold +v,w), +(24) +where proj(·) is the projection function for which we use the Gram-Schmidt orthogonalization +algorithm [46]. The set of Tp columns in Tnew +v,w is then used as a new codebook for agent uv,w +during the next Ns near-RT loops (i.e., ℓv,w + Ns ≤ ℓ < ℓv,w + 2Ns). After these Ns near-RT +loops, where a set of Ns rewards using the new codebook are collected by our PA algorithm, the +agent computes +¯rnew +v,w = 1 +Ns +2Ns−1 +� +n=Ns +r(ℓv,w+n) +uv,w +(p), +(25) +to evaluate the quality of the new codebook. At this point, agent uv,w has evaluated (23) and (25) +from using two different codebooks Told +v,w and Tnew +v,w, respectively, and determines which codebook +to keep by the end of search using the following criterion +T(Nn(ℓv,w+2Ns)) +uv,w += +� +� +� +� +� +Tnew +v,w +if ¯rnew +v,w > ¯rold +v,w, +Told +v,w +otherwise. +(26) + +20 +Algorithm 2: Proposed Codebook Search (CS) Scheme +1 Input: Pilot length Tp, number of consistent PAs required for stability Ncs, codebook evaluation interval Ns, +number of RT loops N, set of agents U +2 Initialize CS run index v = 0, set of agents subject for CS U(v) +cs += ∅, the counter for agent u au = 0, +CSrun = 0, and CSiso = 0; assign identical codebook for all u ∈ U; capture S(0) +u +using (10) +3 for ℓ = 1 to N do +4 +for u ∈ U do +5 +Capture S(ℓ) +u +using (10) +6 +if S(ℓ) +u += S(ℓ−1) +u +then au = au + 1 else au = 0; if au = Ncs then ℓ⋆ +u = ℓ +7 +if CSrun = 0 then +8 +U(v) +cs += {u ∈ U|ℓ⋆ +u < ℓ}; if |U(v) +cs | > 0 then w = 1 and CSrun = 1 +9 +if CSrun = 1 then +10 +if CSiso = 0 then ℓv,w = ℓ; CSiso = 1 +11 +if CSiso = 1 then +12 +if ℓ = ℓv,w + Ns − 1 then compute ¯rold +v,w using (23); apply new codebook Tnew +v,w using (24) +13 +if ℓ = ℓv,w + 2Ns − 1 then +14 +Compute ¯rnew +v,w using (25); decide codebook using (26); w = w + 1 and CSiso = 0 +15 +if w > |U(v) +cs | then v = v + 1; CSrun = 0 +16 Output: Rotated codebook T (N) +u +, ∀u ∈ U +As the CS described above runs for each agent in U (v) +cs , total 2Ns|U(v) +cs | near-RT loops are spent +to complete the CS run v. For every run, each agent tries a new codebook generated using a +random rotation and decides to keep whichever codebook that yields higher reward. The algorithm +starts its very first CS run at ℓ = minu∈U ℓ⋆ +u and continuously conducts each subsequent CS run. +By changing the codebook only when it is determined to be better, the algorithm proceeds to find +the best set of U codebooks that minimizes the degree of PC. Note that, in order to evaluate the +codebooks, our CS scheme utilizes the reward r(ℓ) +u (p), which is obtained during our PA scheme. +Therefore, no additional information needs to be collected the O-DUs to conduct the CS. The +overall procedure for our CS scheme is summarized in Alg 2. + +21 +75 +50 +25 +0 +25 +50 +75 +x (m) +40 +20 +0 +20 +40 +y (m) +O-RUs +Users (i = 0) +Users (i = N) +Fig. 6: Geographical layout of O-RAN CFmMIMO with U = 4, M = 96, and K = 24. O-RUs connected to the same O-DU +have the same color. Each user moves from the initial (circle) to the final position (cross) in 10 seconds. +IV. NUMERICAL EVALUATION +In this section, we evaluate our pilot assignment (PA) scheme under O-RAN CFmMIMO +channel estimation scenarios with various system parameters. We analyze both channel estimation +performance and computational complexity to discuss the scalability and practicality of our +method. In addition, we compare the performance of our proposed approach against different +baselines which include [30], [32] among others. +A. Simulation Setup, Performance Metrics, and Baselines +We consider different combinations of O-DUs (U = 4), single-antenna O-RUs (M = 96), and +single-antenna users (K ∈ {24, 36}) placed in an area of 100 m × 150 m geometry to create +O-RAN CFmMIMO systems. We assume the same number of O-RUs connected to each O-DU +(i.e., |MDU +u | = M +U , ∀u) and the same number of users paired with each agent in the near-RT +RIC (i.e., |KDU +u | = K +U , ∀u). We set channel estimation interval Te = 1 ms, implying our O-RAN +RT loop occurs once every 1 ms. Each scenario is simulated with maximum N = 10000 RT +loops, which corresponds to 10 seconds with Te = 1 ms. We assume Nn = 10 RT loops to occur +per O-RAN near-RT loop and L = 9 internal experience generation per near-RT loop unless +stated otherwise. For mobile scenarios, we generate initial (i = 0) and final (i = N) positions +for each user such that the velocity vk ranges from 0 m/s (or 0 km/h) to 1.4 m/s (or 5 km/h). +Then, for each i = 0, 1, . . . , N, the position of each user is updated according to vk. Such a +mobile scenario for 96 × 24 CFmMIMO (where M × K refers to M O-RUs and K users) with + +22 +U = 4 O-DUs (equivalently, U = 4 agents in the near-RT RIC) is demonstrated in Fig. 6. The +large-scale fading factor β(i) +km, ∀k, m is assumed to follow the 3GPP urban-micro line-of-sight +pathloss model [47] with carrier frequency fc = 2 GHz, O-RU height of 10 m, and user height of +1.5 m. We consider a pilot length of Tp = 4 and a RU cluster size of M UE +k += 8, ∀k unless stated +otherwise. For our codebook search (CS) scheme, we consider an agent to be stable if the PA is +consistent for Ncs = 100 consecutive times and assume the codebook evaluation interval Ns = 5. +We use the same DQN design for all agents: one convolutional neural network (CNN) with 32 +kernels of size |KDU +u | × Tp followed by two fully connected layers of width |KDU +u |Tp. All layers +use ReLU activation and Adam optimizer with learning rate of 0.001. The discount factor for the +weight update is set γ = 0.5. We also set the size of replay memory Dm = 1000 and train the +neural network using Db = 128 samples per minibatch. The train network weights are updated via +SGD and synchronized with the target network whenever 200 and 400 new additional experiences +are stored in the replay memory, respectively. We implement ϵ-greedy action-selection with the +probability of selecting a random action in the ℓ-th near-RT loop computed as ϵℓ = e−(Γ/N)Nnℓ, +where Γ = 15 is the scaling factor. +We now describe the baseline methods to be simulated for performance comparison. We first +consider a random assignment strategy (PA-RA) where pilots are assigned randomly for each +channel estimation. The strategy does not impose any complexity but yields mediocre channel +estimation performance. We also consider an exhaustive method (PA-ES) where the entire T K +p +combinations of pilots are searched to find the PA having the lowest MSE, which is evaluated +using βkm and σ2 assumed to be known a priori. PA-ES provides the best MSE performance but +is considered impractical in terms of computational complexity as the search space exponentially +increases with the number of users. We also consider two PA algorithms in the recent literature: +PA strategies using Tabu-search [30] and Hungarian [32] methods. Tabu-search-based PA (PA-TS) +utilizes the Tabu-search framework to find the MSE-minimizing pilot combination while the PA +using the Hungarian algorithm (PA-HG) iteratively solves a reward matrix to find the PA solution. +Both strategies require prior knowledge of βkm and σ2 and have computational complexity +that becomes prohibitive as the number of users increases. Note that these baseline methods +do not consider practical framework (e.g., distributed or decentralized PA) but simply rely on +a centralized processor, which makes them hard to integrate into O-RAN architecture. Also, +they do not take the user mobility into account and fail to adapt to the change imposed by the + +23 +TABLE I +Comparisons of key properties among different PA algorithms +PA Algorithm +O-RAN +Scalable +Decentralized +Possible without +Adaptive to +integrated +prior channel knowledge +mobility +PA-RA + + + + + +PA-TS [30] + + + + + +PA-HG [32] + + + + + +PA-ES + + + + + +PA-DRL + MSG + + + + + +PA-DRL + MSG + CBS + + + + + +time-varying dynamics. +We next discuss our PA scheme to be simulated for detailed evaluation. We conduct the +learning process described in Sec. III-D with inter-DU message passing (PA-DRL+MSG), i.e., +�p(ℓ) +u +is computed by each O-DU and transferred to the agent. In addition, we apply the CS +scheme described in Sec. III-E along with PA-DRL+MSG (PA-DRL+MSG+CBS) to assess the +improvement brought by adjusting the codebook orientation across O-DUs. As our PA scheme is +specifically tailored to O-RAN architecture, practical implementation with scalable computation +is possible. Since we base our learning on the DRL framework, which offers training that is +adaptive to the dynamic environment, and conduct CS that checks the real-time observation, +our PA scheme can reflect the user mobility. The properties of the algorithms regarding several +practical aspects are summarized in Table I. +We evaluate the performance of our proposed PA scheme over two different metrics: (i) the +sum-MSE defined for the objective function in Pu, i.e., � +k∈K MSE(i) +k , and (ii) the runtime it +takes to obtain the converged MSE. For the numerical results, we run each scenario 50 times +and take their average to make our analysis statistically significant. +B. Performance of O-RAN CFmMIMO +1) Impact of PA on channel estimation: We first demonstrate the impact of PA on channel +estimation in our O-RAN CFmMIMO system. We provide sum-MSE versus signal to noise ratio +(SNR) plots for different values of Tp and K in Fig. 7a where we define SNR as +1 +σ2. +Now we discuss several facts which are observed from the plots in Fig. 7a. First, we see that +Tp = 8 yields lower MSE than Tp = 4. It is expected since the number of users sharing the same +pilot tends to be smaller for larger Tp. Next, for lower SNRs, the MSE gap between PA-RA + +24 +20 +30 +40 +50 +SNR (dB) +10 +1 +10 +0 +Sum-MSE +PA-RA (Tp = 4) +PA-ES (Tp = 4) +PA-RA (Tp = 8) +PA-ES (Tp = 8) +Zero Interference +20 +30 +40 +50 +SNR (dB) +(a) Sum-MSE vs. SNR with K = 24 (left) and K = 36 (right). +0 +2000 +4000 +6000 +8000 +RT Loop Index, i +0.34 +0.36 +0.38 +0.40 +0.42 +0.44 +0.46 +Moving-averaged Sum-MSE +Nnear = 20, L = 4 +Nnear = 10, L = 4 +Nnear = 20, L = 9 +Nnear = 10, L = 9 +Nnear = 20, L = 19 +Nnear = 10, L = 19 +(b) Sum-MSE vs. RT loop with K = 24. +Fig. 7: Sum-MSE vs. SNR plot in terms of Tp and K (left) and sum-MSE vs. RT loop plot in terms of Nn and L (right). +and PA-ES is not significant since the noise dominantly contributes to channel estimation error. +However, as SNR increases, interference due to PC becomes more dominant and forces an error +floor, making the curves almost horizontal. For the case of 50 dB SNR, we find that with Tp = 4 +and K = 24, optimizing PA can reduce the sum-MSE up to 27%. For the remaining experiments, +we use SNR of 50 dB to focus on the interference-limited regime. +2) Impact of O-RAN parameters: We assess the impact of O-RAN-dependent system parameters +on the performance of our PA scheme. The sum-MSE performance curves (moving-averaged +with a window size of 500) of PA-DRL+MSG over the O-RAN RT loop for different values of +Nn and L are shown in Fig. 7b. Recall that Nn is the number of RT loops for a single near-RT +loop, and L is the number of extra experiences generated per near-RT loop by the agent. Both +Nn and L are dependent on the capability of O-RAN in which CFmMIMO network is built. +Now, we make the following observations from Fig. 7b. First, regardless of the parameter +values, our scheme shows stabilized (i.e., converged) sum-MSE performance, which verifies the +effectiveness of our learning when implemented under O-RAN architecture. Second, a lower Nn +yields improved MSE regardless of L. Here, lower Nn implies more near-RT loops during the +given number of RT loops, allowing agents to interact with the environment more frequently +and take more actions to find better solutions. Third, a higher L (more internal loops) allows +us to achieve greater sum-MSE reduction in earlier RT loops, validating that more experiences +collected in replay memory within the same period are beneficial. Thus, with greater size of +datasets available, our scheme is expected to find the PA faster with low sum-MSE. + +25 +0 +2000 +4000 +6000 +8000 +RT Loop Index, i +0.300 +0.325 +0.350 +0.375 +0.400 +0.425 +0.450 +Moving-averaged Sum-MSE +PA-RA +PA-HG (est. pathloss) +PA-TS (est. pathloss) +PA-DRL +PA-HG (true pathloss) +PA-DRL + MSG +PA-TS (true pathloss) +PA-DRL + CBS +PA-DRL + MSG + CBS +PA-ES +(a) K = 24 users +0 +2000 +4000 +6000 +8000 +RT Loop Index, i +0.80 +0.85 +0.90 +0.95 +1.00 +1.05 +1.10 +Moving-averaged Sum-MSE +PA-RA +PA-HG (est. pathloss) +PA-TS (est. pathloss) +PA-HG (true pathloss) +PA-DRL +PA-TS (true pathloss) +PA-DRL + CBS +PA-DRL + MSG +PA-DRL + MSG + CBS +PA-ES +(b) K = 36 users +Fig. 8: Sum-MSE performance of different PA schemes over 24 stationary users (left) and 36 stationary users (right). +C. Performance Comparison Against Different Baselines +Now we assess our proposed PA scheme and compare its performance with several baselines +over two metrics: channel estimation MSE and algorithm runtime. +1) Comparison in MSE: First, we consider static scenarios, i.e., vk = 0, ∀k. The plots showing +sum-MSE performance (moving-averaged with a window size of 500) over RT loops for K = 24 +and K = 36 are presented in Fig. 8a and Figs. 8b, respectively. Note that the PA solutions +obtained by PA-HG, PA-TS, and PA-ES required true pathloss information and were fixed for the +entire RT loops. Among these approaches, it is verified from both figures that PA-ES yields much +better MSE performance than PA-TS and PA-HG. We also considered the case where PA-HG +and PA-TS are conducted using the estimated pathloss, which yields a considerable performance +gap compared to the case of using true pathloss knowledge. Given that these baselines require +prior knowledge (preferably accurate) to achieve the given performance, our learning-based PA +scheme, which does not impose such requirement, is still able to show competitive performance +against them. PA-DRL+MSG clearly outperforms PA-HG and PA-TS with estimated pathloss +and provides comparable performance with the ones with true pathloss. Once we utilize CS +scheme, our proposed PA-DRL+MSG+CBS shows significant improvement and achieves better +performance than PA-ES as a result of jointly optimizing both PA and codebook orientation. +Next, we consider scenarios in which users move over time (i.e., β(i) +km changes over i, and +vk > 0, ∀k ∈ K). Fig. 9 shows the sum-MSE performance (moving-averaged with a window +size of 500) of different PA algorithms with K = 24 evaluated at three different user velocities: + +26 +0 +2000 +4000 +6000 +8000 +RT Loop Index, i +0.34 +0.36 +0.38 +0.40 +0.42 +0.44 +0.46 +Moving-averaged Sum-MSE +PA-DRL + MSG + CBS +PA-HG (true pathloss) +PA-TS (true pathloss) +PA-ES +(a) Velocity = 1 km/h. +0 +2000 +4000 +6000 +8000 +RT Loop Index, i +0.34 +0.36 +0.38 +0.40 +0.42 +0.44 +0.46 +Moving-averaged Sum-MSE +PA-DRL + MSG + CBS +PA-HG (true pathloss) +PA-TS (true pathloss) +PA-ES +(b) Velocity = 3 km/h. +0 +2000 +4000 +6000 +8000 +RT Loop Index, i +0.34 +0.36 +0.38 +0.40 +0.42 +0.44 +0.46 +Moving-averaged Sum-MSE +PA-DRL + MSG + CBS +PA-HG (true pathloss) +PA-TS (true pathloss) +PA-ES +(c) Velocity = 5 km/h. +Fig. 9: MSE performance of different PA schemes over 24 mobile users with different velocities: 1 km/h, 3 km/h, and 5 km/h. +1, 3, and 5 km/h. PA solution obtained by the baselines at the beginning (i.e, i = 0) becomes +less effective as time advances, showing a different degree of steady increase by the velocity. +Unlike these baselines, as our PA and CS schemes make their decisions based on the real-time +observations, in the proposed PA-DRL+MSG+CBS, PAs can be performed in an adaptive manner, +maintaining its performance as shown in Fig. 9. Hence, our scheme can provide competitive +performance with the prior knowledge-constrained baseline methods under a dynamic environment. +Overall, our scheme provides satisfactory performance in MSE as it exploits the decentralized +architecture of O-RAN CFmMIMO via distributed learning and codebook adjustment. +2) Comparison in algorithm runtime: Now, we evaluate and compare the computational +complexity of different PA algorithms. We first provide the runtime measurements of different PA +methods with various number of users K in Fig. 10a. The complexities for PA-TS and PA-HG, +which are respectively O(NtabuK2M) [30] and O(KT 3 +p ) [32], are confirmed by our experimental +result that shows a polynomial increase. Hence, both PA-TS and PA-HG are rendered impractical +when PA needs to perform over a CFmMIMO network with a growing network size. Meanwhile, +our PA algorithm shows a relatively negligible increase, implying its effectiveness in scalability. +The steady runtimes from our PA scheme are due to the utilization of (i) O-RAN architecture +where duration-varing tasks are distributed across the network and (ii) DNNs of fixed size which +only perform a forward computation to determine each pilot update step over near-RT loop. We +observe a slight increase in runtime when we consider inter-DU messages into our PA scheme +because generating a new set of messages imposes extra computations. Note that our CS scheme +barely adds any runtime as it utilizes the rewards already computed during our PA scheme. We +hence conclude that our low-complexity PA scheme is a scalable strategy that supports large-scale + +27 +30 +40 +50 +60 +70 +Number of Users, K +5 +10 +15 +20 +25 +30 +Total Runtime (s) +PA-TS +PA-HG +PA-DRL + MSG + CBS +PA-DRL + MSG +PA-DRL + CBS +PA-DRL +(a) Comparison over K. +5 +10 +Size of Codebook, Tp +2 +4 +6 +8 +Total Runtime (s) +PA-TS +PA-HG +PA-DRL + MSG + CBS +PA-DRL + MSG +PA-DRL + CBS +PA-DRL +10 +15 +20 +Size of RU Cluster +(b) Comparison over Tp (left) and MUE +k +(right). +Fig. 10: Runtime comparison of different PA schemes over different K values (left) and Tp and size of RU cluster values (right). +Until converged +Set time = 2.5 s +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +Sum-MSE +PA-HG +PA-TS +PA-DRL + MSG +PA-DRL + MSG + CBS +Fig. 11: Sum-MSE comparison of different PA schemes with a fixed runtime (deadline). PA-ES is not included since the +considered runtime (i.e., 2.5 seconds) is too short to evaluate the reliable performance of exhaustive search. +CFmMIMO systems. Note that PA-ES, which is the best baseline in MSE minimization, requires +an extreme amount of runtime as it searches over all T K +p +combinations of PA. On the other +hand, PA-RA requires no extra runtime but shows much worse MSE performance than other PA +schemes (Figs. 8a and 8b). +Next, we assess the total runtime required to conduct PA algorithms over different values of +Tp (left) or M UE +k +(right) in Fig. 10b. For varying Tp (the length of pilot), only PA-HG shows +undesirable behavior in complexity since the size of the reward matrix used in the Hungarian +algorithm depends on Tp. With respect to M UE +k +(the size of RU cluster), both PA-TS and PA-HG +display a linear increase. Meanwhile, our proposed scheme provides consistent runtimes for both +parameters, which verifies their scalability to support a network with large system parameters. + +28 +3) Comparison in MSE with channel estimation deadline: We next demonstrate the impact of +having a channel estimation deadline (in terms of runtime) on the MSE performance to consider +practical scenarios where time resource for PA can be strictly limited. In Fig. 11, we provide a +bar chart summarizing the runtime measurements for K = 36 stationary users in two different +cases: (i) PA algorithms run until the MSE performance converges and (ii) algorithms only run +for 2.5 seconds runtime. As expected, when the time constraint is imposed, every PA algorithm +shows degradation in sum-MSE as compared to the case where the algorithms fully run until +converged. Both PA-HG and PA-TS algorithms show significant increase in their MSE since the +amount of runtime allocated is considerably lower than the runtime required for convergence. +Meanwhile, our proposed PA scheme show relatively less increase in MSE due to its scalable +runtime which is not impacted by the time constraint significantly. This result once again confirms +the computational advantage of our PA scheme over the baseline methods. +V. CONCLUSION +In this paper, we developed a learning-based PA scheme for the decentralized CFmMIMO system +framed in O-RAN. We adopted O-RAN as a practical system architecture where distinct network +functions and multi-timescale control loops efficiently govern the framework of our scheme. After +formulating the PA problem and designing the corresponding Markov game model, we developed a +PA algorithm based on the MA-DRL framework. 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TR 36.814 V9.2.0, March 2017. + diff --git a/idE3T4oBgHgl3EQf4wsK/content/tmp_files/load_file.txt b/idE3T4oBgHgl3EQf4wsK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cbb10d2113a68b2dfc5f032e741a125826bd01d1 --- /dev/null +++ b/idE3T4oBgHgl3EQf4wsK/content/tmp_files/load_file.txt @@ -0,0 +1,1214 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf,len=1213 +page_content='1 A Decentralized Pilot Assignment Methodology for Scalable O-RAN Cell-Free Massive MIMO Myeung Suk Oh, Student Member, IEEE, Anindya Bijoy Das, Member, IEEE, Seyyedali Hosseinalipour, Member, IEEE, Taejoon Kim, Senior Member, IEEE, David J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Love, Fellow, IEEE, and Christopher G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Brinton, Senior Member, IEEE Abstract Radio access networks (RANs) in monolithic architectures have limited adaptability to supporting different network scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Recently, open-RAN (O-RAN) techniques have begun adding enormous flexibility to RAN implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' O-RAN is a natural architectural fit for cell-free massive multiple- input multiple-output (CFmMIMO) systems, where many geographically-distributed access points (APs) are employed to achieve ubiquitous coverage and enhanced user performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In this paper, we address the decentralized pilot assignment (PA) problem for scalable O-RAN-based CFmMIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We propose a low-complexity PA scheme using a multi-agent deep reinforcement learning (MA-DRL) framework in which multiple learning agents perform distributed learning over the O-RAN communication architecture to suppress pilot contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Our approach does not require prior channel knowledge but instead relies on real-time interactions made with the environment during the learning procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In addition, we design a codebook search (CS) scheme that exploits the decentralization of our O-RAN CFmMIMO architecture, where different codebook sets can be utilized to further improve PA performance without any significant additional complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Numerical evaluations verify that our proposed scheme provides substantial computational scalability advantages and improvements in channel estimation performance compared to the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Index Terms M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Oh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Das, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Love, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Brinton are with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, 47907 USA (e-mail: {oh223, das207, djlove, cgb}@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Hosseinalipour is with the Department of Electrical Engineering, University at Buffalo, NY, 14260 USA (email: alipour@buffalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Kim is with the Department of Electrical Engineering and Computer Science, the University of Kansas, Lawrence, KS, 66045 USA (email: taejoonkim@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='04774v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='SP] 12 Jan 2023 2 Open-RAN (O-RAN), cell-free massive MIMO, deep reinforcement learning, pilot assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' INTRODUCTION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Open Radio Access Network (O-RAN) Next generation wireless technologies will likely employ many dispersed radio access networks (RANs) for ubiquitous coverage and enhanced user performance [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' However, interconnecting different RANs to create one seamless network requires well-defined network functions and interfaces which are flexible in their integration capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Recently, the evolution of software- defined open RAN (O-RAN) solutions have added enormous flexibility to the implementation of current 5G networks [3]–[5] and development of emerging 6G networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' O-RAN offers software-defined disaggregation on virtual network functions (VNFs) and necessary interfaces to support their coordination, allowing system implementations that are adaptive to various architectural settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' With this openness and flexibility, O-RAN promotes interoperability across different RAN vendors and allows network operators to adapt to different wireless environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' O-RAN adopts the functional split defined in 3GPP [6] and defines three distinct units [7]: the open central unit (O-CU), open distributed unit (O-DU), and open radio unit (O-RU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Moreover, O-RAN operation is divided into three different control loops [7]: the real-time (RT), near-RT, and non-RT loops executing at different time-scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The resulting O-RAN architecture, and standard names of interfaces between these elements which enable practical implementations of many RAN operations, are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' O-RAN offers two types of RAN intelligent controllers (RICs) [7] as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 1a: near-RT RIC and non-RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Each of these RICs handles tasks manageable in different time-scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' O-RAN offers virtualization of both RICs, which promotes flexibility in implementing data-driven intelligence tasks that will be key components of emerging wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Various operations can be implemented via custom third-party applications called xApps/rApps [7], allowing RICs to be much more accessible to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In this work, we will consider the implementation of machine learning (ML) algorithms over these RICs to optimize pilot signal assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Due to these aforementioned advantages offered by O-RAN, a number of opportunities to utilize O-RAN on future wireless technologies seem promising, some of which are: Massive multiple-input multiple-output (MIMO) beamforming (BF): To implement ML-based BF strategies that handle both latency-intensive (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', RT beam selection) and data-intensive 3 Near-RT Control Loop 10ms < < 1s Non-RT RIC rApp xApp xApp Near-RT RIC A1 O-RU RAN Database E2 O-FH O-DU O-DU O-RU O-RU O-RU O-RU O-RU Non-RT Control Loop > 1s RT Control Loop < 10ms rApp O-CU O-CU O-DU F1 (a) O-RAN architecture with different types of control loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' O-DU O-RU User O-Cloud Uplink Pilot Transmission User-centric RU Clusters Backhaul O-FH RIC VNF Database Inter-DU Connection (b) A decentralized CFmMIMO system realized in O-RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 1: Illustrations of O-RAN architecture (left) and decentralized O-RAN CFmMIMO system (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', policy update via real-world dataset) tasks is challenging, and O-RAN provides a platform for realizing their framework [8]–[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' ML tasks are implemented in RICs, and BF operation can be split over O-RU and O-DU (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', option 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='2x [11]) to maximize computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Unmanned aerial vehicle (UAV) network: UAVs are typically deployed in dynamic environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', emergency rescue and aerial surveillance [12]), where the network infrastructure is required to be extremely flexible and adaptive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Flexibility and interoperability offered by O-RAN can be exploited to meet this architectural need [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Localization via channel charting: Channel charting is a data-driven localization technique [15] that maps a user to radio geometry using channel information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For the practical implementation of channel charting, O-RAN can offer a balanced distribution of heavy computational load coming from the data that is consistently collected and updated for each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Cell-free Massive MIMO One innovative idea to address the shortcomings of 5G cellular networks is to remove cell boundaries using many dispersed transmission/reception points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' This idea falls within the academic definition of cell-free massive MIMO (CFmMIMO) [16]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' By deploying many geo-distributed access points (APs), CFmMIMO system alleviates the existing cell-edge problems by substantially improving both the reliability [19] and energy efficiency [20] compared to cellular massive MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' These enhancements are due to the user-centric paradigm offered by CFmMIMO, where a group of APs are dynamically selected to form a cluster to serve each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In early CFmMIMO literature, a system with APs connected to a single processing unit (PU) was considered for centralized operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' However, in a scalable system where the number of users 0User4 and APs grow large, the resulting complexity becomes prohibitive [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Thus, CFmMIMO with multiple decentralized PUs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 1b), each of which is connected to a disjoint subset of APs, has been introduced to consider scalability [21]–[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The decentralization allows the system to scale but still be practical by reducing computational and fronthaul load on each PU [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Nevertheless, implementing centralized CFmMIMO techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', signal adaptation and resource allocation) into a decentralized architecture is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' CFmMIMO Pilot Assignment Problem In CFmMIMO, reliable channel estimation at both transmitter and receiver is absolutely critical to facilitate advanced diversity and signal processing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For channel estimation, a set of orthogonal pilots are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' However, when the number of users grows beyond the number of available pilots, some users must share their pilots with others, leading to pilot contamination (PC) that can significantly degrade the channel estimation performance [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' To cope with PC, various pilot assignment (PA) methods have been studied in the CFmMIMO literature [26]–[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In [26], a greedy PA scheme with iterative pilot updates was proposed to mitigate PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' A dynamic pilot reuse scheme to acquire a set of user-pairs for pilot sharing was proposed in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In [29], a user-group PA strategy, in which the same pilot is assigned to users with minimum overlapping APs, was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Other methods to solve the PA problem include k-means clustering [27], graph coloring [31], tabu-search [30], and Hungarian [32] algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' These prior works [26]–[32], however, require a centralized processing for PA and thus are not scalable computationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' They also utilize closed-form expressions derived from Bayesian estimation, requiring any relevant information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', pathloss) to be known a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For large-scale systems, especially under a dynamic environment, accurate prior information is often not available, underscoring the need to develop a PA scheme that does not require prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Overview of Methodology and Contributions Motivated by the aforementioned challenges, we focus on PA in scalable CFmMIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' As CFmMIMO deploys a large number of APs for ubiquitous coverage, it is crucial to maintain a great level of implementation flexibility and interoperability across different RANs for scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Hence, we propose to design our CFmMIMO system in O-RAN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' As O-RAN keeps balance in operational complexities and computational loads via functional split along the network (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', O-RU/DUs and RICs), O-RAN becomes a natural solution for scalable CFmMIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 5 Based on the O-RAN CFmMIMO system, we formulate a decentralized PA problem and develop a learning-based PA scheme to solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In doing so, we resort to multi-agent deep reinforcement learning (MA-DRL) framework, in which a group of agents individually perform their learning that provides a low-complexity solution without an explicit training stage [33]–[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Our PA scheme is designed to operate in the near-RT RIC of O-RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We summarize the key contributions of our work as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We design our CFmMIMO system based on the O-RAN architecture (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We specifically focus on channel estimation and pilot allocation models considering practical aspects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', fronthaul overhead and operational complexity by each functional unit), which can be adopted to the O-RAN CFmMIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We design a Markov game model (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-C) for our MA-DRL which leads to an efficient solution for our decentralized PA problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In particular, we formulate our reward based on observations that are directly measurable at the O-RUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Thus, our scheme does not require prior knowledge of channel statistics, which is different from previous PA algorithms [26]–[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Leverage the availability of RICs, we propose a novel learning-based PA scheme (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-D) aiming to minimize the total mean squared error (MSE) across the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' By adopting the distributed learning framework of MA-DRL, our scheme provides low-complexity PA solutions and therefore offers scalability to support large-scale systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Utilizing the decentralization of our system, we consider two effective ways to improve the PA performance: (i) inter-DU message passing for observation sharing and (ii) low-complexity codebook search (CS) algorithm (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-E) that jointly operates with our PA scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Numerical results verify that these approaches can further improve the PA performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We show that our PA scheme can maintain its performance over a mobile environment, which is possible due to (i) the DRL framework that naturally performs adaptive learning and (ii) the CS algorithm with iterative greedy search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Previous PA methods only consider a static environment and do not address the user mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We numerically evaluate (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' IV) the performance of our PA scheme against the state-of- the-art [30], [32] in both channel estimation performance and computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The results show that our scheme outperforms the benchmarks in terms of sum-MSE and scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 6 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' SYSTEM MODEL AND PROBLEM FORMULATION In this section, we first describe the CFmMIMO system realized in O-RAN architecture (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' II-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Then, after describing the channel model (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' II-B), we provide details on codebook-based channel estimation (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' II-C) and formulate our decentralized PA problem (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' II-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' CFmMIMO Configuration in O-RAN Architecture We consider M single-antenna O-RUs and U O-DUs collected in sets M = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , M} and U = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , U}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We assume each O-RU is connected to one of the O-DUs in U via an open fronthaul (O-FH) connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We define MDU u ⊆ M as the set of O-RUs connected to O-DU u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We assume inter-DU connections [36] to form RU clusters that are fully user-centric since the users can be served by RUs from different sets of MDU u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We focus our work on the PA task while making an assumption that O-FH and inter-DU connections are error-free with no delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Our O-RAN CFmMIMO system is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Here, we have our O-DUs connected to O-Cloud [7] via backhaul network (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' O-Cloud is the cloud computing platform that supports the virtualized network functions (VNFs) within O-RAN, which include RICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In designing our PA scheme, we specifically focus on the near-RT RIC that communicates with O-DUs via E2 interface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Now, within the near-RT RIC, we assume U independent learning agents, each of which has a one-to-one correspondence to one of the O-DUs in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Note that we assume multiple agents to fully impose decentralization on our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Each agent in near-RT RIC conducts local learning through the O-DU and O-RUs connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In addition, we consider a single non-RT RIC interacting with the near-RT RIC via A1 interface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 1a), which is responsible for learning model updates of near-RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Next, we consider K single-antenna users in a set K = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For each user k, a user-centric RU cluster is formed such that only M UE k ≪ M O-RUs are engaged to serve the user, where we define MUE k ⊂ M to be the set of O-RUs serving user k ∈ K (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', M UE k = |MUE k | where | · | denotes the set cardinality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Each MUE k is assumed to be selected and updated using a procedure independent from our PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', radio resource control (RRC) setup procedure [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We also define KRU m ⊂ K to be the set of users served by O-RU m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Since we have U multiple agents performing PA, each user k ∈ K must belong to one of these agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' To develop user-to-agent pairings, we consider two different types of users: (i) user k whose MUE k is connected to a single O-DU u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', MUE k ⊆ MDU u , which we simply pair that user 7 O-RU 3 O-DU 3 O-RU 1 User 1 User 2 User 3 O-DU 2 O-DU 1 O-RU 2 O-RU 4 O-RU 5 O-RU 6 O-RU 7 O-RU 8 O-RU 9 PA Control Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 2: A list of our defined sets and their visual examples for the given decentralized cell-free O-RAN layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' k to the corresponding agent u, and (ii) user k whose MUE k consists of O-RUs from different O-DUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For the second type, a serving O-DU [36], which can be defined by any reasonable criterion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', the O-DU with the most number of O-RUs serving the user), is determined and paired with the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We define KDU u to be the set of users whose PA is managed by O-DU u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Here we consider a scenario with U = 3, M = 9, and K = 3, and the sets that we have defined are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Each O-DU controls three O-RUs that are closest (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', MDU 1 = {1, 2, 3}), and user-centric RU clusters with M UE k = 4 are formed for each user (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', MUE 1 = {1, 2, 4, 5}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Note that O-RU can serve multiple users (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', KRU 2 = {1, 2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Since each user needs an agent for PA, the user is paired to one of the three O-DUs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', KDU 1 = {1, 2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Time-varying Channel Model We assume a periodic channel estimation with time interval Te and indicate each estimation instance using index i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The channel between user k ∈ K and O-RU m ∈ M during channel estimation instance i is formally expressed as g(i) km = � β(i) kmh(i) km, (1) where h(i) km = µkh(i−1) km + � (1 − µ2 k)n(i) km is the small-scale fading factor following the first-order time-varying Gauss-Markov process for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The perturbation term n(i) km is a zero- mean, unit-variance complex Gaussian random variable independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=') over k, m, and i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', n(i) km ∼ CN(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' At i = 0, we assume h(0) km ∼ CN(0, 1) to be mutually independent from n(1) km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The correlation coefficient µk for user k is defined as µk = J0(2π vk c fcTe) [38], where J0(·) is the Bessel function of the first kind of order zero, vk is the velocity of user k, fc is the carrier frequency, and c = 3 × 108 m/s is the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The 8 Backhaul O-DU u User k O-FH O-RU m RIC (Agent u) Near-RT RT DU-based Channel Estimation PA Channel Estimation (a) DU-based channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Backhaul O-RU m O-DU u RIC (Agent u) User k PA Channel Estimation RU-based Channel Estimation O-FH Near-RT RT (b) RU-based channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 3: A block diagram of two different channel estimation structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' term β(i) km in (1) is the large-scale fading factor inversely proportional to the distance between user k and O-RU m at the channel estimation instance i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Codebook-based Channel Estimation We consider uplink channel estimation with Tp channel uses dedicated for each estima- tion instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' This allows Tp orthogonal pilots to be available for channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For channel estimation, user k ∈ KDU u is assigned with one of the Tp pilots in a codebook T (i) u = {φ(i) u,1, φ(i) u,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , φ(i) u,Tp}, where each φ(i) u,t for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , Tp is a unit-norm complex vector of length Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For each T (i) u , we assume mutual orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Thus, for t, t′ = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , Tp, (φ(i) u,t)Hφ(i) u,t′ = 1 if t = t′, and zero otherwise, where (·)H denotes the conjugate transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We denote the pilot assigned to user k for the channel estimation instance i as x(i) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' To conduct channel estimation, each user k ∈ K transmits the assigned pilot x(i) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The signal vector (of length Tp) received by O-RU m ∈ M is then expressed as y(i) m = X(i)g(i) m + w(i) m = � k∈K g(i) kmx(i) k + w(i) m , (2) where X(i) = [x(i) 1 x(i) 2 · · · x(i) K ] is the Tp × K pilot matrix and g(i) m = [g(i) 1m g(i) 2m · · · g(i) Km]⊤ is the channel vector (of length K) for O-RU m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Here, w(i) m ∼ CN(0, σ2ITp) is the zero-mean complex Gaussian noise vector of length Tp with covariance σ2ITp, where In is the n × n identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We discuss two different channel estimation structures within O-RAN architecture, which we illustrate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' One structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 3a) performs channel estimation at O-DU whereas the estimation occurs at O-RU in the other structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For the DU-based channel estimation, y(i) m from each O-RU m ∈ MDU u must be collected by the O-DU in RT scale, significantly increasing the scheduling and data transfer overhead on O-FH as the number of O-RUs grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Such an increasing overhead is critical for the scalability of CFmMIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' On the other hand, channel estimation at O-RU only requires the pilot information of the served users k KEKDUmmk KEKDUk9 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', {x(i) k }k∈KRU m ) to be informed to each individual O-RU in near-RT scale, which does not involve as much O-FH overhead as the DU-based estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Hence, similar to the work in [26], we assume our channel estimation to take place at O-RUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Next, in case of user-centric RU clustering, each RU m ∈ M only needs to estimate |KRU m | different channels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', {g(i) km}k∈KRU m ) associated with users in KRU m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For estimating the channel, we consider two different techniques called pilot-matching [19] and least-square [39] estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' If we set �g(i) m = [�g(i) km]⊤ k∈KRU m as the |KRU m |-length estimated channel vector from O-RU m during the channel estimation instance i, pilot-matching and least-square estimations are expressed as �g(i) m = ( ¯X(i) m )Hy(i) m (3) and �g(i) m = ( ¯X(i) m )H(X(i)(X(i))H)−1y(i) m , (4) respectively, where ¯X(i) m = [x(i) k ]k∈KRU m is the Tp × |KRU m | pilot matrix of the users served by O-RU m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Now, when some of |KRU m | users share the pilot, ¯X(i) m is not unitary (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', ( ¯X(i) m )H ¯X(i) m ̸= I|KRU m |), so the least-square estimation in (4), which utilizes the pseudo-inverse term (X(i)(X(i))H)−1 to negate the PC, yields better estimation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' However, in the least-square approach, since X(i) needs to be known to every O-RU and the size of X(i) increases linearly with K, the resulting overhead causes significant delay as the number of users grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' This motivates the pilot-matching channel estimation scheme in (3) for scalability [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The estimated channel �g(i) km between O-RU m and user k ∈ KRU m is then expressed as �g(i) km= (x(i) k )Hy(i) m = � k′∈K g(i) k′m(x(i) k )Hx(i) k′ +(x(i) k )Hw(i) m = g(i) km+ � k′∈K k′̸=k g(i) k′m(x(i) k )Hx(i) k′ +(x(i) k )Hw(i) m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' (5) Note that the summation term the in last equality captures the effect of PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Problem Formulation We use MSE of the channel estimation described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' II-C for our PA performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For user k served by the O-RUs in MUE k , we define the MSE of the channel estimate in (5) as MSE(i) k = E � � m∈MUE k ����g(i) km − g(i) km ��� 2 � = � m∈MUE k E �����g(i) km − g(i) km ��� 2� = � m∈MUE k E ���� � k′∈K k′̸=k g(i) k′m(x(i) k )Hx(i) k′ + (x(i) k )Hw(i) m ��� 2 � = � m∈MUE k � k′∈K k′̸=k β(i) k′m ���(x(i) k )Hx(i) k′ ��� 2 + σ2, (6) 10 where the expectation is taken over the channel and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The third equality holds as we substitute �g(i) km with (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Next, the last equality holds since g(i) km and w(i) m are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' across k and m with E[|g(i) km|2] = β(i) km and E[∥w(i) m ∥2 2] = σ2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' From (6), we see that the MSE is directly proportional to the interference caused by PC, and thus can be used as an effective metric to quantify the PA performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Since our system involves U agents, each of which handles the PA of user k ∈ KDU u , we can formulate the PA optimization problem for agent u as (Pu) : min {x(i) k }k∈KDU u � k∈K MSE(i) k (7) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' x(i) k ∈ T (i) u , ∀k ∈ KDU u , (8) ∥φ(i) u,t∥2 2 = 1, � φ(i) u,t �H φ(i) u,t′ = 0 if t ̸= t′, ∀t, t′ = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' (9) If β(i) km, ∀k, m is known, one can directly evaluate � k∈K MSE(i) k using (6) and solve Pu using PA algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', the previous works [26]–[32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' However, in large-scale systems, such prior knowledge is often not available, and one can no longer evaluate the objective function in a straightforward manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Suppose the knowledge is somehow available for the MSE to be evaluated, but some of these algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', PAs with Tabu-search [30] and Hungarian algorithm [32] having the complexities of O(NtabuK2M) and O(KT 3 p ), respectively) still cannot be considered as the complexity becomes prohibitive for a large number of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' To address both issues, we propose to solve Pu via a distributed learning framework, details of which are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The decentralization imposed in this work allows our PA scheme to be much more scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' SCALABLE LEARNING-BASED PILOT ASSIGNMENT SCHEME FOR O-RAN CFMMIMO In this section, we first describe how our proposed PA scheme is framed in O-RAN (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Next, after providing preliminaries on MA-DRL (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-B), we design a Markov game model perceiving our PA problem (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-C), and show that the action selection in our learning framework corresponds to minimizing the PC (Theorem 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Finally, we provide implementation details for our DRL-based PA scheme (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-D) and iterative CS algorithm (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Pilot Assignment Framework in O-RAN Architecture Our learning-based PA scheme for CFmMIMO is designed based on O-RAN architecture defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' II-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Its conceptual block diagram is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Here the PA is conducted under three different O-RAN control loops which have been described earlier in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 11 Updated Codebook Information Observation Non-RT RIC RT loop Near-RT loop Non-RT loop Near-RT RIC O-RU User Pilot Assignment Pilot Sequence Channel Estimation Pilot Assignment Information Weight Update Agent Agent O-DU Codebook Search Inter-DU Message RT Loop Near-RT Loop Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 4: A block diagram of the proposed PA scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 1) RT loop: We assume that a single round of channel estimation steps described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' II-C takes place in each RT loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Hence, we denote the index of each RT loop using the same notation used for indexing the channel estimation instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In each RT loop i, users transmit their assigned pilots, and the O-RU m completes the channel estimation to obtain �g(i) km for k ∈ KRU m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 2) Near-RT loop: Near-RT loop occurs once in every Nn RT loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' During each near-RT loop, O-DU u collects observation data, which we describe later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-C, from the O-RUs in MDU u and transfers it to the near-RT RIC to be used for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' At the same time, each agent u in the near-RT RIC conducts PA on the users in KDU u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We use ℓ = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , ⌊ N Nn⌋ to denote the index of near-RT loop, thus, ℓ-th near-RT loop occurs during the Nnℓ-th RT loop (or the Nnℓ-th channel estimation instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The relationship between i and ℓ is visualized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' To further improve our PA performance, two acceleration techniques are introduced: Inter-DU message passing: We consider inter-DU message passing which occurs at each near-RT loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The inter-DU connection is essential for fully realizing user-centric RU clusters in decentralized CFmMIMO [36], and we exploit this feature to improve our PA performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' With inter-DU messages, we aim to reinforce the data observed by the local group of O-RUs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', O-RUs of MDU u ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The details on inter-DU message passing are provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Codebook searching: We leverage the decentralization of our system and develop a CS algorithm that operates jointly with our PA scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In doing so, we adopt the idea of quasi-orthogonal codebooks [40], [41] to be used across the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In multi-cell systems, where each cell conducts its own PA to the serving users, using non-identical orthogonal codebooks across the cells has shown improved system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Inspired by this, we rotate the codebook of each agent in an iterative manner to find the codebook orientation that yields the minimum 12 MSE of channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The detailed steps of our CS scheme is provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 3) Non-RT loop: The non-RT loop is utilized to handle time-insensitive tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In our PA scheme, the update of the learning parameters for near-RT RIC occurs over this loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Here, the non-RT loop occurs once in every Nnon RT loops, and we denote q = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , ⌊ N Nnon⌋ as the non-RT loop index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' As described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 1a, a near-RT loop duration can be as short as 10 ms while the shortest duration for non-RT loop is a second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Hence, we assume Nnon ≫ Nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Preliminaries on Multi-agent Deep Reinforcement Learning MA-DRL addresses scenarios where multiple agents perform simultaneous decision-making based on a Markov game model [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For our decentralized PA problem, we define MA-DRL using a tuple ({S(ℓ) u }u∈U, {a(ℓ) u }u∈U, {r(ℓ) u }u∈U), where S(ℓ) u , a(ℓ) u , and r(ℓ) u are respectively the state, action, and reward of the agent u during the ℓ-th near-RT loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For each loop ℓ, agent u with a state S(ℓ) u makes an action a(ℓ) u to interact with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Subsequently, the agent makes an observation and computes a reward r(ℓ) u which helps to find the next state S(ℓ+1) u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In the non-RT loop, once an agent has completed multiple interactions with the environment, its policy on action selection for a given state is optimized by updating the weights of its respective deep neural network (DNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Here the action is determined based on the Q-value [43] denoted by Q(S(ℓ) u , a(ℓ) u ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The Q-value quantifies the quality of an agent’s action for a given state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Thus, it is important for the agent to obtain accurate Q-values to make correct decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In DRL, these Q-values are computed via a DNN, the weights of which are trained with experiences so that a correct (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', Q-value-maximizing) action can be selected upon each decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Now, in perceiving our PA task as a multi-agent learning problem, there are two conditions we need to consider [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' First, multiple agents making independent decisions simultaneously implies the environment is never seen as stationary to an action of a single agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Second, due to the decentralized architecture, each agent only obtains a part of the observation available from the entire environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Due to these conditions, in multi-agent learning, careful design of the Markov game model is crucial for achieving performance comparable to centralized learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Markov Game Model for Decentralized Pilot Assignment In our O-RAN CFmMIMO setting, channel estimation is repeated for every RT loop i, forming a periodic interaction with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The near-RT PA corresponds to action selection that affects the environment and resulting observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Based on this, we formally define each component of the tuple presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-B to perceive our PA task as a Markov game model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 13 1) States: To represent the PA status of agent u on users in KDU u at the start of near-RT loop ℓ, we define the state as S(ℓ) u = Φ(ℓ) u which is a |KDU u | × Tp sized matrix where [Φ(ℓ) u ]k,t = � � � � � 1 if x(Nnℓ) k = φ(Nnℓ) u,t , 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' (10) As discussed previously, PC occurs when users share a pilot, and this can be indicated by the ones in each column of Φ(ℓ) u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Hence, Φ(ℓ) u can become an effective means to represent the condition of PA for each agent, and we aim to have the agents accurately perceive the relationship between their PA (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', their actions) and the resulting PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 2) Actions: We consider sequential updates on the pilots, where the pilot of only a single user is changed with every action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' If we consider actions that assign pilots to all |KDU u | users at once, this would lead our action space to take T |KDU u | p possible combinations and suffer from the “curse of dimensionality”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We hence define actions as 2-tuples indicating the user of interest and the pilot to be assigned, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The action of agent u at near-RT PA ℓ is formally defined as a(ℓ) u = (k, t), where k ∈ KDU u and t ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , Tp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' With this setting, there are total |KDU u |Tp possible actions for agent u to take, resulting in a more computationally scalable action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 3) Rewards: We propose to compute the reward of each agent u on the ℓ-th near-RT PA based on the average sum-power of the channel estimates obtained by the O-RUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Note that, for each action (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', near-RT PA) taken by an agent, Nn channel estimations are conducted by O-RU m to acquire a set of �g(i) m for Nnℓ ≤ i < Nn(ℓ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Using this information, O-RU m computes p(ℓ) km = 1 Nn Nn−1 � n=0 ����g(Nnℓ+n) km ��� 2 (11) on user k ∈ KRU m during the near-RT loop ℓ and sends it to the corresponding O-DU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' At the end of this transfer, O-DU u collects different sets of p(ℓ) km from each O-RU m ∈ MDU u (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', {{p(ℓ) km}k∈KRU m }m∈MDU u ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In decentralized PA, each agent u ∈ U is responsible for a disjoint subset of K users, and it is desirable for the agent to have access to p(ℓ) km from all O-RUs associated with the users (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', {{p(ℓ) km}m∈MUE k }k∈KDU u ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' However, as each O-DU u is only connected to O-RUs of MDU u , {{p(ℓ) km}m∈MUE k ∩MDU u }k∈KDU u only gets collected by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Hence, O-DU u ends up computing the observation data to be transferred to the agent u as ¯p(ℓ) u = � k∈KDU u � m∈MUE k ∩MDU u p(ℓ) km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Note that the rest of information required by agent u (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', {{p(ℓ) km}m∈MUE k \\MDU u }k∈KDU u ) has been collected by other O-DUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' As mentioned earlier in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-A, since we consider inter-DU 14 messages, this information can be transferred to each corresponding O-DU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Then, each O-DU u can now compute the reinforced observation data which is expressed as �p(ℓ) u = ¯p(ℓ) u + � k∈KDU u � m∈MUE k \\MDU u p(ℓ) km = � k∈KDU u � m∈MUE k p(ℓ) km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' (12) The observation data computed by O-DU u in (12) is transferred to agent u via backhaul, and the reward for agent u at near-RT loop ℓ is subsequently computed using the mapping function r(ℓ) u (p) = (pmax − p)/(pmax − pmin), (13) where p = �p(ℓ) u by the availability of inter-DU message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The mapping function (13) converts the observation data into a reward range such that lower values of p are rewarded higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Here [pmin, pmax] is the range of observation data, which we assume is set by the non-RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We now show that the learning via our Markov model leads to taking an action that minimizes the degree of PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The basic mechanism of learning we utilize is that, for each given state Su, we want the agent u to select the action that maximizes its Q-value [43], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', a⋆ u = arg max au∈Au Q(Su, au), (14) where Au is the set of all possible actions for agent u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The training in DRL is done by updating the network weights via regression toward the experiences obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The Q-value, which is the numerical output of the trained network, is then expected to follow the average of these experiences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', the Q-value is updated through training to yield Q(Su, au) = E[ru(p)|(Su, au)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For each near-RT loop ℓ, the following theorem shows that, with inter-DU message passing, the action selected via (14) is the best action in terms of minimizing the degree of local PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' With �p(ℓ) u available, for a given state S(ℓ) u , taking the action a(ℓ) u which satisfies (14) is equivalent to finding the action that minimizes the degree of pilot contamination occurring on local users in KDU u during the near-RT loop ℓ, which is expressed as � k∈KDU u � m∈MUE k Nn−1 � n=0 � k′∈K,k′̸=k β(Nnℓ+n) k′m ���(x(Nnℓ) k )Hx(Nnℓ) k′ ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' (15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' First, in terms of the parameters defined in our model, we find the expected reward at near-RT loop ℓ for a given state-action pair (S(ℓ) u , a(ℓ) u ), which is expressed as E[r(ℓ) u (�p(ℓ) u )|(S(ℓ) u , a(ℓ) u )] = pmax − E[�p(ℓ) u ] pmax − pmin , (16) 15 where the equality holds from (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Recalling (14), the learning conducted at each agent u aims to find the action achieving the maximum Q-value Q(Su, au), which we discussed to yield E[ru(p)|(Su, au)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Thus, the action selection mechanism of agent u can be expressed as a(ℓ) u = arg max au∈Au E[r(ℓ) u (�p(ℓ) u )|(S(ℓ) u , au)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' (17) Now combining (16) and (17), we can say that a(ℓ) u = arg min au∈Au � k∈KDU u � m∈MUE k E[p(ℓ) km] = arg min au∈Au 1 Nn � k∈KDU u � m∈MUE k Nn−1 � n=0 E �����g(Nnℓ+n) km ��� 2� , (18) where the first and second equalities are obtained using (12) and (11), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Now, for n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , Nn − 1, using (5) we have E �����g(Nnℓ+n) km ��� 2� = E ����g(Nnℓ+n) km ��� 2� + � k′∈K k′̸=k E ����g(Nnℓ+n) k′m (x(Nnℓ+n) k )Hx(Nnℓ+n) k′ ��� 2� + E ����(x(Nnℓ+n) k )Hw(Nnℓ+n) m ��� 2� = β(Nnℓ+n) km + ξ(ℓ,n) km + σ2, (19) where ξ(ℓ,n) km = � k′∈K k′̸=k β(Nnℓ+n) k′m ��(x(Nnℓ+n) k )Hx(Nnℓ+n) k′ ��2 reflects the PC discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' II-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' By the definition of �g(i) km in (5), taking the expectation of |�g(i) km|2 leaves only the autocorrelation terms for �g(i) km and w(i) m , corresponding to β(Nnℓ+n) km = E[|g(Nnℓ+n) km |]2 and σ2 = E[|(x(Nnℓ+n) k )Hw(Nnℓ+n) m |]2 in (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' This is because the channel and noise are assumed uncorrelated across k and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Now, since (i) ξ(ℓ,n) km is the only term that is impacted by action au, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', β(Nnℓ+n) km and σ2 in (19) are independent from PA and (ii) x(i) k only changes once every Nn RT loops, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', x(Nnℓ+n) k is fixed for n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , Nn − 1, by ignoring 1 Nn as a scaling factor, (18) is equivalent to a(ℓ) u = arg min au∈Au � k∈KDU u � m∈MUE k Nn−1 � n=0 � k′∈K,k′̸=k β(Nnℓ+n) k′m ���(x(Nnℓ) k )Hx(Nnℓ) k′ ��� 2 , (20) which represents the degree of PC at near-RT loop ℓ over the users in KDU u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' ■ From Theorem 1, we conclude that learning based on our Markov games model is equivalent to performing the pilot update which minimizes the interference due to PC at each near-RT PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' According to (15), the PA made at each near-RT loop ℓ couples with the pathloss occurring over the corresponding Nn RT loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Since we do not assume prior knowledge on the pathloss β(i) km, we cannot evaluate the exact MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' However, through the reward we define and the learning mechanism of DRL, we can still design our PA scheme such that the MSE performance is 16 Replay Memory Action State Agent in near-RT RIC Reward Observation Random Action Compute Reward Train DNN Weight Copy Deep Q-Network Target DNN Experience O-DU User Inter-DU Message O-RU Compute Reward Experience Weight Update Evaluate Codebook Rotate Codebook Codebook Decision Codebook Search Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 5: A block diagram overview of our PA scheme, consisting of non-RT DNN training and near-RT PA updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' improved over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For static scenarios, where β(i) km is constant over i, all the actions taken over near-RT loops (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', the entire series of successive pilot updates) are contributing to look for a single optimal PA solution that minimizes the sum-MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' On the other hand, for mobile scenarios, each action is led to focus on minimizing the sum-MSE resulted from the current channel statistics by leveraging the past information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Our PA scheme is designed to cope with time varying small-scale and large-scale fading factors upon continuous training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' MA-DRL-based Pilot Assignment Algorithm Given the setting in the previous subsections, we describe our PA algorithm in detail which uses MA-DRL framework to find the solution to our decentralized PA problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We incorporate MA- DRL using the deep Q-network (DQN), which utilizes neural network layers for approximating Q-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' An individual DQN is implemented at each agent in the near-RT RIC for distributed learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 5 provides an overview of our methodology which is also outlined in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We detail each of the steps in the following: Near-RT PA: At ℓ = 0, each agent u randomly assigns one of the Tp sequences in T (0) u to its associated users in KDU u , from which the state S(0) u is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For each subsequent near-RT loop ℓ, the agent u takes an action a(ℓ) u via an ϵ-greedy method [43] and assigns a different pilot sequence to one of its users, obtaining a new state S(ℓ+1) u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Since Nn RT channel estimations occur during a single loop of near-RT PA, each O-DU u collects necessary information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', {p(ℓ) km}k∈KRU m , from the O-RUs in MDU u and computes �p(ℓ) u with the aid of inter-DU message passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The O-DU transfers �p(ℓ) u to its agent in the near-RT RIC, which computes the reward r(ℓ) u (p) and stores an experience tuple (S(ℓ) u , a(ℓ) u , r(ℓ) u (p), S(ℓ+1) u ) in a replay memory of size Dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' k E K!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='kmm EMDU2 ku(l,l)rint,u17 Algorithm 1: Proposed Pilot Assignment (PA) Scheme 1 Input: Pilot length Tp, number of RT loops N, number of RT loops per near-RT loop Nn, number of internal loops L, set of users managed by O-DU u KDU u , set of O-RUs managed by O-DU u MDU u , set of users served by O-RU m KRU m , set of O-RUs serving the user k MUE k , training period, update period 2 Initialize near-RT loop index ℓ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' randomize the parameter vectors θtr u and θta u 3 Generate codebook T (Nnℓ) u ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' randomly assign {φ(Nnℓ) k }k∈KDU u 4 for ℓ = 0 to N do 5 Compute S(ℓ) u using (10) 6 if ℓ > 0 then 7 Compute r(ℓ−1) u (�p(ℓ−1) u ) using (13);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' store (S(ℓ−1) u , a(ℓ−1) u , r(ℓ−1) u (�p(ℓ−1) u ), S(ℓ) u ) in the memory 8 for l = 0 to L − 1 do 9 Select a(ℓ,l) int,u randomly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' compute S(ℓ,l) int,u using (10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' compute r(ℓ,l) int,u using (22) 10 Store (S(ℓ) u , a(ℓ,l) int,u, r(ℓ,l) int,u, S(ℓ,l) int,u) in the memory 11 if ϵ-greedy then select a(ℓ) u randomly else a(ℓ) u = arg maxau Qθtru(S(ℓ) u , au) 12 Update the PA according to a(ℓ) u 13 for i = 0 to Nn − 1 do 14 User k ∈ KDU u transmits φ(Nnℓ+i) k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' O-RU m ∈ MDU u estimates {�g(i) km}k∈KRU m using (5) 15 if mod(ℓ, training period) = 0 then generate a batch from the memory and train θtr u via SGD on (21) 16 if mod(ℓ, update period) = 0 then set θta u = θtr u 17 Output: Updated pilot sequences {φ(N) k }k∈KDU u Non-RT DNN Training: The learning of each agent u is carried out by two DNNs called the train and target networks [33], [45], where their network parameter vectors are denoted by θtr u and θta u , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Once enough experiences have been collected in the memory, a mini-batch of size Db is randomly selected from the memory and used to update θtr u minimizing the loss: L(θtr u ) = Eℓ � yℓ − Qθtru(S(ℓ) u , a(ℓ) u ) � , (21) where yℓ = r(ℓ) u + γ maxa Qθta u (S(ℓ+1) u , a) with γ being the discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Here Qθ(S, a) represents the Q-value for a given pair of state S and action a computed via a DNN of weight vector θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The update is done using stochastic gradient descent (SGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Here, the weights of θtr u are periodically copied to target network θta u , with the length of this period as a design parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Experience generation: By the O-RAN capability, the value of Nn can vary and impact the 18 rate of experiences being collected to each agent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', the number of experiences collected for a given amount of time varies by Nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' If Nn is too large, a sufficient size of data required to perform effective training may not be collected within a desired time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' To resolve the issue and utilize time more efficiently, we exploit the architecture of O-RAN and introduce an internal experience-generating loop inside the near-RT RIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' This internal loop is executed L times during a single near-RT loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In particular, once an experience is obtained via the ℓ-th near-RT loop, we generate L extra experiences by taking a random action and evaluating the corresponding reward for each internal loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We define the reward by the l-th internal loop of the ℓ-th near-RT PA as r(ℓ,l) int,u(p) = � 1 − κ(ℓ,l) u /κmax � r(ℓ) u (p), (22) where κ(ℓ,l) u = ����Tp t=1 �� k∈KDU u (φ(Nnℓ) u,t )Hx(ℓ,l) k − � |KDU u | Tp ����� is the penalty for having more than necessary number of users sharing the same pilot sequence and κmax = 2|KDU u |(Tp − 1)/Tp is the maximum penalty obtainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Integrating this internal loop alongside near-RT PA, we can generate L more experiences to accelerate the convergence of our scheme and train our DNNs to favor sequence combinations that have more evenly spread number of users across Tp sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Iterative Codebook Search (CS) Algorithm We describe our CS algorithm that is designed to work with the PA scheme in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' As each agent assigns pilots to its local users using the codebook T (i) u , CS is iteratively conducted so that the final set of U codebook sets, when combined with our PA solution, suppresses the PC to the minimum degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We detail each of the steps in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' First, we assign each agent u ∈ U with an identical codebook, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', T (0) 1 = T (0) 2 = · · · = T (0) U , and initiate our PA scheme without CS to ensure that the agents first learn and improve their PA only based on the interference resulted from pilot sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We design our algorithm to begin its iterative CS only after the learning on PA is stabilized so that the PA and CS do not impair each other from converging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We determine the PA of agent u to be stable when the state S(ℓ) u remains unchanged over Ncs near-RT loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Once the agent u has given the same PA for Ncs consecutive times at the end of near-RT loop ℓ⋆ u, the agent is perceived as stable and becomes subject for CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Note that ℓ⋆ u is likely to vary for each agent due to our decentralized PA framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' If we design our agents to conduct CS in parallel, it becomes difficult to accurately evaluate a codebook as multiple actions simultaneously affect the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Hence, we propose to have each agent take a turn and conduct CS while the rest of agents is paused from the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' To 19 implement a such design, we define an operation called the CS run in which an isolated CS is conducted for each agent u ∈ U(v) cs , where U (v) cs is the set of agents subject for CS during the v-th CS run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For each isolated search, the following steps are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Suppose it is the turn of the w-th element of U (v) cs , denoted by uv,w, to perform the isolated CS, where w = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , |U(v) cs |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We first define ℓv,w to be the near-RT loop in which the agent uv,w begins its search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We also let Ns define the number of near-RT loops to be spent for codebook evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' During the first Ns near-RT loops (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', ℓv,w ≤ ℓ < ℓv,w + Ns), the quality of current codebook matrix Told v,w = [φ(Nnℓv,w) uv,w,1 , φ(Nnℓv,w) uv,w,2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , φ(Nnℓv,w) uv,w,Tp ] is evaluated by computing ¯rold v,w = 1 Ns Ns−1 � n=0 r(ℓv,w+n) uv,w (p), (23) which is the average of the most Ns recent rewards collected at agent uv,w via our PA algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Note that (23) represents the quality of PA performed using the codebook T (Nnℓv,w) uv,w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' After obtaining (23), the agent generates a Tp × Tp column-normalized random perturbation matrix Pv,w and computes the rotation matrix as Rv,w = � 1 − η2 uv,wITp + ηuv,wPv,w, where ηuv,w = 1 − ℓv,w−ℓ⋆ uv,w N/Nn−ℓ⋆uv,w is the perturbation degree designed to decrease with ℓv,w so that a converged solution is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Note that larger ηuv,w results in Rv,w with greater perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' After acquiring Rv,w, the agent rotates the current codebook to obtain a new codebook matrix Tnew v,w = proj(Rv,wTold v,w), (24) where proj(·) is the projection function for which we use the Gram-Schmidt orthogonalization algorithm [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The set of Tp columns in Tnew v,w is then used as a new codebook for agent uv,w during the next Ns near-RT loops (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', ℓv,w + Ns ≤ ℓ < ℓv,w + 2Ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' After these Ns near-RT loops, where a set of Ns rewards using the new codebook are collected by our PA algorithm, the agent computes ¯rnew v,w = 1 Ns 2Ns−1 � n=Ns r(ℓv,w+n) uv,w (p), (25) to evaluate the quality of the new codebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' At this point, agent uv,w has evaluated (23) and (25) from using two different codebooks Told v,w and Tnew v,w, respectively, and determines which codebook to keep by the end of search using the following criterion T(Nn(ℓv,w+2Ns)) uv,w = � � � � � Tnew v,w if ¯rnew v,w > ¯rold v,w, Told v,w otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' (26) 20 Algorithm 2: Proposed Codebook Search (CS) Scheme 1 Input: Pilot length Tp, number of consistent PAs required for stability Ncs, codebook evaluation interval Ns, number of RT loops N, set of agents U 2 Initialize CS run index v = 0, set of agents subject for CS U(v) cs = ∅, the counter for agent u au = 0, CSrun = 0, and CSiso = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' assign identical codebook for all u ∈ U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' capture S(0) u using (10) 3 for ℓ = 1 to N do 4 for u ∈ U do 5 Capture S(ℓ) u using (10) 6 if S(ℓ) u = S(ℓ−1) u then au = au + 1 else au = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' if au = Ncs then ℓ⋆ u = ℓ 7 if CSrun = 0 then 8 U(v) cs = {u ∈ U|ℓ⋆ u < ℓ};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' if |U(v) cs | > 0 then w = 1 and CSrun = 1 9 if CSrun = 1 then 10 if CSiso = 0 then ℓv,w = ℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' CSiso = 1 11 if CSiso = 1 then 12 if ℓ = ℓv,w + Ns − 1 then compute ¯rold v,w using (23);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' apply new codebook Tnew v,w using (24) 13 if ℓ = ℓv,w + 2Ns − 1 then 14 Compute ¯rnew v,w using (25);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' decide codebook using (26);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' w = w + 1 and CSiso = 0 15 if w > |U(v) cs | then v = v + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' CSrun = 0 16 Output: Rotated codebook T (N) u , ∀u ∈ U As the CS described above runs for each agent in U (v) cs , total 2Ns|U(v) cs | near-RT loops are spent to complete the CS run v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For every run, each agent tries a new codebook generated using a random rotation and decides to keep whichever codebook that yields higher reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The algorithm starts its very first CS run at ℓ = minu∈U ℓ⋆ u and continuously conducts each subsequent CS run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' By changing the codebook only when it is determined to be better, the algorithm proceeds to find the best set of U codebooks that minimizes the degree of PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Note that, in order to evaluate the codebooks, our CS scheme utilizes the reward r(ℓ) u (p), which is obtained during our PA scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Therefore, no additional information needs to be collected the O-DUs to conduct the CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The overall procedure for our CS scheme is summarized in Alg 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 21 75 50 25 0 25 50 75 x (m) 40 20 0 20 40 y (m) O-RUs Users (i = 0) Users (i = N) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 6: Geographical layout of O-RAN CFmMIMO with U = 4, M = 96, and K = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' O-RUs connected to the same O-DU have the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Each user moves from the initial (circle) to the final position (cross) in 10 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' NUMERICAL EVALUATION In this section, we evaluate our pilot assignment (PA) scheme under O-RAN CFmMIMO channel estimation scenarios with various system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We analyze both channel estimation performance and computational complexity to discuss the scalability and practicality of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In addition, we compare the performance of our proposed approach against different baselines which include [30], [32] among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Simulation Setup, Performance Metrics, and Baselines We consider different combinations of O-DUs (U = 4), single-antenna O-RUs (M = 96), and single-antenna users (K ∈ {24, 36}) placed in an area of 100 m × 150 m geometry to create O-RAN CFmMIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We assume the same number of O-RUs connected to each O-DU (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', |MDU u | = M U , ∀u) and the same number of users paired with each agent in the near-RT RIC (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', |KDU u | = K U , ∀u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We set channel estimation interval Te = 1 ms, implying our O-RAN RT loop occurs once every 1 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Each scenario is simulated with maximum N = 10000 RT loops, which corresponds to 10 seconds with Te = 1 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We assume Nn = 10 RT loops to occur per O-RAN near-RT loop and L = 9 internal experience generation per near-RT loop unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For mobile scenarios, we generate initial (i = 0) and final (i = N) positions for each user such that the velocity vk ranges from 0 m/s (or 0 km/h) to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='4 m/s (or 5 km/h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Then, for each i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' , N, the position of each user is updated according to vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Such a mobile scenario for 96 × 24 CFmMIMO (where M × K refers to M O-RUs and K users) with 22 U = 4 O-DUs (equivalently, U = 4 agents in the near-RT RIC) is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The large-scale fading factor β(i) km, ∀k, m is assumed to follow the 3GPP urban-micro line-of-sight pathloss model [47] with carrier frequency fc = 2 GHz, O-RU height of 10 m, and user height of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='5 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We consider a pilot length of Tp = 4 and a RU cluster size of M UE k = 8, ∀k unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For our codebook search (CS) scheme, we consider an agent to be stable if the PA is consistent for Ncs = 100 consecutive times and assume the codebook evaluation interval Ns = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We use the same DQN design for all agents: one convolutional neural network (CNN) with 32 kernels of size |KDU u | × Tp followed by two fully connected layers of width |KDU u |Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' All layers use ReLU activation and Adam optimizer with learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The discount factor for the weight update is set γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We also set the size of replay memory Dm = 1000 and train the neural network using Db = 128 samples per minibatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The train network weights are updated via SGD and synchronized with the target network whenever 200 and 400 new additional experiences are stored in the replay memory, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We implement ϵ-greedy action-selection with the probability of selecting a random action in the ℓ-th near-RT loop computed as ϵℓ = e−(Γ/N)Nnℓ, where Γ = 15 is the scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We now describe the baseline methods to be simulated for performance comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We first consider a random assignment strategy (PA-RA) where pilots are assigned randomly for each channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The strategy does not impose any complexity but yields mediocre channel estimation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We also consider an exhaustive method (PA-ES) where the entire T K p combinations of pilots are searched to find the PA having the lowest MSE, which is evaluated using βkm and σ2 assumed to be known a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' PA-ES provides the best MSE performance but is considered impractical in terms of computational complexity as the search space exponentially increases with the number of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We also consider two PA algorithms in the recent literature: PA strategies using Tabu-search [30] and Hungarian [32] methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Tabu-search-based PA (PA-TS) utilizes the Tabu-search framework to find the MSE-minimizing pilot combination while the PA using the Hungarian algorithm (PA-HG) iteratively solves a reward matrix to find the PA solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Both strategies require prior knowledge of βkm and σ2 and have computational complexity that becomes prohibitive as the number of users increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Note that these baseline methods do not consider practical framework (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', distributed or decentralized PA) but simply rely on a centralized processor, which makes them hard to integrate into O-RAN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Also, they do not take the user mobility into account and fail to adapt to the change imposed by the 23 TABLE I Comparisons of key properties among different PA algorithms PA Algorithm O-RAN Scalable Decentralized Possible without Adaptive to integrated prior channel knowledge mobility PA-RA \x17 \x13 \x17 \x13 \x17 PA-TS [30] \x17 \x17 \x17 \x17 \x17 PA-HG [32] \x17 \x17 \x17 \x17 \x17 PA-ES \x17 \x17 \x17 \x17 \x17 PA-DRL + MSG \x13 \x13 \x13 \x13 \x13 PA-DRL + MSG + CBS \x13 \x13 \x13 \x13 \x13 time-varying dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We next discuss our PA scheme to be simulated for detailed evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We conduct the learning process described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-D with inter-DU message passing (PA-DRL+MSG), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', �p(ℓ) u is computed by each O-DU and transferred to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In addition, we apply the CS scheme described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' III-E along with PA-DRL+MSG (PA-DRL+MSG+CBS) to assess the improvement brought by adjusting the codebook orientation across O-DUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' As our PA scheme is specifically tailored to O-RAN architecture, practical implementation with scalable computation is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Since we base our learning on the DRL framework, which offers training that is adaptive to the dynamic environment, and conduct CS that checks the real-time observation, our PA scheme can reflect the user mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The properties of the algorithms regarding several practical aspects are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We evaluate the performance of our proposed PA scheme over two different metrics: (i) the sum-MSE defined for the objective function in Pu, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', � k∈K MSE(i) k , and (ii) the runtime it takes to obtain the converged MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For the numerical results, we run each scenario 50 times and take their average to make our analysis statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Performance of O-RAN CFmMIMO 1) Impact of PA on channel estimation: We first demonstrate the impact of PA on channel estimation in our O-RAN CFmMIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We provide sum-MSE versus signal to noise ratio (SNR) plots for different values of Tp and K in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 7a where we define SNR as 1 σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Now we discuss several facts which are observed from the plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' First, we see that Tp = 8 yields lower MSE than Tp = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' It is expected since the number of users sharing the same pilot tends to be smaller for larger Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Next, for lower SNRs, the MSE gap between PA-RA 24 20 30 40 50 SNR (dB) 10 1 10 0 Sum-MSE PA-RA (Tp = 4) PA-ES (Tp = 4) PA-RA (Tp = 8) PA-ES (Tp = 8) Zero Interference 20 30 40 50 SNR (dB) (a) Sum-MSE vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' SNR with K = 24 (left) and K = 36 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 0 2000 4000 6000 8000 RT Loop Index, i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='46 Moving-averaged Sum-MSE Nnear = 20, L = 4 Nnear = 10, L = 4 Nnear = 20, L = 9 Nnear = 10, L = 9 Nnear = 20, L = 19 Nnear = 10, L = 19 (b) Sum-MSE vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' RT loop with K = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 7: Sum-MSE vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' SNR plot in terms of Tp and K (left) and sum-MSE vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' RT loop plot in terms of Nn and L (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' and PA-ES is not significant since the noise dominantly contributes to channel estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' However, as SNR increases, interference due to PC becomes more dominant and forces an error floor, making the curves almost horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For the case of 50 dB SNR, we find that with Tp = 4 and K = 24, optimizing PA can reduce the sum-MSE up to 27%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For the remaining experiments, we use SNR of 50 dB to focus on the interference-limited regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 2) Impact of O-RAN parameters: We assess the impact of O-RAN-dependent system parameters on the performance of our PA scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The sum-MSE performance curves (moving-averaged with a window size of 500) of PA-DRL+MSG over the O-RAN RT loop for different values of Nn and L are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Recall that Nn is the number of RT loops for a single near-RT loop, and L is the number of extra experiences generated per near-RT loop by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Both Nn and L are dependent on the capability of O-RAN in which CFmMIMO network is built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Now, we make the following observations from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' First, regardless of the parameter values, our scheme shows stabilized (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', converged) sum-MSE performance, which verifies the effectiveness of our learning when implemented under O-RAN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Second, a lower Nn yields improved MSE regardless of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Here, lower Nn implies more near-RT loops during the given number of RT loops, allowing agents to interact with the environment more frequently and take more actions to find better solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Third, a higher L (more internal loops) allows us to achieve greater sum-MSE reduction in earlier RT loops, validating that more experiences collected in replay memory within the same period are beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Thus, with greater size of datasets available, our scheme is expected to find the PA faster with low sum-MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 25 0 2000 4000 6000 8000 RT Loop Index, i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='450 Moving-averaged Sum-MSE PA-RA PA-HG (est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' pathloss) PA-TS (est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' pathloss) PA-DRL PA-HG (true pathloss) PA-DRL + MSG PA-TS (true pathloss) PA-DRL + CBS PA-DRL + MSG + CBS PA-ES (a) K = 24 users 0 2000 4000 6000 8000 RT Loop Index, i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='10 Moving-averaged Sum-MSE PA-RA PA-HG (est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' pathloss) PA-TS (est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' pathloss) PA-HG (true pathloss) PA-DRL PA-TS (true pathloss) PA-DRL + CBS PA-DRL + MSG PA-DRL + MSG + CBS PA-ES (b) K = 36 users Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 8: Sum-MSE performance of different PA schemes over 24 stationary users (left) and 36 stationary users (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Performance Comparison Against Different Baselines Now we assess our proposed PA scheme and compare its performance with several baselines over two metrics: channel estimation MSE and algorithm runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 1) Comparison in MSE: First, we consider static scenarios, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', vk = 0, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The plots showing sum-MSE performance (moving-averaged with a window size of 500) over RT loops for K = 24 and K = 36 are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 8a and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 8b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Note that the PA solutions obtained by PA-HG, PA-TS, and PA-ES required true pathloss information and were fixed for the entire RT loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Among these approaches, it is verified from both figures that PA-ES yields much better MSE performance than PA-TS and PA-HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We also considered the case where PA-HG and PA-TS are conducted using the estimated pathloss, which yields a considerable performance gap compared to the case of using true pathloss knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Given that these baselines require prior knowledge (preferably accurate) to achieve the given performance, our learning-based PA scheme, which does not impose such requirement, is still able to show competitive performance against them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' PA-DRL+MSG clearly outperforms PA-HG and PA-TS with estimated pathloss and provides comparable performance with the ones with true pathloss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Once we utilize CS scheme, our proposed PA-DRL+MSG+CBS shows significant improvement and achieves better performance than PA-ES as a result of jointly optimizing both PA and codebook orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Next, we consider scenarios in which users move over time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', β(i) km changes over i, and vk > 0, ∀k ∈ K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 9 shows the sum-MSE performance (moving-averaged with a window size of 500) of different PA algorithms with K = 24 evaluated at three different user velocities: 26 0 2000 4000 6000 8000 RT Loop Index, i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='46 Moving-averaged Sum-MSE PA-DRL + MSG + CBS PA-HG (true pathloss) PA-TS (true pathloss) PA-ES (a) Velocity = 1 km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 0 2000 4000 6000 8000 RT Loop Index, i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='46 Moving-averaged Sum-MSE PA-DRL + MSG + CBS PA-HG (true pathloss) PA-TS (true pathloss) PA-ES (b) Velocity = 3 km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 0 2000 4000 6000 8000 RT Loop Index, i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='46 Moving-averaged Sum-MSE PA-DRL + MSG + CBS PA-HG (true pathloss) PA-TS (true pathloss) PA-ES (c) Velocity = 5 km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 9: MSE performance of different PA schemes over 24 mobile users with different velocities: 1 km/h, 3 km/h, and 5 km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 1, 3, and 5 km/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' PA solution obtained by the baselines at the beginning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e, i = 0) becomes less effective as time advances, showing a different degree of steady increase by the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Unlike these baselines, as our PA and CS schemes make their decisions based on the real-time observations, in the proposed PA-DRL+MSG+CBS, PAs can be performed in an adaptive manner, maintaining its performance as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Hence, our scheme can provide competitive performance with the prior knowledge-constrained baseline methods under a dynamic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Overall, our scheme provides satisfactory performance in MSE as it exploits the decentralized architecture of O-RAN CFmMIMO via distributed learning and codebook adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 2) Comparison in algorithm runtime: Now, we evaluate and compare the computational complexity of different PA algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We first provide the runtime measurements of different PA methods with various number of users K in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The complexities for PA-TS and PA-HG, which are respectively O(NtabuK2M) [30] and O(KT 3 p ) [32], are confirmed by our experimental result that shows a polynomial increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Hence, both PA-TS and PA-HG are rendered impractical when PA needs to perform over a CFmMIMO network with a growing network size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Meanwhile, our PA algorithm shows a relatively negligible increase, implying its effectiveness in scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' The steady runtimes from our PA scheme are due to the utilization of (i) O-RAN architecture where duration-varing tasks are distributed across the network and (ii) DNNs of fixed size which only perform a forward computation to determine each pilot update step over near-RT loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We observe a slight increase in runtime when we consider inter-DU messages into our PA scheme because generating a new set of messages imposes extra computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Note that our CS scheme barely adds any runtime as it utilizes the rewards already computed during our PA scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We hence conclude that our low-complexity PA scheme is a scalable strategy that supports large-scale 27 30 40 50 60 70 Number of Users, K 5 10 15 20 25 30 Total Runtime (s) PA-TS PA-HG PA-DRL + MSG + CBS PA-DRL + MSG PA-DRL + CBS PA-DRL (a) Comparison over K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 5 10 Size of Codebook, Tp 2 4 6 8 Total Runtime (s) PA-TS PA-HG PA-DRL + MSG + CBS PA-DRL + MSG PA-DRL + CBS PA-DRL 10 15 20 Size of RU Cluster (b) Comparison over Tp (left) and MUE k (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 10: Runtime comparison of different PA schemes over different K values (left) and Tp and size of RU cluster values (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Until converged Set time = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='5 s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='00 Sum-MSE PA-HG PA-TS PA-DRL + MSG PA-DRL + MSG + CBS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 11: Sum-MSE comparison of different PA schemes with a fixed runtime (deadline).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' PA-ES is not included since the considered runtime (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=', 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='5 seconds) is too short to evaluate the reliable performance of exhaustive search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' CFmMIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Note that PA-ES, which is the best baseline in MSE minimization, requires an extreme amount of runtime as it searches over all T K p combinations of PA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' On the other hand, PA-RA requires no extra runtime but shows much worse MSE performance than other PA schemes (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 8a and 8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Next, we assess the total runtime required to conduct PA algorithms over different values of Tp (left) or M UE k (right) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 10b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' For varying Tp (the length of pilot), only PA-HG shows undesirable behavior in complexity since the size of the reward matrix used in the Hungarian algorithm depends on Tp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' With respect to M UE k (the size of RU cluster), both PA-TS and PA-HG display a linear increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Meanwhile, our proposed scheme provides consistent runtimes for both parameters, which verifies their scalability to support a network with large system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 28 3) Comparison in MSE with channel estimation deadline: We next demonstrate the impact of having a channel estimation deadline (in terms of runtime) on the MSE performance to consider practical scenarios where time resource for PA can be strictly limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' 11, we provide a bar chart summarizing the runtime measurements for K = 36 stationary users in two different cases: (i) PA algorithms run until the MSE performance converges and (ii) algorithms only run for 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='5 seconds runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' As expected, when the time constraint is imposed, every PA algorithm shows degradation in sum-MSE as compared to the case where the algorithms fully run until converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Both PA-HG and PA-TS algorithms show significant increase in their MSE since the amount of runtime allocated is considerably lower than the runtime required for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Meanwhile, our proposed PA scheme show relatively less increase in MSE due to its scalable runtime which is not impacted by the time constraint significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' This result once again confirms the computational advantage of our PA scheme over the baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' CONCLUSION In this paper, we developed a learning-based PA scheme for the decentralized CFmMIMO system framed in O-RAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We adopted O-RAN as a practical system architecture where distinct network functions and multi-timescale control loops efficiently govern the framework of our scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' After formulating the PA problem and designing the corresponding Markov game model, we developed a PA algorithm based on the MA-DRL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' We also developed a CS scheme that accelerates our learning-based PA in MSE-minimization without any significant additional complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Compared to the state-of-the-art baselines, our approach provided satisfactory performance in terms of both channel estimation MSE and computational scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Furthermore, unlike most of the existing PA strategies, our scheme does not require any prior channel knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' REFERENCES [1] M.' metadata={'source': 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aspects,” Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content=' TR 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='814 V9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} +page_content='0, March 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/idE3T4oBgHgl3EQf4wsK/content/2301.04774v1.pdf'} diff --git a/j9E0T4oBgHgl3EQf7QLf/content/tmp_files/2301.02775v1.pdf.txt b/j9E0T4oBgHgl3EQf7QLf/content/tmp_files/2301.02775v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0ee671484a1219fbc6e02b8b82a86c9e6335ed66 --- /dev/null +++ b/j9E0T4oBgHgl3EQf7QLf/content/tmp_files/2301.02775v1.pdf.txt @@ -0,0 +1,3299 @@ +Springer Nature 2021 LATEX template +The messy death of a multiple star system +and the resulting planetary nebula as +observed by JWST +Orsola De Marco1,2*, Muhammad Akashi3,4, Stavros +Akras5, Javier Alcolea6, Isabel Aleman7, Philippe +Amram8, Bruce Balick9, Elvire De Beck10, Eric G. +Blackman11,12, Henri M. J. Boffin13, Panos Boumis5, Jesse +Bublitz14, Beatrice Bucciarelli15, Valentin Bujarrabal6, Jan +Cami16,17,18, Nicholas Chornay19, You-Hua Chu20, Romano +L.M. Corradi21,22, Adam Frank23, Guillermo +Garc´ıa-Segura24, D. A. Garc´ıa-Hern´andez22,25, Jorge +Garc´ıa-Rojas22,25, Veronica G´omez-Llanos22,25, Denise R. +Gon¸calves26, Mart´ın A. Guerrero27, David +Jones22,25, Amanda I. Karakas28,29, Joel H. Kastner30,31, Sun +Kwok32, Foteini Lykou33,34, Arturo Manchado22,25,35, Mikako +Matsuura36, Iain McDonald37,38, Ana +Monreal-Ibero39, Hektor Monteiro7, Paula Moraga +Baez31, Christophe Morisset24, Brent Miszalski40, Shazrene +S. Mohamed41,42,43,44, Rodolfo Montez Jr.45, Jason +Nordhaus46,47, Claudia Mendes de Oliveira48, Zara +Osborn28,29, Masaaki Otsuka49, Quentin A. Parker50,51, Els +Peeters16,17,18, Bruno C. Quint52, Guillermo Quintana- +Lacaci53, Matt Redman54, Ashley J. Ruiter55, Laurence +Sabin24, Carmen S´anchez Contreras56, Miguel +Santander-Garc´ıa6, Ivo Seitenzahl55, Raghvendra +Sahai57, Noam Soker3, Angela K. Speck58, Letizia +Stanghellini59, Wolfgang Steffen60, Jes´us A. Toal´a61, Toshiya +Ueta62, Griet Van de Steene63, Eva Villaver56, Paolo +Ventura64, Wouter Vlemmings65, Jeremy R. Walsh13, Roger +Wesson36, Hans van Winckel66 and Albert A. Zijlstra38 +1 +arXiv:2301.02775v1 [astro-ph.SR] 7 Jan 2023 + +Springer Nature 2021 LATEX template +2 +JWST PN +1*School of Mathematical and Physical Sciences, Macquarie +University, Sydney, NSW 2109, Australia. +2*Astronomy, Astrophysics and Astrophotonics Research Centre, +Macquarie University, Sydney, NSW 2109, Australia. +3Department of Physics, Technion, Haifa, 3200003, Israel. +4Kinneret College on the Sea of Galilee, Samakh 15132, Israel. +5Institute for Astronomy, Astrophysics, Space Applications and +Remote Sensing, National Observatory of Athens, GR 15236 +Penteli, Greece. +6Observatorio Astron´omico Nacional (OAN/IGN), Alfonso XII, +3, 28014 Madrid, Spain. +7Instituto de F´ısica e Qu´ımica, Universidade Federal de Itajub´a, +Av. BPS 1303, Pinheirinho, Itajub´a 37500-903, Brazil. +8Aix-Marseille Univ., CNRS, CNES, LAM (Laboratoire +d’Astrophysique de Marseille), Marseille, France. +9Astronomy Department, University of Washington, Seattle, WA +98105-1580, USA. +10Department of Space, Earth and Environment, Chalmers +University of Technology, S-41296 Gothenburg, Sweden. +11Department of Physics and Astronomy, University of +Rochester, Rochester, NY 14627, USA. +12Laboratory for Laser Energetics, University of Rochester, +Rochester NY, 14623, USA. +13European Southern Observatory, Karl-Schwarzschild Strasse 2, +D-85748 Garching, Germany. +14Green Bank Observatory, 155 Observatory Road, PO Box 2, +Green Bank, WV 24944, USA. +15INAF - Osservatorio Astrofisico di Torino, Via Osservatorio 20, +10023, Pino Torinese, Italy. +16Department of Physics & Astronomy, University of Western +Ontario, London, ON, N6A 3K7, Canada. +17Institute for Earth and Space Exploration, University of +Western Ontario, London, ON, N6A 3K7, Canada. +18SETI Institute, 399 Bernardo Avenue, Suite 200, Mountain +View, CA 94043, USA. +19Institute of Astronomy, University of Cambridge, Madingley +Road, Cambridge CB3 0HA, UK. +20Institute of Astronomy and Astrophysics, Academia Sinica +(ASIAA), No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan. + +Springer Nature 2021 LATEX template +JWST PN +3 +21GRANTECAN, Cuesta de San Jos´e s/n, E-38712, Bre˜na Baja, +La Palma, Spain. +22Instituto de Astrof´ısica de Canarias, E-38205 La Laguna, +Tenerife, Spain. +23Department of Physics and Astronomy, University of +Rochester, Rochester, NY 14627, USA. +24Instituto de Astronom´ıa, Universidad Nacional Aut´onoma de +M´exico, Km. 107 Carr. Tijuana-Ensenada, 22860, Ensenada, +B. C., Mexico. +25Departamento de Astrof´ısica, Universidad de La Laguna, +E-38206 La Laguna, Tenerife, Spain. +26Observat´orio do Valongo, Universidade Federal do Rio de +Janeiro, Ladeira Pedro Antonio 43, Rio de Janeiro 20080-090, +Brazil. +27Instituto de Astrof´ısica de Andaluc´ıa, IAA-CSIC, Glorieta de la +Astronom´ıa, s/n, E-18008, Granada, Spain. +28School of Physics & Astronomy, Monash University, Clayton +VIC 3800, Australia. +29ARC Centre of Excellence for All Sky Astrophysics in 3 +Dimensions (ASTRO 3D). +30Center for Imaging Science, Rochester Institute of Technology, +Rochester, NY 14623, USA. +31School of Physics and Astronomy and Laboratory for +Multiwavelength Astrophysics, Rochester Institute of Technology, +USA. +32Department of Earth, Ocean, and Atmospheric Sciences, +University of British Columbia, Vancouver, Canada. +33Konkoly Observatory, Research Centre for Astronomy and +Earth Sciences, E¨otv¨os Lor´and Research Network (ELKH), +Konkoly-Thege Mikl´os ´ut 15-17, 1121 Budapest, Hungary. +34CSFK, MTA Centre of Excellence, Konkoly-Thege Mikl´os ´ut +15-17, 1121 Budapest, Hungary. +35Consejo Superior de Investigaciones Cient´ıficas, Spain. +36School of Physics and Astronomy, Cardiff University, The +Parade, Cardiff CF24 3AA, UK. +37Department of Physical Sciences, The Open University, Walton +Hall, Milton Keynes, MK7 6AA, UK. +38Jodrell Bank Centre for Astrophysics, Department of Physics +and Astronomy, The University of Manchester, Oxford Road M13 +9PL Manchester, UK. + +Springer Nature 2021 LATEX template +4 +JWST PN +39Leiden Observatory, Leiden University, Niels Bohrweg 2, NL +2333 CA Leiden, The Netherlands. +40Australian Astronomical Optics, Faculty of Science and +Engineering, Macquarie University, North Ryde, NSW 2113, +Australia. +41Department of Physics, University of Miami, Coral Gables, FL +33124, USA. +42South African Astronomical Observatory, P.O. Box 9, 7935 +Observatory, South Africa. +43Astronomy Department, University of Cape Town, 7701 +Rondebosch, South Africa. +44NITheCS National Institute for Theoretical and Computational +Sciences, South Africa. +45Center for Astrophysics, Harvard & Smithsonian, 60 Garden +Street, Cambridge, MA 02138, USA. +46Center for Computational Relativity and Gravitation, +Rochester Institute of Technology, Rochester, NY 14623, USA. +47National Technical Institute for the Deaf, Rochester Institute of +Technology, Rochester, NY 14623, USA. +48Departamento de Astronomia, Instituto de Astronomia, +Geof´ısica e Ciˆencias Atmosf´ericas da USP, Cidade Universit´aria, +05508-900, S˜ao Paulo, SP, Brazil. +49Okayama Observatory, Kyoto University, Honjo, Kamogata, +Asakuchi, Okayama, 719-0232, Japan. +50Department of Physics, CYM Physics Building, The University +of Hong Kong, Pokfulam, Hong Kong SAR, PRC. +51Laboratory for Space Research, Cyberport 4, Cyberport, Hong +Kong SAR, PRC. +52Rubin Observatory Project Office, 950 N. Cherry Ave., Tucson, +AZ 85719, USA. +53Dept. of Molecular Astrophysics. IFF-CSIC. C/ Serrano 123, +E-28006, Madrid, Spain. +54Centre for Astronomy, School of Physics, National University of +Ireland Galway, Galway H91 CF50, Ireland. +55University of New South Wales, Australian Defence Force +Academy, Canberra, Australian Capital Territory, Australia. +56Centro de Astrobiolog´ıa (CAB), CSIC-INTA, Camino Bajo del +Castillo s/n, ESAC campus, 28692, Villanueva de la Ca˜nada, +Madrid, Spain. + +Springer Nature 2021 LATEX template +JWST PN +5 +57Jet Propulsion Laboratory, California Institute of Technology, +CA 91109, Pasadena, USA. +58University of Texas at San Antonio, Department of Physics and +Astronomy, Applied Engineering and Technology Building, One +UTSA Circle, San Antonio, TX 78249, United States. +59NSF’s NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA. +60ilumbra, AstroPhysical MediaStudio, Hautzenbergstrasse 1, +67661 Kaiserslautern, Germany. +61Instituto de Radioastronom´ıa y Astrof´ısica, UNAM, Antigua +Carretera a P´atzcuaro 8701, Ex-Hda. San Jos´e de la Huerta, +Morelia 58089, Mich., Mexico. +62Department of Physics and Astronomy, University of Denver, +2112 E Wesley Ave., Denver, CO 80208, USA. +63Royal Observatory of Belgium, Astronomy and Astrophysics, +Ringlaan 3, 1180 Brussels, Belgium. +64INAF – Osservatorio Astronomico di Roma, Via Frascati 33, +I-00040, Monte Porzio Catone (RM), Italy. +65Onsala Space Observatory, Department of Space, Earth and +Environment, Chalmers University of Technology, Onsala, +Sweden. +66Institute of Astronomy, KULeuven, Celestijnenlaan 200D, +B-3001 Leuven, Belgium. +*Corresponding author(s). E-mail(s): orsola.demarco@mq.edu.au; +Abstract +Planetary nebulae (PNe), the ejected envelopes of red giant stars, pro- +vide us with a history of the last, mass-losing phases of 90% of stars +initially more massive than the Sun. Here, we analyse James Webb +Space Telescope (JWST) Early Release Observation (ERO) images of +the PN NGC 3132. A structured, extended H2 halo surrounding an +ionised central bubble is imprinted with spiral structures, likely shaped +by a low-mass companion orbiting the central star at ∼40–60 AU. The +images also reveal a mid-IR excess at the central star interpreted as a +dusty disk, indicative of an interaction with another, closer companion. +Including the previously known, A-type visual companion, the progeni- +tor of the NGC 3132 PN must have been at least a stellar quartet. The +JWST images allow us to generate a model of the illumination, ionisa- +tion and hydrodynamics of the molecular halo, demonstrating the power +of JWST to investigate complex stellar outflows. Further, new measure- +ments of the A-type visual companion allow us to derive the value for + +Springer Nature 2021 LATEX template +6 +JWST PN +the mass of the progenitor of a central star to date with excellent pre- +cision: 2.86 ± 0.06 M⊙ . These results serve as pathfinders for future +JWST observations of PNe providing unique insight into fundamental +astrophysical processes including colliding winds, and binary star inter- +actions, with implications for supernovae and gravitational wave systems. +Keywords: stars: AGB and post-AGB, stars: evolution, ISM: jets and +outflows, ISM: molecules, planetary nebulae: individual: NGC 3132 +Main +Introduction +Planetary nebulae (PNe) are the ejected envelopes of intermediate-mass (∼1– +8 M⊙) stars that have recently terminated their asymptotic giant branch +(AGB) stage of evolution. Moving outwards from the hot pre-white dwarf star +(T ∼ 105 K) that is the progeny of the AGB star, the structure of a canonical +quasi-spherical PN consists of a hot, sparse, wind-heated bubble (T ∼ 107K) +surrounded by a dense shell of displaced, ionised AGB gas (T ∼ 104 K), which +in turn may still be surrounded by “pristine,” cold (T ∼ 102 K), molecule- +and dust-rich AGB ejecta. On the other hand, if the progenitor star interacted +with a companion(s) during its post-main sequence evolution, we would expect +departures from spherical symmetry, perhaps including spiral structures and +arcs [e.g., 1–3], the presence of a dense, molecule-rich torus [e.g., 4], one or +more pairs of polar lobes formed by fast, collimated outflows and jets [e.g., +5, 6], and/or a dusty, circumbinary disk [7]. The type of interaction depends +on the orbital radius, and ranges from common envelope evolution for close +binaries [8], to accretion disks and gravitational focussing of the wind for wider +systems [9–11], to displacement of the central star from the geometric centre +of the nebula for the widest systems [12]. +The first Hubble Space Telescope (HST) images of PNe revealed a breath- +taking new world of details and far more complex structures than had been +gleaned from ground-based images [e.g., 13, 14]. The superb spatial resolu- +tion of HST, combined with high-resolution, kinematic mapping, enabled the +construction of detailed 3D, morpho-kinematic models, which, together with +hydrodynamic models [e.g., 15, 16], started to connect our understanding of the +evolution of the structures and kinematics of PNe with their possible binary +star origins [e.g., 17–19]. +The James Webb Space Telescope (JWST), with its superb sensitivity and +high spatial resolution from near- to mid-IR, is now poised to enable a leap +of similar magnitude in our understanding of PNe. This journey began when +JWST released near-IR and mid-IR images of just one PN, NGC 3132, as +part of its ERO program. NGC 3132 is a nearby (D ∼ 750 pc), molecule-rich +[20, 21], ring-like PN, long known to harbour a visual binary comprising the +central (progenitor) star and an A star companion. In this paper we show that + +Springer Nature 2021 LATEX template +JWST PN +7 +the JWST ERO images contain multiple, new lines of evidence that NGC 3132 +is the recent product of a hierarchical multiple progenitor stellar system, which +has experienced both indirect and direct interactions involving one or more +components. Such binary interactions have taken on new importance in the era +of gravitational wave detectors (LIGO [22], LISA [23]) and ambitious transient +surveys [24]. Indeed, PNe like NGC 3132 offer unique insight into the formation +pathways of the close, single and double degenerate binaries that are eventual +gravitational wave sources and (perhaps) type Ia supernova progenitors [25– +27]. +Results +A flocculent molecular halo surrounding an ionised bubble +Figure 1 displays colour overlays of NIRCam and MIRI images of NGC 3132 +that highlight JWST’s clean separation of the PN’s ionised (H ii) and molec- +ular (H2) regions. The full resulting JWST image suite, along with basic +information, is presented in Specification of JWST NIRCam and MIRI imag- +ing and Supplementary Figure 1. The images reveal, for the first time, the +extent and detailed structure of the halo of molecular gas that lies exterior to +the nebula’s central, ionised cavity and its bright and thin, peripheral ellip- +tical ring (cf. [28]). This molecular halo is well detected in rovibrational H2 +emission at 2.12 µm (1–0 S(1)), 4.7 µm (0–0 S(9)), and 7.7 µm (0–0 S(5)) out +to 60 arcsec (∼0.22 pc at the adopted distance of 754 pc, see Properties and +distance of NGC 3132) from the central star. Spatially organised structures — +arcs and patterns of spikes emanating radially outward — are observed in the +halo H2 emission on medium to large scales, while molecular arcs, loops, and +knots are detected on size scales from ∼500 AU down to the limiting (∼75 AU +) resolution of the images. The typical thickness of the bright H2 rings that +surround the nebular core is ∼ 1 − 2 arcsec (∼750-1500 AU), measured at 2.1, +4.7 and 7.7 µm. +Figure 1 conclusively demonstrates that the molecular gas is much clumpier +than the ionised gas component of NGC 3132 (see also Supplementary +Figure 3). In hydrogen recombination lines and [S iii] emission (Figure 1, top- +left), the nebula’s central ionised cavity (within ∼25 arcsec of the central star) +appears as a relatively smooth elliptical region that is bounded by a single, +sharped-edged ring; whereas in H2 (Figure 1, bottom-left), this same central +region appears as a far more complex system of clumpy filaments. The regions +in and around this bright, inner H2 ring system contain as many as 20 dense +clumps (knots) per square arcsec, implying the total number of H2 knots in this +region exceeds 104. The H2 knots in the outer (halo) region are less distinct +and further apart. +The presence of radially-directed spike features in the H2 halo indicates +that direct irradiation by UV photons, leaking through less dense gas between +the inner ring system’s H2 knots, are most likely responsible for the excitation +of the IR H2 lines in the extended halo, although shock excitation cannot be + +Springer Nature 2021 LATEX template +8 +JWST PN +completely ruled out (see [29] and references therein). The relative lack of H2 +halo emission to the East-Northeast and West-Southwest of the central star +then indicates a general lack of central star UV illumination, as opposed to lack +of halo molecular mass in those directions (see Discussion). Measurements of +the extinction of background nebulosity through representative knots suggests +typical knot densities of ∼ 106 cm−3 and masses of ∼ 10−5 M⊙ (see Densities, +masses and excitation of the H2 knots), suggesting a total H2 mass of ∼0.1 M⊙ +in the central ring region. +The system of (broken) concentric arcs revealed in the H2 halo by the +JWST images is similar to those observed in the extended, dusty envelopes +of many AGB stars, proto-PNe and PNe (e.g., [3, 30, 31]). A widely accepted +scenario to explain the formation of such arc systems is the modulation of an +AGB wind by a stellar or substellar companion, creating 3D spiral-like patterns +along the orbital plane [see 1, 32–35, and references therein]. The average +angular distance between the arc structures, 2 arcsec, implies an orbital period +of 290-480 years and an orbital separation of 40-60 AU between the central +star and the companion that shapes the mass loss. Here, we have assumed a +companion mass of 0.2 M⊙, the highest mass main-sequence star that could +hide in the present-day central star’s glare yet still form a visible arc system +(other parameters are an expansion velocity in the range 15-25 km s−1 [31] +and an assumed late-AGB central star mass of ∼0.8 M⊙; likely still 0.1-0.2 M⊙ +larger than the post-AGB mass). The bright A2 V visual companion seen at +∼1300 AU projected separation from the central star cannot be responsible, +suggesting (at least) a triple system in a stable configuration. +The dusty central star +In the MIRI images obtained at wavelengths longer than 10 µm, the faint +central star appears as bright or brighter than its A2 V main sequence visual +companion [36]; see Figure 2. This infrared excess was undetectable in the mid- +infrared at lower spatial resolution (e.g., in WISE images [37]) because of the +surrounding bright nebulosity. The JWST-discovered IR excess indicates that +a considerable amount of warm dust is present around the ultra-hot (∼110 kK) +PN central star. The thermal infrared source appears marginally extended in +the 11.3 and 12.8 µm MIRI images with an apparent size of ∼300 AU (FWHM) +at 12.8 µm (see PSF measurements of the central star). +The bottom panel of Figure 2 displays the central star’s near-IR to mid-IR +spectral energy distribution fitted by a combination of a hot stellar photosphere +represented by a blackbody curve and two curves to fit the infrared data points. +The two curves are generated with a model that follows closely that of [38] for +the Helix nebula. A number of 100 µm grains are taken as blackbody spheres +with temperatures set by absorption and re-emittance of the stellar luminosity +(200 L⊙; a correction factor is then applied to simulate a grain size distribution +between 60 and 1000 µm, as done by [38]). The temperature varies as d0.5, +where d is the distance to the star. The surface density of the disk is taken as +constant. The resulting blackbody radiation is calculated at each radius, and + +Springer Nature 2021 LATEX template +JWST PN +9 +the emission is summed over all radii. A better model will require radiative +transfer, actual dust emissivities, a range of grains sizes, and for the silicate +feature, the inclination of the disk. This will be explored in a future paper. +The best-fit model disk has an inner radius of 55 AU and outer radius +140 AU, and a dust mass of 3 × 1026 g or 2 × 10−7 M⊙ (approximately 0.05 +Earth masses). The dust temperature range (inner to outer radius) is 130 to +80 K. The outer radius of 140 AU, though poorly constrained, is consistent +with the deconvolved half-width of the marginally extended mid-IR source. +These dimensions resemble those inferred for the disk orbiting the central star +of the Helix (35–150 AU; [38]), but the dust mass is somewhat smaller (cfr. +0.13 earth masses). The outer radius could be slightly larger, if the 18 µm flux +is underestimated because of detector saturation. An additional inner, hotter +disk — with radius between 3 and 8 AU, a temperature between 550 to 335 K +(inside to outside) and a very small mass of 2 × 1022 g (approximately 0.02 +times the mass of Ceres) — is needed to fit the 3.5 and 7 µm fluxes. While this +model does not constrain the geometry of the distribution to be that of a disk, +the reasoning behind a disk structure is based on a physical reasoning whereby +only a rotating Keplerian disk can be shown to be stable and relatively long- +lived, while other structures, such as shells, are easily shown to be unstable +[39]. +The A2 V companion is slightly evolved [36] and has a mass of MA2V = +2.40 ± 0.15 M⊙, using the PARSEC isochrones. Its visual companion, the +PN central star, must have descended from a more massive star, as it has +evolved faster. Extrapolating the same PARSEC isochrone gives an initial +main sequence mass for the central star of Mi = 2.86±0.06 M⊙. This is poten- +tially the most precise initial mass for any PN central star or white dwarf yet +determined. We estimate the error to be 0.16 M⊙ if we add systematic effects +between different isochrone models (see Central star system’s masses). +The current (near-final) mass of a PN central star descended from such a +∼2.9 M⊙ progenitor is predicted to be Mf ∼ 0.66 ± 0.05 M⊙ based on initial- +final mass relations [40], albeit with larger systematic uncertainties that are +dependent on details of the mass loss process adopted by the models. It is +noteworthy that photoionisation models of the nebula require a cooler, dimmer +and overall less massive central star (0.58±0.03 M⊙) than what we have found. +We find that we can reconcile the mass of the star today and that of the +photoionisation model, while also matching the nebular abundances and the +nebular age, if we assume that the AGB evolution of a 2.86 M⊙ star, was +interrupted by a binary interaction that ejected the envelope. We conjecture +that the AGB evolution was interrupted at a core mass of 0.61 M⊙, because +for larger values, the C/O ratio of the stellar envelope gas would increase +above unity (counter to the observation of crystalline silicate grains). At larger +masses the N/O ratio would also increase above the observed value of 0.42. + +Springer Nature 2021 LATEX template +10 +JWST PN +Discussion +The first striking discovery of JWST is the presence of the dusty disk around +the ultra-hot central star. This indicates that JWST can accurately detect +dusty disks lighter than Ceres, as far as ∼700 pc away. For our PN, the presence +of such a disk orbiting the PN central star favours a close binary interaction, +where the companion either merged with the primary star, or is still in orbit +but is undetected (mass < 0.2 M⊙; based on an unresolved or barely resolved, +equal-brightness companion); in either case, the companion has donated a sub- +stantial fraction of its angular momentum to the gas [41, 42]. Observationally, +such disks around PN central stars, though rare, appear to be by and large +associated with known or strongly suspected binarity [39] and may be related +to circumbinary disks detected around other classes of post-AGB binary stars +[43]. +An interacting binary scenario is reinforced by the shape of the ionised cav- +ity, which represents the inner, most recent mass-loss phase, when the already +hot central star emitted a fast, tenuous wind. Pairing the JWST images with +spatially resolved spectroscopy we constructed a 3D visualisation of this cav- +ity (see Morpho-kinematic modelling in Supplementary Material). In Figure 3 +we show that this inner cavity is inferred to be an expanding prolate ellipsoid +with its long axis tilted at approximately 30 deg to the line of sight. Its surface +is not smooth and presents instead a number of protuberances, most of which +can be paired via axes passing through, or very near the central star. Prolate +cavities such as these, with misaligned structures, are common in PN and are +likely sculpted by jets from interacting binaries in the earlier, pre-PN phase of +the nebula [44], with additional details added during the interaction between +the AGB wind and post-AGB fast wind and via the process of PN ionisation. +The numerous protuberances clearly evident in the 3D reconstruction could +arise from ionised gas breaking out of the inner cavity through an uneven outer +shell. The apparent pairing between these protuberances may argue instead +for the presence of intermittent and toppling jets [45]. To generate jets over +such a wide range of axes, an interacting binary is not enough, and one would +have to conjecture that the central star is or was a member of not just a close +binary, but of an interacting triple system [46]. Recent studies of interactions +in triple systems [47, 48] also argue for the possibility of interactions yielding +complex ejecta. +Outside the ionised ellipsoid, one encounters material ejected earlier in +the star’s history. The AGB mass loss, at rates of up to ∼10−5 M⊙yr−1 and +speeds of ∼10 km s−1 over a ∼105 yr timescale [49], generates an enormous, +expanding envelope of molecular gas and dust. The H2 halo imaged by JWST +constitutes the most recently ejected (inner) region of this AGB envelope. The +spikes observed in the halo (Figure 1, right panel) show that the inner cavity +is very porous, though less so near the minor axis where the cavity edges are +brightest, densest, and least fractured. +The JWST images motivated 2D hydrodynamic simulations to replicate +these flocculent structures. In Figure 4 we see two time snapshots towards the + +Springer Nature 2021 LATEX template +JWST PN +11 +end of a simulation where an inner, faster wind from the heating central star +and its ionising radiation, plough into the dense AGB (halo) material (see +Methods, Section 66). The fragmentation that happens at the interface of the +swept-up material also creates the variable opacity needed to shield some of +the wind material from ionising radiation, which then quickly recombines and +allows the formation of molecules. Non ionising radiation leaks more readily +because the opacity above 913 ˚A is lower. These photons produce florescence +of H2. +In Figure 4 we see two time snapshots towards the end of the simulation. +In the first panel we see a set of approximately radial spikes, but 200 years +later those straight and thin spikes evolve to thicker and sometimes curved +ones. In the right column of Figure 4 two different parts of the nebula exhibit +thinner and straighter spikes (top-right panel) or thicker, bent ones (bottom- +right panel). Although the entire nebula was ejected and ionized over a short +time interval, there can be a delay in the evolution of a given spike in a specific +part of the nebula, related to the local opacity in the swept-up shell. Figure 4 +suggests that differences of only ∼ 200 years in the timescales of mass ejection +and/or the progress of illumination along specific directions can explain the +marked differences observed in the flocculent structure around the nebula. +The successful modelling of illumination percolating unevenly into the +molecular halo (Figure 4) motivated a further geometric model of the halo, +presented in Figure 5. This Figure compares the extended H2 structures as +imaged by JWST with a model consisting of two thick, concentric, unbroken +but clumpy, shells of material that are illuminated by the central star through +a porous ellipsoid representing the boundary of the ionised cavity, with reduced +opacity in the polar regions. As a result of the uneven illumination the dis- +tribution of H2 material appears fragmented and is generally brighter toward +the polar regions (and suppressed along the equatorial plane) of the central +ellipsoidal, ionised region. The distribution seen in the JWST H2 images could +be reproduced more closely by altering the opacity of the inner ellipsoid. Fly- +though movies of the 3D reconstructions of both the inner ellipsoid (Figures 3) +and the outer H2 halo (Figure 5) can be found following the links. +The arches in the JWST images, are not smeared as is typical of those +seen in projection [e.g., 50], but are instead sharp. This possibly indicates that +these arches are on or near the plane of the sky, indicating that the orbit +of the companion at ∼40-60 AU is closely aligned to the waist of the inner +ellipsoid. This companion cannot partake in the formation of the disk around +the central star, though it may play a secondary role in the shaping of other +PN structures. It is also unlikely to have launched strong jets because at such +distance the accretion rate would be very low. As such, this would be an +additional companion to the inner binary (or triple), making it a tertiary (or +quaternary) companion. +The visual A-type companion would then be a fourth (fifth) member of +the group, an almost complete bystander from the point of view of interac- +tion and shaping, but critically important for this study: Its well measured + +Springer Nature 2021 LATEX template +12 +JWST PN +mass, and slight evolved status, constrained the initial mass of the central star: +(2.86±0.06) M⊙. +To reconstruct the events that lead to the demise of the progenitor of +NGC3132, the PN acts like a murder scene. The A-type companion, could not +have partaken to the interaction that unravelled the AGB star, but was (and +is) certainly present. A second companion at 40-60AU left an indelible trail +of its presence in the form of arcs, but was not close enough to generate the +dusty disk, nor shape the ionised cavity, implying that there must have been +at least another accomplice. This points the finger at a close-by companion, +that is either avoiding detection, or has perished in the interaction (merged). +If the numerous protuberances seen in the ionised cavity come in pairs, then +tumbling jet axes would be needed and this would point the finger to the +presence of a second, close companion [47, 48], which would make the system +a quintet. Even ignoring the putative second, close companion, we can state +with good degree of certainty that the system is at least a quartet. Systems of +four or five stars orbiting within a few ×1000 AU are not impossibly rare for +primary stars in the progenitor mass range of interest here [e.g., HD 104237; +51]; indeed, present estimates indicate that 50% or more of stars of 2-3 M⊙ +are in multiple systems, and of order 2% of A-type stars have four companions +[52]. +JWST is at the starting gate of its promise as an astrophysical pathfinder. +With complementary radio, interferometric and time resolved observations, it +can find the temporal signatures of active convective mass ejection from the +surfaces of AGB stars and the subsequent gravitational influence of companion +stars in dynamically- and thermally-complex outflows. Thus JWST offers the +potential to intimately connect the histories of PNe and the role of close stel- +lar companions to studies of chemical evolution, nebular shaping and binary +interactions for the next century. +Methods +Properties and distance of NGC 3132 +The inner, ionised cavity of NGC 3132 is elliptical in shape, with a major +axis of ∼40 arcsec (0.15 pc) and an electron density of n ∼ 103 cm−3. The +ionization structure and abundances were the subject of a recent study by [53]. +The nebula is also known to be molecule-rich [20]; it is among the brightest +PNe in near-IR H2 emission [21, 54]. +A bright A2 V star is found near the centre of the PN, but is too cool to be +the ionizing star; the actual PN progenitor is much fainter and is located ∼1.7 +arcsec to the South-West of the A star [55, 56]. The A2 V star has the same +radial velocity and extinction as the PN, and its proper motion (µα = −7.747 +mas/yr σµα = 0.026; and µδ = −0.125 mas/yr σµδ = 0.031) agrees with that +of the central star (µα = −7.677 mas/yr σµα = 0.235; and µδ = 0.197 mas/yr +σµδ = 0.275), demonstrating that the PN progenitor and A-type companion +constitute a comoving visual binary. The distance to NGC 3132 is obtained + +Springer Nature 2021 LATEX template +JWST PN +13 +from Gaia DR3 measurements of this visual binary. No Gaia DR3 radial veloc- +ity is available for the optically faint central star (the PN progenitor). However, +the brighter (A-type) visual companion and the PN have the same radial veloc- +ity: (−11.4 ± 1.6) km s−1 for the A star from Gaia, and (−10 ± 3) km s−1 for +the PN from [57]. The A star and PN central star also have compatible Gaia +DR3 proper motions (within 1.5σ). +The brighter, A-type star has a Gaia DR3 geometric distance (median of +geometric distance posterior) of 754 pc, with lower and upper 1σ-like confi- +dence intervals (16th and 87th percentiles of the posterior) of 18 pc and 15 pc +respectively [58]. The fainter central star has a Gaia DR3 geometric distance +of 2124.7 pc, with lower and upper 1σ-like bars of 559.1 pc, and 1464.5 pc. The +quality flags of the astrometric solution for this star are not optimal, most likely +due to the vicinity of the much brighter A-star; in particular, the goodness- +of-fit along the scan is 16.9, while it should be close to unity. We therefore +adopt the Gaia DR3 distance to the central star’s visual A-type companion, +754+15 +−18 pc, as the distance to the PN. +Densities, masses and excitation of the H2 knots +The clumpiness of NGC 3132 in H2 emission links this nebula to other +molecule-rich PNe, such as the Helix Nebula (NGC 7293, [59–62]), Ring Neb- +ula (NGC 6720, [63]), and the hourglass-shaped (bipolar) nebula NGC 2346 +[64], in which the molecular emission seems to be associated with dense knots +that are embedded in or surround the ionised gas. The origin of such H2 knots +in PNe — as overdensities in the former AGB wind, vs. formation in situ +following recombination of H, as the central star enters the cooling track — +remains an open question [65]. In contrast to the Helix Nebula, there is little +evidence for cometary tails emanating from the knots in the inner regions of +NGC 3132. However, NGC 3132’s system of approximately radially-directed +H2 spikes external to the main H2-bright ring system has close analogues in, +e.g., the Ring and Dumbbell Nebulae [21, 63]. +Some H2 knots in NGC 3132 are seen in absorption against the bright back- +ground nebular emission. This extinction is apparent not only in optical (HST) +images but also, surprisingly, even in the JWST NIRCam near-infrared images +(see Supplementary Figure 4). We measured the extinction at 1.87 µm for two +knots seen in absorption against the (Paα) nebula background: the largest knot +on the west side (coordinates 10:07:00.4, −40:26:08.8), and one of the dark- +est on the east side (10:07:02.5, −40:26:00.3). The diameters of these knots +are ∼0.36 arcsec and ∼0.15 arcsec, while their extinction is ∼0.57 mag and +∼0.25 mag (at 1.87 µm), respectively; using the dust extinction law A(λ)/A(V ) +from [66], the corresponding values of A(V ) are 3.9 and 1.7 mag assuming +RV = 3.1. We then estimate the hydrogen column densities N(H) from these +extinction measurements, and convert to the hydrogen density n(H) of the knot +by assuming that the knot diameters are roughly equivalent to their depths +along the line of sight. Using the conversion between A(V ) and N(H) from [67], +where H is the combination of H0, H+ and H2, the estimated column densities + +Springer Nature 2021 LATEX template +14 +JWST PN +are N(H) = 7.3×1021 cm−2 and N(H) = 3.2×1021 cm−2, respectively. For the +adopted distance of 754 pc, the estimated densities are n(H) ∼ 2 × 106 cm−3 +for both knots. These densities suggest knot masses of 10−5 M⊙, similar to the +typical knot (“globule”) masses found in the Helix Nebula [68]. +The critical density of excitation of the 2.12 µm H2 1–0 S(1) line at a kinetic +temperature of 2000 K is 9×105 cm−3 [69], if the collision partner is H. The +critical density is higher for the 1–0 S(1) line than for the 0–0 S(9) 4.69 µm +H2 line [6×104 cm−3; 69, 70]. Hence, the excitation of H2 should be nearly +thermal if the gas temperature is sufficiently high, with the caveat that both +critical densities are higher if the primary collision partner is H2 rather than H. +PSF measurements of the central star +To ascertain whether the mid-IR source associated with the PN central star is +extended, we measured the JWST instrumental point spread function (PSF), +using Gaussian fitting of field stars. We measured Gaussian FWHMs of 0.29, +0.40, 0.44 and 0.58 arcsec at 7.7, 11.3, 12.8 and 18 µm, respectively. We also +measured two compact, slightly resolved galaxies in the field. +We then repeated the procedure for the central star. No fit was possible at +18 µm, due to saturation (see Supplementary Figure 5). At 7.7 µm the central +star is on the edge of the diffraction spike of the A star, and only an upper limit +on FWHM could be obtained. However, measurements of the PN central star +image in the 11.3 and 12.8 µm filters gave consistent results, with measured +FWHMs of 0.55 and 0.60 arcsec, significantly larger than the respective PSFs +and comparable to the two field galaxies. Gaussian deconvolution using the +PSF yields deconvolved FWHM values for the central star of ≤ 0.3 (≤ 230 AU) +at 7 µm, and 0.4 arcsec (300 AU) at 11.3 and 12.8 µm. The extent of the central +star at 18 µm is ≳ 0.9 arcsec in diameter (see Supplemenraty Figures 5 and 7). +Central star system’s masses +We determined the mass of the A-star companion using version 1.2 of the +PARSEC isochrones [71] for solar metallicity, taken as Z = 0.0152. We used +Mbol,0 = (0.34 ± 0.25) mag and the GAIA DR3 spectroscopic temperature +Teff = (9200 ± 200) K, where the errors are conservative. The star is confirmed +to be beginning to turn off the main sequence, in a phase where the luminosity +of (57 ± 15) L⊙ increases by 0.1% per Myr and the temperature decreases +by 7 K per Myr (see Supplementary Figure 6). The isochrones yield an age +of (5.3 ± 0.3) × 108 yr and a mass of MA2V = (2.40 ± 0.15) M⊙. The central +star of the PN is evolving on the same isochrone, but from a more massive +star as it has evolved further. We use the same isochrones to determine the +initial mass of a star on the thermal-pulsing AGB, the phase where the central +star ejected the envelope. This gives an initial mass for the central star of +Mi = (2.86±0.06) M⊙. We have carried out the same isochrone fitting using an +alternative stellar evolutionary model (the DARTMOUTH code; [72]). Both + +Springer Nature 2021 LATEX template +JWST PN +15 +the A2V star mass and the mass of the progenitor of the central star decrease +by 0.15 M⊙. +The final, CS, mass for such a star is 0.66 M⊙. However, we have shown that +such a star would show a high C/O∼2, while the presence of silicate features +in the Spitzer spectrum indicate that C/O≲1. To reconcile the mass and the +abundances we conjecture that the evolution was interrupted by the binary +interaction that formed the disk, when the core mass was 0.61 M⊙. With such +a mass the evolutionary time to the current position on the HR diagram is +in better agreement with the age of the nebula. This mass is also in better +agreement with that derived from the photoionisation model (0.58±0.03) M⊙. +Photoionisation modelling +The stratified ionisation and excitation structure of NGC 3132 is evident in +Fig. 1, wherein the bright rim of ionized gas, as traced by [S iii] and Brα +emission, lies nestled inside the peak H2 emission. However, significant ionised +hydrogen and high-excitation plasma — traced by [Ne ii] and [S iii] emission +in the MIRI F1280W and F1800W filter images, respectively — is observed +beyond the bright inner, elliptical ring. +We constructed a three-dimensional photoionisation model using the code +Mocassin [73]. To constrain the model we used the Multi Unit Spectroscopic +Explorer (MUSE) emission line maps and absolute Hβ flux of [74], the optical +integrated line fluxes from [75], the IR line fluxes from [76], as well as the +velocity-position data obtained from the high-resolution scanning Fabry-Perot +interferometer, SAM-FP, mounted on the SOAR telescope adaptive module. +The observations were taken under photometric conditions. The seeing during +the observations was 0.7 arcsec for the [N ii] observations to 0.9 arcsec for the +Hα one. The FWHM of a Ne calibration lamp lines was 0.586 ˚A or 26.8 km s−1, +which corresponds to a spectral resolution of about 11 200 at Hα. +We determined the density structure by fitting the emission line maps to +the SAM-FP images of [N ii] λ6584 and Hα, using a distance of 754 kpc. +The model adopts as free parameters the temperature and luminosity of the +ionising source, and the elemental abundance of the gas component (assumed +constant throughout the nebula); we assumed that no dust is mixed in the gas. +For the ionising source we use the NLTE model atmospheres of central stars +of planetary nebulae from [77]. +We find that a model invoking an unobscured central star with effective +temperature Teff = 110 kK and luminosity L = 200 L⊙ well matches the +observational data. However, we find that the present-day central star mass +implied by the comparison, between these stellar parameters and the evolution- +ary tracks of [40] (0.58±0.03 M⊙) is inconsistent with the (large) initial mass +inferred from consideration of the presence of the comoving, wide-separation +A-type companion (0.66 M⊙; see Central star system’s masses). Furthermore, +the tracks of [40] indicate that, for this mass, we would have a post-AGB age +of 20 000 yrs, whereas the position-velocity data from the SAM-FP instrument + +Springer Nature 2021 LATEX template +16 +JWST PN +yield an expansion velocity of 25-35 km s−1 implying a much shorter and +inconsistent nebular dynamical age in the range 2200–5700 yrs. +The C/O and N/O abundances of the nebula, as well as the crystalline sil- +icate nature of the dust in the PN, indicate that this object has not undergone +hot bottom burning and that it has not undergone sufficient dredge up to have +increased the C/O ratio above unity. By the time the 2.86-M⊙ star reaches the +tip of the AGB its C/O ratio is approximately 2. It therefore seems that the +mass implied by the initial-to-final mass relation using a main sequence mass +of 2.86 M⊙, is too high. We have two ways to resolve this inconsistency (which +may both be operating). The central star is shielded by dust in the circum- +stellar disk making it appear, to the PN, as a cooler star, and/or the central +star mass is actually smaller than 0.66 M⊙, because the AGB evolution was +interrupted by a binary interaction. +If the stellar ascent of the AGB was interrupted, we can determine the +upper limit for a mass that would produce a nebula with C/O ≲ 1 and +N/O∼0.4. This is 0.61 M⊙. The time for a star of this mass to move from the +AGB to the location on the HR diagram with an approximate temperature and +luminosity (110kK, 200 L⊙) as measured above is ∼10 000 yrs. The time-scale +of the transition from AGB to post-AGB and PN is tightly connected with +the rate at which the envelope is consumed: the results obtained are there- +fore sensitive to the mass-loss description. The time of 10 000 yr, is based on +the classic mass loss rates dictated by Reimers or Blocker [78]. This estimate +must be considered as an upper limit of the duration of this phase; indeed the +recent works on the AGB to post-AGB transition by [79] and [80] showed that +to reproduce the infrared excess of post-AGB stars in the Galaxy and in the +Magellanic Clouds one has to invoke significantly higher mass-loss rates than +those based on the aforementioned formulations, something that would reduce +the time-scales by a factor of ∼5. The timescale of 10 000 yrs is therefore easily +reconciled with the observed timescale of 2200-5700 yrs implied by the nebula. +Hydrodynamic modelling +The hydrodynamic simulation used to interpret the fragmentation and +radial spikes is a 2-dimensional hydrodynamic simulation using the magneto- +hydrodynamic code ZEUS-3D. The computational grid is in spherical coordi- +nates and consists of 800 × 800 equidistant zones in r and θ respectively, with +an angular extent of 90◦. The wind and UV luminosity inputs correspond to a +stellar post-AGB model with 0.677 M⊙ which evolves from an initial 2.5 M⊙ +main sequence star [81]. +At simulation time 0 yr the star has Teff = 10 000 K and the AGB wind +(v = 10 km s−1, ˙M = 10−6 M⊙ yr−1) has a homogeneous distribution outside +of the pre-PN. The pre-PN has had 1000 yr of evolution prior to this moment, +during which time a wide magnetic jet operated with a velocity v = 230 km s−1, +and a mass-loss rate +˙M = 1.3 × 10−7 M⊙ yr−1; this simulation is taken from +Model C6 in [82]. At this time the star starts emitting a fast tenuous wind +with a velocity v from 240 to 14 000 km s−1 and a mass-loss rate, +˙M ranging + +Springer Nature 2021 LATEX template +JWST PN +17 +from 1.06 × 10−7 to 1.13 × 10−10 M⊙ yr−1 over 4000 yrs that sweeps up the +AGB wind material. At the same time (0 yr) the ionisation front propagates +into the medium. +Data availability +HST data are available at HST Legacy Archive (https://hla.stsci.edu). +JWST data were obtained from the Mikulski Archive for Space Tele- +scopes at the Space Telescope Science Institute (https://archive.stsci.edu/). +MUSE data were collected at the European Organisation for Astronomi- +cal Research in the Southern Hemisphere, Chile (ESO Programme 60.A- +9100), presented by Monreal-Ibero et al. (2020) are available at the ESO +Archive (http://archive.eso.org). San Pedro de Martir data is available at +http://kincatpn.astrosen.unam.mx. +Code availability +The +code +MOCASSIN +is +available +at +the +following +URL: +https://mocassin.nebulousresearch.org/. ZEUS3-D is available at the Labora- +tory for Computational Astrophysics [83]). The compiled version of Shape is +available at http://www.astrosen.unam.mx/shape. +Acknowledgements. +We would like to start by acknowledging the Inter- +national Astronomical Union that oversees the work of Commission H3 on +Planetary Nebulae. It is through the coordinating activity of this commit- +tee that this paper came together. SA acknowledges support under the grant +5077 financed by IAASARS/NOA. JA and VB acknowledge support from +EVENTs/NEBULAE WEB research program, Spanish AEI grant PID2019- +105203GB-C21. IA acknowledges the support of CAPES, Brazil (Finance +Code 001). EDB acknowledges financial support from the Swedish National +Space Agency. EB acknowledges NSF grants AST-1813298 and PHY-2020249. +JC and EP acknowledge support from an NSERC Discovery Grant. GG-S +thanks Michael L. Norman and the Laboratory for Computational Astro- +physics for the use of ZEUS-3D. DAGH and AM acknowledge support from +the ACIISI, Gobierno de Canarias and the European Regional Development +Fund (ERDF) under grant with reference PROID2020010051 as well as from +the State Research Agency (AEI) of the Spanish Ministry of Science and +Innovation (MICINN) under grant PID2020-115758GB-I00. JGR acknowl- +edges support from Spanish AEI under Severo Ochoa Centres of Excellence +Programme 2020-2023 (CEX2019-000920-S). JGR and VGLL acknowledge +support from ACIISI and ERDF under grant ProID2021010074. DGR acknowl- +edges the CNPq grant 313016/2020-8. MAG acknowledges support of grant +PGC 2018-102184-B-I00 of the Ministerio de Educaci´on, Innovaci´on y Uni- +versidades cofunded with FEDER funds and from the State Agency for +Research of the Spanish MCIU through the “Center of Excellence Severo +Ochoa” award to the Instituto de Astrof´ısica de Andaluc´ıa (SEV-2017-0709). + +Springer Nature 2021 LATEX template +18 +JWST PN +DJ acknowledges support from the Erasmus+ programme of the European +Union under grant number 2020-1-CZ01-KA203-078200. AK and ZO were +supported by the Australian Research Council Centre of Excellence for All +Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number +CE170100013. This research is/was supported by an Australian Government +Research Training Program (RTP) Scholarship. MM and RW acknowledge +support from STFC Consolidated grant (2422911). CM acknowledges sup- +port from UNAM/DGAPA/PAPIIT under grant IN101220. SM acknowledges +funding from UMiami, the South African National Research Foundation and +the University of Cape Town VC2030 Future Leaders Award. JN acknowl- +edges support from NSF grant AST-2009713. CMdO acknowledges funding +from FAPESP through projects 2017/50277-0, 2019/11910-4 e 2019/26492- +3 and CNPq, process number 309209/2019-6. JHK and PMB acknowledge +support from NSF grant AST-2206033 and a NRAO Student Observing Sup- +port grant to Rochester Institute of Technology. MO was supported by JSPS +Grants-in-Aid for Scientific Research(C) (JP19K03914 and 22K03675). +QAP acknowledges support from the HKSAR Research grants council. +Vera C. Rubin Observatory is a Federal project jointly funded by the National +Science Foundation (NSF) and the Department of Energy (DOE) Office of Sci- +ence, with early construction funding received from private donations through +the LSST Corporation. The NSF-funded LSST (now Rubin Observatory) +Project Office for construction was established as an operating center under +the management of the Association of Universities for Research in Astron- +omy (AURA). The DOE-funded effort to build the Rubin Observatory LSST +Camera (LSSTCam) is managed by SLAC National Accelerator Laboratory +(SLAC). +AJR was supported by the Australian Research Council through award +number FT170100243. LS acknowledges support from PAPIIT UNAM grant +IN110122. CSC’s work is part of I+D+i project PID2019-105203GB-C22 +funded by the Spanish MCIN/AEI/10.13039/501100011033. MSG acknowl- +edges support by the Spanish Ministry of Science and Innovation (MICINN) +through projects AxIN (grant AYA2016-78994-P) and EVENTs/Nebulae-Web +(grant PID2019-105203GB-C21). RS’s contribution to the research described +here was carried out at the Jet Propulsion Laboratory, California Institute of +Technology, under a contract with NASA. J.A.T. would like to thank Mar- +cos Moshisnky Fundation (Mexico) and UNAM PAPIIT project IA101622 +EV acknowledges support from the ”On the rocks II project” funded by +the Spanish Ministerio de Ciencia, Innovaci´on y Universidades under grant +PGC2018-101950-B-I00. AZ acknowledges support from STFC under grant +ST/T000414/1. +This +research +made +use +of +Photutils, +an +Astropy +package +for +detection and photometry of astronomical sources [84], of the Span- +ish Virtual Observatory (https://svo.cab.inta-csic.es) project funded by +MCIN/AEI/10.13039/501100011033/ through grant PID2020-112949GB-I00 +and of the computing facilities available at the Laboratory of Computational + +Springer Nature 2021 LATEX template +JWST PN +19 +Astrophysics of the Universidade Federal de Itajub´a (LAC-UNIFEI, which is +maintained with grants from CAPES, CNPq and FAPEMIG). +Based on observations made with the NASA/ESA Hubble Space Tele- +scope, and obtained from the Hubble Legacy Archive, which is a collaboration +between the Space Telescope Science Institute (STScI/NASA), the Space +Telescope European Coordinating Facility (ST-ECF/ESAC/ESA) and the +Canadian Astronomy Data Centre (CADC/NRC/CSA). +The JWST Early Release Observations and associated materials were +developed, executed, and compiled by the ERO production team: Hannah +Braun, Claire Blome, Matthew Brown, Margaret Carruthers, Dan Coe, Joseph +DePasquale, Nestor Espinoza, Macarena Garcia Marin, Karl Gordon, Alaina +Henry, Leah Hustak, Andi James, Ann Jenkins, Anton Koekemoer, Stephanie +LaMassa, David Law, Alexandra Lockwood, Amaya Moro-Martin, Susan +Mullally, Alyssa Pagan, Dani Player, Klaus Pontoppidan, Charles Proffitt, +Christine Pulliam, Leah Ramsay, Swara Ravindranath, Neill Reid, Massimo +Robberto, Elena Sabbi, Leonardo Ubeda. The EROs were also made possible +by the foundational efforts and support from the JWST instruments, STScI +planning and scheduling, and Data Management teams. +Finally, this work would not have been possible without the collaborative +platforms Slack (slack.com) and Overleaf (overleaf.com). +Author contribution +The following authors have contributed majorly to multiple aspects of the +work that lead to this paper, the writing and the formatting of figures: +De Marco (writing, structure, interpretation, synthesis), Aleman (H2 interpre- +tation), Balick (processing and interpreting images), Garc´ıa-Segura (2D hydro +modelling), Kastner (writing, H2 measurements and interpretation), Matsuura +(imaging, photometry, H2 interpretation), Miszalski (stellar photometry), +Mohamed (hydrodynamics of binaries), Monreal-Ibero (MUSE data analy- +sis), Monteiro (photoionisation and morpho-kinematic models), Moraga Baez +(JWST image production), Morisset (photoionisation modelling), Sahai (disk +model, comparative interpretation), Soker (hydro modelling, interpretation), +Stanghellini (distances, abundance interpretation), Steffen (morpho-kinematic +models), Walsh (spatially resolved spectroscopy), Zijlstra (disk model, H2 +measurements, writing, interpretation). +The following authors have contributed key expertise to aspects of this +paper: Akashi (hydrodynamic modelling and jet interpretation), Alcolea +(CO observations), Akras (H2 interpretation), Amram (space-resolved spec- +troscopy), Blackman (hydrodynamics), Bublitz (HST and radio images of +fast evolving PN), Bucciarelli (Gaia data), Bujarrabal (radio observations, + +Springer Nature 2021 LATEX template +20 +JWST PN +disk observation and interpretation, comparative studies), Chu (disk inter- +pretation), Cami (molecular formation), Corradi (final review, interpreta- +tion), Garc´ıa-Hernandez (IR dust/PAH features and abundances), Garc´ıa- +Rojas (photoionisation modelling), G´omez-Llanos (photoionisation mod- +elling), Gon¸calves (comparative analysis), Guerrero (Xray imaging), Jones +(close binaries), Karakas (final review, stellar nucleosynthesis), Manchado +(nebular morphology, H2 interpretation), McDonald (photometry modelling), +Montez (X-ray and UV imaging), Osborn (binary nucleosynthesis), Otsuka +(IR imaging), Parker (morphology), Peeters (nebular spectroscopy, PAHs), +Ruiter (binary populations), Sabin (abundances), S´anchez Contreras (radio), +Santander-Garc´ıa (nebular evolution), Seitenzahl (star and star nebula asso- +ciation), Speck (dust), Toal´a (morphology), Ueta (nebular imaging), Van de +Steene (IR observations), Ventura (AGB evolution model). +The following authors contributed by commenting on some aspects of +the analysis and manuscript: De Beck, Boffin, Boumis, Chornay, Frank, +Kwok, Lykou, Nordhaus, Oliveira, Quint, Quintana-Lacaci, Redman, Villaver, +Vlemmings, Wesson, and Van Winckel. +Competing interest statement +We declare that no conflict of interest exists between any of the authors and +the content and production of this paper. + +Springer Nature 2021 LATEX template +JWST PN +21 +Figures +Fig. 1 JWST images of the PN NGC 3132. Left column, top and bottom: color overlays +of JWST NIRCam and MIRI images that cleanly distinguish between the PN’s ionized +gas (i.e., H ii region; top panel) and molecular gas (as seen in H2; bottom panel). Note +the sharp contrast between the relatively smooth appearance of the H ii region and the +flocculent structure of the H2 ring system and extended H2 halo. These images are presented +with square-root and log intensity stretches, respectively, from the background sky to peak +intensity levels in each image. Right image: a grey-scale, single filter (F470N), zoomed-in +NIRCam image that more readily displays details of the flocculent H2 halo. North is towards +the top, East is towards the left. +References +[1] Mastrodemos, N. & Morris, M. Bipolar Pre-Planetary Nebulae: Hydro- +dynamics of Dusty Winds in Binary Systems. II. Morphology of the +Circumstellar Envelopes. +Astrophys. J. 523, 357–380 (1999). +https: +//doi.org/10.1086/307717 . +[2] Mohamed, S. & Podsiadlowski, P. Mass Transfer in Mira-type Binaries. +Baltic Astronomy 21, 88–96 (2012) . + +alph +otSpringer Nature 2021 LATEX template +22 +JWST PN +Fig. 2 The dusty central star of the PN NGC 3132. JWST NIRCam F187N (top left) +and MIRI F1280W (top right) images of the central region of NGC 3132. The JWST MIRI +images reveal the detection of a mid-IR excess at the nebula’s true (hot, compact) central +star, which is seen projected ∼1.7′′ (∼1300AU) SW of the main-sequence A-type companion +(which is far brighter shortward of ∼10 µm). North up and East is to the left. Colour bars +indicate surface brightness in log (MJy ster−1). The bottom panel shows the near-IR to +mid-IR spectral energy distribution of the central star of NGC 3132 overlaid with a model +consisting of a combination of a hot blackbody spectrum representing the central star’s +photosphere (blue line) and a dusty circumstellar double disk model to fit the NIR and MIR +data points (red line, with the cooler disk as a dashed line). The wide companion, A star’s +spectral energy distribution is shown as a dotted line. Vertical error bars are set at 10% of +the flux values, while horizzontal bars show the width of the bandpass. +[3] Maercker, M. et al. Unexpectedly large mass loss during the thermal pulse +cycle of the red giant star R Sculptoris. Nature 490, 232–234 (2012). +https://doi.org/10.1038/nature11511, arXiv:1210.3030 [astro-ph.SR]. +[4] Santander-Garc´ıa, M. et al. ALMA high spatial resolution observations +of the dense molecular region of NGC 6302. Astron. Astrophys. 597, A27 +(2017). https://doi.org/10.1051/0004-6361/201629288, arXiv:1609.06455 +[astro-ph.SR]. +[5] Sahai, R. & Trauger, J. T. Multipolar Bubbles and Jets in Low-Excitation +Planetary Nebulae: Toward a New Understanding of the Formation and +Shaping of Planetary Nebulae. Astron. J. 116, 1357–1366 (1998). https: +//doi.org/10.1086/300504 . +[6] Sahai, R., Morris, M. R. & Villar, G. G. Young Planetary Nebulae: Hubble + +F187N +F1280W +15 +15 +1.80 +1.75 +10 +10 +1.75 +1.50 +Dec offset [arcsec] +1.25 +offset [arcsec] +1.70 +5 +1.00 +1.65 +0.75 +1.60 +-5 +0.50 +Dec +-5 +1.55 +0.25 +-10 +-10 +0.00 +1.50 +15, +0.25 +-10 +-15 +15. +10 +-10 +15 +1.45 +15 +10 +5 +0 +15 +RA offset [arcsec] +0 +RA offset [arcsec] +100 +star +dust disk +[mJy] +10 +flux +TTT +CSPN +0.1 +0.01 +1 +10 +100 +wavelength micronSpringer Nature 2021 LATEX template +JWST PN +23 +Earth view +East-West view +North-South view +To Earth +To Earth +Fig. 3 Morpho-kinematic reconstruction of the ionised cavity of PN NGC 3132. Emission +in the [N ii] line as seen from Earth (left image; North is towards the top and East is +towards the left), a view from the East, which we call East-West view (middle image), and +a view from the North which we call North-South view (right image). The colour-coding is +for Doppler-shift as seen from Earth, with blue for material approaching the observer, red +for receding gas and green for no velocity along the observer’s line of sight. We note the +prominent green (zero Doppler shift) belt in the middle image, and the filament that wraps +around the waist of the ellipsoid and which is red-shifted on one side and blue-shifted on +the other. A fly-through movie of this model can be found at this link. +Space Telescope Imaging and a New Morphological Classification System. +Astron. J. 141 (4), 134 (2011). https://doi.org/10.1088/0004-6256/141/ +4/134, arXiv:1101.2214 [astro-ph.GA]. +[7] van Winckel, H. +Post-Agb Stars. +Annu. Rev. Astron. Astrophys. 41, +391–427 (2003). https://doi.org/10.1146/annurev.astro.41.071601.170018 +. +[8] Ivanova, N. et al. Common envelope evolution: where we stand and how +we can move forward. +Astron. Astrophys. 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Zenodo (2022). + +Springer Nature 2021 LATEX template +JWST PN +1 +Supplementary Material +Specifications of JWST NIRCam and MIRI imaging +As part of its ERO program [1], JWST obtained ten images of NGC 3132: six +individual NIRCam images, through filters F090W, F187N, F212N, F356W, +F444W, and F470N, and four MIRI images, through filters F770W, F1130W, +F1280W, F1800W (Supplementary Figure 1). Basic information about these +NIRCam and MIRI filters is presented in Supplementary Table 1. The native +NIRCam field of view is 2.2×2.2 arcmin, with a pixel scale of 0.031 arcsec/pixel +in the range 0.6–2.3 µm and 0.063 arcsec/pixel in the range 2.4–5.0 µm; +the native MIRI field of view is 1.7 × 1.3 arcmin with a pixel scale of +0.11 arcsec/pixel in the range 5-27 µm. The NIRCam instrument provides +Nyquist-sampled imaging at 2 (short wavelength channel) and 4 (long wave- +length channel) microns with a PSF FWHM of ∼2 pixels in both cases. The +MIRI instrument in imaging mode, on the other hand, provides a FWHM of +0.22 arcsec (PSF FWHM of 2 pixels) for wavelengths ≥6.25 µm [2]. The NIR- +Cam imaging of NGC 3132 used 8 dither points with an offset of approximately +6 arcsec, while MIRI imaging used a 1 × 2 tile mosaic with 8 dither points. +The final NIRCam and MIRI images cover areas of approximately 150 × 150 +arcsec2 and 150 ×130 arcsec2, respectively. +Images +were +downloaded +from +the +MAST +archive +(calibration +CRDS VER11.16.3) and are neither continuum subtracted (free-free emis- +sion is included) nor background subtracted; the background regions of the +MIRI images display significant flux that is thermal emission from the (cold) +telescope and sunshade. In addition, although specific lines are targeted by +specific narrow-band filters, additional (albeit weaker) lines may be present +in some bandpasses (e.g., the F187N bandpass, which is dominated by Paα, +is contaminated by He I lines). In order to determine the emission features +present in each band, we have generated a simple model to predict the IR +spectrum of the nebula, and compared it to previously published Spitzer +observations, see Supplementary Figure 2, alongside the JWST bandpasses. +Here we see, for example, that the MIRI F770W image is likely contaminated +with H i and [Ar ii] in certain regions. Spitzer spectroscopy indicates no sign +of PAHs at 6.2, 7.7 and 8.6 µm and only a very weak 11.3 µm feature as +well as cristalline silicates [3, 4]. This is usually associated to the presence of +neutral PAHs [5]. As a result we have not included PAH emission contribution +at 7.7 µm. On the other hand, Spitzer spectra do not cover the region below +5 µm so we have indicated that there could be a weak 3.3 µm feature. +In Supplementary Figure 3 we select three regions of the nebula observed +through the NIRCam filter F212N, which we present, enlarged, in Supple- +mentary Figure 4. The latter Figure presents image sequences consisting of +archival HST images and the new JWST (ERO) images. These sequences illus- +trate the contrast between the smoothness of the emission from ionised gas +vs. the clumpiness of H2 emission, as well as the correspondence between dust +extinction (most evident in the HST images) and the H2 filaments and knots. +arXiv:2301.02775v1 [astro-ph.SR] 7 Jan 2023 + +Springer Nature 2021 LATEX template +2 +JWST PN +Supplementary Table 1 List of JWST filters used in this work +and expected emission in the corresponding band. +Filter name +λ11 +λ21 +Emission features +Date2 +Texp2 +(µm) +(µm) +within bandpass +(sec) +NIRCam +F090W +0.795 +1.005 +[S iii] 9069,9562 ˚A +2022-06-03 +5841 +F187N +1.863 +1.884 +H i Paα +2022-06-03 +9277 +F212N +2.109 +2.134 +H2 (1,0) S(1) +2022-06-03 +9277 +F356W +3.140 +3.980 +H2; Dust; PAHs? +2022-06-03 +1460 +F405N3 +4.028 +4.074 +H i Brα +F470N3 +4.683 +4.733 +H2 (0,0) S(9) +F444W +3.880 +4.986 +H i Brα; H2; Dust? +2022-06-03 +2319×2 +MIRI +F770W +6.6 +8.8 +H2 (0,0) S(5) +2022-06-12 +2708 +F1130W +10.95 +11.65 +PAHs +2022-06-12 +2708 +F1280W +11.6 +14.0 +[Ne ii] 12.8µm; +2022-06-12 +2708 +H2 (0,0) S(2) +F1800W +16.5 +19.5 +Warm dust; +2022-06-12 +2708 +[S iii] 18.6µm +1λ1, λ2: wavelengths at which the bandpass transmission is 50% of +the peak transmission. +2Observing date and exposure time. +3Pupil wheel filter; used in combination with F444W. +Central star magnitudes +JWST has detected the central visual binary from F090W to F1800W (Sup- +plementary Figure 5). At near-infrared wavelengths, the central star is faint +and on the edge of the diffraction spikes from the nearby A-type star. From +7.7 µm longward, the central star is observed to increase in brightness, and at +18 µm it exceeds the flux from the A-type star. +We obtained optical magnitudes of the central star from the Hubble Source +Catalogue version 3 [6]. The F438W, F555W and F814W Wide Field Cam- +era 3 magnitudes in the AB system were converted to Jy. The calibrated 2D +resampled NIRCam and MIRI observations available as i2d pipeline products +were used to perform aperture photometry of the central star using the pho- +tutils package [7]. A circular aperture was centered on the central star with +radii corresponding to the 80% encircled energy radius tabulated in the rele- +vant aperture correction tables, sufficient to include the majority of the central +star flux. The tables JWST nircam apcorr 0004 and JWST miri apcorr 0008 +were sourced from the JWST Calibration Reference Data System. Aperture +photometry of the central source was performed with the error extension of +the image included. Three circular sky apertures of the same radius were +selected nearby the central star to best sample the challenging background. +The background includes artefacts from the diffraction pattern of the A-type +star nearby (more prominent in the NIRCam images) and the structured neb- +ular background from NGC 3132. In the F090W, F212N, F405N and F470N + +Springer Nature 2021 LATEX template +JWST PN +3 +60 +40 +20 +0 +20 +40 +60 +RA offset [arcsec] +60 +40 +20 +0 +20 +40 +60 +Dec offset [arcsec] +F090W, 0.902 m +0.50 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +60 +40 +20 +0 +20 +40 +60 +RA offset [arcsec] +60 +40 +20 +0 +20 +40 +60 +Dec offset [arcsec] +F187N, Pa +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +60 +40 +20 +0 +20 +40 +60 +RA offset [arcsec] +60 +40 +20 +0 +20 +40 +60 +Dec offset [arcsec] +F212N, H2 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +60 +40 +20 +0 +20 +40 +60 +RA offset [arcsec] +60 +40 +20 +0 +20 +40 +60 +Dec offset [arcsec] +F356W, 3.57 m +0.8 +0.6 +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +60 +40 +20 +0 +20 +40 +60 +RA offset [arcsec] +60 +40 +20 +0 +20 +40 +60 +Dec offset [arcsec] +F405N, Br +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +60 +40 +20 +0 +20 +40 +60 +RA offset [arcsec] +60 +40 +20 +0 +20 +40 +60 +Dec offset [arcsec] +F470N, H2 +0.25 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +60 +40 +20 +0 +20 +40 +60 +RA offset [arcsec] +60 +40 +20 +0 +20 +40 +60 +Dec offset [arcsec] +F770W, H2 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +1.4 +1.5 +60 +40 +20 +0 +20 +40 +60 +RA offset [arcsec] +60 +40 +20 +0 +20 +40 +60 +Dec offset [arcsec] +F1130W, 11.3 m +1.40 +1.45 +1.50 +1.55 +1.60 +60 +40 +20 +0 +20 +40 +60 +RA offset [arcsec] +60 +40 +20 +0 +20 +40 +60 +Dec offset [arcsec] +F1280W, [Ne II] +1.45 +1.50 +1.55 +1.60 +1.65 +1.70 +1.75 +1.80 +60 +40 +20 +0 +20 +40 +60 +RA offset [arcsec] +60 +40 +20 +0 +20 +40 +60 +Dec offset [arcsec] +F1800W, [S III] +2.00 +2.05 +2.10 +2.15 +2.20 +2.25 +2.30 +Supplementary Figure 1 JWST NIRCam and MIRI images of NGC 3132. North up and +East is to the left. Colour bars indicate surface brightness in log(MJy ster−1). +filters the dominant background and/or intrinsic faintness of the central star +precluded any meaningful fluxes from being measured. +The sky background was estimated as the average of the median counts +in each sky aperture, where the median was calculated using sigma clipping +with σ = 3. The sky background was scaled to the aperture area A of the +central star aperture before it was subtracted from the aperture sum. The flux +of the central star was calculated as the sky subtracted aperture sum scaled +by the MJy/sr to µJy conversion factor and the aperture correction sourced +from the aperture correction tables. The uncertainty in the flux was estimated +as �σp + 2σsky where σp is the photutils aperture sum err and σsky is Aσ2 +b, +where σb is the average of the standard deviation of counts in each sky aperture. + +Springer Nature 2021 LATEX template +4 +JWST PN +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +Wavelength ( m) +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Fux (Arbitrary units) +[Ca II] +[S III] +[S III] +[C I] +HeII +H I +[Fe II] +H I (Pa ) +H2 +H2 +H2 +H2 +H2; H I +H2 +H2 +H2 +H I +H2 +H I (Br ) +[K III]; H I +H2 +F090W +F187N F212N +F356W +F405N +F444W +F470N +6 +8 +10 +12 +14 +16 +18 +20 +Wavelength ( m) +0 +2 +4 +6 +8 +10 +Fux (Arbitrary units) +H2 +H2 +[Ni II] +H2; [Ar II] +[Na III] +H I +H2 +H2 +[Ar III] +[S IV] +[Ni II] +PAH +H2; HI +[Ne II] +[Ne V] +[Cl II] +[Ne III] +H2 +H I +[S III] +F770W +F1130W +F1280W +F1800W +Supplementary Figure 2 Simulated IR spectrum of NGC 3132, overlaid with the JWST +filters to demonstrate bandpass contamination. Top panel: the simulated spectrum of +NGC 3132 between 0.8 and 5.2 µm (blue line) with, overlaid the JWST bandpasses (labelled +coloured shapes). Bottom panel: the simulated spectrum of NGC 3132 between 5 and 20 µm +(blue line) with, overlaid the JWST bandpasses (labelled coloured shapes). +10h07m06s +03s +00s +06m57s +-40°25'30" +26'00" +30" +27'00" +RA (ICRS) +Dec (ICRS) +Supplementary Figure 3 JWST/NIRCam F212N image of NGC 3132. White squares +indicate positions and sizes of “blowup” regions highlighted in Figure 4. + +Springer Nature 2021 LATEX template +JWST PN +5 +HST F502N +HST F658N +NIRCam F187N +NIRCam F212N +NIRCam F405N +MIRI F1130W +1"/753 AU +1"/753 AU +1"/753 AU +Supplementary Figure 4 The enlarged images of knots in three representative regions, +indicated by white boxes in Figure 3: the West side of the ring (top row), near the centre +of the ring (middle row) and the East side of the ring (bottom row). Filament structures +stand out in the NIRCam F212N images. A few filaments are dusty, as clearly seen in the +HST optical images (first two columns), but also in the NIRCam F187N and F405N images +(third and fifth columns). +However, due to the complex sky background, these uncertainties are likely +underestimates of the true uncertainty, which we estimate to be 5–10% of the +flux. +Supplementary Table 2 gives the measured and dereddened fluxes using +AV = 3.1E(B − V ), where E(B − V ) = 0.09 mag [8], and filter extinction +ratios are from the Spanish Virtual Observatory Filter Profile Service [9, 10]. +In Supplementary Figure 6 we present the PARSEC isochrones with the +derived location of the A2V star. +Filter +Fν (µJy) +Formal +Fν,0 (µJy) +Formal +error∗ +error∗ +WFPC3/F438W +1620.32 +10.21 +2274.64 +14.3 +WFPC3/F555W +1188.50 +7.33 +1556.61 +9.6 +WFPC3/F814W +560.27 +3.29 +654.18 +3.8 +NIRCam/F187N +130.43 +4.65 +135.77 +4.84 +NIRCam/F356W +50.78 +5.31 +51.67 +5.41 +MIRI/F770W +285.68 +13.47 +288.02 +13.58 +MIRI/F1130W +1445.32 +2.46 +1462.85 +2.49 +MIRI/F1280W +1362.75 +5.85 +1373.09 +5.90 +MIRI/F1800W +11344.60 +20.93 +11425.35 +21.08 +∗The actual error on this photometric measurements is likely +∗ closer to 5-10% of the flux values. +Supplementary Table 2 Measured (Fν) and dereddened (Fν,0) fluxes of the central +star. To convert to magnitudes: mag(AB) = −2.5 ∗ log10(Fν [in Jansky]) + 8.90 + +Springer Nature 2021 LATEX template +6 +JWST PN +a. HST F814W +b. NIRCam F090W +c. NIRCam F187N +d. NIRCam F212N +e. NIRCam F356W +f. MIRI F770W +g. MIRI F1130W +h. MIRI F1280W +i. MIRI F1800W +2 arcsec +Supplementary Figure 5 Sections of the JWST images zooming in on the central stars. +NIRCam F090W, F187N and F212N images are deconvolved with simulated PSFs. HST, +NIRCam F356W and MIRI images are not deconvolved. Note that the slight offset of +the positions of the central star in the F090W and F187N images is due to imperfect +deconvolution due to the nearby saturated star. +Extended structure of the central star +Supplementary Figure 7 demonstrates the extended structure of the central +star in the MIRI F1130W and F1800W band images. From the left to right +column, we see the central, the A-type star, an example of a saturated +star, and the simulated PSF. The images are oriented such that North is +145 degrees clockwise. An example image of a saturated star is from the Taran- +tula Nebula region, which was observed as a part of the first image program +(PID 2729). The coordinate of this saturated star is RA=05h38m33.61s and +Dec=−69d04m50.5s. The MIRI PSF is simulated based on Webb PSF software +version 1.1.0 (https://jwst-docs.stsci.edu/jwst-mid-infrared-instrument/miri- +performance/miri-point-spread-functions). The top row shows the images, and +the second and third rows show radial profiles, which are sliced in horizontal +and vertical directions across the peak (blue lines). Dotted lines indicate the + +Springer Nature 2021 LATEX template +JWST PN +7 +4 +3.95 +3.9 +3.85 +1 +1.5 +2 +2.5 +1 +2 +3 +4 +5 +6 +7 +8 +2.6 +2.5 +2.4 +Supplementary Figure 6 PARSEC [11] isochrones with the location of the A2 V star. +The isochrones are labelled in order of descending age : 9, 8, 7, 6, 5.6, 5.3, 5, 4 in units of +108 yr. The dashed lines connect points of constant mass, labelled in solar units. The error +bar on the horizzontal axis is the temperature uncertainty given by Gaia, ±200 K. The error +bar on the vertical axis is based on a conservative 0.25 mag error on the absolute magnitude. +data points that are strongly affected by other factors, such as bright neigh- +bouring stars or detector saturation. The central star is mildly saturated in the +F1800W image, with data quality flags of about 5—8 pixels across the peak, +so that the data within the 10 pixels from the peak of the central star are plot- +ted as dotted lines. On the central star radial profiles of F1800W, green lines +demonstrate the simulated radial profile with 0.44 arcsec radius flat-intensity +‘disk’. This shows that the central star is extended at a scale of ≳0.8 arcsec. +In the radial profiles, the PSF is also plotted as an orange line, which has a +FWHM of 0.58 arcsec at F1800W. +Supplementary Figure 7 demonstrates that the central star is clearly +extended more than the PSF. On the central star radial profiles, green lines +demonstrate the radial profile with a 0.44 arcsec radius, flat-intensity ‘disk’. +In reality, the intensity gradually decreases radially, rather than a flat inten- +sity with a cliff edge, and the tailing of this gradual decrease continues beyond +0.4 arcsec. +Morpho-kinematic modelling +In Figure 3 we have presented the 3D morpho-kinematic reconstruction of the +ionised region of NGC 3132 using the interactive morpho-kinematic modelling +software Shape [12]. In addition to the new images from JWST, spectroscopic +reference data for the reconstruction are position-velocity diagrams from the +San Pedro M´artir Kinematic Catalogue of Galactic Planetary Nebulae ([13, +14]). +An assumption is made on the current velocity field in order to map the +Doppler-shift to a 3D position of that image element. 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structure of the central star +in the MIRI F1130W (top panel) and F1800W (bottom panel) band images. From the left +to right column, the central star, the A-type star, an example of a saturated star, and the +simulated PSF. The top row displays the images, and the second and third row show radial +profiles, which are sliced in the horizontal and vertical directions across the peak (blue +lines). Dotted lines indicate the data points strongly affected by other factors, such as bright +neighbouring stars or detector saturation for the F1800W image. In the radial profiles, the +PSF is also plotted as an orange line.The central star is clearly extended more than the +PSF. On the central star radial profiles of the F1800W image, the green line demonstrates +the radial profile of a 0.44 arcsec-radius, flat-intensity ‘disk’. + +Springer Nature 2021 LATEX template +JWST PN +9 +The velocity field is 1 km s−1 arcsec−1. This value ensures that the cross-section +of the main shell is approximately circular. We estimate the uncertainty to be +of the order of 30%. Note that the stretching of the structures along the line +of sight is proportional to this value. In other words, the shape of the nebula +along the line of sight is linearly related to the component of the velocity +vector. This model of the inner ionised cavity supersedes or rather, completes, +the barrel or “diabolo” model [16] in view of more sensitive spectroscopy that +allowed us to detect the faint, and fast, closed ends of the ellipsoid along its +major axis, which is close to the line of sight. +In Supplementary Figure 8 we show the slit positions and resulting position- +velocity diagram (top row: observed, bottom row: simulated) reconstruction +with Shape. A 3D fly through the 3D volume can be found at this link. The +position-velocity diagrams in Supplementary Figure 8 are effectively 2D ren- +ditions of the spectral line shape as we move along the slit. The slit positions +are indicated. +In the final step of our morpho-kinematic model (Figure 3), we place two +complete shells of non-uniform, filamentary material around the central star +to match the observed size of the H2 halo. The inner shell ranges from 33 to +45 arcsec from the central star, and the second ranges from 60 to 70 arcsec. +These are illuminated through a partially opaque ellipsoid (approximately cor- +responding with the ellipsoid containing the ionised nebula) that has reduced +opacity around the poles and has an overall porous opacity. +At this point we are only interested in testing the possibility that the +opacity of the walls of the central cavity within which the exciting central star +resides, could be the cause of the overall emissivity distribution and features. +The central star is made to radiate as a blackbody. We have then used a +simple proxy for the various radiative processes that are at work here, i.e., +isotropic scattering on dust. Since we expect dust to dominate the transport +of the exciting radiation, this is a reasonable first test, with the details being +irrelevant for this simple geometric simulation. A spherical density modulation +was also imposed with ad hoc spacing responding to the following modulation: +ρ/ρ0 = 0.3 + sin(0.8 r/arcsec)10 sin(1.4 r/arcsec)2. This generates the arch +pattern. +This type of painstaking morpho-kinematic modelling is critically depen- +dent on images at different wavelengths as well as spatially-resolved spec- +troscopy. On the basis of this type of data driven 3D model, we can now +understand structures first revealed by JWST such as the H2 halo and the +dusty central stars. +References +[1] Pontoppidan, K., Blome, C., Braun, H., Brown, M., Carruthers, M., Coe, +D., DePasquale, J., Espinoza, N., Garcia Marin, M., Gordon, K.D., Henry, +A., Hustak, L., James, A., Koekemoer, A.M., LaMassa, S., Law, D., +Lockwood, A., Moro-Martin, A., Mullally, S.E., Pagan, A., Player, D., + +Springer Nature 2021 LATEX template +10 +JWST PN +0 +-20 +-40 +20 +40 +arcsec +0 +-20 +-40 +20 +40 +arcsec +SPM a +SPM b +SPM c +H A +H D +SPM a +SPM b +SPM c +H a +H d +0 +100 +km/s +0 +-20 +-40 +20 +40 +arcsec +0 +-20 +-40 +20 +40 +arcsec +-100 +SPM a +SPM b +SPM c +H A +H D +Supplementary Figure 8 Position velocity data and model to achieve the morpho- +kinematic model of PN NGC 3132. Top figure: a MUSE [N ii] image of NGC 3132 with the +slit positions marked. 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='SR] 7 Jan 2023 Springer Nature 2021 LATEX template 2 JWST PN 1*School of Mathematical and Physical Sciences, Macquarie University, Sydney, NSW 2109, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 2*Astronomy, Astrophysics and Astrophotonics Research Centre, Macquarie University, Sydney, NSW 2109, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 3Department of Physics, Technion, Haifa, 3200003, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 4Kinneret College on the Sea of Galilee, Samakh 15132, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 5Institute for Astronomy, Astrophysics, Space Applications and Remote Sensing, National Observatory of Athens, GR 15236 Penteli, Greece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 6Observatorio Astron´omico Nacional (OAN/IGN), Alfonso XII, 3, 28014 Madrid, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 7Instituto de F´ısica e Qu´ımica, Universidade Federal de Itajub´a, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' BPS 1303, Pinheirinho, Itajub´a 37500-903, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 8Aix-Marseille Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', CNRS, CNES, LAM (Laboratoire d’Astrophysique de Marseille), Marseille, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 9Astronomy Department, University of Washington, Seattle, WA 98105-1580, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 10Department of Space, Earth and Environment, Chalmers University of Technology, S-41296 Gothenburg, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 11Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA.' metadata={'source': 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+page_content=' 16Department of Physics & Astronomy, University of Western Ontario, London, ON, N6A 3K7, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 17Institute for Earth and Space Exploration, University of Western Ontario, London, ON, N6A 3K7, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 18SETI Institute, 399 Bernardo Avenue, Suite 200, Mountain View, CA 94043, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 19Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 20Institute of Astronomy and Astrophysics, Academia Sinica (ASIAA), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Springer Nature 2021 LATEX template JWST PN 3 21GRANTECAN, Cuesta de San Jos´e s/n, E-38712, Bre˜na Baja, La Palma, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 22Instituto de Astrof´ısica de Canarias, E-38205 La Laguna, Tenerife, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 23Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 24Instituto de Astronom´ıa, Universidad Nacional Aut´onoma de M´exico, Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 107 Carr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Tijuana-Ensenada, 22860, Ensenada, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 25Departamento de Astrof´ısica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain.' metadata={'source': 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Hungary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 34CSFK, MTA Centre of Excellence, Konkoly-Thege Mikl´os ´ut 15-17, 1121 Budapest, Hungary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 35Consejo Superior de Investigaciones Cient´ıficas, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 36School of Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 37Department of Physical Sciences, The Open University, Walton Hall, Milton Keynes, MK7 6AA, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 38Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, The University of Manchester, Oxford Road M13 9PL Manchester, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 4 JWST PN 39Leiden Observatory, Leiden University, Niels Bohrweg 2, NL 2333 CA Leiden, The Netherlands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 40Australian Astronomical Optics, Faculty of Science and Engineering, Macquarie University, North Ryde, NSW 2113, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 41Department of Physics, University of Miami, Coral Gables, FL 33124, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 42South African Astronomical Observatory, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Box 9, 7935 Observatory, South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 43Astronomy Department, University of Cape Town, 7701 Rondebosch, South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 44NITheCS National Institute for Theoretical and Computational Sciences, South Africa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 45Center for Astrophysics, Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 46Center for Computational Relativity and Gravitation, Rochester Institute of Technology, Rochester, NY 14623, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 47National Technical Institute for the Deaf, Rochester Institute of Technology, Rochester, NY 14623, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 48Departamento de Astronomia, Instituto de Astronomia, Geof´ısica e Ciˆencias Atmosf´ericas da USP, Cidade Universit´aria, 05508-900, S˜ao Paulo, SP, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 49Okayama Observatory, Kyoto University, Honjo, Kamogata, Asakuchi, Okayama, 719-0232, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 50Department of Physics, CYM Physics Building, The University of Hong Kong, Pokfulam, Hong Kong SAR, PRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 51Laboratory for Space Research, Cyberport 4, Cyberport, Hong Kong SAR, PRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 52Rubin Observatory Project Office, 950 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Cherry 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Galway H91 CF50, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 55University of New South Wales, Australian Defence Force Academy, Canberra, Australian Capital Territory, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 56Centro de Astrobiolog´ıa (CAB), CSIC-INTA, Camino Bajo del Castillo s/n, ESAC campus, 28692, Villanueva de la Ca˜nada, Madrid, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Springer Nature 2021 LATEX template JWST PN 5 57Jet Propulsion Laboratory, California Institute of Technology, CA 91109, Pasadena, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 58University of Texas at San Antonio, Department of Physics and Astronomy, Applied Engineering and Technology Building, One UTSA Circle, San Antonio, TX 78249, United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 59NSF’s NOIRLab, 950 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Cherry Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Tucson, AZ 85719, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 60ilumbra, AstroPhysical MediaStudio, Hautzenbergstrasse 1, 67661 Kaiserslautern, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 61Instituto de Radioastronom´ıa y Astrof´ısica, UNAM, Antigua Carretera a P´atzcuaro 8701, Ex-Hda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' San Jos´e de la Huerta, Morelia 58089, Mich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Mexico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 62Department of Physics and Astronomy, University of Denver, 2112 E Wesley Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Denver, CO 80208, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 63Royal Observatory of Belgium, Astronomy and Astrophysics, Ringlaan 3, 1180 Brussels, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 64INAF – Osservatorio Astronomico di Roma, Via Frascati 33, I-00040, Monte Porzio Catone (RM), Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 65Onsala Space Observatory, Department of Space, Earth and Environment, Chalmers University of Technology, Onsala, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 66Institute of Astronomy, KULeuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' E-mail(s): orsola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='demarco@mq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='au;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Abstract Planetary nebulae (PNe), the ejected envelopes of red giant stars, pro- vide us with a history of the last, mass-losing phases of 90% of stars initially more massive than the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Here, we analyse James Webb Space Telescope (JWST) Early Release Observation (ERO) images of the PN NGC 3132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' A structured, extended H2 halo surrounding an ionised central bubble is imprinted with spiral structures, likely shaped by a low-mass companion orbiting the central star at ∼40–60 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The images also reveal a mid-IR excess at the central star interpreted as a dusty disk, indicative of an interaction with another, closer companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Including the previously known, A-type visual companion, the progeni- tor of the NGC 3132 PN must have been at least a stellar quartet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The JWST images allow us to generate a model of the illumination, ionisa- tion and hydrodynamics of the molecular halo, demonstrating the power of JWST to investigate complex stellar outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Further, new measure- ments of the A-type visual companion allow us to derive the value for Springer Nature 2021 LATEX template 6 JWST PN the mass of the progenitor of a central star to date with excellent pre- cision: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='06 M⊙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' These results serve as pathfinders for future JWST observations of PNe providing unique insight into fundamental astrophysical processes including colliding winds, and binary star inter- actions, with implications for supernovae and gravitational wave systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Keywords: stars: AGB and post-AGB, stars: evolution, ISM: jets and outflows, ISM: molecules, planetary nebulae: individual: NGC 3132 Main Introduction Planetary nebulae (PNe) are the ejected envelopes of intermediate-mass (∼1– 8 M⊙) stars that have recently terminated their asymptotic giant branch (AGB) stage of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Moving outwards from the hot pre-white dwarf star (T ∼ 105 K) that is the progeny of the AGB star, the structure of a canonical quasi-spherical PN consists of a hot, sparse, wind-heated bubble (T ∼ 107K) surrounded by a dense shell of displaced, ionised AGB gas (T ∼ 104 K), which in turn may still be surrounded by “pristine,” cold (T ∼ 102 K), molecule- and dust-rich AGB ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' On the other hand, if the progenitor star interacted with a companion(s) during its post-main sequence evolution, we would expect departures from spherical symmetry, perhaps including spiral structures and arcs [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', 1–3], the presence of a dense, molecule-rich torus [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', 4], one or more pairs of polar lobes formed by fast, collimated outflows and jets [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', 5, 6], and/or a dusty, circumbinary disk [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The type of interaction depends on the orbital radius, and ranges from common envelope evolution for close binaries [8], to accretion disks and gravitational focussing of the wind for wider systems [9–11], to displacement of the central star from the geometric centre of the nebula for the widest systems [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The first Hubble Space Telescope (HST) images of PNe revealed a breath- taking new world of details and far more complex structures than had been gleaned from ground-based images [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The superb spatial resolu- tion of HST, combined with high-resolution, kinematic mapping, enabled the construction of detailed 3D, morpho-kinematic models, which, together with hydrodynamic models [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', 15, 16], started to connect our understanding of the evolution of the structures and kinematics of PNe with their possible binary star origins [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', 17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The James Webb Space Telescope (JWST), with its superb sensitivity and high spatial resolution from near- to mid-IR, is now poised to enable a leap of similar magnitude in our understanding of PNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This journey began when JWST released near-IR and mid-IR images of just one PN, NGC 3132, as part of its ERO program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' NGC 3132 is a nearby (D ∼ 750 pc), molecule-rich [20, 21], ring-like PN, long known to harbour a visual binary comprising the central (progenitor) star and an A star companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In this paper we show that Springer Nature 2021 LATEX template JWST PN 7 the JWST ERO images contain multiple, new lines of evidence that NGC 3132 is the recent product of a hierarchical multiple progenitor stellar system, which has experienced both indirect and direct interactions involving one or more components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Such binary interactions have taken on new importance in the era of gravitational wave detectors (LIGO [22], LISA [23]) and ambitious transient surveys [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Indeed, PNe like NGC 3132 offer unique insight into the formation pathways of the close, single and double degenerate binaries that are eventual gravitational wave sources and (perhaps) type Ia supernova progenitors [25– 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Results A flocculent molecular halo surrounding an ionised bubble Figure 1 displays colour overlays of NIRCam and MIRI images of NGC 3132 that highlight JWST’s clean separation of the PN’s ionised (H ii) and molec- ular (H2) regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The full resulting JWST image suite, along with basic information, is presented in Specification of JWST NIRCam and MIRI imag- ing and Supplementary Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The images reveal, for the first time, the extent and detailed structure of the halo of molecular gas that lies exterior to the nebula’s central, ionised cavity and its bright and thin, peripheral ellip- tical ring (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This molecular halo is well detected in rovibrational H2 emission at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='12 µm (1–0 S(1)), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='7 µm (0–0 S(9)), and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='7 µm (0–0 S(5)) out to 60 arcsec (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='22 pc at the adopted distance of 754 pc, see Properties and distance of NGC 3132) from the central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Spatially organised structures — arcs and patterns of spikes emanating radially outward — are observed in the halo H2 emission on medium to large scales, while molecular arcs, loops, and knots are detected on size scales from ∼500 AU down to the limiting (∼75 AU ) resolution of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The typical thickness of the bright H2 rings that surround the nebular core is ∼ 1 − 2 arcsec (∼750-1500 AU), measured at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='7 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Figure 1 conclusively demonstrates that the molecular gas is much clumpier than the ionised gas component of NGC 3132 (see also Supplementary Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In hydrogen recombination lines and [S iii] emission (Figure 1, top- left), the nebula’s central ionised cavity (within ∼25 arcsec of the central star) appears as a relatively smooth elliptical region that is bounded by a single, sharped-edged ring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' whereas in H2 (Figure 1, bottom-left), this same central region appears as a far more complex system of clumpy filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The regions in and around this bright, inner H2 ring system contain as many as 20 dense clumps (knots) per square arcsec, implying the total number of H2 knots in this region exceeds 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The H2 knots in the outer (halo) region are less distinct and further apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The presence of radially-directed spike features in the H2 halo indicates that direct irradiation by UV photons, leaking through less dense gas between the inner ring system’s H2 knots, are most likely responsible for the excitation of the IR H2 lines in the extended halo, although shock excitation cannot be Springer Nature 2021 LATEX template 8 JWST PN completely ruled out (see [29] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The relative lack of H2 halo emission to the East-Northeast and West-Southwest of the central star then indicates a general lack of central star UV illumination, as opposed to lack of halo molecular mass in those directions (see Discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Measurements of the extinction of background nebulosity through representative knots suggests typical knot densities of ∼ 106 cm−3 and masses of ∼ 10−5 M⊙ (see Densities, masses and excitation of the H2 knots), suggesting a total H2 mass of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1 M⊙ in the central ring region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The system of (broken) concentric arcs revealed in the H2 halo by the JWST images is similar to those observed in the extended, dusty envelopes of many AGB stars, proto-PNe and PNe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', [3, 30, 31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' A widely accepted scenario to explain the formation of such arc systems is the modulation of an AGB wind by a stellar or substellar companion, creating 3D spiral-like patterns along the orbital plane [see 1, 32–35, and references therein].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The average angular distance between the arc structures, 2 arcsec, implies an orbital period of 290-480 years and an orbital separation of 40-60 AU between the central star and the companion that shapes the mass loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Here, we have assumed a companion mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='2 M⊙, the highest mass main-sequence star that could hide in the present-day central star’s glare yet still form a visible arc system (other parameters are an expansion velocity in the range 15-25 km s−1 [31] and an assumed late-AGB central star mass of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='8 M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' likely still 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='2 M⊙ larger than the post-AGB mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The bright A2 V visual companion seen at ∼1300 AU projected separation from the central star cannot be responsible, suggesting (at least) a triple system in a stable configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The dusty central star In the MIRI images obtained at wavelengths longer than 10 µm, the faint central star appears as bright or brighter than its A2 V main sequence visual companion [36];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This infrared excess was undetectable in the mid- infrared at lower spatial resolution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', in WISE images [37]) because of the surrounding bright nebulosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The JWST-discovered IR excess indicates that a considerable amount of warm dust is present around the ultra-hot (∼110 kK) PN central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The thermal infrared source appears marginally extended in the 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='8 µm MIRI images with an apparent size of ∼300 AU (FWHM) at 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='8 µm (see PSF measurements of the central star).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The bottom panel of Figure 2 displays the central star’s near-IR to mid-IR spectral energy distribution fitted by a combination of a hot stellar photosphere represented by a blackbody curve and two curves to fit the infrared data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The two curves are generated with a model that follows closely that of [38] for the Helix nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' A number of 100 µm grains are taken as blackbody spheres with temperatures set by absorption and re-emittance of the stellar luminosity (200 L⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' a correction factor is then applied to simulate a grain size distribution between 60 and 1000 µm, as done by [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The temperature varies as d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='5, where d is the distance to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The surface density of the disk is taken as constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The resulting blackbody radiation is calculated at each radius, and Springer Nature 2021 LATEX template JWST PN 9 the emission is summed over all radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' A better model will require radiative transfer, actual dust emissivities, a range of grains sizes, and for the silicate feature, the inclination of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This will be explored in a future paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The best-fit model disk has an inner radius of 55 AU and outer radius 140 AU, and a dust mass of 3 × 1026 g or 2 × 10−7 M⊙ (approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='05 Earth masses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The dust temperature range (inner to outer radius) is 130 to 80 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The outer radius of 140 AU, though poorly constrained, is consistent with the deconvolved half-width of the marginally extended mid-IR source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' These dimensions resemble those inferred for the disk orbiting the central star of the Helix (35–150 AU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [38]), but the dust mass is somewhat smaller (cfr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='13 earth masses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The outer radius could be slightly larger, if the 18 µm flux is underestimated because of detector saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' An additional inner, hotter disk — with radius between 3 and 8 AU, a temperature between 550 to 335 K (inside to outside) and a very small mass of 2 × 1022 g (approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='02 times the mass of Ceres) — is needed to fit the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='5 and 7 µm fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' While this model does not constrain the geometry of the distribution to be that of a disk, the reasoning behind a disk structure is based on a physical reasoning whereby only a rotating Keplerian disk can be shown to be stable and relatively long- lived, while other structures, such as shells, are easily shown to be unstable [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The A2 V companion is slightly evolved [36] and has a mass of MA2V = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='15 M⊙, using the PARSEC isochrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Its visual companion, the PN central star, must have descended from a more massive star, as it has evolved faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Extrapolating the same PARSEC isochrone gives an initial main sequence mass for the central star of Mi = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='86±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='06 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This is poten- tially the most precise initial mass for any PN central star or white dwarf yet determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We estimate the error to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='16 M⊙ if we add systematic effects between different isochrone models (see Central star system’s masses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The current (near-final) mass of a PN central star descended from such a ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='9 M⊙ progenitor is predicted to be Mf ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='05 M⊙ based on initial- final mass relations [40], albeit with larger systematic uncertainties that are dependent on details of the mass loss process adopted by the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' It is noteworthy that photoionisation models of the nebula require a cooler, dimmer and overall less massive central star (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='58±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='03 M⊙) than what we have found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We find that we can reconcile the mass of the star today and that of the photoionisation model, while also matching the nebular abundances and the nebular age, if we assume that the AGB evolution of a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='86 M⊙ star, was interrupted by a binary interaction that ejected the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We conjecture that the AGB evolution was interrupted at a core mass of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='61 M⊙, because for larger values, the C/O ratio of the stellar envelope gas would increase above unity (counter to the observation of crystalline silicate grains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' At larger masses the N/O ratio would also increase above the observed value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 10 JWST PN Discussion The first striking discovery of JWST is the presence of the dusty disk around the ultra-hot central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This indicates that JWST can accurately detect dusty disks lighter than Ceres, as far as ∼700 pc away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' For our PN, the presence of such a disk orbiting the PN central star favours a close binary interaction, where the companion either merged with the primary star, or is still in orbit but is undetected (mass < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='2 M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' based on an unresolved or barely resolved, equal-brightness companion);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' in either case, the companion has donated a sub- stantial fraction of its angular momentum to the gas [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Observationally, such disks around PN central stars, though rare, appear to be by and large associated with known or strongly suspected binarity [39] and may be related to circumbinary disks detected around other classes of post-AGB binary stars [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' An interacting binary scenario is reinforced by the shape of the ionised cav- ity, which represents the inner, most recent mass-loss phase, when the already hot central star emitted a fast, tenuous wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Pairing the JWST images with spatially resolved spectroscopy we constructed a 3D visualisation of this cav- ity (see Morpho-kinematic modelling in Supplementary Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In Figure 3 we show that this inner cavity is inferred to be an expanding prolate ellipsoid with its long axis tilted at approximately 30 deg to the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Its surface is not smooth and presents instead a number of protuberances, most of which can be paired via axes passing through, or very near the central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Prolate cavities such as these, with misaligned structures, are common in PN and are likely sculpted by jets from interacting binaries in the earlier, pre-PN phase of the nebula [44], with additional details added during the interaction between the AGB wind and post-AGB fast wind and via the process of PN ionisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The numerous protuberances clearly evident in the 3D reconstruction could arise from ionised gas breaking out of the inner cavity through an uneven outer shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The apparent pairing between these protuberances may argue instead for the presence of intermittent and toppling jets [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' To generate jets over such a wide range of axes, an interacting binary is not enough, and one would have to conjecture that the central star is or was a member of not just a close binary, but of an interacting triple system [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Recent studies of interactions in triple systems [47, 48] also argue for the possibility of interactions yielding complex ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Outside the ionised ellipsoid, one encounters material ejected earlier in the star’s history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The AGB mass loss, at rates of up to ∼10−5 M⊙yr−1 and speeds of ∼10 km s−1 over a ∼105 yr timescale [49], generates an enormous, expanding envelope of molecular gas and dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The H2 halo imaged by JWST constitutes the most recently ejected (inner) region of this AGB envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The spikes observed in the halo (Figure 1, right panel) show that the inner cavity is very porous, though less so near the minor axis where the cavity edges are brightest, densest, and least fractured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The JWST images motivated 2D hydrodynamic simulations to replicate these flocculent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In Figure 4 we see two time snapshots towards the Springer Nature 2021 LATEX template JWST PN 11 end of a simulation where an inner, faster wind from the heating central star and its ionising radiation, plough into the dense AGB (halo) material (see Methods, Section 66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The fragmentation that happens at the interface of the swept-up material also creates the variable opacity needed to shield some of the wind material from ionising radiation, which then quickly recombines and allows the formation of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Non ionising radiation leaks more readily because the opacity above 913 ˚A is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' These photons produce florescence of H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In Figure 4 we see two time snapshots towards the end of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In the first panel we see a set of approximately radial spikes, but 200 years later those straight and thin spikes evolve to thicker and sometimes curved ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In the right column of Figure 4 two different parts of the nebula exhibit thinner and straighter spikes (top-right panel) or thicker, bent ones (bottom- right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Although the entire nebula was ejected and ionized over a short time interval, there can be a delay in the evolution of a given spike in a specific part of the nebula, related to the local opacity in the swept-up shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Figure 4 suggests that differences of only ∼ 200 years in the timescales of mass ejection and/or the progress of illumination along specific directions can explain the marked differences observed in the flocculent structure around the nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The successful modelling of illumination percolating unevenly into the molecular halo (Figure 4) motivated a further geometric model of the halo, presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This Figure compares the extended H2 structures as imaged by JWST with a model consisting of two thick, concentric, unbroken but clumpy, shells of material that are illuminated by the central star through a porous ellipsoid representing the boundary of the ionised cavity, with reduced opacity in the polar regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' As a result of the uneven illumination the dis- tribution of H2 material appears fragmented and is generally brighter toward the polar regions (and suppressed along the equatorial plane) of the central ellipsoidal, ionised region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The distribution seen in the JWST H2 images could be reproduced more closely by altering the opacity of the inner ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Fly- though movies of the 3D reconstructions of both the inner ellipsoid (Figures 3) and the outer H2 halo (Figure 5) can be found following the links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The arches in the JWST images, are not smeared as is typical of those seen in projection [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', 50], but are instead sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This possibly indicates that these arches are on or near the plane of the sky, indicating that the orbit of the companion at ∼40-60 AU is closely aligned to the waist of the inner ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This companion cannot partake in the formation of the disk around the central star, though it may play a secondary role in the shaping of other PN structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' It is also unlikely to have launched strong jets because at such distance the accretion rate would be very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' As such, this would be an additional companion to the inner binary (or triple), making it a tertiary (or quaternary) companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The visual A-type companion would then be a fourth (fifth) member of the group, an almost complete bystander from the point of view of interac- tion and shaping, but critically important for this study: Its well measured Springer Nature 2021 LATEX template 12 JWST PN mass, and slight evolved status, constrained the initial mass of the central star: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='86±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='06) M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' To reconstruct the events that lead to the demise of the progenitor of NGC3132, the PN acts like a murder scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The A-type companion, could not have partaken to the interaction that unravelled the AGB star, but was (and is) certainly present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' A second companion at 40-60AU left an indelible trail of its presence in the form of arcs, but was not close enough to generate the dusty disk, nor shape the ionised cavity, implying that there must have been at least another accomplice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This points the finger at a close-by companion, that is either avoiding detection, or has perished in the interaction (merged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' If the numerous protuberances seen in the ionised cavity come in pairs, then tumbling jet axes would be needed and this would point the finger to the presence of a second, close companion [47, 48], which would make the system a quintet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Even ignoring the putative second, close companion, we can state with good degree of certainty that the system is at least a quartet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Systems of four or five stars orbiting within a few ×1000 AU are not impossibly rare for primary stars in the progenitor mass range of interest here [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', HD 104237;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 51];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' indeed, present estimates indicate that 50% or more of stars of 2-3 M⊙ are in multiple systems, and of order 2% of A-type stars have four companions [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' JWST is at the starting gate of its promise as an astrophysical pathfinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' With complementary radio, interferometric and time resolved observations, it can find the temporal signatures of active convective mass ejection from the surfaces of AGB stars and the subsequent gravitational influence of companion stars in dynamically- and thermally-complex outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Thus JWST offers the potential to intimately connect the histories of PNe and the role of close stel- lar companions to studies of chemical evolution, nebular shaping and binary interactions for the next century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Methods Properties and distance of NGC 3132 The inner, ionised cavity of NGC 3132 is elliptical in shape, with a major axis of ∼40 arcsec (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='15 pc) and an electron density of n ∼ 103 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The ionization structure and abundances were the subject of a recent study by [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The nebula is also known to be molecule-rich [20];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' it is among the brightest PNe in near-IR H2 emission [21, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' A bright A2 V star is found near the centre of the PN, but is too cool to be the ionizing star;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' the actual PN progenitor is much fainter and is located ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='7 arcsec to the South-West of the A star [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The A2 V star has the same radial velocity and extinction as the PN, and its proper motion (µα = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='747 mas/yr σµα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='026;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' and µδ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='125 mas/yr σµδ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='031) agrees with that of the central star (µα = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='677 mas/yr σµα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='235;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' and µδ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='197 mas/yr σµδ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='275), demonstrating that the PN progenitor and A-type companion constitute a comoving visual binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The distance to NGC 3132 is obtained Springer Nature 2021 LATEX template JWST PN 13 from Gaia DR3 measurements of this visual binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' No Gaia DR3 radial veloc- ity is available for the optically faint central star (the PN progenitor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' However, the brighter (A-type) visual companion and the PN have the same radial veloc- ity: (−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='6) km s−1 for the A star from Gaia, and (−10 ± 3) km s−1 for the PN from [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The A star and PN central star also have compatible Gaia DR3 proper motions (within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='5σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The brighter, A-type star has a Gaia DR3 geometric distance (median of geometric distance posterior) of 754 pc, with lower and upper 1σ-like confi- dence intervals (16th and 87th percentiles of the posterior) of 18 pc and 15 pc respectively [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The fainter central star has a Gaia DR3 geometric distance of 2124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='7 pc, with lower and upper 1σ-like bars of 559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1 pc, and 1464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='5 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The quality flags of the astrometric solution for this star are not optimal, most likely due to the vicinity of the much brighter A-star;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' in particular, the goodness- of-fit along the scan is 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='9, while it should be close to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We therefore adopt the Gaia DR3 distance to the central star’s visual A-type companion, 754+15 −18 pc, as the distance to the PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Densities, masses and excitation of the H2 knots The clumpiness of NGC 3132 in H2 emission links this nebula to other molecule-rich PNe, such as the Helix Nebula (NGC 7293, [59–62]), Ring Neb- ula (NGC 6720, [63]), and the hourglass-shaped (bipolar) nebula NGC 2346 [64], in which the molecular emission seems to be associated with dense knots that are embedded in or surround the ionised gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The origin of such H2 knots in PNe — as overdensities in the former AGB wind, vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' formation in situ following recombination of H, as the central star enters the cooling track — remains an open question [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In contrast to the Helix Nebula, there is little evidence for cometary tails emanating from the knots in the inner regions of NGC 3132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' However, NGC 3132’s system of approximately radially-directed H2 spikes external to the main H2-bright ring system has close analogues in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', the Ring and Dumbbell Nebulae [21, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Some H2 knots in NGC 3132 are seen in absorption against the bright back- ground nebular emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This extinction is apparent not only in optical (HST) images but also, surprisingly, even in the JWST NIRCam near-infrared images (see Supplementary Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We measured the extinction at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='87 µm for two knots seen in absorption against the (Paα) nebula background: the largest knot on the west side (coordinates 10:07:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='4, −40:26:08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='8), and one of the dark- est on the east side (10:07:02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='5, −40:26:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The diameters of these knots are ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='36 arcsec and ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='15 arcsec, while their extinction is ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='57 mag and ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='25 mag (at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='87 µm), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' using the dust extinction law A(λ)/A(V ) from [66], the corresponding values of A(V ) are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='9 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='7 mag assuming RV = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We then estimate the hydrogen column densities N(H) from these extinction measurements, and convert to the hydrogen density n(H) of the knot by assuming that the knot diameters are roughly equivalent to their depths along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Using the conversion between A(V ) and N(H) from [67], where H is the combination of H0, H+ and H2, the estimated column densities Springer Nature 2021 LATEX template 14 JWST PN are N(H) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3×1021 cm−2 and N(H) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='2×1021 cm−2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' For the adopted distance of 754 pc, the estimated densities are n(H) ∼ 2 × 106 cm−3 for both knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' These densities suggest knot masses of 10−5 M⊙, similar to the typical knot (“globule”) masses found in the Helix Nebula [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The critical density of excitation of the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='12 µm H2 1–0 S(1) line at a kinetic temperature of 2000 K is 9×105 cm−3 [69], if the collision partner is H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The critical density is higher for the 1–0 S(1) line than for the 0–0 S(9) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='69 µm H2 line [6×104 cm−3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Hence, the excitation of H2 should be nearly thermal if the gas temperature is sufficiently high, with the caveat that both critical densities are higher if the primary collision partner is H2 rather than H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' PSF measurements of the central star To ascertain whether the mid-IR source associated with the PN central star is extended, we measured the JWST instrumental point spread function (PSF), using Gaussian fitting of field stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We measured Gaussian FWHMs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='29, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='40, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='44 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='58 arcsec at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='7, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='8 and 18 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We also measured two compact, slightly resolved galaxies in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We then repeated the procedure for the central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' No fit was possible at 18 µm, due to saturation (see Supplementary Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' At 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='7 µm the central star is on the edge of the diffraction spike of the A star, and only an upper limit on FWHM could be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' However, measurements of the PN central star image in the 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='8 µm filters gave consistent results, with measured FWHMs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='55 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='60 arcsec, significantly larger than the respective PSFs and comparable to the two field galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Gaussian deconvolution using the PSF yields deconvolved FWHM values for the central star of ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3 (≤ 230 AU) at 7 µm, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='4 arcsec (300 AU) at 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='8 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The extent of the central star at 18 µm is ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='9 arcsec in diameter (see Supplemenraty Figures 5 and 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Central star system’s masses We determined the mass of the A-star companion using version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='2 of the PARSEC isochrones [71] for solar metallicity, taken as Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='0152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We used Mbol,0 = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='25) mag and the GAIA DR3 spectroscopic temperature Teff = (9200 ± 200) K, where the errors are conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The star is confirmed to be beginning to turn off the main sequence, in a phase where the luminosity of (57 ± 15) L⊙ increases by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1% per Myr and the temperature decreases by 7 K per Myr (see Supplementary Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The isochrones yield an age of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3) × 108 yr and a mass of MA2V = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='15) M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The central star of the PN is evolving on the same isochrone, but from a more massive star as it has evolved further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We use the same isochrones to determine the initial mass of a star on the thermal-pulsing AGB, the phase where the central star ejected the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This gives an initial mass for the central star of Mi = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='86±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='06) M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We have carried out the same isochrone fitting using an alternative stellar evolutionary model (the DARTMOUTH code;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [72]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Both Springer Nature 2021 LATEX template JWST PN 15 the A2V star mass and the mass of the progenitor of the central star decrease by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='15 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The final, CS, mass for such a star is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='66 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' However, we have shown that such a star would show a high C/O∼2, while the presence of silicate features in the Spitzer spectrum indicate that C/O≲1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' To reconcile the mass and the abundances we conjecture that the evolution was interrupted by the binary interaction that formed the disk, when the core mass was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='61 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' With such a mass the evolutionary time to the current position on the HR diagram is in better agreement with the age of the nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This mass is also in better agreement with that derived from the photoionisation model (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='58±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='03) M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Photoionisation modelling The stratified ionisation and excitation structure of NGC 3132 is evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 1, wherein the bright rim of ionized gas, as traced by [S iii] and Brα emission, lies nestled inside the peak H2 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' However, significant ionised hydrogen and high-excitation plasma — traced by [Ne ii] and [S iii] emission in the MIRI F1280W and F1800W filter images, respectively — is observed beyond the bright inner, elliptical ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We constructed a three-dimensional photoionisation model using the code Mocassin [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' To constrain the model we used the Multi Unit Spectroscopic Explorer (MUSE) emission line maps and absolute Hβ flux of [74], the optical integrated line fluxes from [75], the IR line fluxes from [76], as well as the velocity-position data obtained from the high-resolution scanning Fabry-Perot interferometer, SAM-FP, mounted on the SOAR telescope adaptive module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The observations were taken under photometric conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The seeing during the observations was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='7 arcsec for the [N ii] observations to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='9 arcsec for the Hα one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The FWHM of a Ne calibration lamp lines was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='586 ˚A or 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='8 km s−1, which corresponds to a spectral resolution of about 11 200 at Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We determined the density structure by fitting the emission line maps to the SAM-FP images of [N ii] λ6584 and Hα, using a distance of 754 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The model adopts as free parameters the temperature and luminosity of the ionising source, and the elemental abundance of the gas component (assumed constant throughout the nebula);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' we assumed that no dust is mixed in the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' For the ionising source we use the NLTE model atmospheres of central stars of planetary nebulae from [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We find that a model invoking an unobscured central star with effective temperature Teff = 110 kK and luminosity L = 200 L⊙ well matches the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' However, we find that the present-day central star mass implied by the comparison, between these stellar parameters and the evolution- ary tracks of [40] (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='58±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='03 M⊙) is inconsistent with the (large) initial mass inferred from consideration of the presence of the comoving, wide-separation A-type companion (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='66 M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' see Central star system’s masses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Furthermore, the tracks of [40] indicate that, for this mass, we would have a post-AGB age of 20 000 yrs, whereas the position-velocity data from the SAM-FP instrument Springer Nature 2021 LATEX template 16 JWST PN yield an expansion velocity of 25-35 km s−1 implying a much shorter and inconsistent nebular dynamical age in the range 2200–5700 yrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The C/O and N/O abundances of the nebula, as well as the crystalline sil- icate nature of the dust in the PN, indicate that this object has not undergone hot bottom burning and that it has not undergone sufficient dredge up to have increased the C/O ratio above unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' By the time the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='86-M⊙ star reaches the tip of the AGB its C/O ratio is approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' It therefore seems that the mass implied by the initial-to-final mass relation using a main sequence mass of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='86 M⊙, is too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We have two ways to resolve this inconsistency (which may both be operating).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The central star is shielded by dust in the circum- stellar disk making it appear, to the PN, as a cooler star, and/or the central star mass is actually smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='66 M⊙, because the AGB evolution was interrupted by a binary interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' If the stellar ascent of the AGB was interrupted, we can determine the upper limit for a mass that would produce a nebula with C/O ≲ 1 and N/O∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='61 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The time for a star of this mass to move from the AGB to the location on the HR diagram with an approximate temperature and luminosity (110kK, 200 L⊙) as measured above is ∼10 000 yrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The time-scale of the transition from AGB to post-AGB and PN is tightly connected with the rate at which the envelope is consumed: the results obtained are there- fore sensitive to the mass-loss description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The time of 10 000 yr, is based on the classic mass loss rates dictated by Reimers or Blocker [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This estimate must be considered as an upper limit of the duration of this phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' indeed the recent works on the AGB to post-AGB transition by [79] and [80] showed that to reproduce the infrared excess of post-AGB stars in the Galaxy and in the Magellanic Clouds one has to invoke significantly higher mass-loss rates than those based on the aforementioned formulations, something that would reduce the time-scales by a factor of ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The timescale of 10 000 yrs is therefore easily reconciled with the observed timescale of 2200-5700 yrs implied by the nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Hydrodynamic modelling The hydrodynamic simulation used to interpret the fragmentation and radial spikes is a 2-dimensional hydrodynamic simulation using the magneto- hydrodynamic code ZEUS-3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The computational grid is in spherical coordi- nates and consists of 800 × 800 equidistant zones in r and θ respectively, with an angular extent of 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The wind and UV luminosity inputs correspond to a stellar post-AGB model with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='677 M⊙ which evolves from an initial 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='5 M⊙ main sequence star [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' At simulation time 0 yr the star has Teff = 10 000 K and the AGB wind (v = 10 km s−1, ˙M = 10−6 M⊙ yr−1) has a homogeneous distribution outside of the pre-PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The pre-PN has had 1000 yr of evolution prior to this moment, during which time a wide magnetic jet operated with a velocity v = 230 km s−1, and a mass-loss rate ˙M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3 × 10−7 M⊙ yr−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' this simulation is taken from Model C6 in [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' At this time the star starts emitting a fast tenuous wind with a velocity v from 240 to 14 000 km s−1 and a mass-loss rate, ˙M ranging Springer Nature 2021 LATEX template JWST PN 17 from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='06 × 10−7 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='13 × 10−10 M⊙ yr−1 over 4000 yrs that sweeps up the AGB wind material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' At the same time (0 yr) the ionisation front propagates into the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Data availability HST data are available at HST Legacy Archive (https://hla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' JWST data were obtained from the Mikulski Archive for Space Tele- scopes at the Space Telescope Science Institute (https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='stsci.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' San Pedro de Martir data is available at http://kincatpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='astrosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Code availability The code MOCASSIN is available at the following URL: https://mocassin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='nebulousresearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' ZEUS3-D is available at the Labora- tory for Computational Astrophysics [83]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The compiled version of Shape is available at http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='astrosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='unam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='mx/shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We would like to start by acknowledging the Inter- national Astronomical Union that oversees the work of Commission H3 on Planetary Nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' It is through the coordinating activity of this commit- tee that this paper came together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' SA acknowledges support under the grant 5077 financed by IAASARS/NOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' JA and VB acknowledge support from EVENTs/NEBULAE WEB research program, Spanish AEI grant PID2019- 105203GB-C21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' IA acknowledges the support of CAPES, Brazil (Finance Code 001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' EDB acknowledges financial support from the Swedish National Space Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' EB acknowledges NSF grants AST-1813298 and PHY-2020249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' JC and EP acknowledge support from an NSERC Discovery Grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' GG-S thanks Michael L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Norman and the Laboratory for Computational Astro- physics for the use of ZEUS-3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' DAGH and AM acknowledge support from the ACIISI, Gobierno de Canarias and the European Regional Development Fund (ERDF) under grant with reference PROID2020010051 as well as from the State Research Agency (AEI) of the Spanish Ministry of Science and Innovation (MICINN) under grant PID2020-115758GB-I00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' JGR acknowl- edges support from Spanish AEI under Severo Ochoa Centres of Excellence Programme 2020-2023 (CEX2019-000920-S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' JGR and VGLL acknowledge support from ACIISI and ERDF under grant ProID2021010074.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' DGR acknowl- edges the CNPq grant 313016/2020-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' MAG acknowledges support of grant PGC 2018-102184-B-I00 of the Ministerio de Educaci´on, Innovaci´on y Uni- versidades cofunded with FEDER funds and from the State Agency for Research of the Spanish MCIU through the “Center of Excellence Severo Ochoa” award to the Instituto de Astrof´ısica de Andaluc´ıa (SEV-2017-0709).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 18 JWST PN DJ acknowledges support from the Erasmus+ programme of the European Union under grant number 2020-1-CZ01-KA203-078200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' AK and ZO were supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This research is/was supported by an Australian Government Research Training Program (RTP) Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' MM and RW acknowledge support from STFC Consolidated grant (2422911).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' CM acknowledges sup- port from UNAM/DGAPA/PAPIIT under grant IN101220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' SM acknowledges funding from UMiami, the South African National Research Foundation and the University of Cape Town VC2030 Future Leaders Award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' JN acknowl- edges support from NSF grant AST-2009713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' CMdO acknowledges funding from FAPESP through projects 2017/50277-0, 2019/11910-4 e 2019/26492- 3 and CNPq, process number 309209/2019-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' JHK and PMB acknowledge support from NSF grant AST-2206033 and a NRAO Student Observing Sup- port grant to Rochester Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' MO was supported by JSPS Grants-in-Aid for Scientific Research(C) (JP19K03914 and 22K03675).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' QAP acknowledges support from the HKSAR Research grants council.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Vera C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Rubin Observatory is a Federal project jointly funded by the National Science Foundation (NSF) and the Department of Energy (DOE) Office of Sci- ence, with early construction funding received from private donations through the LSST Corporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The NSF-funded LSST (now Rubin Observatory) Project Office for construction was established as an operating center under the management of the Association of Universities for Research in Astron- omy (AURA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The DOE-funded effort to build the Rubin Observatory LSST Camera (LSSTCam) is managed by SLAC National Accelerator Laboratory (SLAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' AJR was supported by the Australian Research Council through award number FT170100243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' LS acknowledges support from PAPIIT UNAM grant IN110122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' CSC’s work is part of I+D+i project PID2019-105203GB-C22 funded by the Spanish MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='13039/501100011033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' MSG acknowl- edges support by the Spanish Ministry of Science and Innovation (MICINN) through projects AxIN (grant AYA2016-78994-P) and EVENTs/Nebulae-Web (grant PID2019-105203GB-C21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' RS’s contribution to the research described here was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' would like to thank Mar- cos Moshisnky Fundation (Mexico) and UNAM PAPIIT project IA101622 EV acknowledges support from the ”On the rocks II project” funded by the Spanish Ministerio de Ciencia, Innovaci´on y Universidades under grant PGC2018-101950-B-I00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' AZ acknowledges support from STFC under grant ST/T000414/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This research made use of Photutils, an Astropy package for detection and photometry of astronomical sources [84], of the Span- ish Virtual Observatory (https://svo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='cab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='inta-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='es) project funded by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='13039/501100011033/ through grant PID2020-112949GB-I00 and of the computing facilities available at the Laboratory of Computational Springer Nature 2021 LATEX template JWST PN 19 Astrophysics of the Universidade Federal de Itajub´a (LAC-UNIFEI, which is maintained with grants from CAPES, CNPq and FAPEMIG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Based on observations made with the NASA/ESA Hubble Space Tele- scope, and obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESAC/ESA) and the Canadian Astronomy Data Centre (CADC/NRC/CSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The JWST Early Release Observations and associated materials were developed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' executed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' and compiled by the ERO production team: Hannah Braun,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Claire Blome,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Matthew Brown,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Margaret Carruthers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Dan Coe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Joseph DePasquale,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Nestor Espinoza,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Macarena Garcia Marin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Karl Gordon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Alaina Henry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Leah Hustak,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Andi James,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Ann Jenkins,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Anton Koekemoer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Stephanie LaMassa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' David Law,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Alexandra Lockwood,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Amaya Moro-Martin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Susan Mullally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Alyssa Pagan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Dani Player,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Klaus Pontoppidan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Charles Proffitt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Christine Pulliam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Leah Ramsay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Swara Ravindranath,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Neill Reid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Massimo Robberto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Elena Sabbi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Leonardo Ubeda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The EROs were also made possible by the foundational efforts and support from the JWST instruments, STScI planning and scheduling, and Data Management teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Finally, this work would not have been possible without the collaborative platforms Slack (slack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='com) and Overleaf (overleaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Author contribution The following authors have contributed majorly to multiple aspects of the work that lead to this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' the writing and the formatting of figures: De Marco (writing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' structure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' interpretation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' synthesis),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Aleman (H2 interpre- tation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Balick (processing and interpreting images),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Garc´ıa-Segura (2D hydro modelling),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Kastner (writing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' H2 measurements and interpretation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Matsuura (imaging,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' photometry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' H2 interpretation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Miszalski (stellar photometry),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Mohamed (hydrodynamics of binaries),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Monreal-Ibero (MUSE data analy- sis),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Monteiro (photoionisation and morpho-kinematic models),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Moraga Baez (JWST image production),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Morisset (photoionisation modelling),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Sahai (disk model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' comparative interpretation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Soker (hydro modelling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' interpretation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Stanghellini (distances,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' abundance interpretation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Steffen (morpho-kinematic models),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Walsh (spatially resolved spectroscopy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Zijlstra (disk model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' H2 measurements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' writing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' interpretation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The following authors have contributed key expertise to aspects of this paper: Akashi (hydrodynamic modelling and jet interpretation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Alcolea (CO observations),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Akras (H2 interpretation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Amram (space-resolved spec- troscopy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Blackman (hydrodynamics),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Bublitz (HST and radio images of fast evolving PN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Bucciarelli (Gaia data),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Bujarrabal (radio observations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 20 JWST PN disk observation and interpretation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' comparative studies),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Chu (disk inter- pretation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Cami (molecular formation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Corradi (final review,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' interpreta- tion),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Garc´ıa-Hernandez (IR dust/PAH features and abundances),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Garc´ıa- Rojas (photoionisation modelling),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' G´omez-Llanos (photoionisation mod- elling),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Gon¸calves (comparative analysis),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Guerrero (Xray imaging),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Jones (close binaries),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Karakas (final review,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' stellar nucleosynthesis),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Manchado (nebular morphology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' H2 interpretation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' McDonald (photometry modelling),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Montez (X-ray and UV imaging),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Osborn (binary nucleosynthesis),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Otsuka (IR imaging),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Parker (morphology),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Peeters (nebular spectroscopy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' PAHs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Ruiter (binary populations),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Sabin (abundances),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' S´anchez Contreras (radio),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Santander-Garc´ıa (nebular evolution),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Seitenzahl (star and star nebula asso- ciation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Speck (dust),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Toal´a (morphology),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Ueta (nebular imaging),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Van de Steene (IR observations),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Ventura (AGB evolution model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The following authors contributed by commenting on some aspects of the analysis and manuscript: De Beck, Boffin, Boumis, Chornay, Frank, Kwok, Lykou, Nordhaus, Oliveira, Quint, Quintana-Lacaci, Redman, Villaver, Vlemmings, Wesson, and Van Winckel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Competing interest statement We declare that no conflict of interest exists between any of the authors and the content and production of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Springer Nature 2021 LATEX template JWST PN 21 Figures Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 1 JWST images of the PN NGC 3132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Left column, top and bottom: color overlays of JWST NIRCam and MIRI images that cleanly distinguish between the PN’s ionized gas (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', H ii region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' top panel) and molecular gas (as seen in H2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' bottom panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Note the sharp contrast between the relatively smooth appearance of the H ii region and the flocculent structure of the H2 ring system and extended H2 halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' These images are presented with square-root and log intensity stretches, respectively, from the background sky to peak intensity levels in each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Right image: a grey-scale, single filter (F470N), zoomed-in NIRCam image that more readily displays details of the flocculent H2 halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' North is towards the top, East is towards the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' References [1] Mastrodemos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' & Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Bipolar Pre-Planetary Nebulae: Hydro- dynamics of Dusty Winds in Binary Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Morphology of the Circumstellar Envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 523, 357–380 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1086/307717 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [2] Mohamed, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' & Podsiadlowski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Mass Transfer in Mira-type Binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Baltic Astronomy 21, 88–96 (2012) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' alph otSpringer Nature 2021 LATEX template 22 JWST PN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 2 The dusty central star of the PN NGC 3132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' JWST NIRCam F187N (top left) and MIRI F1280W (top right) images of the central region of NGC 3132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The JWST MIRI images reveal the detection of a mid-IR excess at the nebula’s true (hot, compact) central star, which is seen projected ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='7′′ (∼1300AU) SW of the main-sequence A-type companion (which is far brighter shortward of ∼10 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' North up and East is to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Colour bars indicate surface brightness in log (MJy ster−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The bottom panel shows the near-IR to mid-IR spectral energy distribution of the central star of NGC 3132 overlaid with a model consisting of a combination of a hot blackbody spectrum representing the central star’s photosphere (blue line) and a dusty circumstellar double disk model to fit the NIR and MIR data points (red line, with the cooler disk as a dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The wide companion, A star’s spectral energy distribution is shown as a dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Vertical error bars are set at 10% of the flux values, while horizzontal bars show the width of the bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [3] Maercker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Unexpectedly large mass loss during the thermal pulse cycle of the red giant star R Sculptoris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Nature 490, 232–234 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1038/nature11511, arXiv:1210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3030 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [4] Santander-Garc´ıa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' ALMA high spatial resolution observations of the dense molecular region of NGC 6302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 597, A27 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1051/0004-6361/201629288, arXiv:1609.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='06455 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [5] Sahai, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' & Trauger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Multipolar Bubbles and Jets in Low-Excitation Planetary Nebulae: Toward a New Understanding of the Formation and Shaping of Planetary Nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 116, 1357–1366 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1086/300504 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [6] Sahai, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Morris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' & Villar, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Young Planetary Nebulae: Hubble F187N F1280W 15 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='75 10 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='50 Dec offset [arcsec] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='25 offset [arcsec] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='70 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='60 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='50 Dec 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='25 10 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='50 15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='25 10 15 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 10 10 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='45 15 10 5 0 15 RA offset [arcsec] 0 RA offset [arcsec] 100 star dust disk [mJy] 10 flux TTT CSPN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='01 1 10 100 wavelength micronSpringer Nature 2021 LATEX template JWST PN 23 Earth view East-West view North-South view To Earth To Earth Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 3 Morpho-kinematic reconstruction of the ionised cavity of PN NGC 3132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Emission in the [N ii] line as seen from Earth (left image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' North is towards the top and East is towards the left), a view from the East, which we call East-West view (middle image), and a view from the North which we call North-South view (right image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The colour-coding is for Doppler-shift as seen from Earth, with blue for material approaching the observer, red for receding gas and green for no velocity along the observer’s line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We note the prominent green (zero Doppler shift) belt in the middle image, and the filament that wraps around the waist of the ellipsoid and which is red-shifted on one side and blue-shifted on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' A fly-through movie of this model can be found at this link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Space Telescope Imaging and a New Morphological Classification System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Post-Agb Stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 41, 391–427 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1146/annurev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='071601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='170018 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [8] Ivanova, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Common envelope evolution: where we stand and how we can move forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 21, 59 (2013).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Numerical Simulations of Wind Accretion in Symbiotic Binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 700, 1148–1160 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1088/0004-637X/700/2/1148, arXiv:0905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3542 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 24 JWST PN Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 4 The physical interpretation of the flocculent H2 structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Left column: a hydrody- namic simulation showing the formation of nebular structures external to the main ionised region, compared with (right column) two quadrants of the JWST images of NGC 3132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Top row: the simulation snapshot at 3800 yr from the on-set of ionisation is compared with similar straight spikes in one region of the nebula (top right image North is to the left, East is towards the top) while, bottom row, at 4000 yr the spikes thicken and bend as is also seen in a different part of the nebula (North to the top, and East to the left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This demonstrates temporal evolution in different parts of the nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [12] Soker, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Visual Wide Binaries and the Structure of Planetary Nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 118 (5), 2424–2429 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1086/301090, arXiv:astro-ph/9907305 [astro-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [13] Balick, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' FLIERs and Other Microstructures in Planetary Nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Images of Elliptical PNs from the Hubble Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 116, 360–371 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1086/300429 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [14] Sahai, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' & Trauger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Multipolar Bubbles and Jets in Low-Excitation Planetary Nebulae: Toward a New Understanding of the Formation and Shaping of Planetary Nebulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 116 (3), 1357–1366 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Springer Nature 2021 LATEX template JWST PN 25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 5 Approximate illumination model of the H2 halo of PN NGC 3132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Left panel: the JWST colour composite image showing the H2 extended structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Right panel the pro- jected Shape image after assuming two concentric, thick uninterrupted shells of material, illuminated by the central star, through a porous ellipsoid with reduced opacity in the polar regions.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 513 (3), 4405–4430 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1093/mnras/stac749, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='08318 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [35] Decin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' (Sub)stellar companions shape the winds of evolved stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Science 369 (6510), 1497–1500 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' abb1229, arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='11694 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [36] M´endez, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' A-type 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1093/mnras/stx3338, arXiv:1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='08582 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [41] Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 914 (2), 111 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3847/ 1538-4357/abfc4e, arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='12831 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='SR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 32 JWST PN [83] Clarke, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' A Consistent Method of Characteristics for Multidimen- sional Magnetohydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 457, 291 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1086/176730 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [84] Bradley, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' astropy/photutils:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Zenodo (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Springer Nature 2021 LATEX template JWST PN 1 Supplementary Material Specifications of JWST NIRCam and MIRI imaging As part of its ERO program [1], JWST obtained ten images of NGC 3132: six individual NIRCam images, through filters F090W, F187N, F212N, F356W, F444W, and F470N, and four MIRI images, through filters F770W, F1130W, F1280W, F1800W (Supplementary Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Basic information about these NIRCam and MIRI filters is presented in Supplementary Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The native NIRCam field of view is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='2×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='2 arcmin, with a pixel scale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='031 arcsec/pixel in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='6–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3 µm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='063 arcsec/pixel in the range 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='4–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='0 µm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' the native MIRI field of view is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='7 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3 arcmin with a pixel scale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='11 arcsec/pixel in the range 5-27 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The NIRCam instrument provides Nyquist-sampled imaging at 2 (short wavelength channel) and 4 (long wave- length channel) microns with a PSF FWHM of ∼2 pixels in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The MIRI instrument in imaging mode, on the other hand, provides a FWHM of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='22 arcsec (PSF FWHM of 2 pixels) for wavelengths ≥6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='25 µm [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The NIR- Cam imaging of NGC 3132 used 8 dither points with an offset of approximately 6 arcsec, while MIRI imaging used a 1 × 2 tile mosaic with 8 dither points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The final NIRCam and MIRI images cover areas of approximately 150 × 150 arcsec2 and 150 ×130 arcsec2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Images were downloaded from the MAST archive (calibration CRDS VER11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3) and are neither continuum subtracted (free-free emis- sion is included) nor background subtracted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' the background regions of the MIRI images display significant flux that is thermal emission from the (cold) telescope and sunshade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In addition, although specific lines are targeted by specific narrow-band filters, additional (albeit weaker) lines may be present in some bandpasses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', the F187N bandpass, which is dominated by Paα, is contaminated by He I lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In order to determine the emission features present in each band, we have generated a simple model to predict the IR spectrum of the nebula, and compared it to previously published Spitzer observations, see Supplementary Figure 2, alongside the JWST bandpasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Here we see, for example, that the MIRI F770W image is likely contaminated with H i and [Ar ii] in certain regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Spitzer spectroscopy indicates no sign of PAHs at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='2, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='6 µm and only a very weak 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3 µm feature as well as cristalline silicates [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This is usually associated to the presence of neutral PAHs [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' As a result we have not included PAH emission contribution at 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' On the other hand, Spitzer spectra do not cover the region below 5 µm so we have indicated that there could be a weak 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3 µm feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In Supplementary Figure 3 we select three regions of the nebula observed through the NIRCam filter F212N, which we present, enlarged, in Supple- mentary Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The latter Figure presents image sequences consisting of archival HST images and the new JWST (ERO) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' These sequences illus- trate the contrast between the smoothness of the emission from ionised gas vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' the clumpiness of H2 emission, as well as the correspondence between dust extinction (most evident in the HST images) and the H2 filaments and knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='02775v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='SR] 7 Jan 2023 Springer Nature 2021 LATEX template 2 JWST PN Supplementary Table 1 List of JWST filters used in this work and expected emission in the corresponding band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Filter name λ11 λ21 Emission features Date2 Texp2 (µm) (µm) within bandpass (sec) NIRCam F090W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='795 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='005 [S iii] 9069,9562 ˚A 2022-06-03 5841 F187N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='863 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='884 H i Paα 2022-06-03 9277 F212N 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='109 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='134 H2 (1,0) S(1) 2022-06-03 9277 F356W 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='140 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='980 H2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Dust;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' PAHs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 2022-06-03 1460 F405N3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='028 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='074 H i Brα F470N3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='683 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='733 H2 (0,0) S(9) F444W 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='880 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='986 H i Brα;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' H2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Dust?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 2022-06-03 2319×2 MIRI F770W 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='8 H2 (0,0) S(5) 2022-06-12 2708 F1130W 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='95 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='65 PAHs 2022-06-12 2708 F1280W 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='6 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='0 [Ne ii] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='8µm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 2022-06-12 2708 H2 (0,0) S(2) F1800W 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='5 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='5 Warm dust;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 2022-06-12 2708 [S iii] 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='6µm 1λ1, λ2: wavelengths at which the bandpass transmission is 50% of the peak transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 2Observing date and exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 3Pupil wheel filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' used in combination with F444W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Central star magnitudes JWST has detected the central visual binary from F090W to F1800W (Sup- plementary Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' At near-infrared wavelengths, the central star is faint and on the edge of the diffraction spikes from the nearby A-type star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' From 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='7 µm longward, the central star is observed to increase in brightness, and at 18 µm it exceeds the flux from the A-type star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We obtained optical magnitudes of the central star from the Hubble Source Catalogue version 3 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The F438W, F555W and F814W Wide Field Cam- era 3 magnitudes in the AB system were converted to Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The calibrated 2D resampled NIRCam and MIRI observations available as i2d pipeline products were used to perform aperture photometry of the central star using the pho- tutils package [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' A circular aperture was centered on the central star with radii corresponding to the 80% encircled energy radius tabulated in the rele- vant aperture correction tables, sufficient to include the majority of the central star flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The tables JWST nircam apcorr 0004 and JWST miri apcorr 0008 were sourced from the JWST Calibration Reference Data System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Aperture photometry of the central source was performed with the error extension of the image included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Three circular sky apertures of the same radius were selected nearby the central star to best sample the challenging background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The background includes artefacts from the diffraction pattern of the A-type star nearby (more prominent in the NIRCam images) and the structured neb- ular background from NGC 3132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In the F090W, F212N, F405N and F470N Springer Nature 2021 LATEX template JWST PN 3 60 40 20 0 20 40 60 RA offset [arcsec] 60 40 20 0 20 40 60 Dec offset [arcsec] F090W, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='902 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='00 0.' metadata={'source': 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+page_content=' North up and East is to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Colour bars indicate surface brightness in log(MJy ster−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' filters the dominant background and/or intrinsic faintness of the central star precluded any meaningful fluxes from being measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The sky background was estimated as the average of the median counts in each sky aperture, where the median was calculated using sigma clipping with σ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The sky background was scaled to the aperture area A of the central star aperture before it was subtracted from the aperture sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The flux of the central star was calculated as the sky subtracted aperture sum scaled by the MJy/sr to µJy conversion factor and the aperture correction sourced from the aperture correction tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The uncertainty in the flux was estimated as �σp + 2σsky where σp is the photutils aperture sum err and σsky is Aσ2 b, where σb is the average of the standard deviation of counts in each sky aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 4 JWST PN 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='0 Wavelength ( m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='00 Fux (Arbitrary units) [Ca II] [S III] [S III] [C I] HeII H I [Fe II] H I (Pa ) H2 H2 H2 H2 H2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' H I H2 H2 H2 H I H2 H I (Br ) [K III];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' H I H2 F090W F187N F212N F356W F405N F444W F470N 6 8 10 12 14 16 18 20 Wavelength ( m) 0 2 4 6 8 10 Fux (Arbitrary units) H2 H2 [Ni II] H2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' [Ar II] [Na III] H I H2 H2 [Ar III] [S IV] [Ni II] PAH H2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' HI [Ne II] [Ne V] [Cl II] [Ne III] H2 H I [S III] F770W F1130W F1280W F1800W Supplementary Figure 2 Simulated IR spectrum of NGC 3132, overlaid with the JWST filters to demonstrate bandpass contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Top panel: the simulated spectrum of NGC 3132 between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='8 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='2 µm (blue line) with, overlaid the JWST bandpasses (labelled coloured shapes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Bottom panel: the simulated spectrum of NGC 3132 between 5 and 20 µm (blue line) with, overlaid the JWST bandpasses (labelled coloured shapes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' 10h07m06s 03s 00s 06m57s 40°25\'30" 26\'00" 30" 27\'00" RA (ICRS) Dec (ICRS) Supplementary Figure 3 JWST/NIRCam F212N image of NGC 3132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' White squares indicate positions and sizes of “blowup” regions highlighted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Springer Nature 2021 LATEX template JWST PN 5 HST F502N HST F658N NIRCam F187N NIRCam F212N NIRCam F405N MIRI F1130W 1"/753 AU 1"/753 AU 1"/753 AU Supplementary Figure 4 The enlarged images of knots in three representative regions, indicated by white boxes in Figure 3: the West side of the ring (top row), near the centre of the ring (middle row) and the East side of the ring (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Filament structures stand out in the NIRCam F212N images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' A few filaments are dusty, as clearly seen in the HST optical images (first two columns), but also in the NIRCam F187N and F405N images (third and fifth columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' However, due to the complex sky background, these uncertainties are likely underestimates of the true uncertainty, which we estimate to be 5–10% of the flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Supplementary Table 2 gives the measured and dereddened fluxes using AV = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1E(B − V ), where E(B − V ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='09 mag [8], and filter extinction ratios are from the Spanish Virtual Observatory Filter Profile Service [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In Supplementary Figure 6 we present the PARSEC isochrones with the derived location of the A2V star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Filter Fν (µJy) Formal Fν,0 (µJy) Formal error∗ error∗ WFPC3/F438W 1620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='32 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='21 2274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='64 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3 WFPC3/F555W 1188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='33 1556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='61 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='6 WFPC3/F814W 560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='29 654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='8 NIRCam/F187N 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='43 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='65 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='77 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='84 NIRCam/F356W 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='78 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='31 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='67 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='41 MIRI/F770W 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='68 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='47 288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='02 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='58 MIRI/F1130W 1445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='46 1462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='85 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='49 MIRI/F1280W 1362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='85 1373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='09 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='90 MIRI/F1800W 11344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='60 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='93 11425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='35 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='08 ∗The actual error on this photometric measurements is likely ∗ closer to 5-10% of the flux values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Supplementary Table 2 Measured (Fν) and dereddened (Fν,0) fluxes of the central star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' To convert to magnitudes: mag(AB) = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='5 ∗ log10(Fν [in Jansky]) + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='90 Springer Nature 2021 LATEX template 6 JWST PN a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' HST F814W b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' NIRCam F090W c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' NIRCam F187N d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' NIRCam F212N e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' NIRCam F356W f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' MIRI F770W g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' MIRI F1130W h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' MIRI F1280W i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' MIRI F1800W 2 arcsec Supplementary Figure 5 Sections of the JWST images zooming in on the central stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' NIRCam F090W, F187N and F212N images are deconvolved with simulated PSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' HST, NIRCam F356W and MIRI images are not deconvolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Note that the slight offset of the positions of the central star in the F090W and F187N images is due to imperfect deconvolution due to the nearby saturated star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Extended structure of the central star Supplementary Figure 7 demonstrates the extended structure of the central star in the MIRI F1130W and F1800W band images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' From the left to right column, we see the central, the A-type star, an example of a saturated star, and the simulated PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The images are oriented such that North is 145 degrees clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' An example image of a saturated star is from the Taran- tula Nebula region, which was observed as a part of the first image program (PID 2729).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The coordinate of this saturated star is RA=05h38m33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='61s and Dec=−69d04m50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='5s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The MIRI PSF is simulated based on Webb PSF software version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='0 (https://jwst-docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='edu/jwst-mid-infrared-instrument/miri- performance/miri-point-spread-functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The top row shows the images, and the second and third rows show radial profiles, which are sliced in horizontal and vertical directions across the peak (blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Dotted lines indicate the Springer Nature 2021 LATEX template JWST PN 7 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='9 3.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The isochrones are labelled in order of descending age : 9, 8, 7, 6, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='6, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3, 5, 4 in units of 108 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The dashed lines connect points of constant mass, labelled in solar units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The error bar on the horizzontal axis is the temperature uncertainty given by Gaia, ±200 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The error bar on the vertical axis is based on a conservative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='25 mag error on the absolute magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' data points that are strongly affected by other factors, such as bright neigh- bouring stars or detector saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The central star is mildly saturated in the F1800W image, with data quality flags of about 5—8 pixels across the peak, so that the data within the 10 pixels from the peak of the central star are plot- ted as dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' On the central star radial profiles of F1800W, green lines demonstrate the simulated radial profile with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='44 arcsec radius flat-intensity ‘disk’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This shows that the central star is extended at a scale of ≳0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='8 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In the radial profiles, the PSF is also plotted as an orange line, which has a FWHM of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='58 arcsec at F1800W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Supplementary Figure 7 demonstrates that the central star is clearly extended more than the PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' On the central star radial profiles, green lines demonstrate the radial profile with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='44 arcsec radius, flat-intensity ‘disk’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In reality, the intensity gradually decreases radially, rather than a flat inten- sity with a cliff edge, and the tailing of this gradual decrease continues beyond 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='4 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Morpho-kinematic modelling In Figure 3 we have presented the 3D morpho-kinematic reconstruction of the ionised region of NGC 3132 using the interactive morpho-kinematic modelling software Shape [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In addition to the new images from JWST, spectroscopic reference data for the reconstruction are position-velocity diagrams from the San Pedro M´artir Kinematic Catalogue of Galactic Planetary Nebulae ([13, 14]).' metadata={'source': 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Supplementary Figure 7 Demonstration of the extended structure of the central star in the MIRI F1130W (top panel) and F1800W (bottom panel) band images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' From the left to right column, the central star, the A-type star, an example of a saturated star, and the simulated PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The top row displays the images, and the second and third row show radial profiles, which are sliced in the horizontal and vertical directions across the peak (blue lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Dotted lines indicate the data points strongly affected by other factors, such as bright neighbouring stars or detector saturation for the F1800W image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In the radial profiles, the PSF is also plotted as an orange line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='The central star is clearly extended more than the PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' On the central star radial profiles of the F1800W image, the green line demonstrates the radial profile of a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='44 arcsec-radius, flat-intensity ‘disk’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Springer Nature 2021 LATEX template JWST PN 9 The velocity field is 1 km s−1 arcsec−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This value ensures that the cross-section of the main shell is approximately circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We estimate the uncertainty to be of the order of 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Note that the stretching of the structures along the line of sight is proportional to this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In other words, the shape of the nebula along the line of sight is linearly related to the component of the velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This model of the inner ionised cavity supersedes or rather, completes, the barrel or “diabolo” model [16] in view of more sensitive spectroscopy that allowed us to detect the faint, and fast, closed ends of the ellipsoid along its major axis, which is close to the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In Supplementary Figure 8 we show the slit positions and resulting position- velocity diagram (top row: observed, bottom row: simulated) reconstruction with Shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' A 3D fly through the 3D volume can be found at this link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The position-velocity diagrams in Supplementary Figure 8 are effectively 2D ren- ditions of the spectral line shape as we move along the slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The slit positions are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' In the final step of our morpho-kinematic model (Figure 3), we place two complete shells of non-uniform, filamentary material around the central star to match the observed size of the H2 halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The inner shell ranges from 33 to 45 arcsec from the central star, and the second ranges from 60 to 70 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' These are illuminated through a partially opaque ellipsoid (approximately cor- responding with the ellipsoid containing the ionised nebula) that has reduced opacity around the poles and has an overall porous opacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' At this point we are only interested in testing the possibility that the opacity of the walls of the central cavity within which the exciting central star resides, could be the cause of the overall emissivity distribution and features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The central star is made to radiate as a blackbody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' We have then used a simple proxy for the various radiative processes that are at work here, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', isotropic scattering on dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Since we expect dust to dominate the transport of the exciting radiation, this is a reasonable first test, with the details being irrelevant for this simple geometric simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' A spherical density modulation was also imposed with ad hoc spacing responding to the following modulation: ρ/ρ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='3 + sin(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='8 r/arcsec)10 sin(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='4 r/arcsec)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This generates the arch pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' This type of painstaking morpho-kinematic modelling is critically depen- dent on images at different wavelengths as well as spatially-resolved spec- troscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' On the basis of this type of data driven 3D model, we can now understand structures first revealed by JWST such as the H2 halo and the dusty central stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' References [1] Pontoppidan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Blome, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Braun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Brown, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Carruthers, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Coe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', DePasquale, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Espinoza, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Garcia Marin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Gordon, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Henry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Hustak, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', James, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Koekemoer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', LaMassa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Law, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Lockwood, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Moro-Martin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Mullally, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Pagan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Player, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Springer Nature 2021 LATEX template 10 JWST PN 0 20 40 20 40 arcsec 0 20 40 20 40 arcsec SPM a SPM b SPM c H A H D SPM a SPM b SPM c H a H d 0 100 km/s 0 20 40 20 40 arcsec 0 20 40 20 40 arcsec 100 SPM a SPM b SPM c H A H D Supplementary Figure 8 Position velocity data and model to achieve the morpho- kinematic model of PN NGC 3132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Top figure: a MUSE [N ii] image of NGC 3132 with the slit positions marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The three slits from [13] are marked in grey and the two slits from [14] are in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Bottom figure, top row: observed position-velocity diagrams for the [N ii] nebular line along the different slits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' Bottom figure, bottom row: corresponding modelled position-velocity diagrams using Shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' The heavy horizontal lines in the last three columns are the continuum emission from the A2 V star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' cepSpringer Nature 2021 LATEX template JWST PN 11 Proffitt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Pulliam, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Ramsay, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Ravindranath, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Reid, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Rob- berto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Sabbi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Ubeda, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=': The JWST Early Release Observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=' arXiv e-prints, 2207–13067 (2022) [2] Bouchet, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Garc´ıa-Mar´ın, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Lagage, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Amiaux, J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Gastaud, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Glasse, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Gordon, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Gougnaud, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} +page_content=', Guillard, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9E0T4oBgHgl3EQf7QLf/content/2301.02775v1.pdf'} 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b/jNE4T4oBgHgl3EQfTAww/content/tmp_files/2301.05003v1.pdf.txt @@ -0,0 +1,560 @@ +arXiv:2301.05003v1 [math.FA] 12 Jan 2023 +RANK-ONE PERTURBATIONS AND NORM-ATTAINING +OPERATORS +MINGU JUNG, GONZALO MART´INEZ-CERVANTES, +AND ABRAHAM RUEDA ZOCA +Abstract. The main goal of this article is to show that for every (re- +flexive) infinite-dimensional Banach space X there exists a reflexive Ba- +nach space Y and T, R ∈ L(X, Y ) such that R is a rank-one operator, +∥T + R∥ > ∥T ∥ but T + R does not attain its norm. +This answers +a question posed by S. Dantas and the first two authors. +Further- +more, motivated by the parallelism exhibited in the literature between +the V -property introduced by V.A. Khatskevich, M.I. Ostrovskii and +V.S. Shulman and the weak maximizing property introduced by R.M. +Aron, D. Garc´ıa, D. Pellegrino and E.V. Teixeira, we also study the rela- +tionship between these two properties and norm-attaining perturbations +of operators. +1. Introduction +Given real Banach spaces X and Y , we denote by L(X, Y ) (resp., K(X, Y )) +the space of all (bounded linear) operators (resp., compact operators) from +X into Y . When X = Y , we simply write L(X) (resp., K(X)). As usual, +the notations BX and SX stand for the closed unit ball of X and the unit +sphere of X, respectively. +A well-known result of J. Lindenstrauss states that if X is reflexive then, +for every Banach space Y , every operator T ∈ L(X, Y ) can be approximated +by norm-attaining operators. Recall that an operator T ∈ L(X, Y ) is said +to be norm-attaining if ∥T∥ = ∥T(x)∥ for some x ∈ BX. Indeed, J. Lin- +denstrauss showed that there exists a compact operator K ∈ K(X, Y ) such +that ∥K∥ is arbitrarily small and T + K attains its norm (see Theorem 1 +and the succeeding remark in [8]), so every operator can be approximated +by norm-attaining compact perturbations of itself. +Pairs of classical Banach spaces quite often satisfy a stronger condition. +Namely, J. Kover [7] proved that for a Hilbert space H, if a compact per- +turbation of an operator T ∈ L(H) has norm strictly greater than the norm +of T, then this perturbation attains its norm, i.e. if ∥T + K∥ > ∥T∥ where +K ∈ K(H) then T + K attains its norm. +2020 Mathematics Subject Classification. 46B10, 46B20, 46B28. +Key words and phrases. Norm-attaining operators; Compact perturbation; Reflexivity; +Weak Maximizing Property; V -pairs. +1 + +2 +JUNG, MART´INEZ-CERVANTES, AND RUEDA ZOCA +Before going beyond in the exposition of literature, let us fix the follow- +ing notation: a pair (X, Y ) of Banach spaces has the compact perturbation +property (for short, CPP) if for any T ∈ L(X, Y ) and K ∈ K(X, Y ) the in- +equality ∥T + K∥ > ∥T∥ implies that T + K is norm-attaining. Notice that +the CPP of the pair (X, Y ) forces the domain space X to be reflexive. We +say that X has the CPP if the pair (X, X) has the CPP. With this notation +in mind, Kover’s result says nothing but that every Hilbert space has the +CPP. +A large class of pairs of Banach spaces enjoying the CPP is given by +the class of V -pairs, which was introduced and intensively developed in +the papers [6, 11]: a bounded linear operator T ∈ L(X, Y ) is said to be +a V -operator if there is a norm-one operator S ∈ L(Y, X) such that the +spectral radius of TS coincides with ∥T∥. If every operator in L(X, Y ) is +a V -operator, then the pair (X, Y ) is said to be a V -pair or to have the +V -property. If X = Y , then X is said to be a V -space or to have the V - +property. Among others, it is proved in [6, Proposition 5] that an operator +having a strictly singular hump is a V -operator if and only if it is norm- +attaining. Thus, every V -pair satisfies the CPP. Moreover, [6, Theorem 1] +generalizes the aforementioned result of Kover to ℓp(Xn) for any sequence +of finite dimensional spaces (Xn) and 1 < p < ∞. +Quite recently R.M. Aron, D. Garc´ıa, D. Pellegrino and E.V. Teixeira [1] +introduced another related property, the so-called weak maximizing prop- +erty. A pair (X, Y ) of Banach spaces is said to have the weak maximizing +property (for short, WMP) if every operator from X into Y with a non- +weakly null maximizing sequence is norm-attaining. Note from [1, Proposi- +tion 2.2] (see also [12, Theorem 1]) that for 1 < p < ∞, 1 ⩽ q < ∞, and +arbitrary index sets Γ1, Γ2, the pair (ℓp(Γ1), ℓq(Γ2)) has the WMP. Further- +more, the WMP implies the CPP [1, Proposition 2.4], which improves the +previous results of Kover. Although there are pairs of Banach spaces with +the CPP failing the WMP (see [2, Proposition 3.6]), it seems to be an open +problem whether the CPP for a pair of reflexive Banach spaces implies the +WMP [2, Question 4.4]. For further results and more examples of pairs with +the WMP we refer the reader to [2, 4]. +Due to the aforementioned results it is natural to wonder whether every +reflexive Banach space X has the CPP. Nevertheless, it is implicitly proved +in [11, Theorem 2] that for every infinite-dimensional Banach space X there +exists an equivalent norm ||| · ||| and a rank-one operator R : X −→ X with +|||I + R||| > |||I||| but with I + R failing to attain its norm in L((X, ||| · |||)). +In particular, (X, ||| · |||) fails the CPP (see Proposition 3.1). +Note that if a Banach space X is such that every operator in L(X) attains +its norm, then X clearly has the CPP. In other words, if X fails the CPP, +then there exists a non-norm attaining operator on X. By the result just +mentioned in [11], we conclude that for any infinite-dimensional Banach +space X, there exists a renorming � +X of X for which not every operator in + +RANK-ONE PERTURBATIONS AND NORM-ATTAINING OPERATORS +3 +L( � +X) is norm-attaining. In fact, it seems to be an open problem whether +there exists a reflexive infinite-dimensional Banach space X such that every +operator in L(X) attains its norm (see, for instance, [6, Problem 8]). +The main goal of this paper is to show that for every infinite-dimensional +space X there exists a reflexive Banach space Y such that the pair (X, Y ) +fails the CPP. Namely, the main theorem of the paper reads as follows. +Theorem 1.1. Let X be an infinite-dimensional reflexive Banach space. +Then there exists a reflexive Banach space Y and T, R ∈ L(X, Y ) with R a +rank-one operator such that ∥T + R∥ > ∥T∥ but with T + R not attaining +its norm. In particular, the pair (X, Y ) fails the CPP. +The aim of Section 2 is to prove Theorem 1.1. Our original motivation +for Theorem 1.1 comes from the study of the WMP and the V -property. +There is a deep parallelism exhibited in the literature between these two +properties; compare [2, 4] to [6, 11]. However, up to our knowledge, it is +not known how these two concepts are related to each other and quite often +if a question is open for one property it is also open for the other. In [2, +Question 4.3] it is asked whether if a reflexive Banach space X satisfies +that the pair (X, Y ) has the WMP for every Banach space Y , then X must +be finite-dimensional. We would like to mention that, to the best of our +knowledge, the same question was open if we replace the WMP with the +CPP. +In particular, Theorem 1.1 gives a positive answer to [2, Question 4.3] and +exhibits another common behaviour between the WMP and the V -property. +Observe that the WMP and the V -property implies the CPP. We know +that the WMP (and therefore the CPP) does not imply the V -property +(the pair (ℓp, ℓq) with 1 < p < q < ∞ has the WMP while it is not a +V -pair). +Nevertheless, we do not know whether the V -property implies +the WMP. This situation also serves as motivation for our investigation +of all the aforementioned properties. In Section 3, we first observe from +the argument of Ostrovskii [11] that the CPP is still a quite restrictive +property: an infinite-dimensional Banach space X with the CPP must be +isometric to ℓp for some 1 < p < ∞ if X has a symmetric basis (Proposition +3.1). Moreover, we see that the existence of non-norm-attaining operators +between Banach spaces X and Y produces pairs of Banach spaces failing +the CPP (Proposition 3.2) and, as a consequence, the pair (Lp[0, 1], Lq[0, 1]) +fails the CPP whenever p > 2 or q < 2 (Corollary 3.6). We also generalize in +Proposition 3.9 the fact that the pair (ℓp, ℓq) has the WMP for 1 < p < ∞ +and 1 ⩽ q < ∞. Finally, we prove that a pair (X, Y ) has the V -property if +and only if every operator from X into Y is norm-attaining provided that +Y has the Dunford-Pettis property (Proposition 3.10), which covers some +results in previous sections. + +4 +JUNG, MART´INEZ-CERVANTES, AND RUEDA ZOCA +2. Proof of Theorem 1.1 +As we have indicated in the introduction, the aim of this section is to +prove Theorem 1.1, for which we will start by considering the Banach space +ℓ∞ of all bounded sequences instead of reflexive ones. +Theorem 2.1. Let X be a reflexive Banach space. If the pair (X, ℓ∞) has +the CPP, then X is finite-dimensional. Namely, if X is infinite-dimensional +then there exists S, R ∈ L(X, ℓ∞) with R a rank-one operator such that +∥S + R∥ > ∥S∥ but with S + R not attaining its norm. +For the proof we need a lemma of geometric nature for points of Fr´echet +differentiability. Let us recall that a point x ∈ SX is said to be a point of +Fr´echet-differentiability of X if the norm ∥ · ∥ : X −→ R is Fr´echet differen- +tiable at x. See [3, Chapter 8] for background about Fr´echet differentiability +in Banach spaces. +Lemma 2.2. Let X be a Banach space. Let x ∈ SX be a point of Fr´echet +differentiability of X and (xn) be a sequence of points of BX such that ∥x − +xn∥ → 2. Then lim sup +n→∞ +∥x + xn∥ < 2. +Proof. Since x is a point of Fr´echet differentiability, by ˇSmulyan Lemma [3, +Lemma 8.4] there exists f ∈ SX∗ with f(x) = 1 and with the following +property: for every ε > 0 there exists δ > 0 such that if g ∈ BX∗ and +g(x) > 1 − δ then ∥g − f∥ < ε. +Assume to the contrary that lim sup +n→∞ +∥x + xn∥ = 2. Take a subsequence +(xnk) such that ∥x + xnk∥ → 2. +Fix ε > 0 and take the δ > 0 associated to f above, which we can assume +to satisfy ε < 1−δ (observe that if δ satisfies the above condition, any δ′ < δ +will also satisfy the condition). +As ∥x − xnk∥ → 2, we can pick k ∈ N such that ∥x ± xnk∥ > 2 − δ. Find +f ± ∈ BX∗ such that f ±(x ± xnk) > 2 − δ. This implies that f ±(x) > 1 − δ +and ±f ±(xnk) > 1 − δ. +Observe, on the one hand, that ∥f + − f −∥ ⩾ (f + − f −)(xnk) > 2(1 − δ). +On the other hand, Smulyan test implies that ∥f + − f −∥ < 2ε, so 1− δ < ε, +a contradiction. +Proof of Theorem 2.1. We suppose by contradiction that the Banach space +X is infinite-dimensional. Let x0 ∈ SX be a point of Fr´echet differentiability +of BX (such point exists because of the reflexivity of X [3, Corollary 11.10]) +and let x∗ +0 ∈ SX∗ such that x∗ +0(x0) = 1. +Set Y := ker(x∗ +0). Since Y is 1-codimensional we get that Y is infinite- +dimensional, so Josefson-Nissenzweig Theorem guarantees the existence of +a weak∗-null sequence (y∗ +n) in SY ∗. Set a norm-one extension x∗ +n ∈ SX∗ of +y∗ +n given by Hahn-Banach Theorem for every n ∈ N. Since y∗ +n ∈ SY ∗ we can +find xn ∈ SY such that y∗ +n(xn) = 1. In particular, x∗ +n(xn) = y∗ +n(xn) = 1 for +every n ∈ N. + +RANK-ONE PERTURBATIONS AND NORM-ATTAINING OPERATORS +5 +If ∥x0 − xn∥ → 2, then using Lemma 2.2, we have lim sup +n→∞ ∥x0 + xn∥ < 2. +Otherwise, passing to a subsequence, we may assume that sup +n∈N +∥x0−xn∥ < 2. +Thus, in any case, we may assume by passing to a subsequence that either +sup +n∈N +∥x0 + xn∥ < 2 or sup +n∈N +∥x0 − xn∥ < 2. +Set ǫ ∈ {−1, +1} such that +sup +n∈N +∥x0 + ǫxn∥ < 2. +Passing again to a further subsequence if necessary, we can suppose that +(x∗ +n(x0)) converges to some α ∈ R. Since every element of X is of the form +ax0 + y with y ∈ Y = ker(x∗ +0), it follows that (x∗ +n − αx∗ +0) is weak∗-null. +Define now gn := (1 − ǫα)x∗ +0 + ǫx∗ +n for every n ∈ N. Observe that +lim +n→∞ gn(x0) = lim +n→∞(1 − ǫα)x∗ +0(x0) + ǫx∗ +n(x0) = 1 − ǫα + ǫα = 1. +Moreover, since xn ∈ Y = ker(x∗ +0), we have x∗ +0(xn) = 0 for every n ∈ N and +therefore +lim +n→∞ gn(ǫxn) = lim +n→∞(ǫ(1 − ǫα)x∗ +0(xn) + ǫ2x∗ +n(xn)) = 1. +Thus, +L := lim sup +n→∞ ∥gn∥ ⩾ +lim +n→∞ gn(x0 + ǫxn) +supn∈N ∥x0 + ǫxn∥ = +2 +supn∈N ∥x0 + ǫxn∥ > 1. +For every n ∈ N take vn ∈ SX such that gn(vn) > ∥gn∥ − 1 +n. Take a +subsequence (gnk) with ∥gnk∥ → L. Note that gnk(vnk) → L as k → ∞. +Passing to a further subsequence, we may assume that |gnk(vnk) − L| < 1 +k +for every k ∈ N. Notice that for each k ∈ N +L − 1 +k < gnk(vnk) ⩽ ∥gnk∥ ⩽ gnk(vnk) + 1 +nk +⩽ gnk(vnk) + 1 +k < L + 2 +k. +This, in particular, shows that ∥gnk∥ ̸= 0. Put αk := ∥gnk∥−1(L − 1 +k) for +every k ∈ N; then αk ∈ (0, 1) and αk → 1 as k → ∞. +Define T ∈ L(X, ℓ∞) by the equation +T(x) := (αkgnk(x))k∈N +and observe that ∥T∥ ⩽ L because αk∥gnk∥ = L − 1 +k < L for every k ∈ N. +On the other hand, observe that ∥T(vnk)∥ ⩾ αkgnk(vnk) > αk(L− 1 +k), which +implies that ∥T∥ = L. +We claim that T does not attain its norm. Assume to the contrary that +there exists u0 ∈ SX such that ∥T(u0)∥ = sup +k∈N +|αkgnk(u0)| = L. Observe +that +gn = (1 − ǫα)x∗ +0 + ǫx∗ +n = x∗ +0 + ǫ(x∗ +n − αx∗ +0) → x∗ +0 +in the weak∗-topology since (x∗ +n−αx∗ +0) is weak∗-null. Consequently gn(u0) → +x∗ +0(u0) and, since |x∗ +0(u0)| ⩽ 1, we can find k0 ∈ N such that |gnk(u0)| < +L+1 +2 +< L for k ⩾ k0. This would imply that ∥T∥ = +max +1⩽k⩽k0−1{αk∥gnk(u0)∥} = + +6 +JUNG, MART´INEZ-CERVANTES, AND RUEDA ZOCA +L. However, αk∥gnk(u0)∥ ⩽ αk∥gnk∥ = L − 1 +k < L for each fixed k, which +leads to a contradiction. +Now notice that +T(x) := (αkgnk(x))k∈N = (αk(1 − ǫα)x∗ +0(x) + ǫαkx∗ +n(x))k∈N = R(x) + S(x), +where S(x) = (ǫαkx∗ +n(x))k∈N and R(x) = (1−ǫα)x∗ +0(x)(αk)k∈N is a rank-one +operator and +∥S∥ = 1 < ∥S + R∥ = ∥T∥ = L, +which concludes the proof. +Remark 2.3. As a consequence of Theorem 2.1, if X is an infinite-dimensional +reflexive Banach space, then the pair (X, ℓ∞) has neither the V -property nor +the WMP. Notice that the argument in Theorem 2.1 also applies to the pair +(X, c). That is, (X, c) does not have the CPP unless X has finite dimension. +Nevertheless, the pair (X, c0) has the WMP for any reflexive Banach space +X [2, Proposition 3.6] while c0 and c are isomorphic. Thus, the CPP is not +an isomorphic property. +Now we prove that we can replace ℓ∞ with a suitable reflexive Banach +space Y in Theorem 2.1. +Proof of Theorem 1.1. By Theorem 2.1, there exist T ∈ L(X, ℓ∞) and K ∈ +K(X, ℓ∞) such that ∥T + K∥ > ∥T∥ = 1 while T + K does not attain its +norm. Actually, K can be chosen to be a rank-one operator. Consider �T +and �K in L(X, ℓ∞ ⊕∞ X) given by +�T(x) = (T(x), x), +�K(x) = (K(x), 0) for every x ∈ X. +Note that ∥ �T + �K∥ = ∥T + K∥ > ∥T∥ = ∥ �T∥, �K is a rank one operator, +and �T + �K does not attain its norm. Let Z := span{ �T(X) ∪ �K(X)} ⊆ +ℓ∞ ⊕∞ X. Observe that �T has a closed range which implies that �T(X) is +isomorphic to a quotient space of X; hence �T(X) is reflexive. It follows that +Z = span{ �T(X) ∪ �K(X)} = �T(X) ⊕ �K(X) is a reflexive Banach space since +�T(X) and �K(X) are reflexive. Considering �T + �K as an operator from X +into Z, we conclude that the pair (X, Z) fails the CPP. +3. Interrelations between the WMP, V -property and CPP +The aim of this section is to make an intensive study of the WMP, V -pairs +and the CPP. As it was said in the introduction, it is known that the CPP +is the weakest one among all the above properties. However, the CPP itself +is still very restrictive, as the following two results shows. +It is worth mentioning that Ostrovskii [11, Theorem 2] proved that for +any infinite-dimensional Banach space X, there exists a renorming � +X of X +such that in � +X there is a projection onto a subspace of codimension 1 that +does not attain its norm (in particular, with norm strictly bigger than one). +From this, we can conclude that � +X does not have the CPP since there is a + +RANK-ONE PERTURBATIONS AND NORM-ATTAINING OPERATORS +7 +rank-one operator Q on � +X such that I � +X − Q does not attain its norm. He +also proved that if X is a Banach space with a symmetric basis and has the +V -property, then X is isometric to ℓp for some 1 < p < ∞ [11, Theorem 3]. +In fact, his argument shows the following: if X is a Banach space with a +symmetric basis and has the CPP, then X is isometric to ℓp. We summarize +these comments in the following proposition. +Proposition 3.1. Let X be an infinite-dimensional Banach space. +(1) Then there is a renorming � +X of X which does not have the CPP. +(2) If X has a symmetric basis and has the CPP, then X is isometric +to ℓp for some 1 < p < ∞. +Another manifestation of the severe restriction that CPP on a pair (X, Y ) +imposes on the spaces X and Y is that we can always find a pair of Banach +spaces which fails to have the CPP from the existence of non-norm-attaining +operators as follows. It is an analogue of [2, Main Theorem]. +Proposition 3.2. Let X and Y be Banach spaces, and suppose that there +exists a non-norm-attaining operator T ∈ L(X, Y ). Then the pair (X ⊕p +R, Y ⊕q R) fails to have the CPP whenever 1 ⩽ q < p ⩽ ∞. +Proof. Let T ∈ L(X, Y ) be a non-norm-attaining operator with ∥T∥ = 1. +Assume first that 1 ⩽ q < p < ∞. Consider �T, R ∈ L(X ⊕p R, Y ⊕q R) given +by �T(x, a) = (T(x), 0) and R(x, a) = (0, a), respectively. It is clear that R is +compact (indeed, of finite rank) and ∥ �T∥ = ∥T∥. Moreover, by [2, Lemma +2.1] and the argument of the proof of [2, Main Theorem], we have that +∥ �T + R∥ = sup{((1 − tp)q/p + tq)1/q : t ∈ [0, 1]} > 1 = ∥ �T∥, +where the hypothesis that 1 ⩽ q < p < ∞ is used. However, it is not difficult +to check that �T + R is not norm-attaining; hence the pair (X ⊕p R, Y ⊕q R) +fails to have the CPP. In the case p = ∞ we have that ∥ �T + R∥ = 21/q > +1 = ∥ �T∥, so the conclusion holds. +Remark 3.3. Under the same hypothesis, it is observed in [2, Main Theorem] +that the pair (X⊕∞R, Y ) fails to have the WMP. However, we cannot expect +the same result for the case of the CPP. In fact, there exists a non-norm- +attaining operator from X into c0 whenever X is an infinite-dimensional +Banach space X [9, Lemma 2.2], while (X, c0) has the CPP for any reflexive +Banach space X [2, Proposition 3.6]. +Remark 3.4. Observe that the Schur property of a Banach space Y implies +that (X, Y ) has the CPP for every reflexive space X (in fact, it has the +WMP [2, Theorem 3.5]). However, the converse is not true as the space c0 +does not have the Schur property. +As an application of Proposition 3.2, the pair (Lp[0, 1], Lq[0, 1]) fails the +CPP whenever p > 2 or q < 2. To verify this we need the following lemma, +which is a version of [2, Proposition 2.2] and [6, Proposition 4] for the CPP. + +8 +JUNG, MART´INEZ-CERVANTES, AND RUEDA ZOCA +Lemma 3.5. Let X and Y be Banach spaces. Suppose that the pair (X, Y ) +has the CPP. Then for any subspaces X1 ⊆ X and Y1 ⊆ Y , the pair +(X/X1, Y1) has the CPP. +Proof. Let π : X → X/X1 be the canonical quotient map and ι be the inclu- +sion from Y1 into Y . Suppose that T ∈ L(X/X1, Y1) and K ∈ K(X/X1, Y1) +satisfy that ∥T +K∥ > ∥T∥. Then it is clear that ∥ιTπ+ιKπ∥ = ∥T +K∥ > +∥T∥ = ∥ιTπ∥. By the assumption, the perturbation ιTπ + ιKπ attains its +norm at some x0 ∈ BX. From this, we have that T + K attains its norm at +π(x0); hence the pair (X/X1, Y1) has the CPP. +Arguing as in the proof of [2, Theorem 3.2] with the aid of Lemma 3.5, +we obtain the following desired result. +Corollary 3.6. If p > 2 or q < 2, then the pair (Lp[0, 1], Lq[0, 1]) fails the +CPP. +Note that the pair (ℓp, ℓq) with 1 < p < q < ∞ has the WMP (and +therefore the CPP) while it is not a V -pair. We do not know whether the +reverse implication holds for every pair of Banach spaces: +Question 3.7. Does every V -pair have the WMP? +A natural idea to prove that every V -pair has the WMP would be to try +to show that every V -operator with a non-weakly null maximizing sequence +is norm-attaining. However, the following remark shows that there are V - +operators with non-weakly null maximizing sequences which do not attain +their norm. +Remark 3.8. Set X = ℓ2 ⊕∞ R. Take a non-norm-attaining operator T ∈ +L(ℓ2) with ∥T∥ = 1 and define �T ∈ L(X, ℓ2) given by �T(x, a) = T(x). It is +immediate that �T does not attain its norm (which is one) and that it has a +non-weakly null maximizing sequence (take ((xn, 1)) with (xn) a maximizing +sequence for T). Nevertheless, we claim that �T is a V -operator. Since T is +a V -operator, there exists a norm-one operator B ∈ L(ℓ2) such that TB − I +is not bounded below. Define �B ∈ L(ℓ2, X) such that �B(x) = (B(x), 0). +It is immediate that �T �B − I is not bounded below and therefore �T is a +V -operator as desired. +Despite the fact that �T is a V -operator, Proposition 3.2 implies that the +pair (X, ℓ2) does not have the CPP (so, it is not a V -pair). +In spite of the fact that the properties WMP and being a V -pair are not +equivalent properties on a pair (X, Y ) of Banach spaces, they are closely +related properties (as both imply, for instance, the CPP) and, because of +that, it is expectable that some results which holds true for V -pairs can be +translated to a WMP version and vice-versa. This is what we will do in the +following proposition. +Observe that, for each 1-complemented subspace X1 of X, the pair (X1, Y ) +has the WMP (resp., is a V -pair) whenever (X, Y ) has the WMP (resp., is + +RANK-ONE PERTURBATIONS AND NORM-ATTAINING OPERATORS +9 +a V -pair) (see [2, Proposition 2.2], resp., [6, Proposition 4]). However, it +is unknown if the same remains true for any subspace of the domain space +X. +In this regard, the following result extends [4, (c) of Corollary 3.8], +where the authors proved the WMP of a pair (X, Y ) for X = (�∞ +n=1 En)p +and Y = (�∞ +n=1 Fn)q, where dim(En), dim(Fn) < ∞, 1 < p < ∞ and +1 ⩽ q < ∞. +Proposition 3.9. Let (En) and (Fn) be sequences of finite-dimensional +spaces. +If X ⊆ (�∞ +n=1 En)p and Y ⊆ (�∞ +n=1 Fn)q for 1 < p < ∞ and +1 ⩽ q < ∞, then the pair (X, Y ) has the WMP. +Proof. The proof is motivated by the result of Ostrovskii [11, Lemma 1]. +Let T ∈ L(X, Y ), ∥T∥ = 1 and let (xn) ⊆ SX be a maximizing sequence for +T. Suppose that (xn) converges weakly to x0 ̸= 0. We aim to show that T +attains its norm. +Case 1: 1 < p ⩽ q < ∞. For simplicity, consider the norm ∥ · ∥r on R2 +given by ∥(a, b)∥r := (|a|r + |b|r)1/r for 1 < r < ∞. Then +(1) For a weakly null sequence (wn) in X and v ∈ X, we have +∥wn + v∥ = ∥(∥wn∥, ∥v∥)∥p + o(1) +(2) ∥(a, b)∥r < ∥(c, d)∥r if 0 ⩽ a ⩽ c and 0 ⩽ b ⩽ d, and at least one of +the inequalities is strict. +Put wn = xn − x0. Then (wn) is weakly null (so, (T(wn)) is weakly null in +Y ); hence +∥T(xn)∥ = ∥T(wn) + T(x0)∥ = ∥(∥T(wn)∥, ∥T(x0)∥)∥q + o(1) +⩽ ∥(∥wn∥, ∥T(x0)∥)∥q + o(1). +On the other hand, +∥T(xn)∥ = ∥wn + x0∥ + o(1) = ∥(∥wn∥, ∥x0∥)∥p + o(1). +Passing to a subsequence, ∥wn∥ → α. Thus, we have +∥(α, ∥x0∥)∥p ⩽ ∥(α, ∥T(x0)∥)∥q ⩽ ∥(α, ∥T(x0)∥)∥p. +This shows that ∥T(x0)∥ = ∥x0∥. Since x0 ̸= 0, we conclude that T attains +its norm at x0. +Case 2: 1 ⩽ q < p < ∞. Let us denote by δZ(t) and ρZ(t) the modulus of +AUC and AUS of a Banach space Z, respectively. Note from [10, Theorem +3.1] that +ρX(t) ⩽ ρ(�∞ +n=1 En)p(t) = (1 + tp)1/p − 1 +< (1 + tq)1/q − 1 = δ(�∞ +n=1 Fn)q(t) ⩽ δY (t) +for 0 < t < 1. By [5, Proposition 2.3], we conclude that every operator from +X into Y is compact; hence T attains its norm. + +10 +JUNG, MART´INEZ-CERVANTES, AND RUEDA ZOCA +As we already observed from Theorem 2.1 and Remark 2.3, the pairs (X, c) +and (X, ℓ∞) cannot be V -pairs (since they fail to have the CPP). This is +also covered by the following result since c and ℓ∞ (in general, C(K)-spaces) +have the Dunford-Pettis property. Recall that a Banach space X is said to +have the Dunford-Pettis property if every weakly compact operator from X +into any Banach space is completely continuous. +Proposition 3.10. Let X be a Banach space and Y a Banach space with +Dunford-Pettis property. If the pair (X, Y ) is a V -pair, then every operator +in L(X, Y ) is norm-attaining. +Proof. Let T ∈ L(X, Y ) with ∥T∥ = 1. +As (X, Y ) is a V -pair, there is +B ∈ L(Y, X) with ∥B∥ = 1 and a sequence (yn) ⊆ SY such that ∥(TB − +I)(yn)∥ → 0. Since X is reflexive, passing to a subsequence, we may assume +that (B(yn)) converges weakly to some x0 ∈ BX. It follows that (TB(yn)) +converges weakly to T(x0) in Y , which in turn implies that (yn) converges +weakly to T(x0). As Y has the Dunford-Pettis property, B is completely +continuous; hence (B(yn)) converges in norm to BT(x0). Note then that +(TB(yn)) converges in norm to TBT(x0); hence, (yn) converges in norm to +TBT(x0). This shows that ∥TBT(x0)∥ = 1 and T attains its norm at x0. +Unlike the WMP, it follows from Proposition 3.10 that for any infinite- +dimensional Banach space X, the pair (X, c0) is not a V -pair, which stresses +the fact that the implication “WMP ⇒ V -property” is false. +Acknowledgment. The research of this paper started during a stay of the +authors in IMAC Castell´on in June 2022. The authors are deeply grateful to +the whole institute, and very specially to Sheldon Dantas, for the hospitality +received during this stay. +Mingu Jung was supported by a KIAS Individual Grant (MG086601) at +Korea Institute for Advanced Study. The last two authors were partially +supported by Agencia Estatal de Investigaci´on and EDRF/FEDER “A way +of making Europe” (MCIN/AEI/10.13039/501100011033) through grants +PID2021-122126NB-C32 and PID2021-122126NB-C31 (Rueda Zoca). The +research of Abraham Rueda Zoca was also supported by Junta de Andaluc´ıa +Grants FQM-0185 and PY20 00255. +References +[1] R.M. Aron, D. Garc´ıa, D. Pellegrino, and E. V. Teixeira, Reflexivity +and non-weakly null maximizing sequences, Proc. Amer. Math. Soc. 148 (2020), +no. 2, 741–750. +[2] S. Dantas, M. Jung, and G. Mart´ınez-Cervantes, Some remarks on the +weak maximizing property, J. Math. Anal. Appl. 504 (2021), article 125433. +[3] M. Fabian, P. Habala, P. H´ajek, V. Montesinos, J. Pelant, and V. +Zizler, Functional Analysis and Infinite dimensional Geometry, CMS Books in +Mathematics, Springer-Verlag, New York, 2001. +[4] L.C. Garc´ıa-Lirola and C. Petitjean, On the weak maximizing properties, +Banach J. Math. Anal. 15, 55 (2021). + +RANK-ONE PERTURBATIONS AND NORM-ATTAINING OPERATORS +11 +[5] W.B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman, Almost +Fr´echet differentiability of Lipschitz mappings between infinite-dimensional Ba- +nach spaces, Proc. Lond. Math. Soc. 3 (84), 711–746 (2002) +[6] V.A. Khatskevich, M.I. Ostrovskii, and V.S. Shulman, Extremal Problems +for Operators in Banach Spaces Arising in the Study of Linear Operator Pencils, +Integr. Equ. Oper. Theory 51 (2005), 109–119. +[7] J. Kover, Compact perturbations and norm-attaining operators, Quaest. Math. +28 (2005), no. 4, 401–408. +[8] J. Lindenstrauss, On operators which attain their norm, Israel J. Math., 1 +(1963), 139–144. +[9] M. Mart´ın, J. Mer´ı, and R. Pay´a, On the intrinsic and the spatial numerical +range, J. Math. Anal. Appl. 318 (2006), 175–189. +[10] V.D. Milman, Geometric theory of Banach spaces II. Geometry of the unit ball. +Uspehi Mat. Nauk 26 73–149 (1971). +[11] M.I. Ostrovskii, Extremal Problems for Operators in Banach Spaces Arising in +the Study of Linear Operator Pencils, II, Integr. Equ. Oper. Theory 51 (2005), +553–564. +[12] D. Pellegrino and E.V. Teixeira, Norm optimization problem for linear oper- +ators in classical Banach spaces, Bull. Braz. Math. Soc. (NS) 40 (2009), 417–431. +(Jung) School of Mathematics, Korea Institute for Advanced Study, 02455 +Seoul, Republic of Korea +ORCID: 0000-0003-2240-2855 +Email address: jmingoo@kias.re.kr +URL: https://clemg.blog/ +(G. Mart´ınez-Cervantes) Universidad de Alicante, Departamento de Matem´aticas, +Facultad de Ciencias, 03080 Alicante, Spain +ORCID: 0000-0002-5927-5215 +Email address: gonzalo.martinez@ua.es +(Rueda Zoca) Universidad de Granada, Facultad de Ciencias. Departamento +de An´alisis Matem´atico, 18071-Granada, Spain +ORCID: 0000-0003-0718-1353 +Email address: abrahamrueda@ugr.es +URL: https://arzenglish.wordpress.com + diff --git a/jNE4T4oBgHgl3EQfTAww/content/tmp_files/load_file.txt b/jNE4T4oBgHgl3EQfTAww/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8eb0178e48f4795f5d76bee1bd3f055855a80ce1 --- /dev/null +++ b/jNE4T4oBgHgl3EQfTAww/content/tmp_files/load_file.txt @@ -0,0 +1,449 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf,len=448 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='05003v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='FA] 12 Jan 2023 RANK-ONE PERTURBATIONS AND NORM-ATTAINING OPERATORS MINGU JUNG, GONZALO MART´INEZ-CERVANTES, AND ABRAHAM RUEDA ZOCA Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' The main goal of this article is to show that for every (re- flexive) infinite-dimensional Banach space X there exists a reflexive Ba- nach space Y and T, R ∈ L(X, Y ) such that R is a rank-one operator, ∥T + R∥ > ∥T ∥ but T + R does not attain its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' This answers a question posed by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Dantas and the first two authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Further- more, motivated by the parallelism exhibited in the literature between the V -property introduced by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Khatskevich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Ostrovskii and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Shulman and the weak maximizing property introduced by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Aron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Garc´ıa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Pellegrino and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Teixeira, we also study the rela- tionship between these two properties and norm-attaining perturbations of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Introduction Given real Banach spaces X and Y , we denote by L(X, Y ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=', K(X, Y )) the space of all (bounded linear) operators (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=', compact operators) from X into Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' When X = Y , we simply write L(X) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=', K(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' As usual, the notations BX and SX stand for the closed unit ball of X and the unit sphere of X, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' A well-known result of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Lindenstrauss states that if X is reflexive then, for every Banach space Y , every operator T ∈ L(X, Y ) can be approximated by norm-attaining operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Recall that an operator T ∈ L(X, Y ) is said to be norm-attaining if ∥T∥ = ∥T(x)∥ for some x ∈ BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Indeed, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Lin- denstrauss showed that there exists a compact operator K ∈ K(X, Y ) such that ∥K∥ is arbitrarily small and T + K attains its norm (see Theorem 1 and the succeeding remark in [8]), so every operator can be approximated by norm-attaining compact perturbations of itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Pairs of classical Banach spaces quite often satisfy a stronger condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Namely, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Kover [7] proved that for a Hilbert space H, if a compact per- turbation of an operator T ∈ L(H) has norm strictly greater than the norm of T, then this perturbation attains its norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' if ∥T + K∥ > ∥T∥ where K ∈ K(H) then T + K attains its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' 46B10, 46B20, 46B28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Norm-attaining operators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Compact perturbation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Reflexivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Weak Maximizing Property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' V -pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' 1 2 JUNG, MART´INEZ-CERVANTES, AND RUEDA ZOCA Before going beyond in the exposition of literature, let us fix the follow- ing notation: a pair (X, Y ) of Banach spaces has the compact perturbation property (for short, CPP) if for any T ∈ L(X, Y ) and K ∈ K(X, Y ) the in- equality ∥T + K∥ > ∥T∥ implies that T + K is norm-attaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Notice that the CPP of the pair (X, Y ) forces the domain space X to be reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' We say that X has the CPP if the pair (X, X) has the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' With this notation in mind, Kover’s result says nothing but that every Hilbert space has the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' A large class of pairs of Banach spaces enjoying the CPP is given by the class of V -pairs, which was introduced and intensively developed in the papers [6, 11]: a bounded linear operator T ∈ L(X, Y ) is said to be a V -operator if there is a norm-one operator S ∈ L(Y, X) such that the spectral radius of TS coincides with ∥T∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' If every operator in L(X, Y ) is a V -operator, then the pair (X, Y ) is said to be a V -pair or to have the V -property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' If X = Y , then X is said to be a V -space or to have the V - property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Among others, it is proved in [6, Proposition 5] that an operator having a strictly singular hump is a V -operator if and only if it is norm- attaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Thus, every V -pair satisfies the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Moreover, [6, Theorem 1] generalizes the aforementioned result of Kover to ℓp(Xn) for any sequence of finite dimensional spaces (Xn) and 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Quite recently R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Aron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Garc´ıa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Pellegrino and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Teixeira [1] introduced another related property, the so-called weak maximizing prop- erty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' A pair (X, Y ) of Banach spaces is said to have the weak maximizing property (for short, WMP) if every operator from X into Y with a non- weakly null maximizing sequence is norm-attaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Note from [1, Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='2] (see also [12, Theorem 1]) that for 1 < p < ∞, 1 ⩽ q < ∞, and arbitrary index sets Γ1, Γ2, the pair (ℓp(Γ1), ℓq(Γ2)) has the WMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Further- more, the WMP implies the CPP [1, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='4], which improves the previous results of Kover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Although there are pairs of Banach spaces with the CPP failing the WMP (see [2, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='6]), it seems to be an open problem whether the CPP for a pair of reflexive Banach spaces implies the WMP [2, Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' For further results and more examples of pairs with the WMP we refer the reader to [2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Due to the aforementioned results it is natural to wonder whether every reflexive Banach space X has the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Nevertheless, it is implicitly proved in [11, Theorem 2] that for every infinite-dimensional Banach space X there exists an equivalent norm ||| · ||| and a rank-one operator R : X −→ X with |||I + R||| > |||I||| but with I + R failing to attain its norm in L((X, ||| · |||)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' In particular, (X, ||| · |||) fails the CPP (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Note that if a Banach space X is such that every operator in L(X) attains its norm, then X clearly has the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' In other words, if X fails the CPP, then there exists a non-norm attaining operator on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' By the result just mentioned in [11], we conclude that for any infinite-dimensional Banach space X, there exists a renorming � X of X for which not every operator in RANK-ONE PERTURBATIONS AND NORM-ATTAINING OPERATORS 3 L( � X) is norm-attaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' In fact, it seems to be an open problem whether there exists a reflexive infinite-dimensional Banach space X such that every operator in L(X) attains its norm (see, for instance, [6, Problem 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' The main goal of this paper is to show that for every infinite-dimensional space X there exists a reflexive Banach space Y such that the pair (X, Y ) fails the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Namely, the main theorem of the paper reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let X be an infinite-dimensional reflexive Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Then there exists a reflexive Banach space Y and T, R ∈ L(X, Y ) with R a rank-one operator such that ∥T + R∥ > ∥T∥ but with T + R not attaining its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' In particular, the pair (X, Y ) fails the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' The aim of Section 2 is to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Our original motivation for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1 comes from the study of the WMP and the V -property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' There is a deep parallelism exhibited in the literature between these two properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' compare [2, 4] to [6, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' However, up to our knowledge, it is not known how these two concepts are related to each other and quite often if a question is open for one property it is also open for the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' In [2, Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='3] it is asked whether if a reflexive Banach space X satisfies that the pair (X, Y ) has the WMP for every Banach space Y , then X must be finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' We would like to mention that, to the best of our knowledge, the same question was open if we replace the WMP with the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' In particular, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1 gives a positive answer to [2, Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='3] and exhibits another common behaviour between the WMP and the V -property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Observe that the WMP and the V -property implies the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' We know that the WMP (and therefore the CPP) does not imply the V -property (the pair (ℓp, ℓq) with 1 < p < q < ∞ has the WMP while it is not a V -pair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Nevertheless, we do not know whether the V -property implies the WMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' This situation also serves as motivation for our investigation of all the aforementioned properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' In Section 3, we first observe from the argument of Ostrovskii [11] that the CPP is still a quite restrictive property: an infinite-dimensional Banach space X with the CPP must be isometric to ℓp for some 1 < p < ∞ if X has a symmetric basis (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Moreover, we see that the existence of non-norm-attaining operators between Banach spaces X and Y produces pairs of Banach spaces failing the CPP (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='2) and, as a consequence, the pair (Lp[0, 1], Lq[0, 1]) fails the CPP whenever p > 2 or q < 2 (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' We also generalize in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='9 the fact that the pair (ℓp, ℓq) has the WMP for 1 < p < ∞ and 1 ⩽ q < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Finally, we prove that a pair (X, Y ) has the V -property if and only if every operator from X into Y is norm-attaining provided that Y has the Dunford-Pettis property (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='10), which covers some results in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' 4 JUNG, MART´INEZ-CERVANTES, AND RUEDA ZOCA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1 As we have indicated in the introduction, the aim of this section is to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1, for which we will start by considering the Banach space ℓ∞ of all bounded sequences instead of reflexive ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let X be a reflexive Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' If the pair (X, ℓ∞) has the CPP, then X is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Namely, if X is infinite-dimensional then there exists S, R ∈ L(X, ℓ∞) with R a rank-one operator such that ∥S + R∥ > ∥S∥ but with S + R not attaining its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' For the proof we need a lemma of geometric nature for points of Fr´echet differentiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let us recall that a point x ∈ SX is said to be a point of Fr´echet-differentiability of X if the norm ∥ · ∥ : X −→ R is Fr´echet differen- tiable at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' See [3, Chapter 8] for background about Fr´echet differentiability in Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let X be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let x ∈ SX be a point of Fr´echet differentiability of X and (xn) be a sequence of points of BX such that ∥x − xn∥ → 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Then lim sup n→∞ ∥x + xn∥ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Since x is a point of Fr´echet differentiability, by ˇSmulyan Lemma [3, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='4] there exists f ∈ SX∗ with f(x) = 1 and with the following property: for every ε > 0 there exists δ > 0 such that if g ∈ BX∗ and g(x) > 1 − δ then ∥g − f∥ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Assume to the contrary that lim sup n→∞ ∥x + xn∥ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Take a subsequence (xnk) such that ∥x + xnk∥ → 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Fix ε > 0 and take the δ > 0 associated to f above, which we can assume to satisfy ε < 1−δ (observe that if δ satisfies the above condition, any δ′ < δ will also satisfy the condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' As ∥x − xnk∥ → 2, we can pick k ∈ N such that ∥x ± xnk∥ > 2 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Find f ± ∈ BX∗ such that f ±(x ± xnk) > 2 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' This implies that f ±(x) > 1 − δ and ±f ±(xnk) > 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Observe, on the one hand, that ∥f + − f −∥ ⩾ (f + − f −)(xnk) > 2(1 − δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' On the other hand, Smulyan test implies that ∥f + − f −∥ < 2ε, so 1− δ < ε, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' We suppose by contradiction that the Banach space X is infinite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let x0 ∈ SX be a point of Fr´echet differentiability of BX (such point exists because of the reflexivity of X [3, Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='10]) and let x∗ 0 ∈ SX∗ such that x∗ 0(x0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Set Y := ker(x∗ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Since Y is 1-codimensional we get that Y is infinite- dimensional, so Josefson-Nissenzweig Theorem guarantees the existence of a weak∗-null sequence (y∗ n) in SY ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Set a norm-one extension x∗ n ∈ SX∗ of y∗ n given by Hahn-Banach Theorem for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Since y∗ n ∈ SY ∗ we can find xn ∈ SY such that y∗ n(xn) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' In particular, x∗ n(xn) = y∗ n(xn) = 1 for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' RANK-ONE PERTURBATIONS AND NORM-ATTAINING OPERATORS 5 If ∥x0 − xn∥ → 2, then using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='2, we have lim sup n→∞ ∥x0 + xn∥ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Otherwise, passing to a subsequence, we may assume that sup n∈N ∥x0−xn∥ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Thus, in any case, we may assume by passing to a subsequence that either sup n∈N ∥x0 + xn∥ < 2 or sup n∈N ∥x0 − xn∥ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Set ǫ ∈ {−1, +1} such that sup n∈N ∥x0 + ǫxn∥ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Passing again to a further subsequence if necessary, we can suppose that (x∗ n(x0)) converges to some α ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Since every element of X is of the form ax0 + y with y ∈ Y = ker(x∗ 0), it follows that (x∗ n − αx∗ 0) is weak∗-null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Define now gn := (1 − ǫα)x∗ 0 + ǫx∗ n for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Observe that lim n→∞ gn(x0) = lim n→∞(1 − ǫα)x∗ 0(x0) + ǫx∗ n(x0) = 1 − ǫα + ǫα = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Moreover, since xn ∈ Y = ker(x∗ 0), we have x∗ 0(xn) = 0 for every n ∈ N and therefore lim n→∞ gn(ǫxn) = lim n→∞(ǫ(1 − ǫα)x∗ 0(xn) + ǫ2x∗ n(xn)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Thus, L := lim sup n→∞ ∥gn∥ ⩾ lim n→∞ gn(x0 + ǫxn) supn∈N ∥x0 + ǫxn∥ = 2 supn∈N ∥x0 + ǫxn∥ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' For every n ∈ N take vn ∈ SX such that gn(vn) > ∥gn∥ − 1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Take a subsequence (gnk) with ∥gnk∥ → L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Note that gnk(vnk) → L as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Passing to a further subsequence, we may assume that |gnk(vnk) − L| < 1 k for every k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Notice that for each k ∈ N L − 1 k < gnk(vnk) ⩽ ∥gnk∥ ⩽ gnk(vnk) + 1 nk ⩽ gnk(vnk) + 1 k < L + 2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' This, in particular, shows that ∥gnk∥ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Put αk := ∥gnk∥−1(L − 1 k) for every k ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' then αk ∈ (0, 1) and αk → 1 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Define T ∈ L(X, ℓ∞) by the equation T(x) := (αkgnk(x))k∈N and observe that ∥T∥ ⩽ L because αk∥gnk∥ = L − 1 k < L for every k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' On the other hand, observe that ∥T(vnk)∥ ⩾ αkgnk(vnk) > αk(L− 1 k), which implies that ∥T∥ = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' We claim that T does not attain its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Assume to the contrary that there exists u0 ∈ SX such that ∥T(u0)∥ = sup k∈N |αkgnk(u0)| = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Observe that gn = (1 − ǫα)x∗ 0 + ǫx∗ n = x∗ 0 + ǫ(x∗ n − αx∗ 0) → x∗ 0 in the weak∗-topology since (x∗ n−αx∗ 0) is weak∗-null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Consequently gn(u0) → x∗ 0(u0) and, since |x∗ 0(u0)| ⩽ 1, we can find k0 ∈ N such that |gnk(u0)| < L+1 2 < L for k ⩾ k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' This would imply that ∥T∥ = max 1⩽k⩽k0−1{αk∥gnk(u0)∥} = 6 JUNG, MART´INEZ-CERVANTES, AND RUEDA ZOCA L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' However, αk∥gnk(u0)∥ ⩽ αk∥gnk∥ = L − 1 k < L for each fixed k, which leads to a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Now notice that T(x) := (αkgnk(x))k∈N = (αk(1 − ǫα)x∗ 0(x) + ǫαkx∗ n(x))k∈N = R(x) + S(x), where S(x) = (ǫαkx∗ n(x))k∈N and R(x) = (1−ǫα)x∗ 0(x)(αk)k∈N is a rank-one operator and ∥S∥ = 1 < ∥S + R∥ = ∥T∥ = L, which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' As a consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1, if X is an infinite-dimensional reflexive Banach space, then the pair (X, ℓ∞) has neither the V -property nor the WMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Notice that the argument in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1 also applies to the pair (X, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' That is, (X, c) does not have the CPP unless X has finite dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Nevertheless, the pair (X, c0) has the WMP for any reflexive Banach space X [2, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='6] while c0 and c are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Thus, the CPP is not an isomorphic property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Now we prove that we can replace ℓ∞ with a suitable reflexive Banach space Y in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1, there exist T ∈ L(X, ℓ∞) and K ∈ K(X, ℓ∞) such that ∥T + K∥ > ∥T∥ = 1 while T + K does not attain its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Actually, K can be chosen to be a rank-one operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Consider �T and �K in L(X, ℓ∞ ⊕∞ X) given by �T(x) = (T(x), x), �K(x) = (K(x), 0) for every x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Note that ∥ �T + �K∥ = ∥T + K∥ > ∥T∥ = ∥ �T∥, �K is a rank one operator, and �T + �K does not attain its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let Z := span{ �T(X) ∪ �K(X)} ⊆ ℓ∞ ⊕∞ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Observe that �T has a closed range which implies that �T(X) is isomorphic to a quotient space of X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' hence �T(X) is reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' It follows that Z = span{ �T(X) ∪ �K(X)} = �T(X) ⊕ �K(X) is a reflexive Banach space since �T(X) and �K(X) are reflexive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Considering �T + �K as an operator from X into Z, we conclude that the pair (X, Z) fails the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Interrelations between the WMP, V -property and CPP The aim of this section is to make an intensive study of the WMP, V -pairs and the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' As it was said in the introduction, it is known that the CPP is the weakest one among all the above properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' However, the CPP itself is still very restrictive, as the following two results shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' It is worth mentioning that Ostrovskii [11, Theorem 2] proved that for any infinite-dimensional Banach space X, there exists a renorming � X of X such that in � X there is a projection onto a subspace of codimension 1 that does not attain its norm (in particular, with norm strictly bigger than one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' From this, we can conclude that � X does not have the CPP since there is a RANK-ONE PERTURBATIONS AND NORM-ATTAINING OPERATORS 7 rank-one operator Q on � X such that I � X − Q does not attain its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' He also proved that if X is a Banach space with a symmetric basis and has the V -property, then X is isometric to ℓp for some 1 < p < ∞ [11, Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' In fact, his argument shows the following: if X is a Banach space with a symmetric basis and has the CPP, then X is isometric to ℓp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' We summarize these comments in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let X be an infinite-dimensional Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' (1) Then there is a renorming � X of X which does not have the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' (2) If X has a symmetric basis and has the CPP, then X is isometric to ℓp for some 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Another manifestation of the severe restriction that CPP on a pair (X, Y ) imposes on the spaces X and Y is that we can always find a pair of Banach spaces which fails to have the CPP from the existence of non-norm-attaining operators as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' It is an analogue of [2, Main Theorem].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let X and Y be Banach spaces, and suppose that there exists a non-norm-attaining operator T ∈ L(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Then the pair (X ⊕p R, Y ⊕q R) fails to have the CPP whenever 1 ⩽ q < p ⩽ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let T ∈ L(X, Y ) be a non-norm-attaining operator with ∥T∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Assume first that 1 ⩽ q < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Consider �T, R ∈ L(X ⊕p R, Y ⊕q R) given by �T(x, a) = (T(x), 0) and R(x, a) = (0, a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' It is clear that R is compact (indeed, of finite rank) and ∥ �T∥ = ∥T∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Moreover, by [2, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1] and the argument of the proof of [2, Main Theorem], we have that ∥ �T + R∥ = sup{((1 − tp)q/p + tq)1/q : t ∈ [0, 1]} > 1 = ∥ �T∥, where the hypothesis that 1 ⩽ q < p < ∞ is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' However, it is not difficult to check that �T + R is not norm-attaining;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' hence the pair (X ⊕p R, Y ⊕q R) fails to have the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' In the case p = ∞ we have that ∥ �T + R∥ = 21/q > 1 = ∥ �T∥, so the conclusion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Under the same hypothesis, it is observed in [2, Main Theorem] that the pair (X⊕∞R, Y ) fails to have the WMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' However, we cannot expect the same result for the case of the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' In fact, there exists a non-norm- attaining operator from X into c0 whenever X is an infinite-dimensional Banach space X [9, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='2], while (X, c0) has the CPP for any reflexive Banach space X [2, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Observe that the Schur property of a Banach space Y implies that (X, Y ) has the CPP for every reflexive space X (in fact, it has the WMP [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' However, the converse is not true as the space c0 does not have the Schur property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' As an application of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='2, the pair (Lp[0, 1], Lq[0, 1]) fails the CPP whenever p > 2 or q < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' To verify this we need the following lemma, which is a version of [2, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='2] and [6, Proposition 4] for the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' 8 JUNG, MART´INEZ-CERVANTES, AND RUEDA ZOCA Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let X and Y be Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Suppose that the pair (X, Y ) has the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Then for any subspaces X1 ⊆ X and Y1 ⊆ Y , the pair (X/X1, Y1) has the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let π : X → X/X1 be the canonical quotient map and ι be the inclu- sion from Y1 into Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Suppose that T ∈ L(X/X1, Y1) and K ∈ K(X/X1, Y1) satisfy that ∥T +K∥ > ∥T∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Then it is clear that ∥ιTπ+ιKπ∥ = ∥T +K∥ > ∥T∥ = ∥ιTπ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' By the assumption, the perturbation ιTπ + ιKπ attains its norm at some x0 ∈ BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' From this, we have that T + K attains its norm at π(x0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' hence the pair (X/X1, Y1) has the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Arguing as in the proof of [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='2] with the aid of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='5, we obtain the following desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' If p > 2 or q < 2, then the pair (Lp[0, 1], Lq[0, 1]) fails the CPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Note that the pair (ℓp, ℓq) with 1 < p < q < ∞ has the WMP (and therefore the CPP) while it is not a V -pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' We do not know whether the reverse implication holds for every pair of Banach spaces: Question 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Does every V -pair have the WMP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' A natural idea to prove that every V -pair has the WMP would be to try to show that every V -operator with a non-weakly null maximizing sequence is norm-attaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' However, the following remark shows that there are V - operators with non-weakly null maximizing sequences which do not attain their norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Set X = ℓ2 ⊕∞ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Take a non-norm-attaining operator T ∈ L(ℓ2) with ∥T∥ = 1 and define �T ∈ L(X, ℓ2) given by �T(x, a) = T(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' It is immediate that �T does not attain its norm (which is one) and that it has a non-weakly null maximizing sequence (take ((xn, 1)) with (xn) a maximizing sequence for T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Nevertheless, we claim that �T is a V -operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Since T is a V -operator, there exists a norm-one operator B ∈ L(ℓ2) such that TB − I is not bounded below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Define �B ∈ L(ℓ2, X) such that �B(x) = (B(x), 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' It is immediate that �T �B − I is not bounded below and therefore �T is a V -operator as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Despite the fact that �T is a V -operator, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='2 implies that the pair (X, ℓ2) does not have the CPP (so, it is not a V -pair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' In spite of the fact that the properties WMP and being a V -pair are not equivalent properties on a pair (X, Y ) of Banach spaces, they are closely related properties (as both imply, for instance, the CPP) and, because of that, it is expectable that some results which holds true for V -pairs can be translated to a WMP version and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' This is what we will do in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Observe that, for each 1-complemented subspace X1 of X, the pair (X1, Y ) has the WMP (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=', is a V -pair) whenever (X, Y ) has the WMP (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=', is RANK-ONE PERTURBATIONS AND NORM-ATTAINING OPERATORS 9 a V -pair) (see [2, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='2], resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=', [6, Proposition 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' However, it is unknown if the same remains true for any subspace of the domain space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' In this regard, the following result extends [4, (c) of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='8], where the authors proved the WMP of a pair (X, Y ) for X = (�∞ n=1 En)p and Y = (�∞ n=1 Fn)q, where dim(En), dim(Fn) < ∞, 1 < p < ∞ and 1 ⩽ q < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let (En) and (Fn) be sequences of finite-dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' If X ⊆ (�∞ n=1 En)p and Y ⊆ (�∞ n=1 Fn)q for 1 < p < ∞ and 1 ⩽ q < ∞, then the pair (X, Y ) has the WMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' The proof is motivated by the result of Ostrovskii [11, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let T ∈ L(X, Y ), ∥T∥ = 1 and let (xn) ⊆ SX be a maximizing sequence for T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Suppose that (xn) converges weakly to x0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' We aim to show that T attains its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Case 1: 1 < p ⩽ q < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' For simplicity, consider the norm ∥ · ∥r on R2 given by ∥(a, b)∥r := (|a|r + |b|r)1/r for 1 < r < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Then (1) For a weakly null sequence (wn) in X and v ∈ X, we have ∥wn + v∥ = ∥(∥wn∥, ∥v∥)∥p + o(1) (2) ∥(a, b)∥r < ∥(c, d)∥r if 0 ⩽ a ⩽ c and 0 ⩽ b ⩽ d, and at least one of the inequalities is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Put wn = xn − x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Then (wn) is weakly null (so, (T(wn)) is weakly null in Y );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' hence ∥T(xn)∥ = ∥T(wn) + T(x0)∥ = ∥(∥T(wn)∥, ∥T(x0)∥)∥q + o(1) ⩽ ∥(∥wn∥, ∥T(x0)∥)∥q + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' On the other hand, ∥T(xn)∥ = ∥wn + x0∥ + o(1) = ∥(∥wn∥, ∥x0∥)∥p + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Passing to a subsequence, ∥wn∥ → α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Thus, we have ∥(α, ∥x0∥)∥p ⩽ ∥(α, ∥T(x0)∥)∥q ⩽ ∥(α, ∥T(x0)∥)∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' This shows that ∥T(x0)∥ = ∥x0∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Since x0 ̸= 0, we conclude that T attains its norm at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Case 2: 1 ⩽ q < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let us denote by δZ(t) and ρZ(t) the modulus of AUC and AUS of a Banach space Z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Note from [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1] that ρX(t) ⩽ ρ(�∞ n=1 En)p(t) = (1 + tp)1/p − 1 < (1 + tq)1/q − 1 = δ(�∞ n=1 Fn)q(t) ⩽ δY (t) for 0 < t < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' By [5, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='3], we conclude that every operator from X into Y is compact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' hence T attains its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' 10 JUNG, MART´INEZ-CERVANTES, AND RUEDA ZOCA As we already observed from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='1 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='3, the pairs (X, c) and (X, ℓ∞) cannot be V -pairs (since they fail to have the CPP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' This is also covered by the following result since c and ℓ∞ (in general, C(K)-spaces) have the Dunford-Pettis property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Recall that a Banach space X is said to have the Dunford-Pettis property if every weakly compact operator from X into any Banach space is completely continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let X be a Banach space and Y a Banach space with Dunford-Pettis property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' If the pair (X, Y ) is a V -pair, then every operator in L(X, Y ) is norm-attaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Let T ∈ L(X, Y ) with ∥T∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' As (X, Y ) is a V -pair, there is B ∈ L(Y, X) with ∥B∥ = 1 and a sequence (yn) ⊆ SY such that ∥(TB − I)(yn)∥ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Since X is reflexive, passing to a subsequence, we may assume that (B(yn)) converges weakly to some x0 ∈ BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' It follows that (TB(yn)) converges weakly to T(x0) in Y , which in turn implies that (yn) converges weakly to T(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' As Y has the Dunford-Pettis property, B is completely continuous;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' hence (B(yn)) converges in norm to BT(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Note then that (TB(yn)) converges in norm to TBT(x0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' hence, (yn) converges in norm to TBT(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' This shows that ∥TBT(x0)∥ = 1 and T attains its norm at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Unlike the WMP, it follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='10 that for any infinite- dimensional Banach space X, the pair (X, c0) is not a V -pair, which stresses the fact that the implication “WMP ⇒ V -property” is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' The research of this paper started during a stay of the authors in IMAC Castell´on in June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' The authors are deeply grateful to the whole institute, and very specially to Sheldon Dantas, for the hospitality received during this stay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Mingu Jung was supported by a KIAS Individual Grant (MG086601) at Korea Institute for Advanced Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' The last two authors were partially supported by Agencia Estatal de Investigaci´on and EDRF/FEDER “A way of making Europe” (MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='13039/501100011033) through grants PID2021-122126NB-C32 and PID2021-122126NB-C31 (Rueda Zoca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' The research of Abraham Rueda Zoca was also supported by Junta de Andaluc´ıa Grants FQM-0185 and PY20 00255.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='kr URL: https://clemg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='blog/ (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Mart´ınez-Cervantes) Universidad de Alicante, Departamento de Matem´aticas, Facultad de Ciencias, 03080 Alicante, Spain ORCID: 0000-0002-5927-5215 Email address: gonzalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='martinez@ua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='es (Rueda Zoca) Universidad de Granada, Facultad de Ciencias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content=' Departamento de An´alisis Matem´atico, 18071-Granada, Spain ORCID: 0000-0003-0718-1353 Email address: abrahamrueda@ugr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='es URL: https://arzenglish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='wordpress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE4T4oBgHgl3EQfTAww/content/2301.05003v1.pdf'} diff --git a/pNFAT4oBgHgl3EQfeR0E/content/tmp_files/2301.08574v1.pdf.txt b/pNFAT4oBgHgl3EQfeR0E/content/tmp_files/2301.08574v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f1e75ab51400924b9600dc9119c9515f8f1931d9 --- /dev/null +++ b/pNFAT4oBgHgl3EQfeR0E/content/tmp_files/2301.08574v1.pdf.txt @@ -0,0 +1,845 @@ +arXiv:2301.08574v1 [math-ph] 20 Jan 2023 +ON POINCAR´E–BIRKHOFF–WITT BASIS +OF QUANTUM GENERAL LINEAR SUPERALGEBRA +ALEXANDER V. RAZUMOV +ABSTRACT. We give a detailed derivation of the commutation relations for the Poincar´e– +Birkhoff–Witt generators of the quantum superalgebra Uq(glM|N). +CONTENTS +1. +Introduction +1 +2. +Lie superalgebra glM|N +2 +3. +Quantum superalgebra Uq(glM|N) +3 +4. +Poincar´e–Birkhoff–Witt basis of Uq(glM|N) +5 +5. +Conclusions +14 +Acknowledgments +14 +References +14 +1. INTRODUCTION +The functional relations are an effective method for investigation of quantum inte- +grable systems. To derive them it is convenient to use the quantum algebraic approach. +Previously, what we call quantum algebra was usually called quantum group. In fact, +this object is an associative algebra, which in a sense is a deformation of the universal +enveloping algebra of a Lie algebra. Nowadays, the term quantum algebra is more com- +monly used, and we adhere to this terminology. The general notion of a quantum algebra +Uq(g), used in the present paper, was proposed by Drinfeld and Jimbo [1, 2] for the case +when g is a Kac–Moody algebra with a symmetrizable generalized Cartan matrix. +The derivation of the functional relations based on the quantum algebraic approach +was given in the papers [3, 4, 5, 6, 7] for the loop Lie algebra g = L(sl2), in the papers +[8, 9, 7] for g = L(sl3), and in the paper [10] we gave the derivation for g = L(slM) with +an arbitrary M.1 The derivation of the functional relations given in the papers [7, 10] +is based on the results of the papers [12, 13, 14]. In the paper [12], using the commuta- +tion relations for the Poincar´e–Birkhoff–Witt generators of the quantum algebra Uq(glM) +presented in the paper [15], we found their action in the Verma Uq(glM)-module. Using +some limiting procedure, we found a set of q-oscillator modules over the positive Borel +subalgebra of Uq(glM). This modules, via Jimbo’s homomorphism were used to construct +the corresponding modules over the positive Borel subalgebra of Uq(L(slM)), which are +used to construct Q-operators.2 Finally, we derived the corresponding functional rela- +tions in the paper [10]. Here, to analyze the tensor products of the q-oscillator modules, +we used their ℓ-weights found in the papers [13, 14]. +1See also the paper [11], where some functional relations for g = L(slM) were presented without +derivation. +2For the terminology used for integrability objects, we refer to the papers [5, 7, 10]. +1 + +2 +A. V. RAZUMOV +By generalizing the defining relations of quantum algebra appropriately, one arrives +at quantum algebras associated with the Lie superalgebras [16]. It would interesting to +generalize the procedure of constructing the functional relations to the case of quantum +superalgebras.3 It seems that the right choice is to start with the quantum superalgebra +Uq(L(slM|N)). Here the very first step should be derivation of the commutation relations +for the Poincar´e–Birkhoff–Witt generators of the quantum algebra Uq(glM|N). Actually, +the commutation relations for this case already were presented in the papers [18, 19, 20] +without proof. There is some disagreement between these papers. This fact prompted us +to rederive the results of the papers [18, 19, 20]. +The structure of the paper is as follows. In section 2 we remind the necessary facts on +the Lie superalgebra glM|N. In section 3 we define the quantum superalgebra Uq(glM|N). +The detailed proof of the commutation relation is given in section 4. +We fix the deformation parameter ¯h in such a way that q = exp(¯h) is not a root of unity +and assume that +qν = exp(¯hν) +for any ν ∈ C. We define q-numbers by the equation +[ν]q = qν − q−ν +q − q−1 , +ν ∈ C +2. LIE SUPERALGEBRA glM|N +We fix two positive integers M and N such that M, N ≥ 1 and M ̸= N, and denote by +CM|N the superspace4 formed by (M + N)-tuples of complex numbers with the following +grading. An element of CM|N is even if its last N components are zero, and odd if its first +M components are zero. For simplicity, we denote the Lie superalgebra gl(CM|N) as +glM|N. We denote by vi, i = 1, . . . , M + N, the elements of the standard basis of CM|N. By +definition, +[vi] = 0, +i = 1, . . . , M, +[vi] = 1, +i = M + 1, . . . , N. +It is convenient to use the notation +[i] = [vi], +i = 1, . . . , M + N. +The elements Eij ∈ glM|N, i, j = 1, . . . , M + N, defined by the equation +Eijvk = viδjk, +form a basis of the Lie superalgebra glM|N. It is clear that the matrices of Eij with respect +to the standard basis of CM|N are the usual matrix units, and we have +EijEkl = δjkEil. +It is also evident that +[Eij] = [i] + [j]. +As the Cartan subalgebra k of the Lie superalgebra glM|N we take the subalgebra span- +ned by the elements Ki = Eii, i = 1, . . . , M + N, which form its basis. +Denote by +(Ξi)i=1,...,M+N the dual basis of the space k∗. For X = ∑M+N +i=1 +ciKi ∈ k we have +[X, Eij] = (ci − cj) Eij = ⟨Ξi − Ξj, X⟩ Eij. +Hence, Eij, i ̸= j, is a root vector corresponding to the root Ξi − Ξj and the root system of +glM|N is the set +∆ = {Ξi − Ξj | i, j = 1, . . . , M + N, i ̸= j}. +3The first results in this direction were obtained in the paper [17]. +4See appendix A of the paper [21] for a minimal set of definitions and notation. + +ON POINCAR´E–BIRKHOFF–WITT BASIS OF QUANTUM GENERAL LINEAR SUPERALGEBRA +3 +We choose as the system of simple roots the set +Π = {Ξi − Ξi+1 | i = 1, . . . , M + N − 1}, +then the system of positive roots corresponding to Π is +∆+ = {αij = Ξi − Ξj | 1 ≤ i < j ≤ M + N}. +Certainly, the corresponding system of negative roots is ∆− = −∆+. Denoting +αi = αi, i+1 = Ξi − Ξi+1, +i = 1, . . . , M + N − 1, +we obtain +αij = +j−1 +∑ +k=1 +αk, +1 ≤ i < j ≤ M + N. +We define a strict partial order ≺ on k∗ as follows. Given α, β ∈ k∗, we assume that β ≺ α +if and only if α − β is the sum of positive roots. +Define a nondegenerate symmetric bilinear form (· | ·) on k∗ by the equation +(Ξi | Ξj) = (−1)[i]δij = diδij, +where +di = (−1)[i]. +We see that +(αij | αmn) = diδim − djδjm − diδin + djδjn, +Below we often use the relations +(αij | αjn) = −dj, +(αij | αmi) = −di, +(2.1) +(αij | αin) = di, +j ̸= n, +(αij | αij) = di + dj, +(αij | αmj) = dj, +i ̸= m. +(2.2) +In fact, these are all nonzero cases. +3. QUANTUM SUPERALGEBRA Uq(glM|N) +We define the quantum superalgebra Uq(glM|N) as a unital associative C-superalgebra +generated by the elements5 +Ei, +Fi, +i = 1, . . . , M + N − 1, +qX, +X ∈ k, +which obey the corresponding defining relations. The Z2-grading of the quantum super- +algebra Uq(glM|N) is defined on generators as +[qX] = 0, +[Ei] = [Fi] = +� +0, +i ̸= M, +1, +i = M. +Before giving the explicit form of the defining relations, introduce the notion of the +q-supercommutator. The abelian group +Q = +M+N−1 +� +i=1 +Z αi. +is called the root lattice of the Lie superalgebra glM|N. Assuming that +qX ∈ Uq(glM|N)0, +Ei ∈ Uq(glM|N)αi, +Fi ∈ Uq(glM|N)−αi, +5We use capital letters to distinguish between generators of the quantum superalgebra Uq(glM|N) and +the quantum superalgebra Uq(L(slM|N)). + +4 +A. V. RAZUMOV +we endow Uq(glM|N) with a Q-grading. Now, for any elements X ∈ Uq(glM|N)α and +Y ∈ Uq(glM|N)β we define the q-supercommutator by the equation +[[X, Y]] = XY − (−1)[X][Y]q−(α|β)YX = XY − (−1)[α][β]q−(α|β)YX +if α, β ≻ 0, by the equation +[[X, Y]] = XY − (−1)[X][Y]q(α|β)YX = XY − (−1)[α][β]q(α|β)XY +if α, β ≺ 0, and by the equation +[[X, Y]] = XY − (−1)[X][Y]YX = XY − (−1)[α][β]YX +if α ≺ 0 and β ≻ 0, or α ≻ 0 and β ≺ 0. +The defining relations of the quantum superalgebra Uq(glM|N) have the form [16] +q0 = 1, +qX1qX2 = qX1+X2, +(3.1) +qXEiq−X = q⟨αi, X⟩Ei, +qXFiq−X = q−⟨αi, X⟩Fi, +(3.2) +[[Ei, Fj]] = δij +qdiKi−di+1Ki+1 − q−diKi+di+1Ki+1 +qi − q−1 +i +, +(3.3) +where i, j = 1, . . . , M + N − 1. Here and below we use the notation +qi = qdi = q(−1)[i]. +It is useful to have in mind that +[2]qi = qi + q−1 +i += q + q−1 = [2]q +and +(qi − q−1 +i +) = di(q − q−1) = (−1)[i](q − q−1). +(3.4) +There are also the following Serre relations +[[Ei, Ej]] = 0, +[[Fj, Fi]] = 0, +(αi | αj) = 0, +(3.5) +[[[[Ei−1, Ei]], Ei]] = 0, +[[Fi, [[ Fi, Fi−1]]]] = 0, +(αi | αi) ̸= 0, +(3.6) +[[Ei, [[Ei, Ei+1]]]] = 0, +[[[[Fi+1, Fi]], Fi]] = 0, +(αi | αi) ̸= 0, +(3.7) +[[[[[[EM−1, EM]], EM+1]], EM]] = 0, +[[FM, [[FM+1, [[FM, FM−1]]]]]] = 0. +(3.8) +Let us rewrite the defining relations (3.5)–(3.7) in a more familiar form. The relations +(3.5) are equivalent to the equations +EiEj = EjEi, +FjFi = FiFj. +|i − j| > 1, +(3.9) +E2 +M = 0, +F2 +M = 0, +(3.10) +and the relations (3.6)–(3.7) are equivalent to +Ei−1E2 +i − [2]qEiEi−1Ei + E2 +i Ei−1 = 0, +F2 +i Fi−1 − [2]qFiFi−1Fi + Fi−1F2 +i = 0, +(3.11) +E2 +i Ei+1 − [2]qEiEi+1Ei + Ei+1E2 +i = 0, +Fi+1F2 +i − [2]qFiFi+1Fi + F2 +i Fi+1 = 0. +(3.12) +where i ̸= M. + +ON POINCAR´E–BIRKHOFF–WITT BASIS OF QUANTUM GENERAL LINEAR SUPERALGEBRA +5 +4. POINCAR´E–BIRKHOFF–WITT BASIS OF Uq(glM|N) +An element a of Uq(glM|N) is called a root vector corresponding to a root γ of glM|N +if a ∈ Uq(glM|N)γ. In particular, Ei and Fi are root vectors corresponding to the roots αi +and −αi. It is possible to construct linearly independent root vectors corresponding to all +roots of glM|N. To this end, being inspired by M. Jimbo [22], we introduce elements Eij +and Fij, 1 ≤ i < j ≤ M + N, with the help of the relations +Ei, i+1 = Ei, +Fi, i+1 = Fi, +(4.1) +Ei, j+1 = [[Eij, Ej, j+1]], +Fi, j+1 = [[Fj, j+1, Fij]], +j > i. +(4.2) +Explicitly, the last two equations look as +Ei, j+1 = Eij Ej, j+1 − qj Ej, j+1 Eij, +Fi,j+1 = Fj, j+1 Fij − q−1 +j +Fij Fj, j+1. +Note that we have +[Eij] = [i] + [j], +in particular, +[Ei] = [Fi] = [i] + [i + 1]. +We also see that +[Eij] = 0 +if and only if j < M or i > M, +[Eij] = 1 +if and only if i ≤ M < j. +It is clear that the vectors Eij and Fij correspond to the roots αij and −αij respectively. +These vectors are linearly independent, and together with the elements qX, X ∈ k, are +called Cartan–Weyl generators of Uq(glM|N). +It appears that the ordered monomials +constructed from the Cartan–Weyl generators form a Poincar´e–Birkhoff–Witt basis of +Uq(glM|N). In this paper we choose the following total order for monomials. First, we +endow the set of the pairs (i, j), where 1 ≤ i < j ≤ M + N, with the lexicographical +order. It means that (i, j) ≺ (m, n) if i < m, or if i = m and j < n.6 Now we say that a +monomial is ordered if it has the form +Fi1j1 · · · Fir jr qX Em1n1 . . . Emsns, +(4.3) +where (i1, j1) ≼ · · · ≼ (ir, jr), (m1, n1) ≼ · · · ≼ (ms, ns) and X is an arbitrary element +of k. In the present paper we only show that any monomial can be written as a finite sum +of monomials of the form (4.3). To prove that they form a basis of Uq(glM|N) one can use +arguments similar to those used in the paper [15] for the the case of the quantum algebra +Uq(glM). +We present the relations necessary for ordering as a sequence of propositions. First +consider the ordering of qX with Eij and Fij. +Proposition 4.1. For any 1 ≤ j < n ≤ M + N and i = 1, . . . , M + N, we have +qνKiEjn q−νKi = (qνδij + q−νδin)Ejn, +qνKiFjn q−νKi = (q−νδij + qνδik)Ejn. +(4.4) +Proof. It is evident that +⟨αj, Ki⟩ = δij − δi, j+1, +and it follows from the defining relation (3.2) that +qνKi +i +Ej q−νKi +i += qν(δij−δi, j+1)Ej, +qνKi +i +Fj q−νKi +i += q−ν(δij−δi, j+1)Fj. +6Note that if we define an ordering of positive roots so that αij ≺ αmn if (i, j) ≺ (m, n) we get a normal +ordering in the sense of [23, 24], see also [25]. + +6 +A. V. RAZUMOV +which follow from (3.2). +qνKiEjnq−νKi = qν ∑n−1 +m=j(δim−δi, m+1)Ejn = qν(δij−δin)Ejn = (qνδij + q−νδin)Ejn +Thus, the first equation of (4.4) is true. The proof of the second equations is similar. +□ +Now we consider the ordering of the root vectors Eij, 1 ≤ i < j ≤ M + N, and Fij, +1 ≤ i < j ≤ M + N. We divide the set of pairs ((i, j), (m, n)), where 1 ≤ i < j ≤ +M + N, 1 ≤ m < n ≤ M + N and (i, j) ≺ (m, n), into six branches Ca, a = I, . . . , VI. +The conditions defining the branches are given in table 1. In the same table we put the +([i] + [j])([m] + [n]) +(αij | αmn) +CI +i = m < j < n +[i] + [j] +(−1)[i] +CII +i < m < n < j +[m] + [n] +0 +CIII +i < m < j = n +[m] + [j] +(−1)[j] +CIV +i < m < j < n +[m] + [j] +0 +CV +i < j = m < n +0 +−(−1)[j] +CVI +i < j < m < n +0 +0 +TABLE 1. +information necessary to construct the corresponding q-supercommutators. To fill table 1 +it is sufficient to use the relations +a2 = a, +a + a = 0, +a ∈ Z2, +and equations (2.1) and (2.2). +Proposition 4.2. For any ((i, j), (m, n)) ∈ CVI one has +[[Eij, Emn]] = EijEmn − EmnEij = 0, +[[Fmn, Fij]] = FmnFij − FijFmn = 0. +(4.5) +Proof. The statement of the proposition is a direct consequence of the Serre relations (3.9). +□ +Proposition 4.3. For any ((i, j), (m, n)) ∈ CV one has +[[Eij, Emn]] = EijEjn − qjEjnEij = Ein, +[[Fmn, Fij]] = FjnFij − q−1 +j +FijFjn = Fin. +Proof. The proposition can be proved by induction over n. For n = j + 1 we have just +the definition (4.2). Assume that the statement of the proposition is valid for some given +n > j, then we have +EijEjn − qjEjnEij = Ein. +Using this equation and proposition 4.2, we get +[[Eij, Ej, n+1]] = EijEj, n+1 − qjEj, n+1Eij = Eij(EjnEn, n+1 − qnEn, n+1Ejn) +− qj(EjnEn, n+1 − qnEn, n+1Ejn)Eij = (EijEjn − qjEjnEij)En, n+1 +− qnEn, n+1(EijEjn − qjEjnEij) = [[Ein, En,n+1]] = Ei, n+1. +Thus, the first equation of the proposition is true. The second one can be proved in the +same way. +□ + +ON POINCAR´E–BIRKHOFF–WITT BASIS OF QUANTUM GENERAL LINEAR SUPERALGEBRA +7 +Proposition 4.4. For any ((i, j), (m, n)) ∈ CII one has +Eij, Emn]] = EijEmn − (−1)[m]+[n]EmnEij = 0, +[[Fmn, Fij]] = FmnFij − (−1)[m]+[n]FijFmn = 0. +Proof. Let us first prove that +[[Em−1, m+2, Em, m+1]] = 0 +(4.6) +for any 2 ≤ m ≤ M + N − 2. It is easy to see that for m = M, equation (4.6) is just the +Serre relation (3.8). If m ̸= M, we have7 +[[Em−1, m+2, Em, m+1]] = [[[[[[Em−1, m, Em, m+1]], Em+1, m+2]], Em, m+1]] += Em−1EmEm+1Em − qmEm−1Em+1E2 +m − qmEmEm+1Em−1Em ++ q2 +mEm+1EmEm−1Em − EmEm−1EmEm+1 + qmEmEm−1Em+1Em ++ qmE2 +mEm+1Em−1 − q2 +mEmEm+1EmEm−1. +Using the first equations of (3.11) and (3.12), we obtain +[[Em−1, m+2, Em, m+1]] += [2]−1 +q Em+1(E2 +mEm−1 + Em+1E2 +m) − qmEm−1Em+1E2 +m − qmEmEm−1Em+1Em ++ [2]−1 +q q2 +mEm+1(E2 +mEm−1 + Em−1E2 +m) − [2]−1 +q (E2 +mEm−1 + Em−1E2 +m)Em+1 ++ qmEmEm−1Em+1Em + qmE2 +mEm+1Em−1 − [2]−1 +q q2 +m(E2 +mEm+1 + Em+1E2 +m)Em−1. +The Serre relations (3.5) give Em−1Em+1 = Em+1Em−1, and we see that equation (4.6) is +true for any admissible value of m. +Assume that +[[Eij, Em, m+1]] = 0 +(4.7) +for some 2 ≤ i < m < j − 1 ≤ M + N − 1. We have +[[Ei−1, j, Em, m+1]] = [[Ei−1,iEij − qiEijEi−1, i, Em, m+1]] += Ei−1, i[[Eij, Em, m+1]] − qi[[Eij, Em, m+1]]Ei−1, i = 0. +If equation (4.7) is valid for some 1 ≤ i < m < j − 1 ≤ M + N − 2, then +[[Ei, j+1, Em, m+1]] = [[EijEj, j+1 − qjEj, j+1Eij, Em, m+1]] += [[Eij, Em, m+1]]Ej, j+1 − qjEj, j+1[[Eij, Em, m+1]] = 0. +Thus, equation (4.7) is valid for any admissible i, j and m. +Finally, assume that the equation +[[Eij, Emn]] = EijEmn − (−1)[m]+[n]EmnEij = 0 +is valid for some 1 ≤ i < m < n < j − 1 ≤ M + N − 1, then we have +[[Eij, Em, n+1]] = [[Eij, EmnEn, n+1 − qnEn, n+1Emn]] += [[Eij, Emn]]En, n+1 − (−1)[n]+[n+1]qnEn, n+1[[Eij, Emn]] = 0. +Now, it is clear that the first equation of the proposition is valid. The second equation +of the proposition can be proved in a similar way. +□ +7It is clear that either m < M or m > M, so that qm = qm+1. + +8 +A. V. RAZUMOV +Proposition 4.5. For any ((i, j), (m, n)) ∈ CI one has +[[Eij, Emn]] = EijEin − (−1)[i]+[j]q−1 +i +EinEij = 0, +(4.8) +[[Fmn, Fij]] = FinFij − (−1)[i]+[j]qiFijFin = 0. +(4.9) +For any ((i, j), (m, n)) ∈ CIII one has +[[Eij, Emn]] = EijEmj − (−1)[m]+[j]q−1 +j +EmjEij = 0, +(4.10) +[[Fmn, Fij]] = FmjFij − (−1)[m]+[j]qjFijFmj = 0. +(4.11) +Proof. Let us consider the case ((i, j), (m, n)) ∈ CI and prove equation (4.8). First we +demonstrate that +[[Ei, i+1, Ei, i+2]] = 0 +(4.12) +for any 1 ≤ i ≤ M + N − 2. We have +[[Ei, i+1, Ei, i+2]] = [[Ei, i+1, [[Ei, i+1, Ei+1, i+2]]]] = [[Ei, [[Ei, Ei+1]]]]. +Hence, for i ̸= M, the equation (4.12) is equivalent to the first of the Serre relations (3.7). +For i = M we obtain +[[Ei, i+1, Ei, i+2]] = [[EM, M+1, [[EM, M+1, EM+1, M+2]]]] += [[EM, EMEM+1 − qM+1EM+1EM]] = −(qM+1 − q−1 +M )EMEM+1EM = 0. +Thus, equation (4.12) is valid for any 1 ≤ i ≤ M + N − 2. Assume that +[[Ei, i+1, Ein]] = Ei, i+1Ein − (−1)[i]+[i+1]q−1 +i +EinEi, i+1 = 0 +(4.13) +for some 1 ≤ i < n − 1 ≤ M + N − 2. Using equation (4.13), we obtain +[[Ei, i+1, Ei, n+1]] = [[Ei, i+1, EinEn, n+1 − qnEn, n+1Ein]] = Ei, i+1EinEn, n+1 +− qnEi, i+1En, n+1Ein − (−1)[i]+[i+1]q−1 +i +(EinEn, n+1Ei, i+1 − qnEn, n+1EinEi, i+1) = 0. +we obtain that [[Ei, i+1, Ei, n+1]] = 0. It follows that +[[Ei, i+1, Ein]] = 0 +for any 1 ≤ i < n − 1 ≤ M + N − 1. Now, assume that +[[Eij, Ein]] = EijEin − (−1)[i]+[j]q−1 +i +EinEij = 0 +for some 1 ≤ i < j + 1 < n ≤ M + N. Using proposition 4.4, we get +[[Ei, j+1, Ein]] = [[EijEj, j+1 − qjEj, j+1Eij, Ein]] += (−1)[j]+[j+1][[Eij, Ein]]Ej, j+1 − qjEj, j+1[[Eij, Ein]] = 0. +Thus, for ((i, j), (m, n)) ∈ CI, equation (4.8) is true. Equation (4.9) can be proved in the +same way. In the case when ((i, j), (m, n)) ∈ CIII, one can prove equations (4.10) and +(4.11) in a similar way. +□ +It follows from the above proposition that if ((i, j), (m, n)) ∈ CI, then +[[Eij, [[Eij, Ejn]]]] = 0, +[[[[Fjn, Fij]], Fij]] = 0, +(4.14) +and if ((i, j), (m, n)) ∈ CIII, then +[[[[Eim, Emn]], Emn]] = 0, +[[Fmn, [[Fmn, Fim]]]] = 0. +(4.15) +These relations are a generalization of the Serre relations (3.6) and (3.7). + +ON POINCAR´E–BIRKHOFF–WITT BASIS OF QUANTUM GENERAL LINEAR SUPERALGEBRA +9 +Note that the quantum supergroup Uq(glM|N) has two natural subgroups isomorphic +to Uq(glM) and Uq(glN). The former is generated by Ei, Fi, i = 1, . . . , M − 1, and qX, where +X belongs to the linear span of the elements Ki, i = 1, . . . M, and the latter is generated +by Ei, Fi, i = M + 1, . . . , M + N − 1, and qX, where X belongs to the linear span of the +elements Ki, i = M + 1, . . . M + N. It is clear that [i] + [j] = 0 iff Eij belongs to one of +these two subgroups. Each of them has no zero divisors, see the paper [15]. Hence, for +any element Eij belonging to them one has E2 +ij ̸= 0. In other words, if [i] + [j] = 1 then +E2 +ij ̸= 0. +Proposition 4.6. For all 1 ≤ i < j ≤ M + N such that [i] + [j] = 1 one has +1 +2 [[Eij, Eij]] = E2 +ij = 0. +(4.16) +Proof. In fact, we should demonstrate that if i ≤ M < j, then +E2 +ij = 0. +(4.17) +First, we show that +E2 +Mj = 0 +(4.18) +for all j > M. It is certainly the case, at least for j = M + 1. Using the fact that qj = q−1 +for any j > M, we obtain +E2 +M, j+1 = (EMjEj, j+1 − qjEj, j+1EMj)2 += EMjEj, j+1EMjEj, j+1 − q−1EMjE2 +j, j+1EMj + q−2Ej, j+1EMjEj, j+1EMj. +(4.19) +It follows from the first relation of (4.15) that +[[[[EMj, Ej, j+1]], Ej, j+1]] = 0, +or, in a more explicit form, +EMjE2 +j, j+1 − [2]qEj, j+1EMjEj, j+1 + E2 +j, j+1EMj = 0, +Multiplying this equation from the left and from the right by EMj, we obtain +−[2]qEMjEj, j+1EMjEj, j+1 + EMjE2 +j, j+1EMj = 0, +(4.20) +EMjE2 +j, j+1EMj − [2]qEj, j+1EMjEj, j+1EMj = 0. +(4.21) +It follows that +EMjEj, j+1EMjEj, j+1 = Ej, j+1EMjEj, j+1EMj. +Using this equation in (4.19), we get +E2 +M, j+1 = −q−1( − [2]qEMjEj, j+1EMjEj, j+1 + EMjE2 +j, j+1EMj). +Now equation (4.21) implies that (4.18) for all M < j ≤ M + N. +Further, we assume that (4.16) is true for some 1 < i < M and M < j ≤ M + N, then +we have +E2 +i−1, j = (Ei−1, iEij − qEijEi−1, i)2 += Ei−1, iEijEi−1, iEij − qEijE2 +i−1, iEij + q2EijEi−1, iEijEi−1, i. +(4.22) +Here we take into account that di = 1 for any i < M. It follows from the first relation of +(4.14) that +[[Ei−1, i, [[Ei−1, i, Eij]]]] = 0, + +10 +A. V. RAZUMOV +or, in a more explicit form, +E2 +i−1, iEij − [2]q Ei−1, iEijEi−1, i + EijE2 +i−1, i = 0. +Multiplying this equation from the left and from the right by Eij, we obtain +EijE2 +i−1, iEij − [2]q EijEi−1, iEijEi−1, i = 0, +(4.23) +−[2]q Ei−1, iEijEi−1, iEij + EijE2 +i−1, iEij = 0. +(4.24) +It follows that +EijEi−1, iEijEi−1, i = Ei−1, iEijEi−1, iEij. +Using this equation in (4.22), we come to +E2 +i−1, j = −q(−[2]q Ei−1, iEijEi−1, iEij + EijE2 +i−1, iEij). +Now equation (4.24) gives +E2 +i−1, j = 0. +Thus, we see that the statement of the proposition is always true. +□ +Proposition 4.7. For any ((i, j), (m, n)) ∈ CIV one has +[[Eij, Emn]] = EijEmn − (−1)[m]+[j]EmnEij = −(qm − q−1 +m )EmjEin, +(4.25) +[[Fmn, Fij]] = FmnFij − (−1)[m]+[j]FijFmn = (qm − q−1 +m )FinFmj. +(4.26) +Proof. Using proposition 4.3, we get +[[Eij, Emn]] = (EimEmj − qmEmjEim)Emn − (−1)[m]+[j]Emn(EimEmj − qmEmjEim) += EimEmjEmn − (−1)[m]+[j]EmnEimEmj +− qm(EmjEimEmn − (−1)[m]+[j]EmnEmjEim). +(4.27) +Proposition 4.5 implies +[[Emj, Emn]] = EmjEmn − (−1)[m]+[j]q−1 +m EmnEmj = 0. +Hence, we have +EmjEmn = (−1)[m]+[j]q−1 +m EmnEmj, +EmnEmj = (−1)[m]+[j]qmEmjEmn +Using these equations in (4.27), we obtain +[[Eij, Emn]] = (−1)[m]+[j]q−1 +m (EimEmn − qmEmnEim)Emj +− qmEmj(EimEmn − qmEmnEim) = (−1)[m]+[j]q−1 +m EinEmj − qmEmjEin. +Finally, it follows from proposition 4.4 that +EinEmj = (−1)[m]+[j]EmjEin, +therefore, +[[Eij, Emn]] = −(qm − q−1 +m )EmjEin. +Thus, equation (4.25) is true. In the same way one can prove equation (4.26). +□ +Proposition 4.8. For any ((i, j), (m, n)) ∈ CV one has +[[Eij, Fmn]] = EijFjn − FjnEij = 0, +[[Emn, Fij]] = EjnFij − FijEjn = 0. +For any ((i, j), (m, n)) ∈ CVI one has +[[Eij, Fmn]] = EijFmn − FmnEij = 0, +[[Emn, Fij]] = EmnFij − FijEmn = 0. + +ON POINCAR´E–BIRKHOFF–WITT BASIS OF QUANTUM GENERAL LINEAR SUPERALGEBRA +11 +Proof. The statement of the proposition is a direct consequence of the defining relation +(3.3). +□ +Proposition 4.9. For any ((i, j), (m, n)) ∈ CII one has +[[Eij, Fmn]] = EijFmn − (−1)[m]+[n]FmnEij = 0, +(4.28) +[[Emn, Fij]] = EmnFij − (−1)[m]+[n]FijEmn = 0. +(4.29) +Proof. Let 1 < k ≤ M + N − 2. Prove equation (4.28) for i = k − 1, m = k, n = k + 1 and +j = k + 2. We have +Ek−1, k+2 = Ek−1EkEk+1 − qk+1Ek−1Ek+1Ek − qkEk+1Ek+1Ek−1 + qkqk+1Ek+1EkEk−1. +It follows that +[[Eij, Fmn]] = Ek−1[[Ek, Fk]]Ek+1 − qk+1Ek−1Ek+1[[Ek, Fk]] +− qk[[Ek, Fk]]Ek+1Ek−1 + qkqk+1Ek+1[[Ek, Fk]]Ek−1. +Using the defining relation (3.3) and proposition 4.1, we obtain +[[Ek−1, k+2, Fk, k+1]] = 0. +Now, let 1 < i < k < j − 1 ≤ M + N − 1 and +[[Eij, Fk, k+1]] = EijFk, k+1 − (−1)[k]+[k+1]Fk, k+1Eij = 0. +(4.30) +We have +[[Ei−1, j, Fk, k+1]] = [[Ei−1, iEij − qiEijEi−1, i, Fk, k+1]] = Ei−1,iEijFk, k+1 +− (−1)[k]+[k+1]Fk, k+1Ei−1, iEij − qj(EijEi−1, iFk, k+1 − (−1)[k]+[k+1]Fk, k+1EijEi−1, i). +It follows from proposition 4.8 that +[[Ei−1, j, Fk, k+1]] = 0. +Further, let 1 ≤ i < k < j − 1 ≤ M + N − 2 and equation (4.30) is true. We obtain +[[Ei, j+1, Fk, k+1]] = [[EijEj, j+1 − qjEj, j+1Eij, Fk, k+1]] = EijEj, j+1Fk, k+1 +− (−1)[k]+[k+1]Fk, k+1EijEj, j+1 − (Ej,j+1EijFk, k+1 − (−1)[k]+[k+1]qjFk, k+1Ej, j+1Eij), +and proposition 4.8 implies that +[[Ei, j+1, Fk, k+1]] = 0. +Hence, we have +[[Eij, Fk, k+1]] = 0 +for all possible i, j and k. Assume now that for some 1 ≤ i < m < n < M + N − 1 we +have +[[Eij, Fmn]] = EijEmn − (−1)[m]+[n]EmnEij = 0. +Then, we obtain +[[Eij, Fm, n+1]] = [[Eij, Fn, n+1Fmn − q−1 +n FmnFn, n+1]] = EijFn, n+1Fmn +− (−1)[m]+[n+1]Fn, n+1FnmEij − q−1 +n (EijFmnFn, n+1 − (−1)[m]+[n+1]FmnFn, n+1Eij) = 0. +Thus, equation (4.28) is true. In the same way one can prove equation (4.29). +□ + +12 +A. V. RAZUMOV +Proposition 4.10. For any ((i, j), (m, n)) ∈ CI one has +[[Eij, Fin]] = EijFin − (−1)[i]+[j]FinEij = −(−1)[i]+[j]q−diKi+djKjFjn, +(4.31) +[[Ein, Fij]] = EinFij − (−1)[i]+[j]FijEin = −(−1)[i]+[j]Ejn qdiKi−djKj. +(4.32) +For any ((i, j), (m, n)) ∈ CIII one has +[[Eij, Fmn]] = EijFmj − (−1)[m]+[j]FmjEij = q−dmKm+djKjEim, +(4.33) +[[Emn, Fij]] = EmjFij − (−1)[m]+[j]FijEmj = FimqdmKm−djKj. +(4.34) +Proof. We first prove equation (4.31) for j = i + 1. Using proposition 4.3, we obtain +[[Ei, i+1, Fin]] = [[Ei, i+1, Fi+1, nFi, i+1 − q−1 +i+1Fi, i+1Fi+1, n]] += Ei, i+1Fi+1, nFi, i+1 − (−1)[i]+[i+1]Fi+1, nFi, i+1Ei, i+1 +− q−1 +i+1(Ei, i+1Fi, i+1Fi+1, n − (−1)[i]+[i+1]Fi, i+1Fi+1, nEi, i+1). +Further, proposition 4.8 gives +[[Ei, i+1, Fin]] = Fi+1, n[[Ei, Fi]] − q−1 +i+1[[Ei, Fi]]Fi+1, n += (qi − q−1 +i +)−1(Fi+1, n(qdiKi−di+1Ki+1 − q−diKi+di+1Ki+1) +− q−1 +i+1(qdiKi−di+1Ki+1 − q−diKi+di+1Ki+1)Fi+1, n). +and, using proposition 4.1, we come to +[[Ei, i+1, Fin]] = −(qi+1 − q−1 +i+1)(qi − q−1 +i +)−1q−diKi+di+1Ki+1Fi+1, n. +Finally, it follows from (3.4) that +[[Ei, i+1, Fin]] = −(−1)−[i]+[i+1]q−diKi+di+1Ki+1Fi+1, n. +Now, let 1 ≤ i < j < n − 1 ≤ M + N − 1 and equation +[[Eij, Fin]] = −(−1)[i]+[j]q−diKi+djKjFjn +(4.35) +be true. Using proposition 4.3, we obtain +[[Ei, j+1, Fin]] = [[EijEj, j+1 − qjEj, j+1Eij, Fin]] += EijEj, j+1Fin − (−1)[i]+[j+1]FinEijEj, j+1 +− qj(Ej, j+1EijFin − (−1)[i]+[j+1]FinEj, j+1Eij). +It follows from proposition 4.4, equation (4.35) and proposition 4.1 that +[[Ei, j+1, Fin]] = (−1)[j]+[j+1][[Eij, Fin]]Ej, j+1 − qjEj, j+1[[Eij, Fin]] += (−1)[i]+[j]q−diKi+djKj[[Ej, j+1, Fjn]] = −(−1)[i]+[j+1]q−diKi+dj+1Kj+1Fj+1, n. +We see that equation (4.31) is always true. In the same way one can prove equations +(4.32), (4.33) and (4.34). +□ +Proposition 4.11. For any 1 ≤ i < j ≤ M + N we have +[[Eij, Fij]] = EijFij − (−1)[i]+[j]FijEij = qdiKi−djKj − q−diKi+djKj +qi − q−1 +i +. + +ON POINCAR´E–BIRKHOFF–WITT BASIS OF QUANTUM GENERAL LINEAR SUPERALGEBRA +13 +Proof. The statement of the proposition is certainly true for j = i + 1. Let us consider the +case when j − i > 1. It follows from proposition 4.3 that +[[Eij, Fij]] = [[Eij, Fi+1, jFi, i+1 − q−1 +i+1Fi, i+1Fi+1, j]] += EijFi+1, jFi, i+1 − (−1)[i]+[j]Fi+1, jFi, i+1Eij +− q−1 +i+1(EijFi, i+1Fi+1, j − (−1)[i]+[j]Fi, i+1Fi+1, jEij). +(4.36) +Equation (4.32) implies +Fi, i+1Eij = (−1)[i]+[i+1]EijFi, i+1 + Ei+1, jqdiKi−di+1Ki+1, +EijFi, i+1 = (−1)[i]+[i+1]Fi, i+1Eij − (−1)[i]+[i+1]Ei+1, jqdiKi−di+1Ki+1. +Using these equations in (4.36), we obtain +[[Eij, Fij]] = [[Eij, Fi+1, j]]Fi, i+1 + q−1 +i+1(−1)[i]+[i+1]Fi, i+1[[Eij, Fi+1, j]] ++ q−1 +i+1(−1)[i]+[i+1]Ei+1, jqdiKi−di+1Ki+1Fi+1, j +− (−1)[i]+[j]Fi+1, jEi+1,jqdiKi−di+1Ki+1. +(4.37) +We have +[[Eij, Fi+1, j]] = q−di+1Ki+1+djKjEi, i+1, +see proposition 4.11. Taking this equation, and proposition 4.1 and equation (3.4) into +account, we come to the equation +[[Eij, Fij]] = q−di+1Ki+1+djKj[[Ei, i+1, Fi, i+1]] ++ (−1)[i]+[i+1][[Ei+1, j, Fi+1, j]]qdiKi−di+1Ki+1 = qdiKi−djKj − q−diKi+djKj +qi − q−1 +i +. +That was to be proved. +□ +Proposition 4.12. For any ((i, j), (m, n)) ∈ CIV one has +[[Eij, Fmn]] = EijFmn − (−1)[m]+[j]FmnEij = −(qj − q−1 +j )q−dmKm+djKjFjnEim, +[[Emn, Fij]] = EmnFij − (−1)[m]+[j]FijEmn = (qj − q−1 +j )FimEjnqdmKm−djKj. +Proof. It follows from propositions 4.3 and 4.9 that +[[Eij, Fmn]] = [[EimEmj − qmEmjEim, Fmn]] += EimEmjFmn − (−1)[m]+[j]FmnEimEmj − qm(EmjEimFmn − (−1)[m]+[j]FmnEmjEim) += Eim[[Emj, Fmn]] − qm[[Emj, Fmn]]Eim. +Now, using equation (4.31), proposition 4.1 and proposition 4.8, we get +[[Eij, Fmn]] = (−1)[m]+[j](qm − q−1 +m )q−dmKm+djKjFjnEim. +Now, taking into account equations (3.4), we see that the first equation of the proposition +is true. The second equation can be proved similarly. +□ +One can get convinced that the propositions 4.1–4.12 allow us to reduce any monomial +on the Poincar´e–Birkhoff–Witt generators to the ordered form (4.3). + +14 +A. V. RAZUMOV +5. CONCLUSIONS +We have derived the commutation relations for the Poincar´e–Birkhoff–Witt generators +of the quantum algebra Uq(glM|N). Our results do not fully coincide with the results of +the papers [18, 19, 20]. We are planning to use the obtained relations for constructing of +q-oscillator representations of the positive Borel subalgebra of the quantum superalgebra +Uq(glM|N). +Acknowledgments. This work was supported in part by the RFBR grant # 20-51-12005. +REFERENCES +[1] V. G. Drinfeld, Hopf algebras and the quantum Yang-–Baxter equation (in Russian), Dokl. Akad. Nauk +SSSR 283 (1985), 1060–1064. +[2] M. Jimbo, A q-difference analogue of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63–69. +[3] V. V. Bazhanov, S. L. Lukyanov, and A. B. Zamolodchikov, Integrable structure of conformal field theory +III. The Yang–Baxter relation, Commun. Math. Phys. 200 (1999), 297–324, arXiv:hep-th/9805008. +[4] H. Boos, F. G¨ohmann, A. Kl¨umper, Kh. S. Nirov, and A. V. 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Surveys 44 (1989), 257–258. +INSTITUTE FOR HIGH ENERGY PHYSICS, NRC “KURCHATOV INSTITUTE”, 142281 PROTVINO, MOS- +COW REGION, RUSSIA +Email address: Alexander.Razumov@ihep.ru + diff --git a/pNFAT4oBgHgl3EQfeR0E/content/tmp_files/load_file.txt b/pNFAT4oBgHgl3EQfeR0E/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d7ab45e2fb7d5f50dc93fd095cc583a84436ef5e --- /dev/null +++ b/pNFAT4oBgHgl3EQfeR0E/content/tmp_files/load_file.txt @@ -0,0 +1,674 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf,len=673 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='08574v1 [math-ph] 20 Jan 2023 ON POINCAR´E–BIRKHOFF–WITT BASIS OF QUANTUM GENERAL LINEAR SUPERALGEBRA ALEXANDER V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' RAZUMOV ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We give a detailed derivation of the commutation relations for the Poincar´e– Birkhoff–Witt generators of the quantum superalgebra Uq(glM|N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' CONTENTS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Lie superalgebra glM|N 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Quantum superalgebra Uq(glM|N) 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Poincar´e–Birkhoff–Witt basis of Uq(glM|N) 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Conclusions 14 Acknowledgments 14 References 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' INTRODUCTION The functional relations are an effective method for investigation of quantum inte- grable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' To derive them it is convenient to use the quantum algebraic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Previously, what we call quantum algebra was usually called quantum group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' In fact, this object is an associative algebra, which in a sense is a deformation of the universal enveloping algebra of a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Nowadays, the term quantum algebra is more com- monly used, and we adhere to this terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The general notion of a quantum algebra Uq(g), used in the present paper, was proposed by Drinfeld and Jimbo [1, 2] for the case when g is a Kac–Moody algebra with a symmetrizable generalized Cartan matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The derivation of the functional relations based on the quantum algebraic approach was given in the papers [3, 4, 5, 6, 7] for the loop Lie algebra g = L(sl2), in the papers [8, 9, 7] for g = L(sl3), and in the paper [10] we gave the derivation for g = L(slM) with an arbitrary M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='1 The derivation of the functional relations given in the papers [7, 10] is based on the results of the papers [12, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' In the paper [12], using the commuta- tion relations for the Poincar´e–Birkhoff–Witt generators of the quantum algebra Uq(glM) presented in the paper [15], we found their action in the Verma Uq(glM)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Using some limiting procedure, we found a set of q-oscillator modules over the positive Borel subalgebra of Uq(glM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' This modules, via Jimbo’s homomorphism were used to construct the corresponding modules over the positive Borel subalgebra of Uq(L(slM)), which are used to construct Q-operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='2 Finally, we derived the corresponding functional rela- tions in the paper [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Here, to analyze the tensor products of the q-oscillator modules, we used their ℓ-weights found in the papers [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' 1See also the paper [11], where some functional relations for g = L(slM) were presented without derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' 2For the terminology used for integrability objects, we refer to the papers [5, 7, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' 1 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' RAZUMOV By generalizing the defining relations of quantum algebra appropriately, one arrives at quantum algebras associated with the Lie superalgebras [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It would interesting to generalize the procedure of constructing the functional relations to the case of quantum superalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='3 It seems that the right choice is to start with the quantum superalgebra Uq(L(slM|N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Here the very first step should be derivation of the commutation relations for the Poincar´e–Birkhoff–Witt generators of the quantum algebra Uq(glM|N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Actually, the commutation relations for this case already were presented in the papers [18, 19, 20] without proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' There is some disagreement between these papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' This fact prompted us to rederive the results of the papers [18, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' In section 2 we remind the necessary facts on the Lie superalgebra glM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' In section 3 we define the quantum superalgebra Uq(glM|N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The detailed proof of the commutation relation is given in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We fix the deformation parameter ¯h in such a way that q = exp(¯h) is not a root of unity and assume that qν = exp(¯hν) for any ν ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We define q-numbers by the equation [ν]q = qν − q−ν q − q−1 , ν ∈ C 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' LIE SUPERALGEBRA glM|N We fix two positive integers M and N such that M, N ≥ 1 and M ̸= N, and denote by CM|N the superspace4 formed by (M + N)-tuples of complex numbers with the following grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' An element of CM|N is even if its last N components are zero, and odd if its first M components are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For simplicity, we denote the Lie superalgebra gl(CM|N) as glM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We denote by vi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' , M + N, the elements of the standard basis of CM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' By definition, [vi] = 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' , M, [vi] = 1, i = M + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It is convenient to use the notation [i] = [vi], i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' , M + N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The elements Eij ∈ glM|N, i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' , M + N, defined by the equation Eijvk = viδjk, form a basis of the Lie superalgebra glM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It is clear that the matrices of Eij with respect to the standard basis of CM|N are the usual matrix units, and we have EijEkl = δjkEil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It is also evident that [Eij] = [i] + [j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' As the Cartan subalgebra k of the Lie superalgebra glM|N we take the subalgebra span- ned by the elements Ki = Eii, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' , M + N, which form its basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Denote by (Ξi)i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=',M+N the dual basis of the space k∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For X = ∑M+N i=1 ciKi ∈ k we have [X, Eij] = (ci − cj) Eij = ⟨Ξi − Ξj, X⟩ Eij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Hence, Eij, i ̸= j, is a root vector corresponding to the root Ξi − Ξj and the root system of glM|N is the set ∆ = {Ξi − Ξj | i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' , M + N, i ̸= j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' 3The first results in this direction were obtained in the paper [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' 4See appendix A of the paper [21] for a minimal set of definitions and notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' ON POINCAR´E–BIRKHOFF–WITT BASIS OF QUANTUM GENERAL LINEAR SUPERALGEBRA 3 We choose as the system of simple roots the set Π = {Ξi − Ξi+1 | i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' , M + N − 1}, then the system of positive roots corresponding to Π is ∆+ = {αij = Ξi − Ξj | 1 ≤ i < j ≤ M + N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Certainly, the corresponding system of negative roots is ∆− = −∆+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Denoting αi = αi, i+1 = Ξi − Ξi+1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' , M + N − 1, we obtain αij = j−1 ∑ k=1 αk, 1 ≤ i < j ≤ M + N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We define a strict partial order ≺ on k∗ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Given α, β ∈ k∗, we assume that β ≺ α if and only if α − β is the sum of positive roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Define a nondegenerate symmetric bilinear form (· | ·) on k∗ by the equation (Ξi | Ξj) = (−1)[i]δij = diδij, where di = (−1)[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We see that (αij | αmn) = diδim − djδjm − diδin + djδjn, Below we often use the relations (αij | αjn) = −dj, (αij | αmi) = −di, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='1) (αij | αin) = di, j ̸= n, (αij | αij) = di + dj, (αij | αmj) = dj, i ̸= m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='2) In fact, these are all nonzero cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' QUANTUM SUPERALGEBRA Uq(glM|N) We define the quantum superalgebra Uq(glM|N) as a unital associative C-superalgebra generated by the elements5 Ei, Fi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' , M + N − 1, qX, X ∈ k, which obey the corresponding defining relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The Z2-grading of the quantum super- algebra Uq(glM|N) is defined on generators as [qX] = 0, [Ei] = [Fi] = � 0, i ̸= M, 1, i = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Before giving the explicit form of the defining relations, introduce the notion of the q-supercommutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The abelian group Q = M+N−1 � i=1 Z αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' is called the root lattice of the Lie superalgebra glM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Assuming that qX ∈ Uq(glM|N)0, Ei ∈ Uq(glM|N)αi, Fi ∈ Uq(glM|N)−αi, 5We use capital letters to distinguish between generators of the quantum superalgebra Uq(glM|N) and the quantum superalgebra Uq(L(slM|N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' RAZUMOV we endow Uq(glM|N) with a Q-grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Now, for any elements X ∈ Uq(glM|N)α and Y ∈ Uq(glM|N)β we define the q-supercommutator by the equation [[X, Y]] = XY − (−1)[X][Y]q−(α|β)YX = XY − (−1)[α][β]q−(α|β)YX if α, β ≻ 0, by the equation [[X, Y]] = XY − (−1)[X][Y]q(α|β)YX = XY − (−1)[α][β]q(α|β)XY if α, β ≺ 0, and by the equation [[X, Y]] = XY − (−1)[X][Y]YX = XY − (−1)[α][β]YX if α ≺ 0 and β ≻ 0, or α ≻ 0 and β ≺ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The defining relations of the quantum superalgebra Uq(glM|N) have the form [16] q0 = 1, qX1qX2 = qX1+X2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='1) qXEiq−X = q⟨αi, X⟩Ei, qXFiq−X = q−⟨αi, X⟩Fi, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='2) [[Ei, Fj]] = δij qdiKi−di+1Ki+1 − q−diKi+di+1Ki+1 qi − q−1 i , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='3) where i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' , M + N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Here and below we use the notation qi = qdi = q(−1)[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It is useful to have in mind that [2]qi = qi + q−1 i = q + q−1 = [2]q and (qi − q−1 i ) = di(q − q−1) = (−1)[i](q − q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='4) There are also the following Serre relations [[Ei, Ej]] = 0, [[Fj, Fi]] = 0, (αi | αj) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='5) [[[[Ei−1, Ei]], Ei]] = 0, [[Fi, [[ Fi, Fi−1]]]] = 0, (αi | αi) ̸= 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='6) [[Ei, [[Ei, Ei+1]]]] = 0, [[[[Fi+1, Fi]], Fi]] = 0, (αi | αi) ̸= 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='7) [[[[[[EM−1, EM]], EM+1]], EM]] = 0, [[FM, [[FM+1, [[FM, FM−1]]]]]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='8) Let us rewrite the defining relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='5)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='7) in a more familiar form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='5) are equivalent to the equations EiEj = EjEi, FjFi = FiFj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' |i − j| > 1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='9) E2 M = 0, F2 M = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='10) and the relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='6)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='7) are equivalent to Ei−1E2 i − [2]qEiEi−1Ei + E2 i Ei−1 = 0, F2 i Fi−1 − [2]qFiFi−1Fi + Fi−1F2 i = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='11) E2 i Ei+1 − [2]qEiEi+1Ei + Ei+1E2 i = 0, Fi+1F2 i − [2]qFiFi+1Fi + F2 i Fi+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='12) where i ̸= M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' ON POINCAR´E–BIRKHOFF–WITT BASIS OF QUANTUM GENERAL LINEAR SUPERALGEBRA 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' POINCAR´E–BIRKHOFF–WITT BASIS OF Uq(glM|N) An element a of Uq(glM|N) is called a root vector corresponding to a root γ of glM|N if a ∈ Uq(glM|N)γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' In particular, Ei and Fi are root vectors corresponding to the roots αi and −αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It is possible to construct linearly independent root vectors corresponding to all roots of glM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' To this end, being inspired by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Jimbo [22], we introduce elements Eij and Fij, 1 ≤ i < j ≤ M + N, with the help of the relations Ei, i+1 = Ei, Fi, i+1 = Fi, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='1) Ei, j+1 = [[Eij, Ej, j+1]], Fi, j+1 = [[Fj, j+1, Fij]], j > i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='2) Explicitly, the last two equations look as Ei, j+1 = Eij Ej, j+1 − qj Ej, j+1 Eij, Fi,j+1 = Fj, j+1 Fij − q−1 j Fij Fj, j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Note that we have [Eij] = [i] + [j], in particular, [Ei] = [Fi] = [i] + [i + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We also see that [Eij] = 0 if and only if j < M or i > M, [Eij] = 1 if and only if i ≤ M < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It is clear that the vectors Eij and Fij correspond to the roots αij and −αij respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' These vectors are linearly independent, and together with the elements qX, X ∈ k, are called Cartan–Weyl generators of Uq(glM|N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It appears that the ordered monomials constructed from the Cartan–Weyl generators form a Poincar´e–Birkhoff–Witt basis of Uq(glM|N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' In this paper we choose the following total order for monomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' First, we endow the set of the pairs (i, j), where 1 ≤ i < j ≤ M + N, with the lexicographical order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It means that (i, j) ≺ (m, n) if i < m, or if i = m and j < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='6 Now we say that a monomial is ordered if it has the form Fi1j1 · · · Fir jr qX Em1n1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Emsns, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='3) where (i1, j1) ≼ · · · ≼ (ir, jr), (m1, n1) ≼ · · · ≼ (ms, ns) and X is an arbitrary element of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' In the present paper we only show that any monomial can be written as a finite sum of monomials of the form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' To prove that they form a basis of Uq(glM|N) one can use arguments similar to those used in the paper [15] for the the case of the quantum algebra Uq(glM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We present the relations necessary for ordering as a sequence of propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' First consider the ordering of qX with Eij and Fij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For any 1 ≤ j < n ≤ M + N and i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' , M + N, we have qνKiEjn q−νKi = (qνδij + q−νδin)Ejn, qνKiFjn q−νKi = (q−νδij + qνδik)Ejn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It is evident that ⟨αj, Ki⟩ = δij − δi, j+1, and it follows from the defining relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='2) that qνKi i Ej q−νKi i = qν(δij−δi, j+1)Ej, qνKi i Fj q−νKi i = q−ν(δij−δi, j+1)Fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' 6Note that if we define an ordering of positive roots so that αij ≺ αmn if (i, j) ≺ (m, n) we get a normal ordering in the sense of [23, 24], see also [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' RAZUMOV which follow from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' qνKiEjnq−νKi = qν ∑n−1 m=j(δim−δi, m+1)Ejn = qν(δij−δin)Ejn = (qνδij + q−νδin)Ejn Thus, the first equation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='4) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The proof of the second equations is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' □ Now we consider the ordering of the root vectors Eij, 1 ≤ i < j ≤ M + N, and Fij, 1 ≤ i < j ≤ M + N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We divide the set of pairs ((i, j), (m, n)), where 1 ≤ i < j ≤ M + N, 1 ≤ m < n ≤ M + N and (i, j) ≺ (m, n), into six branches Ca, a = I, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' , VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The conditions defining the branches are given in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' In the same table we put the ([i] + [j])([m] + [n]) (αij | αmn) CI i = m < j < n [i] + [j] (−1)[i] CII i < m < n < j [m] + [n] 0 CIII i < m < j = n [m] + [j] (−1)[j] CIV i < m < j < n [m] + [j] 0 CV i < j = m < n 0 −(−1)[j] CVI i < j < m < n 0 0 TABLE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' information necessary to construct the corresponding q-supercommutators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' To fill table 1 it is sufficient to use the relations a2 = a, a + a = 0, a ∈ Z2, and equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For any ((i, j), (m, n)) ∈ CVI one has [[Eij, Emn]] = EijEmn − EmnEij = 0, [[Fmn, Fij]] = FmnFij − FijFmn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The statement of the proposition is a direct consequence of the Serre relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For any ((i, j), (m, n)) ∈ CV one has [[Eij, Emn]] = EijEjn − qjEjnEij = Ein, [[Fmn, Fij]] = FjnFij − q−1 j FijFjn = Fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The proposition can be proved by induction over n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For n = j + 1 we have just the definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Assume that the statement of the proposition is valid for some given n > j, then we have EijEjn − qjEjnEij = Ein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Using this equation and proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='2, we get [[Eij, Ej, n+1]] = EijEj, n+1 − qjEj, n+1Eij = Eij(EjnEn, n+1 − qnEn, n+1Ejn) − qj(EjnEn, n+1 − qnEn, n+1Ejn)Eij = (EijEjn − qjEjnEij)En, n+1 − qnEn, n+1(EijEjn − qjEjnEij) = [[Ein, En,n+1]] = Ei, n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Thus, the first equation of the proposition is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The second one can be proved in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' □ ON POINCAR´E–BIRKHOFF–WITT BASIS OF QUANTUM GENERAL LINEAR SUPERALGEBRA 7 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For any ((i, j), (m, n)) ∈ CII one has Eij, Emn]] = EijEmn − (−1)[m]+[n]EmnEij = 0, [[Fmn, Fij]] = FmnFij − (−1)[m]+[n]FijFmn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Let us first prove that [[Em−1, m+2, Em, m+1]] = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='6) for any 2 ≤ m ≤ M + N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It is easy to see that for m = M, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='6) is just the Serre relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' If m ̸= M, we have7 [[Em−1, m+2, Em, m+1]] = [[[[[[Em−1, m, Em, m+1]], Em+1, m+2]], Em, m+1]] = Em−1EmEm+1Em − qmEm−1Em+1E2 m − qmEmEm+1Em−1Em + q2 mEm+1EmEm−1Em − EmEm−1EmEm+1 + qmEmEm−1Em+1Em + qmE2 mEm+1Em−1 − q2 mEmEm+1EmEm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Using the first equations of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='12), we obtain [[Em−1, m+2, Em, m+1]] = [2]−1 q Em+1(E2 mEm−1 + Em+1E2 m) − qmEm−1Em+1E2 m − qmEmEm−1Em+1Em + [2]−1 q q2 mEm+1(E2 mEm−1 + Em−1E2 m) − [2]−1 q (E2 mEm−1 + Em−1E2 m)Em+1 + qmEmEm−1Em+1Em + qmE2 mEm+1Em−1 − [2]−1 q q2 m(E2 mEm+1 + Em+1E2 m)Em−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The Serre relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='5) give Em−1Em+1 = Em+1Em−1, and we see that equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='6) is true for any admissible value of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Assume that [[Eij, Em, m+1]] = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='7) for some 2 ≤ i < m < j − 1 ≤ M + N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We have [[Ei−1, j, Em, m+1]] = [[Ei−1,iEij − qiEijEi−1, i, Em, m+1]] = Ei−1, i[[Eij, Em, m+1]] − qi[[Eij, Em, m+1]]Ei−1, i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' If equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='7) is valid for some 1 ≤ i < m < j − 1 ≤ M + N − 2, then [[Ei, j+1, Em, m+1]] = [[EijEj, j+1 − qjEj, j+1Eij, Em, m+1]] = [[Eij, Em, m+1]]Ej, j+1 − qjEj, j+1[[Eij, Em, m+1]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Thus, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='7) is valid for any admissible i, j and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Finally, assume that the equation [[Eij, Emn]] = EijEmn − (−1)[m]+[n]EmnEij = 0 is valid for some 1 ≤ i < m < n < j − 1 ≤ M + N − 1, then we have [[Eij, Em, n+1]] = [[Eij, EmnEn, n+1 − qnEn, n+1Emn]] = [[Eij, Emn]]En, n+1 − (−1)[n]+[n+1]qnEn, n+1[[Eij, Emn]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Now, it is clear that the first equation of the proposition is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The second equation of the proposition can be proved in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' □ 7It is clear that either m < M or m > M, so that qm = qm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' 8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' RAZUMOV Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For any ((i, j), (m, n)) ∈ CI one has [[Eij, Emn]] = EijEin − (−1)[i]+[j]q−1 i EinEij = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='8) [[Fmn, Fij]] = FinFij − (−1)[i]+[j]qiFijFin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='9) For any ((i, j), (m, n)) ∈ CIII one has [[Eij, Emn]] = EijEmj − (−1)[m]+[j]q−1 j EmjEij = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='10) [[Fmn, Fij]] = FmjFij − (−1)[m]+[j]qjFijFmj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Let us consider the case ((i, j), (m, n)) ∈ CI and prove equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' First we demonstrate that [[Ei, i+1, Ei, i+2]] = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='12) for any 1 ≤ i ≤ M + N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We have [[Ei, i+1, Ei, i+2]] = [[Ei, i+1, [[Ei, i+1, Ei+1, i+2]]]] = [[Ei, [[Ei, Ei+1]]]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Hence, for i ̸= M, the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='12) is equivalent to the first of the Serre relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For i = M we obtain [[Ei, i+1, Ei, i+2]] = [[EM, M+1, [[EM, M+1, EM+1, M+2]]]] = [[EM, EMEM+1 − qM+1EM+1EM]] = −(qM+1 − q−1 M )EMEM+1EM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Thus, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='12) is valid for any 1 ≤ i ≤ M + N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Assume that [[Ei, i+1, Ein]] = Ei, i+1Ein − (−1)[i]+[i+1]q−1 i EinEi, i+1 = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='13) for some 1 ≤ i < n − 1 ≤ M + N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Using equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='13), we obtain [[Ei, i+1, Ei, n+1]] = [[Ei, i+1, EinEn, n+1 − qnEn, n+1Ein]] = Ei, i+1EinEn, n+1 − qnEi, i+1En, n+1Ein − (−1)[i]+[i+1]q−1 i (EinEn, n+1Ei, i+1 − qnEn, n+1EinEi, i+1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' we obtain that [[Ei, i+1, Ei, n+1]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It follows that [[Ei, i+1, Ein]] = 0 for any 1 ≤ i < n − 1 ≤ M + N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Now, assume that [[Eij, Ein]] = EijEin − (−1)[i]+[j]q−1 i EinEij = 0 for some 1 ≤ i < j + 1 < n ≤ M + N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Using proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='4, we get [[Ei, j+1, Ein]] = [[EijEj, j+1 − qjEj, j+1Eij, Ein]] = (−1)[j]+[j+1][[Eij, Ein]]Ej, j+1 − qjEj, j+1[[Eij, Ein]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Thus, for ((i, j), (m, n)) ∈ CI, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='8) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='9) can be proved in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' In the case when ((i, j), (m, n)) ∈ CIII, one can prove equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='11) in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' □ It follows from the above proposition that if ((i, j), (m, n)) ∈ CI, then [[Eij, [[Eij, Ejn]]]] = 0, [[[[Fjn, Fij]], Fij]] = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='14) and if ((i, j), (m, n)) ∈ CIII, then [[[[Eim, Emn]], Emn]] = 0, [[Fmn, [[Fmn, Fim]]]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='15) These relations are a generalization of the Serre relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' ON POINCAR´E–BIRKHOFF–WITT BASIS OF QUANTUM GENERAL LINEAR SUPERALGEBRA 9 Note that the quantum supergroup Uq(glM|N) has two natural subgroups isomorphic to Uq(glM) and Uq(glN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The former is generated by Ei, Fi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' , M − 1, and qX, where X belongs to the linear span of the elements Ki, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' M, and the latter is generated by Ei, Fi, i = M + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' , M + N − 1, and qX, where X belongs to the linear span of the elements Ki, i = M + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' M + N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It is clear that [i] + [j] = 0 iff Eij belongs to one of these two subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Each of them has no zero divisors, see the paper [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Hence, for any element Eij belonging to them one has E2 ij ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' In other words, if [i] + [j] = 1 then E2 ij ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For all 1 ≤ i < j ≤ M + N such that [i] + [j] = 1 one has 1 2 [[Eij, Eij]] = E2 ij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' In fact, we should demonstrate that if i ≤ M < j, then E2 ij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='17) First, we show that E2 Mj = 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='18) for all j > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It is certainly the case, at least for j = M + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Using the fact that qj = q−1 for any j > M, we obtain E2 M, j+1 = (EMjEj, j+1 − qjEj, j+1EMj)2 = EMjEj, j+1EMjEj, j+1 − q−1EMjE2 j, j+1EMj + q−2Ej, j+1EMjEj, j+1EMj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='19) It follows from the first relation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='15) that [[[[EMj, Ej, j+1]], Ej, j+1]] = 0, or, in a more explicit form, EMjE2 j, j+1 − [2]qEj, j+1EMjEj, j+1 + E2 j, j+1EMj = 0, Multiplying this equation from the left and from the right by EMj, we obtain −[2]qEMjEj, j+1EMjEj, j+1 + EMjE2 j, j+1EMj = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='20) EMjE2 j, j+1EMj − [2]qEj, j+1EMjEj, j+1EMj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='21) It follows that EMjEj, j+1EMjEj, j+1 = Ej, j+1EMjEj, j+1EMj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Using this equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='19), we get E2 M, j+1 = −q−1( − [2]qEMjEj, j+1EMjEj, j+1 + EMjE2 j, j+1EMj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Now equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='21) implies that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='18) for all M < j ≤ M + N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Further, we assume that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='16) is true for some 1 < i < M and M < j ≤ M + N, then we have E2 i−1, j = (Ei−1, iEij − qEijEi−1, i)2 = Ei−1, iEijEi−1, iEij − qEijE2 i−1, iEij + q2EijEi−1, iEijEi−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='22) Here we take into account that di = 1 for any i < M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It follows from the first relation of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='14) that [[Ei−1, i, [[Ei−1, i, Eij]]]] = 0, 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' RAZUMOV or, in a more explicit form, E2 i−1, iEij − [2]q Ei−1, iEijEi−1, i + EijE2 i−1, i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Multiplying this equation from the left and from the right by Eij, we obtain EijE2 i−1, iEij − [2]q EijEi−1, iEijEi−1, i = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='23) −[2]q Ei−1, iEijEi−1, iEij + EijE2 i−1, iEij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='24) It follows that EijEi−1, iEijEi−1, i = Ei−1, iEijEi−1, iEij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Using this equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='22), we come to E2 i−1, j = −q(−[2]q Ei−1, iEijEi−1, iEij + EijE2 i−1, iEij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Now equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='24) gives E2 i−1, j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Thus, we see that the statement of the proposition is always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For any ((i, j), (m, n)) ∈ CIV one has [[Eij, Emn]] = EijEmn − (−1)[m]+[j]EmnEij = −(qm − q−1 m )EmjEin, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='25) [[Fmn, Fij]] = FmnFij − (−1)[m]+[j]FijFmn = (qm − q−1 m )FinFmj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='26) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Using proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='3, we get [[Eij, Emn]] = (EimEmj − qmEmjEim)Emn − (−1)[m]+[j]Emn(EimEmj − qmEmjEim) = EimEmjEmn − (−1)[m]+[j]EmnEimEmj − qm(EmjEimEmn − (−1)[m]+[j]EmnEmjEim).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='27) Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='5 implies [[Emj, Emn]] = EmjEmn − (−1)[m]+[j]q−1 m EmnEmj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Hence, we have EmjEmn = (−1)[m]+[j]q−1 m EmnEmj, EmnEmj = (−1)[m]+[j]qmEmjEmn Using these equations in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='27), we obtain [[Eij, Emn]] = (−1)[m]+[j]q−1 m (EimEmn − qmEmnEim)Emj − qmEmj(EimEmn − qmEmnEim) = (−1)[m]+[j]q−1 m EinEmj − qmEmjEin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Finally, it follows from proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='4 that EinEmj = (−1)[m]+[j]EmjEin, therefore, [[Eij, Emn]] = −(qm − q−1 m )EmjEin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Thus, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='25) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' In the same way one can prove equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For any ((i, j), (m, n)) ∈ CV one has [[Eij, Fmn]] = EijFjn − FjnEij = 0, [[Emn, Fij]] = EjnFij − FijEjn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For any ((i, j), (m, n)) ∈ CVI one has [[Eij, Fmn]] = EijFmn − FmnEij = 0, [[Emn, Fij]] = EmnFij − FijEmn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' ON POINCAR´E–BIRKHOFF–WITT BASIS OF QUANTUM GENERAL LINEAR SUPERALGEBRA 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The statement of the proposition is a direct consequence of the defining relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For any ((i, j), (m, n)) ∈ CII one has [[Eij, Fmn]] = EijFmn − (−1)[m]+[n]FmnEij = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='28) [[Emn, Fij]] = EmnFij − (−1)[m]+[n]FijEmn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='29) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Let 1 < k ≤ M + N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Prove equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='28) for i = k − 1, m = k, n = k + 1 and j = k + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We have Ek−1, k+2 = Ek−1EkEk+1 − qk+1Ek−1Ek+1Ek − qkEk+1Ek+1Ek−1 + qkqk+1Ek+1EkEk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It follows that [[Eij, Fmn]] = Ek−1[[Ek, Fk]]Ek+1 − qk+1Ek−1Ek+1[[Ek, Fk]] − qk[[Ek, Fk]]Ek+1Ek−1 + qkqk+1Ek+1[[Ek, Fk]]Ek−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Using the defining relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='3) and proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='1, we obtain [[Ek−1, k+2, Fk, k+1]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Now, let 1 < i < k < j − 1 ≤ M + N − 1 and [[Eij, Fk, k+1]] = EijFk, k+1 − (−1)[k]+[k+1]Fk, k+1Eij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='30) We have [[Ei−1, j, Fk, k+1]] = [[Ei−1, iEij − qiEijEi−1, i, Fk, k+1]] = Ei−1,iEijFk, k+1 − (−1)[k]+[k+1]Fk, k+1Ei−1, iEij − qj(EijEi−1, iFk, k+1 − (−1)[k]+[k+1]Fk, k+1EijEi−1, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It follows from proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='8 that [[Ei−1, j, Fk, k+1]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Further, let 1 ≤ i < k < j − 1 ≤ M + N − 2 and equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='30) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We obtain [[Ei, j+1, Fk, k+1]] = [[EijEj, j+1 − qjEj, j+1Eij, Fk, k+1]] = EijEj, j+1Fk, k+1 − (−1)[k]+[k+1]Fk, k+1EijEj, j+1 − (Ej,j+1EijFk, k+1 − (−1)[k]+[k+1]qjFk, k+1Ej, j+1Eij), and proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='8 implies that [[Ei, j+1, Fk, k+1]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Hence, we have [[Eij, Fk, k+1]] = 0 for all possible i, j and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Assume now that for some 1 ≤ i < m < n < M + N − 1 we have [[Eij, Fmn]] = EijEmn − (−1)[m]+[n]EmnEij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Then, we obtain [[Eij, Fm, n+1]] = [[Eij, Fn, n+1Fmn − q−1 n FmnFn, n+1]] = EijFn, n+1Fmn − (−1)[m]+[n+1]Fn, n+1FnmEij − q−1 n (EijFmnFn, n+1 − (−1)[m]+[n+1]FmnFn, n+1Eij) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Thus, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='28) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' In the same way one can prove equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' □ 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' RAZUMOV Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For any ((i, j), (m, n)) ∈ CI one has [[Eij, Fin]] = EijFin − (−1)[i]+[j]FinEij = −(−1)[i]+[j]q−diKi+djKjFjn, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='31) [[Ein, Fij]] = EinFij − (−1)[i]+[j]FijEin = −(−1)[i]+[j]Ejn qdiKi−djKj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='32) For any ((i, j), (m, n)) ∈ CIII one has [[Eij, Fmn]] = EijFmj − (−1)[m]+[j]FmjEij = q−dmKm+djKjEim, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='33) [[Emn, Fij]] = EmjFij − (−1)[m]+[j]FijEmj = FimqdmKm−djKj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='34) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We first prove equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='31) for j = i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Using proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='3, we obtain [[Ei, i+1, Fin]] = [[Ei, i+1, Fi+1, nFi, i+1 − q−1 i+1Fi, i+1Fi+1, n]] = Ei, i+1Fi+1, nFi, i+1 − (−1)[i]+[i+1]Fi+1, nFi, i+1Ei, i+1 − q−1 i+1(Ei, i+1Fi, i+1Fi+1, n − (−1)[i]+[i+1]Fi, i+1Fi+1, nEi, i+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Further, proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='8 gives [[Ei, i+1, Fin]] = Fi+1, n[[Ei, Fi]] − q−1 i+1[[Ei, Fi]]Fi+1, n = (qi − q−1 i )−1(Fi+1, n(qdiKi−di+1Ki+1 − q−diKi+di+1Ki+1) − q−1 i+1(qdiKi−di+1Ki+1 − q−diKi+di+1Ki+1)Fi+1, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' and, using proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='1, we come to [[Ei, i+1, Fin]] = −(qi+1 − q−1 i+1)(qi − q−1 i )−1q−diKi+di+1Ki+1Fi+1, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Finally, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='4) that [[Ei, i+1, Fin]] = −(−1)−[i]+[i+1]q−diKi+di+1Ki+1Fi+1, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Now, let 1 ≤ i < j < n − 1 ≤ M + N − 1 and equation [[Eij, Fin]] = −(−1)[i]+[j]q−diKi+djKjFjn (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='35) be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Using proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='3, we obtain [[Ei, j+1, Fin]] = [[EijEj, j+1 − qjEj, j+1Eij, Fin]] = EijEj, j+1Fin − (−1)[i]+[j+1]FinEijEj, j+1 − qj(Ej, j+1EijFin − (−1)[i]+[j+1]FinEj, j+1Eij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It follows from proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='4, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='35) and proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='1 that [[Ei, j+1, Fin]] = (−1)[j]+[j+1][[Eij, Fin]]Ej, j+1 − qjEj, j+1[[Eij, Fin]] = (−1)[i]+[j]q−diKi+djKj[[Ej, j+1, Fjn]] = −(−1)[i]+[j+1]q−diKi+dj+1Kj+1Fj+1, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We see that equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='31) is always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' In the same way one can prove equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='32), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='33) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For any 1 ≤ i < j ≤ M + N we have [[Eij, Fij]] = EijFij − (−1)[i]+[j]FijEij = qdiKi−djKj − q−diKi+djKj qi − q−1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' ON POINCAR´E–BIRKHOFF–WITT BASIS OF QUANTUM GENERAL LINEAR SUPERALGEBRA 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The statement of the proposition is certainly true for j = i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Let us consider the case when j − i > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It follows from proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='3 that [[Eij, Fij]] = [[Eij, Fi+1, jFi, i+1 − q−1 i+1Fi, i+1Fi+1, j]] = EijFi+1, jFi, i+1 − (−1)[i]+[j]Fi+1, jFi, i+1Eij − q−1 i+1(EijFi, i+1Fi+1, j − (−1)[i]+[j]Fi, i+1Fi+1, jEij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='36) Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='32) implies Fi, i+1Eij = (−1)[i]+[i+1]EijFi, i+1 + Ei+1, jqdiKi−di+1Ki+1, EijFi, i+1 = (−1)[i]+[i+1]Fi, i+1Eij − (−1)[i]+[i+1]Ei+1, jqdiKi−di+1Ki+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Using these equations in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='36), we obtain [[Eij, Fij]] = [[Eij, Fi+1, j]]Fi, i+1 + q−1 i+1(−1)[i]+[i+1]Fi, i+1[[Eij, Fi+1, j]] + q−1 i+1(−1)[i]+[i+1]Ei+1, jqdiKi−di+1Ki+1Fi+1, j − (−1)[i]+[j]Fi+1, jEi+1,jqdiKi−di+1Ki+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='37) We have [[Eij, Fi+1, j]] = q−di+1Ki+1+djKjEi, i+1, see proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Taking this equation, and proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='1 and equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='4) into account, we come to the equation [[Eij, Fij]] = q−di+1Ki+1+djKj[[Ei, i+1, Fi, i+1]] + (−1)[i]+[i+1][[Ei+1, j, Fi+1, j]]qdiKi−di+1Ki+1 = qdiKi−djKj − q−diKi+djKj qi − q−1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' That was to be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' For any ((i, j), (m, n)) ∈ CIV one has [[Eij, Fmn]] = EijFmn − (−1)[m]+[j]FmnEij = −(qj − q−1 j )q−dmKm+djKjFjnEim, [[Emn, Fij]] = EmnFij − (−1)[m]+[j]FijEmn = (qj − q−1 j )FimEjnqdmKm−djKj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' It follows from propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='9 that [[Eij, Fmn]] = [[EimEmj − qmEmjEim, Fmn]] = EimEmjFmn − (−1)[m]+[j]FmnEimEmj − qm(EmjEimFmn − (−1)[m]+[j]FmnEmjEim) = Eim[[Emj, Fmn]] − qm[[Emj, Fmn]]Eim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Now, using equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='31), proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='1 and proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='8, we get [[Eij, Fmn]] = (−1)[m]+[j](qm − q−1 m )q−dmKm+djKjFjnEim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Now, taking into account equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='4), we see that the first equation of the proposition is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' The second equation can be proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' □ One can get convinced that the propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='12 allow us to reduce any monomial on the Poincar´e–Birkhoff–Witt generators to the ordered form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' RAZUMOV 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' CONCLUSIONS We have derived the commutation relations for the Poincar´e–Birkhoff–Witt generators of the quantum algebra Uq(glM|N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Our results do not fully coincide with the results of the papers [18, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' We are planning to use the obtained relations for constructing of q-oscillator representations of the positive Borel subalgebra of the quantum superalgebra Uq(glM|N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' This work was supported in part by the RFBR grant # 20-51-12005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' REFERENCES [1] V.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Razumov, Universal R-matrix and functional relations, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' 26 (2014), 1430005 (66pp), arXiv:1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='1631 [math-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content=' [6] Kh.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='Razumov@ihep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} +page_content='ru' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNFAT4oBgHgl3EQfeR0E/content/2301.08574v1.pdf'} diff --git a/rtE0T4oBgHgl3EQfawAZ/content/tmp_files/2301.02338v1.pdf.txt b/rtE0T4oBgHgl3EQfawAZ/content/tmp_files/2301.02338v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8d0433675786715fcbc4fec1bd6e7e7122522389 --- /dev/null +++ b/rtE0T4oBgHgl3EQfawAZ/content/tmp_files/2301.02338v1.pdf.txt @@ -0,0 +1,1272 @@ +Draft version January 9, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +GOALS-JWST: Pulling Back the Curtain on the AGN and Star Formation in VV 114 +J. Rich,1 S. Aalto,2 A.S. Evans,3, 4 V. Charmandaris,5, 6, 7 G. C. Privon,3, 8 T. Lai,9 H. Inami,10 S. Linden,11 +L. Armus,9 T. Diaz-Santos,6, 7 P. Appleton,9 L. Barcos-Mu˜noz,4 T. B¨oker,12 K. L. Larson,13 D. R. Law,14 +M. A. Malkan,15 A. M. Medling,16, 17 Y. Song,4, 3 V. U,18 P. van der Werf,19 T. Bohn,10 M. J. I. Brown,20 +L. Finnerty,15 C. Hayward,21 J. Howell,22 K. Iwasawa,23, 24 F. Kemper,25, 24, 26 J. Marshall,27 J. M. Mazzarella,22 +J. McKinney,28 F. Muller-Sanchez,29 E.J. Murphy,30 D. Sanders,31 B. T. Soifer,32 S. Stierwalt,33 and J. Surace22 +1The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena, CA 91101 +2Department of Space, Earth and Environment, Chalmers University of Technology, 412 96 Gothenburg, Sweden +3National Radio Astronomy Observatory, 520 Edgemont Rd, Charlottesville, VA, 22903, USA +4Department of Astronomy, University of Virginia, 530 McCormick Road, Charlottesville, VA 22903, USA +5Department of Physics, University of Crete, Heraklion, 71003, Greece +6Institute of Astrophysics, Foundation for Research and Technology-Hellas (FORTH), Heraklion, 70013, Greece +7School of Sciences, European University Cyprus, Diogenes street, Engomi, 1516 Nicosia, Cyprus +8Department of Astronomy, University of Florida, P.O. Box 112055, Gainesville, FL 32611, USA +9IPAC, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA +10Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan +11Department of Astronomy, University of Massachusetts at Amherst, Amherst, MA 01003, USA +12European Space Agency, Space Telescope Science Institute, Baltimore, MD 21218, USA +13AURA for the European Space Agency (ESA), Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA +14Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA +15Department of Physics & Astronomy, 430 Portola Plaza, University of California, Los Angeles, CA 90095, USA +16Department of Physics & Astronomy and Ritter Astrophysical Research Center, University of Toledo, Toledo, OH 43606,USA +17ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D); Australia +18Department of Physics and Astronomy, 4129 Frederick Reines Hall, University of California, Irvine, CA 92697, USA +19Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands +20School of Physics and Astronomy, Monash University, Clayton, VIC 3800, Australia +21Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA +22IPAC, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125 +23Institut de Ci`encies del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Mart´ı i Franqu`es, 1, 08028 Barcelona, Spain +24ICREA, Pg. Llu´ıs Companys 23, 08010 Barcelona, Spain +25Institut de Ciencies de l’Espai (ICE, CSIC), Can Magrans, s/n, 08193 Bellaterra, Barcelona, Spain +26Institut d’Estudis Espacials de Catalunya (IEEC), E-08034 Barcelona, Spain +27Glendale Community College, 1500 N. Verdugo Rd., Glendale, CA 91208 +28Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA. +29Department of Physics and Materials Science, The University of Memphis, 3720 Alumni Avenue, Memphis, TN 38152, USA +30National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA +31Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822 +32Division of Physics, Mathematics and Astronomy, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125 +33Physics Department, 1600 Campus Road, Occidental College, Los Angeles, CA 90041, USA +Submitted to ApJ Letters +ABSTRACT +We present results from the James Webb Space Telescope (JWST) Director’s Discretionary Time +Early Release Science (ERS) program 1328 targeting the nearby, Luminous Infrared Galaxy (LIRG), +VV 114. +We use the MIRI and NIRSpec instruments to obtain integral-field spectroscopy of the +heavily obscured Eastern nucleus (V114E) and surrounding regions. +The spatially resolved, high- +resolution, spectra reveal the physical conditions in the gas and dust over a projected area of 2-3 kpc +that includes the two brightest IR sources, the NE and SW cores. Our observations show for the +first time spectroscopic evidence that the SW core hosts an AGN as evidenced by its very low 6.2µm +arXiv:2301.02338v1 [astro-ph.GA] 6 Jan 2023 + +2 +and 3.3µm PAH equivalent widths (0.12 and 0.017 µm respectively) and mid and near-IR colors. +Our observations of the NE core show signs of deeply embedded star formation including absorption +features due to aliphatic hydrocarbons, large quantities of amorphous silicates, as well as HCN due +to cool gas along the line of sight. We detect elevated [Fe II]/Pfα consistent with extended shocks +coincident with enhanced emission from warm H2, far from the IR-bright cores and clumps. We also +identify broadening and multiple kinematic components in both H2 and fine structure lines caused by +outflows and previously identified tidal features. +Keywords: galaxies: star formation, interactions, evolution infrared: galaxies +1. INTRODUCTION +VV 114 (Arp236, IC1623) is an interacting system un- +dergoing vigorous starburst activity. With an infrared +luminosity of LIR ∼ 4.5 × 1011 L⊙, and a distance of 80 +Mpc, it is one of the brightest objects in the IRAS Bright +Galaxy Sample (Soifer et al. 1987). It appears to be an +early-stage merger of two galaxies that are aligned east- +west with a projected nuclear separation of ∼8 kpc, des- +ignated in the literature as VV 114E and VV 114W. At +optical wavelengths, VV 114 shows a highly disturbed +morphology with very faint tidal tails extending over 25 +kpc from the center (Arp 1966). The western compo- +nent, VV 114W, is more extended than the eastern one, +and dominates the emission at short wavelengths. Much +of the mid-infrared emission is diffuse and extended over +several kpc with some indication of an AGN based on +the mid-infrared color of the more compact nuclear re- +gion in VV 114E (Le Floc’h et al. 2002). ALMA observa- +tions show abundant cold, dense gas (traced by e.g. CO, +HCO, HCN), evidence for shocked gas in the overlap re- +gion between the two galaxies (traced by Methanol), a +molecular outflow, and a possible buried AGN in VV +114E (Iono et al. 2013; Saito et al. 2015, 2017, 2018). +The majority of the IR emission and by extension to- +tal energy output of the system is dominated by VV +114E and the extreme UV/optical to IR ratios of VV +114 make it a more plausible analog to high-z IR lumi- +nous mergers (Charmandaris et al. 2004; Howell et al. +2010). +Mid-infrared spectra centered on VV 114E taken with +Spitzer IRS (∼ 10′′ × 36′′) show a moderately strong +9.7µm Silicate absorption (s9.7µm = −0.98), a 6.2µm +PAH equivalent width intermediate to LIRGs dominated +by star formation or AGN (EQW6.2µm = 0.30µm), and +a ratio of H2 to PAH luminosity slightly above values +associated with photodissociation region emission (Guil- +lard et al. 2012a; Stierwalt et al. 2013, 2014). No coronal +lines (e.g. [Ne V]) were detected in the Spitzer spectra, +and fine structure line flux ratios were consistent with +LIRGs primarily dominated by star formation with some +composite AGN/starburst activity (Inami et al. 2013). +Observations with Chandra, XMM, and NuSTAR of VV +114E indicate that the X-ray emission appears to be gen- +erated primarily through star fomation, with little to +no X-ray emission coming from an AGN (Grimes et al. +2006; Garofali et al. 2020; Ricci et al. 2021). +Visible +wavelength integral field spectroscopy indicates a mix +of star formation and shock emission, the latter indi- +cated by elevated emission line ratios and line widths +across both galaxies (Rich et al. 2011, 2015). Finally, +in a companion paper to the present study, Evans et al. +(2022) propose the presence of a reddened starburst and +an AGN in the bright NE and SW cores, respectively, +of VV 114E based on broadband JWST mid-IR colors. +In this paper we present the James Webb Space Tele- +scope (JWST) combined MIRI/NIRSpec integral field +spectroscopic observations of the nucleus of VV 114E +and surrounding regions taken as part of the Early Re- +lease Science (ERS) program 1328 (Co-PIs: L. Armus +and A. Evans). The data allow us to resolve the proper- +ties of the two brightest sources, the NE and SW cores, +as well as star clusters and diffuse emission surrounding +the eastern nucleus. +Throughout the paper we adopt a cosmology of Ho=70 +km s−1 Mpc−1, ΩM=0.28, and ΩΛ=0.72. The redshift +of VV 114 (z=0.0202) corresponds to an angular scale +of 400 pc/1′′(Wright 2006). +2. OBSERVATIONS, DATA REDUCTION +VV 114 was observed with MIRI Rieke et al. (2015); +Labiano et al. (2021) on 2 July 2022 (MIRI imaging, +Bouchet et al. 2015), on 5 July 2022 (MRS spectroscopy, +Wells et al. 2015), and by NIRSpec (Jakobsen et al. +2022) on 19 July 2022 (IFU spectroscopy, B¨oker et al. +2022). We include MIRI imaging in this paper to indi- +cate the locations of our spectral extraction apertures. +The full analysis of these imaging data products are de- +scribed in Evans et al. (2022). +2.1. MIRI MRS Data +The MIRI MRS observations include three grating set- +tings (SHORT, MEDIUM, and LONG) in order to cover + +3 +WFC3 F435W/F814W +2 kpc +MIRI F770W +1h07m47.8s +47.6s +47.4s +47.2s +-17°30'20" +22" +24" +26" +28" +Right Ascension +Declination +2 kpc +aa bb cc +dd ee +ff +gg hh +ii +jj +MIRI F770W Sub +5 +10 +15 +20 +25 +Rest Wavelength [ m] +10 +1 +101 +103 +105 +107 +109 +1011 +Jy (relative) +c(SW) +d +a(NE) +h +j +g +b +i +e +f +6.2 m PAH +9.7 m Si +[Fe II] +[Ar II] +Pf +[Ar III] +H2 S(3) +[S IV] +H2 S(2) +[Ne II] +[Cl II] +[Ne III] +H2 S(1) +[S III] +[Fe III] +Figure 1. Images and spectra of VV 114. Top-Left: HST 435W/814W color image, Middle-Left: JWST F770W image with +the same FOV, the green box corresponds to the MIRI SUB128 subarray FOV, as shown in the Bottom-Left panel, Bottom-Left: +F770W SUB128 image with MIRI MRS channel 1 FOV (dark blue Box), NIRSpec FOV (light blue box) and our ten 0.4′′ radius +extraction apertures. Right: Full MIRI spectra of the 10 regions marked in the F770W subarray image. Spectra are shown at +rest frame wavelengths assuming a systemic velocity of 6056 km/s and sorted from top to bottom in order of decreasing 6.2 µm +PAH equivalent width. +the entire wavelength range accessible to the four IFU +channels (4.9-27.9µm). A four-point dither pattern was +employed to recover extended emission and avoid sat- +uration, with separate off-source observations for back- +ground subtraction. The field of view (FOV) and posi- +tion angle (PA) vary by channel, but the FOV with full +wavelength coverage is defined by channel 1, ∼ 5′′ × 4′′ +at PA=255° as shown in Fig. 1. +Our procedure for reducing MRS data for the ERS +1328 targets is described in detail in U et al. (2022), +but a brief summary is given here: uncalibrated data +are processed using the most recently available de- +velopmental release of the JWST Science Calibration +Pipeline (Bushouse et al. 2022), version 1.8.3, using +calibration reference data system (CRDS) context file +jwst 0963.pmap. The resulting data products generated +by the pipeline are 12 background-subtracted, fringe- +corrected, wavelength and flux calibrated data cubes– +one for every combination of channel and grating set- +ting combining the four dither pointings. These cubes +are used to generate 1-d spectra of regions of interest +and 2-d maps of the properties of particular emission +line features. +We extract spectra from 10 regions of interest within +the channel 1 FOV to investigate IR-bright sources re- +solved in the MIRI imaging as well as diffuse emission + +4 +surrounding those sources and in the eastern nucleus +(Fig. 1). We chose five locations coincident with the +two brightest IR sources, the NE and SW cores (a, c), +and several bright clumps identified in the MIRI imaging +data and in both the submm and radio (d, e, f) (Saito +et al. 2018; Evans et al. 2022; Song et al. 2022). We +also chose five regions intended to capture diffuse emis- +sion between (b) and around (g, h, i, j) the bright clumps +coincident with some tidal and shock features previously +identified using ALMA (Saito et al. 2017, 2018). +Spectra are extracted from each of the 12 data cubes +using apertures with a radius of 0.4′′(160 pc). This re- +sults in twelve 1-d spectra for each extracted region, +with some overlap in adjacent wavelength regions. The +12 individual bands for each spectra have slight off- +sets in flux (a few percent) which are multiplicatively +scaled, trimmed, and stitched in order to create continu- +ously smooth 1-d spectra over the full MIRI wavelength +range. +This process begins by using the overlapping +wavelength region to scale Channel 4 MEDIUM spec- +trum to the longest wavelength Channel 4 LONG spec- +trum (∼ 23 − 25µm), trimming the overlapping values +from the noisier spectrum of the two, and continuing the +process to channels at shorter and shorter wavelengths. +Finally, a wavelength dependent aperture correction is +applied to each spectrum (U et al. 2022). +2.2. NIRSpec IFU Data +NIRSpec IFU observations were taken with three +grating +and +filter +combinations: +G140H/F100LP, +G235H/F170LP, and G395H/F290LP to cover the spec- +tral range from 0.97-5.3 µm. Calculations in this paper +were made using the wavelength region covered by the +G235H/F170LP and G395H/F290LP settings, ∼1.7-5.3 +µm. This wavelength range allows us to measure the +AGN-sensitive 3.3µm PAH feature. Again a four-point +dither pattern was employed to completely sample the +PSF and to avoid saturation, with an additional “Leak- +cal” image taken for each grating setting. The FOV of +the combined dither pattern is ∼ 3.6′′ × 3.8′′ centered +on the SW Core at a PA of ∼ 32◦ (Fig. 1). +We reduce the NIRSpec data in a similar fashion to +the MRS data, using calibration reference data sys- +tem (CRDS) context file jwst 1009.pmap. Uncalibrated +four-point dither pattern science images and Leakcal +images, one for each of the two NIRSpec chips, are +downloaded using MAST resulting in 10 image files. +These are first put through Stage 1 processing with the +Detector1 pipeline, which applies detector-level calibra- +tions and produces count rate files calculated from the +non-destructive “ramp” readouts. These rate files are +then processed using the Spec2 pipeline, which applies +Figure 2. Two dimensional images of flux (a, d), relative +velocity (b, e), and FWHM (c, f) generated from the spaxel- +by-spaxel fits to the [Ne II] (12.8 µm) and H2 S(2) (12.3 µm) +emission lines. The FWHM shown is corrected for instru- +mental broadening as described in the text. The relative ve- +locity map is generated by subtracting a systemic velocity of +6056 km/s from each spaxel. The broadest H2 FWHM in the +NW is just outside the Ch1 FOV but falls within the Ch3 +FOV, and corresponds to the eastern edge of the shocked +“overlap” region observed in Saito et al. (2017). +The ele- +vated FWHM in [Ne II] is near apertures that show evidence +of shocks and double peaked emission line profiles (g, h, i, j). +physical corrections and flux and wavelength calibra- +tions. At this step in the overall pipeline, the Leakcal +files are also used to correct for any stray light that may +fall on the detector due to failed open MSA shutters. +Finally we run the Spec3 pipeline step, which produces +a final combined data cube sampled with 0.1′′ spaxels. +For our analysis in this paper, we extract from the final +data cube spectra from the two brightest IR sources, the +NE and SW cores (a and c). We matched our apertures +to the MIRI MRS extraction radius of 0.4′′ centered at +the same two locations. For the G395H/F290LP setting +this produces flux and wavelength calibrated 1-d spectra +covering 2.87-5.27µm, with a gap in coverage from 4.06- +4.18µm in the middle of the spectrum due to the gap be- +tween the two NIRSpec chips. For G235H/F170LP the +range is 1.66-3.17µm with a gap from 2.40-2.45µm. We +use the overlapping wavelength range between the NIR- +Spec and MIRI data to scale the G395H/F290LP spec- +trum to match the MIRI spectrum, and the overlapping +region between the two NIRSpec settings to scale and +stitch the shorter wavelength spectra (G235H/F170LP) +to the longer wavelength NIRSpec spectra. +3. RESULTS + +10-19 +100 +100 +200 +3005 +Table 1. Spectral Feature Strengths, Fluxes, and FWHM +Region ID +EQW3.3 +EQW6.2 +s9.7 +[Fe II] 5.34 +FWHM +Pfα +FWHM +a (NE Core) +0.121±0.001 +0.264±0.018 +-2.45±0.03 +7.41±0.22 +194±13 +1.56±0.62 +142±37 +b +0.514±0.017 +-1.13±0.03 +11.6±0.29 +186±12 +2.83±0.58 +148±21 +c (SW Core) +0.017±0.001 +0.106±0.002 +-1.06±0.01 +7.49±0.40 +178±14 +4.76±0.92 +154±20 +d +0.199±0.015 +-1.44±0.02 +9.87±0.38 +166±13 +4.92±0.88 +135±18 +e +0.652±0.015 +-1.17±0.04 +7.64±0.19 +185±12 +2.23±0.37 +152±18 +f +0.720±0.062 +-0.90±0.04 +3.09±0.13 +153±13 +2.55±0.37 +120±13 +g +0.491±0.036 +-1.05±0.02 +5.03±0.20 +221±14 +0.79±0.16 +192±25 +h +0.34±0.17 +-0.75±0.04 +2.32±0.57 +147±29 +0.24±0.17 +160±73 +i +0.56±0.14 +-0.73±0.05 +2.78±0.39 +231±26 +1.08±0.36 +224±44 +j +0.359±0.045 +-0.82±0.01 +2.30±0.14 +270±17 +0.24±0.07 +243±43 +Values of the 3.3 and 6.2µm EQW, 9.7µm silicate strength, as well as line fluxes (10−18 W/m−2 ) and FWHM (km +s−1, corrected for instrumental broadening) for emission line features measured in our extracted regions that were +used in our analysis and discussion. This table is a subset of the total measurements, a machine readable version +of the full table is available online. +We use the 1-d aperture extracted spectra to measure +emission and absorption features that trace the physical +properties of the gas and dust. Several of our apertures +are centered on bright, unresolved mid-IR sources seen +in the MIRI imaging observations (Evans et al. 2022), +including the bright NE and SW cores (a & c), a source +directly SE of the SW core (d), and a deeply embedded +star cluster (f) with Mass M∼ 106 M⊙, age t ∼ 1 − 2 +Myr, and extinction AV ∼ 8 (see Linden et al. 2022). +The remaining apertures trace diffuse emission generally +showing elevated EQW6.2µm and strong H2 emission. +3.1. Emission Line Properties +We perform fits to features in each 1-d MIRI spec- +trum using the “lmfit” package (Newville et al. 2014) +with resulting values given in Table 1. Atomic and H2 +emission lines are fit with a single Gaussian component +combined with a polynomial fit to the local continuum +over a range of 0.1µm. The resulting Gaussian param- +eters are used to derive the observed flux of each line. +The width of the Gaussian fit is used to determine the +intrinsic FWHM of each emission line by subtracting in +quadrature the instrumental resolution of MRS at the +observed wavelength (Labiano et al. 2021). +Several fine structure lines and H2 lines are well de- +tected and resolved, as well as the hydrogen recombina- +tion lines Pfund α and Humphreys α and β. Emission +line ratios show variation between apertures indicative +of both widespread star formation and shock excitation. +We do not detect any of the high-ionization coronal lines +typically found in mid-IR spectra of AGN dominated +galaxies (e.g. +[Ne V]; Genzel et al. 1998; Lutz et al. +2000; Sturm et al. 2002; Weedman et al. 2005; Armus +et al. 2007) in contrast to JWST observations of NGC +7469 that show nine well-detected coronal lines excited +by high-energy photons from the central AGN (Armus +et al. 2022). +Emission line ratios sensitive to AGN activity (e.g [S +IV]/[Ne II], [O IV]/[Ne II]) show values indicative of a +composite of AGN and starburst activity (Inami et al. +2013). Apertures g, h and j show elevated values of [Fe +II]/Pfα (> 5) and g & j show [Fe II] FWHM higher than +the other apertures (200-300 km/s). +Apertures a, b, c, d, and e have fine structure lines +([Ar II], [Ar III], [Ne II], [Cl II], [Ne III], [S IV]) with +FWHM ranging between 150-200 km/s and show no +trend with emission line ionization potential. In the NE +core (a) the H2 FWHM is ∼50 km/s narrower than the +fine structure lines, while in the SW core (c) the H2 +and fine structure line widths are similar. This trend is +reversed in aperture f where the H2 FWHM are ∼200 +km/s vs. ∼100 km/s for the fine structure lines. The +broadest emission line widths in our data are observed +in the spectrum of aperture i, which samples the diffuse +H2 bright gas near the western edge of our FOV close +to the “overlap” region defined in (Saito et al. 2017). +In order to assess the morphology and kinematics of +the region observed with MIRI/MRS, we have also car- +ried out spaxel-by-spaxel fits of the [Ne II] 12.8 µm fine +structure emission line and H2 S(2) 12.3 µm molecu- +lar emission line. These two emission lines are gener- +ally quite luminous and are well detected across nearly +every spaxel. +The lines are covered by Channel 3B +(13.29-15.52µm) which provides a wider field of view +(7′′ × 6′′) while still maintaining relatively high spectral +resolution (R∼2800-3000) with somewhat larger spax- +els (∼0.2′′). Fits to the two emission lines are carried +out on a spaxel-by-spaxel basis using the Channel 3B + +6 +6.00 +6.25 +6.50 +6.75 +7.00 +7.25 +7.50 +7.75 +8.00 +100 +101 +Relative Flux (Jy) +6.85 m Aliphatic C-H +7.25 m Aliphatic C-H +a +c +f +14 +16 +18 +20 +22 +100 +101 +16 m Crystalline Si +Amorph. Si +a +c +f +13.6 +13.7 +13.8 +13.9 +14.0 +14.1 +14.2 +Rest Wavelength [ m] +0.80 +0.85 +0.90 +0.95 +1.00 +1.05 +1.10 +1.15 +1.20 +13.7 m C2H2 +14.04 m HCN +a +c +f +NE Core +Figure 3. Expanded view of spectral features present only +in the NE nucleus (region a) indicative of a highly embedded +source, compared with spectra of the SW nucleus (c) and +region f (embedded cluster). Dashed vertical lines in each +panel correspond to the following features Top panel: Ab- +sorption features due to aliphatic hydrocarbons at 6.85 and +7.25 µm. Spectra are normalized to 1 Jy at 6.7 µm. Middle +panel: crystalline silicate absorption features at 16 and 19 +µm combined with amorphous silicate absorption at 18 µm. +Spectra are normalized to 1 Jy at 14.5 µm. Bottom Panel: +C2H2 and HCN absorption. Spectra are normalized to 1 Jy +at 13.8 µm. +sub-band data cube, with a single Gaussian component +fit to each line independently, in the same manner as +the aperture extracted spectra. The resulting maps are +shown in Figure 2. +The variation in the flux and FWHM of both the [Ne +II] and H2 lines agrees with the values measured in our +extracted apertures. The broadest FWHM in both lines +lies outside the Channel 1 FOV, our closest aperture +extractions are i and j. The increase in H2 FWHM at the +NW corner of our map corresponds to a portion of the +“overlap” region observed in Saito et al. (2017) with a +similar increase in FWHM seen at submm wavelengths. +3.2. PAH Equivalent Width +We measure the equivalent width of the 3.3 µm and +6.2 µm PAH emission features (EQW3.3µm, EQW6.2µm) +in the NIRSpec and MIRI spectra by applying the same +method to both. First, portions of the spectrum adja- +cent to each PAH feature are used to perform a linear +interpolation of the continuum. +We then integrate a +spline fit from 3.20–3.28 and 5.95–6.55 µm (rest frame, +EQW3.3µm, EQW6.2µm) to calculate the PAH feature +flux. The integrated flux is divided by the continuum +flux density at the wavelength of the peak of each PAH +feature. +The EQW6.2µm values range from 0.11–0.72 µm, +bracketing the published Spitzer IRS value of 0.3 µm +(Stierwalt et al. 2013). The NE core has an EQW6.2µm +of 0.264µm and a relatively high EQW3.3µm of 0.121µm, +while the SW core has a low EQW6.2µm of 0.106µm and +a very low values of EQW3.3µm of 0.016µm, indicative +of an AGN (e.g. Imanishi et al. 2010; Petric et al. 2011, +see discussion). +The largest EQW6.2µm of 0.72 µm is measured in +source f, a very young, highly enshrouded star cluster +revealed by JWST NIRCam (Linden et al. 2022). The +remaining apertures have a range of values from ∼ 0.20– +0.65µm tracing the presence of extended star formation +and the influence of the AGN in the SW core. +3.3. Silicate Absorption Strength +To compare with previously published values we also +calculate the apparent 9.7µm silicate absorption feature +strength in a manner consistent with Spoon et al. (2007) +measurements of PAH-dominated sources. We assume +an extrapolated power law fit for the continuum, con- +strained by portions of the spectrum at 5.5µm and 14µm +and take the natural logarithm of the ratio of the ob- +served and interpolated continuum flux density at 9.7µm +(s9.7µm). +The majority of the s9.7µm values measured range from +-0.73 to -1.44, bracketing the published Spitzer IRS mea- +surement of -0.98, with the exception of the NE core (a). + +7 +Visual inspection of the NE core’s spectrum indicates a +much deeper silicate absorption feature, measured to be +s9.7µm=-2.45. For absorption dominated sources Spoon +et al. (2007) used a spline fit to determine a continuum. +If we assume the NE core is absorption dominated and +apply the same process, the measured s9.7µm would be +stronger (-2.87). Several other features unique to the +spectrum of the NE source are shown in Fig. 3 including +aliphatic hydrocarbon absorption and crystalline silicate +absorption, both observed in ULIRGs with deep 9.7µm +silicate absorption (Spoon et al. 2007). We note that +these strong absorption features are not seen in either +the SW (c) core or the embedded cluster (f). +3.4. Molecular Absorption Features +The +MIRI +spectra +allow +high +resolution +vibration−rotation spectroscopy of gas-phase molecules +towards dust-enshrouded regions in the VV 114 system. +We report the detection in absorption of the ν2 14.04 +µm bending mode of hydrogen cyanide (HCN) and ten- +tatively also the ν5 13.7 µm bending mode of acetylene +(C2H2) towards the NE Core (a). +C2H2 is a key in- +gredient in the gas-phase formation of large molecules +such as HC3N, and HCN is one of the most abundant +nitrogen bearing molecules in dense (n > 1 × 104 cm−3) +molecular clouds. +These lines have been previously detected by Spitzer +towards dust enshrouded young stellar objects (YSOs) +(e.g. Lahuis & van Dishoeck 2000) and towards lumi- +nous and obscured infrared galaxies (LIRGs) (Lahuis +et al. 2007). Lahuis et al. (2007) find HCN column den- +sities ranging between N(HCN)=1−12×1016 cm−2 and +warm gas with excitation temperatures Tex=230-700 K. +Owing to the higher spatial and spectral resolution, the +JWST MIRI spectrum of the NE core is more complex +than those found by Lahuis et al. (2007) using Spitzer. +We perform preliminary fits using the methodology +of Lahuis & van Dishoeck (2000) which assumes LTE. +Our preliminary results are consistent with an excitation +temperature of 300-500 K and an N(HCN) of 1−5×1016 +cm−2. +For an HCN abundance (with respect to H2) +of 10−8-10−7 (e.g. +Schilke et al. 1992; Lahuis & van +Dishoeck 2000; Harada et al. 2013), this would be con- +sistent with a high obscuration with N(H2) 1023-1024 +cm−2. The HCN and C2H2 spectra require further anal- +ysis, including multiple temperature component model- +ing and inclusion of non-LTE effects (Buiten et al. in +prep). +It is also possible that the nuclear obscuration is even +higher with column densities in excess of N(H2) 1025 +cm−2–the so called Compact Obscured Nuclei (CONs, +e.g. Aalto et al. 2015). Such objects are characterized by +high mm and submm continuum surface brightness and +luminous emission from mm-wave rotational transitions +of HCN within the vibrationally excited ladder (HCN- +vib) (e.g. Sakamoto et al. 2010; Aalto et al. 2015, 2019; +Sakamoto et al. 2021; Falstad et al. 2021). +For such +deeply enshrouded objects, the HCN 14 µm line, or the +silicate absorption, may not trace the full N(H2), but +only its surface. The mm/submm HCN-vib line (and +the mm/submm continuum) would probe deeper, re- +vealing the full obscuring column. Falstad et al. (2021) +propose that CONs have surface brightness of HCN-vib +of Σ(HCN-vib)> 1 L⊙pc−2. A recent ALMA study by +Saito et al. (2018) does not detect HCN-vib emission +towards the NE Core of VV 114 with a limit of Σ(HCN- +vib)< 0.12 L⊙pc−2. The relatively faint mm continuum +found by Saito et al. (2017) is consistent with the HCN- +vib non-detection. This suggests that the NE Core ei- +ther does not fulfil the CON criteria of Falstad et al. +(2021) or that the radius of the CON region is smaller +than 12 pc. +4. DISCUSSION +The JWST spectra and imaging resolve a blend of +diffuse emission from star formation and shocks, several +reddened star-forming knots, and an AGN. Although +the resolved spectra cover a much smaller region, our +results confirm the analysis of MIRI imaging data in +Evans et al. (2022). The SW core (c) has an elevated +continuum at ∼ 5µm, consistent with the presence of +dust in thermal equilibrium at temperatures near the +sublimation temperature of the silicate grains (∼1200 +K). This elevated continuum has been demonstrated by +Laurent et al. (2000); Petric et al. (2011) (and refer- +ences therein) as a telltale sign of a dust enshrouded, +optically thick AGN. We also find PAH emission in all +of our extracted apertures and widespread diffuse emis- +sion in our emission line maps. The various emission +and absorption features allow us to directly probe the +nature of the bright cores and diffuse emission as well +as the kinematics of the gas in VV 114E. +4.1. Evidence for AGN Activity +Previous multiwavelength observations of VV 114 +hinted at an AGN contribution in the X-ray, mid-IR, +and submm (e.g. Le Floc’h et al. 2002; Grimes et al. +2006; Iono et al. 2013). As the brightest compact sources +from the mid-IR to radio, the NE and SW cores are the +most likely to harbor an AGN, with Iono et al. (2013) +identifying the NE core as a potential AGN. We do not +detect coronal lines in either the NE or SW core, but +this does not rule out an AGN; the ULIRG Mrk 231 is +a well-known optically classified Seyfert 1 with no ob- +served coronal lines in the mid-IR. Instead, very low + +8 +Figure 4. NIRSpec spectra and PAH diagnostic plots used (a): NIRSpec Spectra of the NE and SW cores in VV 114. The SW +core shows a strongly rising continuum characteristic of hot dust associated with an AGN. The 3.3 µm PAH feature is detected +in both spectra, but is significantly weaker in the SW core. (b) & (c): expanded region and fit to the 3.3 µm PAH feature in +the NE (b) and SW (c) cores. (d): Equivalent width of the 6.2µm PAH feature plotted vs. Silicate strength. Measurements +for our MIRI apertures plotted as colored circles. Values measured with Spitzer/IRS for ULIRGs and LIRGs from the GOALS +sample are plotted as “+” symbols, with VV 114E and three comparative ULIRGs denoted with red diamonds (Stierwalt et al. +2013). Starburst galaxy M82 is shown as a blue square (Spoon et al. 2007). (e): Equivalent widths of 6.2µm and 3.3µm PAH +features for the SW and NE cores (Inami et al. 2018). (f): Flux Density ratio and 3.3µm PAH EQW for the SW and NE cores +(Inami et al. 2018). The SW core lies near AGN-dominated ULIRGs in both diagnostic plots. +PAH equivalent width and silicate absorption indicate +the presence of the AGN in Mrk 231 (Armus et al. 2007; +Imanishi et al. 2010; Inami et al. 2013; Stierwalt et al. +2014). +To compare with other AGN and starburst dominated +LIRGs we plot our measured values on several PAH +diagnostic diagrams (Fig. +4). +We first plot s9.7 vs. +EQW6.2µm (Fig. 4d): the apertures extracted from dif- +fuse emission (with the exception of d) as well as the +star forming clump in aperture f have values consistent +with starburst galaxies and lie near the literature val- +ues for M82 (Spoon et al. 2007). The SW core shows +contribution from an AGN (e.g. Spoon et al. 2007; Mar- +shall et al. 2018) and lies near Mrk 273, a ULIRG with +a Seyfert 2 nucleus (Armus et al. 2007; Stierwalt et al. +2013; U et al. 2013). Aperture d is directly adjacent to +the SW core, in fact partially overlapping, and is likely +showing some contribution from the AGN that is more +clearly seen in the SW core. The NE core has a corre- +spondingly higher obscuration and lies near the values +observed for the ULIRG Arp 220, a deeply embedded +starburst that also has C2H2 and HCN absorption crys- +talline silicate features in its mid-IR spectrum (Spoon +et al. 2006; Lahuis et al. 2007). + +10- +(a) +a(NE) +(d) +NE Core +.3 +b +c(SW) +NE Core +SW Core +d +S9.7 +SW Core +Mrk 231 +0 +100 +EQW6.2μm +0.8- +0.7 +PAH) +0.6 +0. +10-3 +u 0.4 +2 +2.5 +4.5 +2.0 +3.0 +3.5 +4.0 +5.0 +Wavelength [um] +NE Core +0.2 +Q +NE Core +SW Core +asw Core +le-12 +0.0- +2.0. + Spline Fit +Spline Fit +(b) +(c) +6.0- +0.00 +0.02 +0.08 +0.10 +0.12 +0.14 +0.16 +0.04 +0.06 +Local Continuum +Local Continuum +EQW(3.3μm PAH) +1.8- +5.5 +_un +0.16- +(f) +5.0- +1.4- +AH) +0.14- +cm- +cm + NE Core +0.12 +P +4.5 +1.2 - +μm +I-s +0.10- +s +30.081 +1.0- +4.0. +3 + Starburst dominated +0.06. +W +AGN dominated +0.8 +0.6 +0.02 + SW Core +3.0- +0.00- +100 +101 +0.4 +F(4.3μm)/F(2.8μm) +3.0 +3.1 +3.2 +3.3 +3.2 +3.5 +3.0 +3.1 +3.3 +3.4 +3.4 +3.5 +Wavelength [um] +Wavelength [um]9 +The spectrum of the SW core at shorter IR wave- +lengths is consistent with the heating and processing of +dust by an AGN, which reduces the 3.3µm PAH EQW +(Fig. +4a-c) as well as the 6.2µm EQW (EQW3.3 < +0.04µm and EQW6.2 < 0.20µm, Imanishi et al. 2010; +Petric et al. 2011). The 3.3µm PAH feature is directly +adjacent to a broad absorption feature caused by H2O +ice that may impact our continuum measurement and +in turn the measured EQW3.3µm (Imanishi et al. 2008; +Inami et al. 2018). Moreover, the attenuation of the con- +tinuum and the 3.3µm feature may be different depend- +ing on the geometry of the dust and PAH emisison (Lai +et al. 2020). We make a simple estimate of the impact by +performing a power law fit to the 1.7-5.0µm spectrum +and divide our integrated flux by the continuum flux +density at 3.3µm. This decreases the EQW3.3µm slightly +from 0.12µm and 0.017µm to 0.11µm and 0.015µm for +the NE and SW core, but does not affect our conclusions +regarding the nature the NE and SW cores. +Inami et al. (2018) used AKARI to propose revised +AGN diagnostics including the 3.3 µm PAH feature as +well as the Fν(4.3)/Fν(2.8) flux density ratio. +When +placed on a plot of EQW6.2µm vs. EQW3.3µm (Fig. 4e, +following Inami et al. 2018), the SW core lies in a region +populated by known AGN while the high EQW3.3µm of +the NE core is more consistent with ULIRGs dominated +by star formation. We also place the NE and SW core +on the EQW3.3µm vs. flux density ratio diagnostic pro- +posed by Inami et al. (2018) in Fig. 4f. The SW core +again clearly lies in the AGN-dominated portion of the +diagram. +To estimate the contribution of the AGN in the SW +core to the total luminosity of VV 114 we take the es- +timation of LIR ∼ 5 ± 0.5 × 1010L⊙ by Evans et al. +(2022), about 12% of the total LIR, as an upper limit. +The presence of PAH features in the SW core indicates +a blended contribution to the IR of star formation and +AGN. Sources with similar EQW3.3 and colors have a +bolometric AGN contribution of ∼ 30 − 50% (D´ıaz- +Santos et al. 2017; Inami et al. 2018), which means the +contribution of the AGN to the total luminosity of the +entire VV 114 system is ∼ 5%. +Although Iono et al. (2013) identified the NE core +as potentially harboring an AGN due to the enhanced +HCN/HCO+ ratio, an analysis of X-ray and millime- +ter data by Privon et al. (2020) showed that in fact the +HCN/HCO+ is not a robust indicator of total AGN lu- +minosity or its fractional contribution to infrared lumi- +nosity. +Our findings regarding the nature of the two +bright cores in VV 114E are not in agreement with the +analysis of JWST MIRI data by Donnan et al. (2022) +who propose the NE core as a potential AGN host based +on its compact nature. +Interestingly, the NE core does have a somewhat low +EQW6.2µm despite a strong EQW3.3µm, placing it in a +region of Fig. 4e with few other galaxies. The galaxies +with the most similar values to the NE core are II Zw 96 +and IRAS F19297-0406, which also have similar s9.7µm +when comparing the values measured using Spitzer for +all three galaxies (Stierwalt et al. 2013). II Zw 96 is an +unusually compact and powerful starburst (Inami et al. +2010, 2022) and IRAS F19297-0406 has a powerful star- +burst driven outflow (Soto et al. 2012; Veilleux et al. +2013), physical conditions which may warm dust in a +way that lowers EQW6.2µm. The curious nature of both +the NE and SW cores warrants a more thorough follow- +up analysis decomposing the relative contribution of the +stellar, PAH, dust, and AGN components of the SED. +4.2. Shocks and Tidal Features +Evidence for extended shocks in VV 114 has pre- +viously been suggested at visible wavelengths via en- +hanced [O I] and [S II] emission and broadened line pro- +files across VV 114 (Rich et al. 2011, 2015) and in the +submm from the presence of methanol (CH3OH) in the +“overlap” region between VV 114W & E. Apertures f, g, +h, and i lie closest to the overlap region and both emis- +sion line ratios and molecular and atomic line widths +show evidence of shocked gas. +The atomic lines in aperture f all have narrow line +widths (∼100-150 km/s) and ratios typical of buried, +young star-forming regions as this source was revealed +to be in (Linden et al. 2022). Aperture i encompasses a +fainter star forming knot seen in both MIRI and NIR- +CAM imaging, and several emission line profiles show +a double peaked profile. The broader H2 lines in aper- +tures f and i (∼200-300 km/s) are similar to the values +extracted just to the north in apertures g and h, which +show elevated [Fe II]/Pfα values indicative of shocks, +also seen in MRS observations of NGC 7469 (U et al. +2022). +These regions are consistent with ALMA observations +of VV 114 where the elevated FWHM of 150-300 km/s +in the molecular gas (e.g. CO(1-0), CH3OH) is found +in the “overlap” region and is suggested to be the prod- +uct of both shocks, and overlapping kinematic compo- +nents due to the merger (Saito et al. 2017, 2018) and +may be similar in nature to the shocked bridge seen +in Stephan’s Quintet (Guillard et al. 2012b; Appleton +et al. 2017). However, the clusters in this region of VV +114 are 1-2 orders of magnitude more massive and ∼2-3 +times dustier than those seen in the bridge of Stephan’s +Quintet (Fedotov et al. 2011; Linden et al. 2022). + +10 +Resolved emission line profiles in aperture j show a +strong indication of both blue and red-shifted wings, po- +tentially due to projection effects of the tidal arm that +extends from VV 114W across the IR bright cores south- +ward beyond the FOV of our aperture extractions (Saito +et al. 2015). These values are supported by examining +the [Ne II] emission line profiles which fall in Channel +3 and has a wider FOV than Channel 1. Looking sev- +eral arcseconds southeast of the SW core, the [Ne II] +line profiles show significant broadening and wings in +several spaxels (Fig. 2). We also see enhanced H2/[Ne +II] in this region and at the furthest SE region of our +spaxel-by-spaxel maps. +The value of [Fe II]/Pfα ∼ 9 +in aperture j also indicates the presence of shocked gas +which likely extends beyond the Channel 1 FOV and fol- +lows the elevated emission line ratios caused by shocks +seen in visible light IFU observations (Rich et al. 2011). +Aperture b appears to be dominated by diffuse emis- +sion when examining the MIRI and NIRCAM images, +but does not display the same characteristics of shock +excitation as the other apertures that trace diffuse gas. +Previous observations of shock excitation in VV 114 +have suggested both galactic winds and tidally driven +gas flows as sources of shock excitation (Rich et al. 2011; +Saito et al. 2017). The JWST data show some shocked +gas coincident with tidal features previously observed +in the submm and radio as well as kinematic profiles +potentially associated with galactic winds. +Follow-up +work mapping the two dimensional kinematics of the +molecular and atomic gas, as well as the temperature +and distribution of H2 gas using these data will provide +a more complete picture of the shock excitation in VV +114, especially when combined with wide field optical +IFU observations from MUSE. +4.3. Star Formation Rates +If we assume that the atomic and fine structure line +emission is dominated by star formation in the aper- +tures with bright star-forming clumps (a, e, f), we can +calculate star formation rates using a hydrogen recom- +bination line flux or Neon emission. If we scale either +Pfund or Humphreys α to Hα assuming Case B recombi- +nation (Hummer & Storey 1987) and use equation (2) in +Murphy et al. (2011) the estimated SFR for each spec- +trum is ∼ 1M⊙/yr. Using the [Ne II] and [Ne III] fluxes +and equation (3) in Ho & Keto (2007) yields an SFR +from ∼ 1.5 − 2.5M⊙/yr. +These values are consistent +with those found by Song et al. (2022) using the radio +emission of bright knots measured with the VLA and +amount to ∼ 2 − 3% of the total SFR per aperture. +5. CONCLUSIONS +Our analysis of MIRI MRS and NIRSpec IFU spec- +troscopy of VV 114E shows emission and absorption fea- +tures that allow us to resolve variations in the proper- +ties of the IR bright nuclear cores, unresolved clumps, +and diffuse gas. +The integrated properties of the +two bright cores and the diffuse gas agree with past +multi-wavelength observations of VV 114E that show +widespread star formation and diffuse emission from +shocks, and reveal spectroscopic evidence of an AGN +in the SW core. More specifically: +• The SW core harbors an AGN as indicated by +its extremely low 3.3 and 6.2 µm PAH equiva- +lent widths and strong 3-5µm continuum, consis- +tent with AGN-dominated LIRGs. The SW core is +also surrounded by star forming knots and diffuse +emission, which is evident in the atomic line ratios +at longer wavelengths. The AGN in the SW core +likely accounts for ∼ 5% of the total luminosity of +VV 114. +• The NE core is deeply embedded, its mid-IR +spectrum displays strong 9.7µm silicate absorp- +tion, crystalline silicate features, aliphatic hydro- +carbons, and HCN absorption. Using the 14µm +HCN absorption line we calculate a temperature +and column density of 300-500 K and N(HCN) of +1 − 5 × 1016 cm−2. Our data show no evidence +of an AGN impacting the atomic or molecular gas +at mid-IR wavelengths and we conclude that this +source is a deeply buried star forming region. +• The diffuse gas NW of the nuclear region shows el- +evated [Fe II]/Pfα and higher H2 line widths along +with double peaked profiles caused by shocked +gas in the overlap region previously observed by +ALMA. +A fit to [Ne II] across the MIRI FOV +reveals broader emission profiles to the south of +the nucleus consistent with the extended tidal fea- +ture observed by ALMA, as well as shocked gas +∼1.5′′ SE of the SW Core that likely extends be- +yond the Channel 1 FOV, consistent with previous +visible light IFU observations. +These early 3D spectral data highlight the power of +combining the NIRSpec and MIRI data to elucidate the +nature of complex, obscured star formation and AGN +in the local Universe. Taken together the spectroscopic +datasets from both JWST instruments are extremely +rich and will facilitate detailed and thorough analysis in +future papers in this series. +ACKNOWLEDGEMENTS + +11 +We thank the referee for their helpful comments. +This work is based on observations made with the +NASA/ESA/CSA JWST. The research was supported +by NASA grant JWST-ERS-01328. The data were ob- +tained from the Mikulski Archive for Space Telescopes +at the Space Telescope Science Institute, which is op- +erated by the Association of Universities for Research +in Astronomy, Inc., under NASA contract NAS 5-03127 +for JWST. The specific observations analyzed can be +accessed via +10.17909/yqk1-jr92. +VU acknowledges +funding support from NASA Astrophysics Data Anal- +ysis Program (ADAP) grant 80NSSC20K0450. +The +Flatiron Institute is supported by the Simons Founda- +tion. +AMM acknowledges support from the National +Science Foundation under Grant No. +2009416. +ASE +and SL acknowledge support from NASA grants HST- +GO15472 and HST-GO16914. YS was funded in part by +the NSF through the Grote Reber Fellowship Program +administered by Associated Universities, Inc./National +Radio Astronomy Observatory. +The National Radio +Astronomy Observatory is a facility of the National +Science Foundation operated under cooperative agree- +ment by Associated Universities, Inc. +F.M-S. ac- +knowledges support from NASA through ADAP award +80NSSC19K1096. SA gratefully acknowledges support +from an ERC Advanced Grant 789410, from the Swedish +Research Council and from the Knut and Alice Wallen- +berg (KAW) Foundation. SA gratefully acknowledges +John Black for helpful discussions. +KI acknowledges +support by the Spanish MCIN under grant PID2019- +105510GB-C33/AEI/10.13039/501100011033. +HI and +TB acknowledge support from JSPS KAKENHI grant +No. +JP21H01129 and the Ito Foundation for Pro- +motion of Science. +This work was also partly sup- +ported by the Spanish program Unidad de Excelen- +cia Mara de Maeztu CEX2020-001058-M, financed by +MCIN/AEI/10.13039/501100011033. +The computa- +tions presented here were conducted through Carnegie’s +partnership in the Resnick High Performance Comput- +ing Center, a facility supported by Resnick Sustainabil- +ity Institute at the California Institute of Technology. +Finally, this research has made use of the NASA/IPAC +Extragalactic Database (NED) which is operated by the +Jet Propulsion Laboratory, California Institute of Tech- +nology, under contract with the National Aeronautics +and Space Administration. +Facilities: JWST (NIRCam, NIRSpec and MIRI) +REFERENCES +Aalto, S., Mart´ın, S., Costagliola, F., et al. 2015, A&A, +584, A42 +Aalto, S., Muller, S., K¨onig, S., et al. 2019, A&A, 627, A147 +Appleton, P. 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+page_content=' 530 McCormick Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Charlottesville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' VA 22903,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' USA 5Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' University of Crete,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Heraklion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 71003,' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Hiroshima University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 1-3-1 Kagamiyama,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Higashi-Hiroshima,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Hiroshima 739-8526,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Japan 11Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' University of Massachusetts at Amherst,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' USA 13AURA for the European Space Agency (ESA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Space Telescope Science Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 3700 San Martin Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Baltimore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' MD 21218,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' USA 14Space Telescope Science Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 3700 San Martin Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Baltimore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' MD 21218,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' USA 15Department of Physics & Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 430 Portola Plaza,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Los Angeles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' CA 90095,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' USA 16Department of Physics & Astronomy and Ritter Astrophysical Research Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' University of Toledo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Toledo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' OH 43606,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='USA 17ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Australia 18Department of Physics and Astronomy, 4129 Frederick Reines Hall, University of California, Irvine, CA 92697, USA 19Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands 20School of Physics and Astronomy, Monash University, Clayton, VIC 3800, Australia 21Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA 22IPAC, California Institute of Technology, 1200 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' California Blvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Pasadena, CA 91125 23Institut de Ci`encies del Cosmos (ICCUB), Universitat de Barcelona (IEEC-UB), Mart´ı i Franqu`es, 1, 08028 Barcelona, Spain 24ICREA, Pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Llu´ıs Companys 23, 08010 Barcelona, Spain 25Institut de Ciencies de l’Espai (ICE, CSIC), Can Magrans, s/n, 08193 Bellaterra, Barcelona, Spain 26Institut d’Estudis Espacials de Catalunya (IEEC), E-08034 Barcelona, Spain 27Glendale Community College, 1500 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Verdugo Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Glendale, CA 91208 28Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 29Department of Physics and Materials Science, The University of Memphis, 3720 Alumni Avenue, Memphis, TN 38152, USA 30National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA 31Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822 32Division of Physics, Mathematics and Astronomy, California Institute of Technology, 1200 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' California Blvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Pasadena, CA 91125 33Physics Department, 1600 Campus Road, Occidental College, Los Angeles, CA 90041, USA Submitted to ApJ Letters ABSTRACT We present results from the James Webb Space Telescope (JWST) Director’s Discretionary Time Early Release Science (ERS) program 1328 targeting the nearby, Luminous Infrared Galaxy (LIRG), VV 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We use the MIRI and NIRSpec instruments to obtain integral-field spectroscopy of the heavily obscured Eastern nucleus (V114E) and surrounding regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The spatially resolved, high- resolution, spectra reveal the physical conditions in the gas and dust over a projected area of 2-3 kpc that includes the two brightest IR sources, the NE and SW cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Our observations show for the first time spectroscopic evidence that the SW core hosts an AGN as evidenced by its very low 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='02338v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='GA] 6 Jan 2023 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm PAH equivalent widths (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='12 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='017 µm respectively) and mid and near-IR colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Our observations of the NE core show signs of deeply embedded star formation including absorption features due to aliphatic hydrocarbons, large quantities of amorphous silicates, as well as HCN due to cool gas along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We detect elevated [Fe II]/Pfα consistent with extended shocks coincident with enhanced emission from warm H2, far from the IR-bright cores and clumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We also identify broadening and multiple kinematic components in both H2 and fine structure lines caused by outflows and previously identified tidal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Keywords: galaxies: star formation, interactions, evolution infrared: galaxies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' INTRODUCTION VV 114 (Arp236, IC1623) is an interacting system un- dergoing vigorous starburst activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' With an infrared luminosity of LIR ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='5 × 1011 L⊙, and a distance of 80 Mpc, it is one of the brightest objects in the IRAS Bright Galaxy Sample (Soifer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' It appears to be an early-stage merger of two galaxies that are aligned east- west with a projected nuclear separation of ∼8 kpc, des- ignated in the literature as VV 114E and VV 114W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' At optical wavelengths, VV 114 shows a highly disturbed morphology with very faint tidal tails extending over 25 kpc from the center (Arp 1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The western compo- nent, VV 114W, is more extended than the eastern one, and dominates the emission at short wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Much of the mid-infrared emission is diffuse and extended over several kpc with some indication of an AGN based on the mid-infrared color of the more compact nuclear re- gion in VV 114E (Le Floc’h et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' ALMA observa- tions show abundant cold, dense gas (traced by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' CO, HCO, HCN), evidence for shocked gas in the overlap re- gion between the two galaxies (traced by Methanol), a molecular outflow, and a possible buried AGN in VV 114E (Iono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2015, 2017, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The majority of the IR emission and by extension to- tal energy output of the system is dominated by VV 114E and the extreme UV/optical to IR ratios of VV 114 make it a more plausible analog to high-z IR lumi- nous mergers (Charmandaris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Mid-infrared spectra centered on VV 114E taken with Spitzer IRS (∼ 10′′ × 36′′) show a moderately strong 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7µm Silicate absorption (s9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7µm = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='98), a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm PAH equivalent width intermediate to LIRGs dominated by star formation or AGN (EQW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='30µm), and a ratio of H2 to PAH luminosity slightly above values associated with photodissociation region emission (Guil- lard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2012a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Stierwalt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2013, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' No coronal lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' [Ne V]) were detected in the Spitzer spectra, and fine structure line flux ratios were consistent with LIRGs primarily dominated by star formation with some composite AGN/starburst activity (Inami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Observations with Chandra, XMM, and NuSTAR of VV 114E indicate that the X-ray emission appears to be gen- erated primarily through star fomation, with little to no X-ray emission coming from an AGN (Grimes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Garofali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Ricci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Visible wavelength integral field spectroscopy indicates a mix of star formation and shock emission, the latter indi- cated by elevated emission line ratios and line widths across both galaxies (Rich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2011, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Finally, in a companion paper to the present study, Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2022) propose the presence of a reddened starburst and an AGN in the bright NE and SW cores, respectively, of VV 114E based on broadband JWST mid-IR colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' In this paper we present the James Webb Space Tele- scope (JWST) combined MIRI/NIRSpec integral field spectroscopic observations of the nucleus of VV 114E and surrounding regions taken as part of the Early Re- lease Science (ERS) program 1328 (Co-PIs: L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Armus and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Evans).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The data allow us to resolve the proper- ties of the two brightest sources, the NE and SW cores, as well as star clusters and diffuse emission surrounding the eastern nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Throughout the paper we adopt a cosmology of Ho=70 km s−1 Mpc−1, ΩM=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='28, and ΩΛ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The redshift of VV 114 (z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0202) corresponds to an angular scale of 400 pc/1′′(Wright 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' OBSERVATIONS, DATA REDUCTION VV 114 was observed with MIRI Rieke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Labiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2021) on 2 July 2022 (MIRI imaging, Bouchet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2015), on 5 July 2022 (MRS spectroscopy, Wells et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2015), and by NIRSpec (Jakobsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2022) on 19 July 2022 (IFU spectroscopy, B¨oker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We include MIRI imaging in this paper to indi- cate the locations of our spectral extraction apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The full analysis of these imaging data products are de- scribed in Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' MIRI MRS Data The MIRI MRS observations include three grating set- tings (SHORT, MEDIUM, and LONG) in order to cover 3 WFC3 F435W/F814W 2 kpc MIRI F770W 1h07m47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='8s 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='6s 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='4s 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2s 17°30\'20" 22" 24" 26" 28" Right Ascension Declination 2 kpc aa bb cc dd ee ff gg hh ii jj MIRI F770W Sub 5 10 15 20 25 Rest Wavelength [ m] 10 1 101 103 105 107 109 1011 Jy (relative) c(SW) d a(NE) h j g b i e f 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2 m PAH 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7 m Si [Fe II] [Ar II] Pf [Ar III] H2 S(3) [S IV] H2 S(2) [Ne II] [Cl II] [Ne III] H2 S(1) [S III] [Fe III] Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Images and spectra of VV 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Top-Left: HST 435W/814W color image, Middle-Left: JWST F770W image with the same FOV, the green box corresponds to the MIRI SUB128 subarray FOV, as shown in the Bottom-Left panel, Bottom-Left: F770W SUB128 image with MIRI MRS channel 1 FOV (dark blue Box), NIRSpec FOV (light blue box) and our ten 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='4′′ radius extraction apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Right: Full MIRI spectra of the 10 regions marked in the F770W subarray image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Spectra are shown at rest frame wavelengths assuming a systemic velocity of 6056 km/s and sorted from top to bottom in order of decreasing 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2 µm PAH equivalent width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' the entire wavelength range accessible to the four IFU channels (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='9-27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='9µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' A four-point dither pattern was employed to recover extended emission and avoid sat- uration, with separate off-source observations for back- ground subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The field of view (FOV) and posi- tion angle (PA) vary by channel, but the FOV with full wavelength coverage is defined by channel 1, ∼ 5′′ × 4′′ at PA=255° as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Our procedure for reducing MRS data for the ERS 1328 targets is described in detail in U et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2022), but a brief summary is given here: uncalibrated data are processed using the most recently available de- velopmental release of the JWST Science Calibration Pipeline (Bushouse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2022), version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3, using calibration reference data system (CRDS) context file jwst 0963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='pmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The resulting data products generated by the pipeline are 12 background-subtracted, fringe- corrected, wavelength and flux calibrated data cubes– one for every combination of channel and grating set- ting combining the four dither pointings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' These cubes are used to generate 1-d spectra of regions of interest and 2-d maps of the properties of particular emission line features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We extract spectra from 10 regions of interest within the channel 1 FOV to investigate IR-bright sources re- solved in the MIRI imaging as well as diffuse emission 4 surrounding those sources and in the eastern nucleus (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We chose five locations coincident with the two brightest IR sources, the NE and SW cores (a, c), and several bright clumps identified in the MIRI imaging data and in both the submm and radio (d, e, f) (Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We also chose five regions intended to capture diffuse emis- sion between (b) and around (g, h, i, j) the bright clumps coincident with some tidal and shock features previously identified using ALMA (Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2017, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Spectra are extracted from each of the 12 data cubes using apertures with a radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='4′′(160 pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' This re- sults in twelve 1-d spectra for each extracted region, with some overlap in adjacent wavelength regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The 12 individual bands for each spectra have slight off- sets in flux (a few percent) which are multiplicatively scaled, trimmed, and stitched in order to create continu- ously smooth 1-d spectra over the full MIRI wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' This process begins by using the overlapping wavelength region to scale Channel 4 MEDIUM spec- trum to the longest wavelength Channel 4 LONG spec- trum (∼ 23 − 25µm), trimming the overlapping values from the noisier spectrum of the two, and continuing the process to channels at shorter and shorter wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Finally, a wavelength dependent aperture correction is applied to each spectrum (U et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' NIRSpec IFU Data NIRSpec IFU observations were taken with three grating and filter combinations: G140H/F100LP, G235H/F170LP, and G395H/F290LP to cover the spec- tral range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='97-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Calculations in this paper were made using the wavelength region covered by the G235H/F170LP and G395H/F290LP settings, ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' This wavelength range allows us to measure the AGN-sensitive 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm PAH feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Again a four-point dither pattern was employed to completely sample the PSF and to avoid saturation, with an additional “Leak- cal” image taken for each grating setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The FOV of the combined dither pattern is ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='6′′ × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='8′′ centered on the SW Core at a PA of ∼ 32◦ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We reduce the NIRSpec data in a similar fashion to the MRS data, using calibration reference data sys- tem (CRDS) context file jwst 1009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='pmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Uncalibrated four-point dither pattern science images and Leakcal images, one for each of the two NIRSpec chips, are downloaded using MAST resulting in 10 image files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' These are first put through Stage 1 processing with the Detector1 pipeline, which applies detector-level calibra- tions and produces count rate files calculated from the non-destructive “ramp” readouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' These rate files are then processed using the Spec2 pipeline, which applies Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Two dimensional images of flux (a, d), relative velocity (b, e), and FWHM (c, f) generated from the spaxel- by-spaxel fits to the [Ne II] (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='8 µm) and H2 S(2) (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 µm) emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The FWHM shown is corrected for instru- mental broadening as described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The relative ve- locity map is generated by subtracting a systemic velocity of 6056 km/s from each spaxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The broadest H2 FWHM in the NW is just outside the Ch1 FOV but falls within the Ch3 FOV, and corresponds to the eastern edge of the shocked “overlap” region observed in Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The ele- vated FWHM in [Ne II] is near apertures that show evidence of shocks and double peaked emission line profiles (g, h, i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' physical corrections and flux and wavelength calibra- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' At this step in the overall pipeline, the Leakcal files are also used to correct for any stray light that may fall on the detector due to failed open MSA shutters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Finally we run the Spec3 pipeline step, which produces a final combined data cube sampled with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='1′′ spaxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' For our analysis in this paper, we extract from the final data cube spectra from the two brightest IR sources, the NE and SW cores (a and c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We matched our apertures to the MIRI MRS extraction radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='4′′ centered at the same two locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' For the G395H/F290LP setting this produces flux and wavelength calibrated 1-d spectra covering 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='87-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='27µm, with a gap in coverage from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='06- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='18µm in the middle of the spectrum due to the gap be- tween the two NIRSpec chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' For G235H/F170LP the range is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='66-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='17µm with a gap from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='40-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='45µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We use the overlapping wavelength range between the NIR- Spec and MIRI data to scale the G395H/F290LP spec- trum to match the MIRI spectrum, and the overlapping region between the two NIRSpec settings to scale and stitch the shorter wavelength spectra (G235H/F170LP) to the longer wavelength NIRSpec spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' RESULTS 10-19 100 100 200 3005 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Spectral Feature Strengths, Fluxes, and FWHM Region ID EQW3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 EQW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2 s9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7 [Fe II] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='34 FWHM Pfα FWHM a (NE Core) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='121±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='264±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='018 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='45±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='03 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='41±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='22 194±13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='56±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='62 142±37 b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='514±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='017 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='13±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='03 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='29 186±12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='83±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='58 148±21 c (SW Core) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='017±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='106±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='04 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='32±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='57 147±29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='17 160±73 i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='56±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='73±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='39 231±26 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='36 224±44 j 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='359±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='30±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='14 270±17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='07 243±43 Values of the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm EQW, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7µm silicate strength, as well as line fluxes (10−18 W/m−2 ) and FWHM (km s−1, corrected for instrumental broadening) for emission line features measured in our extracted regions that were used in our analysis and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' This table is a subset of the total measurements, a machine readable version of the full table is available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We use the 1-d aperture extracted spectra to measure emission and absorption features that trace the physical properties of the gas and dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Several of our apertures are centered on bright, unresolved mid-IR sources seen in the MIRI imaging observations (Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2022), including the bright NE and SW cores (a & c), a source directly SE of the SW core (d), and a deeply embedded star cluster (f) with Mass M∼ 106 M⊙, age t ∼ 1 − 2 Myr, and extinction AV ∼ 8 (see Linden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The remaining apertures trace diffuse emission generally showing elevated EQW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm and strong H2 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Emission Line Properties We perform fits to features in each 1-d MIRI spec- trum using the “lmfit” package (Newville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2014) with resulting values given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Atomic and H2 emission lines are fit with a single Gaussian component combined with a polynomial fit to the local continuum over a range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='1µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The resulting Gaussian param- eters are used to derive the observed flux of each line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The width of the Gaussian fit is used to determine the intrinsic FWHM of each emission line by subtracting in quadrature the instrumental resolution of MRS at the observed wavelength (Labiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Several fine structure lines and H2 lines are well de- tected and resolved, as well as the hydrogen recombina- tion lines Pfund α and Humphreys α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Emission line ratios show variation between apertures indicative of both widespread star formation and shock excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We do not detect any of the high-ionization coronal lines typically found in mid-IR spectra of AGN dominated galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' [Ne V];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Genzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Lutz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Weedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Armus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2007) in contrast to JWST observations of NGC 7469 that show nine well-detected coronal lines excited by high-energy photons from the central AGN (Armus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Emission line ratios sensitive to AGN activity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='g [S IV]/[Ne II], [O IV]/[Ne II]) show values indicative of a composite of AGN and starburst activity (Inami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Apertures g, h and j show elevated values of [Fe II]/Pfα (> 5) and g & j show [Fe II] FWHM higher than the other apertures (200-300 km/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Apertures a, b, c, d, and e have fine structure lines ([Ar II], [Ar III], [Ne II], [Cl II], [Ne III], [S IV]) with FWHM ranging between 150-200 km/s and show no trend with emission line ionization potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' In the NE core (a) the H2 FWHM is ∼50 km/s narrower than the fine structure lines, while in the SW core (c) the H2 and fine structure line widths are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' This trend is reversed in aperture f where the H2 FWHM are ∼200 km/s vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' ∼100 km/s for the fine structure lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The broadest emission line widths in our data are observed in the spectrum of aperture i, which samples the diffuse H2 bright gas near the western edge of our FOV close to the “overlap” region defined in (Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' In order to assess the morphology and kinematics of the region observed with MIRI/MRS, we have also car- ried out spaxel-by-spaxel fits of the [Ne II] 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='8 µm fine structure emission line and H2 S(2) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 µm molecu- lar emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' These two emission lines are gener- ally quite luminous and are well detected across nearly every spaxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The lines are covered by Channel 3B (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='29-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='52µm) which provides a wider field of view (7′′ × 6′′) while still maintaining relatively high spectral resolution (R∼2800-3000) with somewhat larger spax- els (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Fits to the two emission lines are carried out on a spaxel-by-spaxel basis using the Channel 3B 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='25 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='50 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='00 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='75 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='00 100 101 Relative Flux (Jy) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='85 m Aliphatic C-H 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='25 m Aliphatic C-H a c f 14 16 18 20 22 100 101 16 m Crystalline Si Amorph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Si a c f 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2 Rest Wavelength [ m] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='20 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7 m C2H2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='04 m HCN a c f NE Core Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Expanded view of spectral features present only in the NE nucleus (region a) indicative of a highly embedded source, compared with spectra of the SW nucleus (c) and region f (embedded cluster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Dashed vertical lines in each panel correspond to the following features Top panel: Ab- sorption features due to aliphatic hydrocarbons at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='85 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='25 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Spectra are normalized to 1 Jy at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Middle panel: crystalline silicate absorption features at 16 and 19 µm combined with amorphous silicate absorption at 18 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Spectra are normalized to 1 Jy at 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Bottom Panel: C2H2 and HCN absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Spectra are normalized to 1 Jy at 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='8 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' sub-band data cube, with a single Gaussian component fit to each line independently, in the same manner as the aperture extracted spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The resulting maps are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The variation in the flux and FWHM of both the [Ne II] and H2 lines agrees with the values measured in our extracted apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The broadest FWHM in both lines lies outside the Channel 1 FOV, our closest aperture extractions are i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The increase in H2 FWHM at the NW corner of our map corresponds to a portion of the “overlap” region observed in Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2017) with a similar increase in FWHM seen at submm wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' PAH Equivalent Width We measure the equivalent width of the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 µm and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2 µm PAH emission features (EQW3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm, EQW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm) in the NIRSpec and MIRI spectra by applying the same method to both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' First, portions of the spectrum adja- cent to each PAH feature are used to perform a linear interpolation of the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We then integrate a spline fit from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='20–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='28 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='95–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='55 µm (rest frame, EQW3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm, EQW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm) to calculate the PAH feature flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The integrated flux is divided by the continuum flux density at the wavelength of the peak of each PAH feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The EQW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm values range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='11–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='72 µm, bracketing the published Spitzer IRS value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 µm (Stierwalt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The NE core has an EQW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='264µm and a relatively high EQW3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='121µm, while the SW core has a low EQW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='106µm and a very low values of EQW3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='016µm, indicative of an AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Imanishi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Petric et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2011, see discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The largest EQW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='72 µm is measured in source f, a very young, highly enshrouded star cluster revealed by JWST NIRCam (Linden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The remaining apertures have a range of values from ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='20– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='65µm tracing the presence of extended star formation and the influence of the AGN in the SW core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Silicate Absorption Strength To compare with previously published values we also calculate the apparent 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7µm silicate absorption feature strength in a manner consistent with Spoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2007) measurements of PAH-dominated sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We assume an extrapolated power law fit for the continuum, con- strained by portions of the spectrum at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='5µm and 14µm and take the natural logarithm of the ratio of the ob- served and interpolated continuum flux density at 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7µm (s9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The majority of the s9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7µm values measured range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='73 to -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='44, bracketing the published Spitzer IRS mea- surement of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='98, with the exception of the NE core (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 7 Visual inspection of the NE core’s spectrum indicates a much deeper silicate absorption feature, measured to be s9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7µm=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' For absorption dominated sources Spoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2007) used a spline fit to determine a continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' If we assume the NE core is absorption dominated and apply the same process, the measured s9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7µm would be stronger (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Several other features unique to the spectrum of the NE source are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 3 including aliphatic hydrocarbon absorption and crystalline silicate absorption, both observed in ULIRGs with deep 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7µm silicate absorption (Spoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We note that these strong absorption features are not seen in either the SW (c) core or the embedded cluster (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Molecular Absorption Features The MIRI spectra allow high resolution vibration−rotation spectroscopy of gas-phase molecules towards dust-enshrouded regions in the VV 114 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We report the detection in absorption of the ν2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='04 µm bending mode of hydrogen cyanide (HCN) and ten- tatively also the ν5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7 µm bending mode of acetylene (C2H2) towards the NE Core (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' C2H2 is a key in- gredient in the gas-phase formation of large molecules such as HC3N, and HCN is one of the most abundant nitrogen bearing molecules in dense (n > 1 × 104 cm−3) molecular clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' These lines have been previously detected by Spitzer towards dust enshrouded young stellar objects (YSOs) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Lahuis & van Dishoeck 2000) and towards lumi- nous and obscured infrared galaxies (LIRGs) (Lahuis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Lahuis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2007) find HCN column den- sities ranging between N(HCN)=1−12×1016 cm−2 and warm gas with excitation temperatures Tex=230-700 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Owing to the higher spatial and spectral resolution, the JWST MIRI spectrum of the NE core is more complex than those found by Lahuis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2007) using Spitzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We perform preliminary fits using the methodology of Lahuis & van Dishoeck (2000) which assumes LTE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Our preliminary results are consistent with an excitation temperature of 300-500 K and an N(HCN) of 1−5×1016 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' For an HCN abundance (with respect to H2) of 10−8-10−7 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Schilke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Lahuis & van Dishoeck 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Harada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2013), this would be con- sistent with a high obscuration with N(H2) 1023-1024 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The HCN and C2H2 spectra require further anal- ysis, including multiple temperature component model- ing and inclusion of non-LTE effects (Buiten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' It is also possible that the nuclear obscuration is even higher with column densities in excess of N(H2) 1025 cm−2–the so called Compact Obscured Nuclei (CONs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Aalto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Such objects are characterized by high mm and submm continuum surface brightness and luminous emission from mm-wave rotational transitions of HCN within the vibrationally excited ladder (HCN- vib) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Sakamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Aalto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2015, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Sakamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Falstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' For such deeply enshrouded objects, the HCN 14 µm line, or the silicate absorption, may not trace the full N(H2), but only its surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The mm/submm HCN-vib line (and the mm/submm continuum) would probe deeper, re- vealing the full obscuring column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Falstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2021) propose that CONs have surface brightness of HCN-vib of Σ(HCN-vib)> 1 L⊙pc−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' A recent ALMA study by Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2018) does not detect HCN-vib emission towards the NE Core of VV 114 with a limit of Σ(HCN- vib)< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='12 L⊙pc−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The relatively faint mm continuum found by Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2017) is consistent with the HCN- vib non-detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' This suggests that the NE Core ei- ther does not fulfil the CON criteria of Falstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2021) or that the radius of the CON region is smaller than 12 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' DISCUSSION The JWST spectra and imaging resolve a blend of diffuse emission from star formation and shocks, several reddened star-forming knots, and an AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Although the resolved spectra cover a much smaller region, our results confirm the analysis of MIRI imaging data in Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The SW core (c) has an elevated continuum at ∼ 5µm, consistent with the presence of dust in thermal equilibrium at temperatures near the sublimation temperature of the silicate grains (∼1200 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' This elevated continuum has been demonstrated by Laurent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Petric et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2011) (and refer- ences therein) as a telltale sign of a dust enshrouded, optically thick AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We also find PAH emission in all of our extracted apertures and widespread diffuse emis- sion in our emission line maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The various emission and absorption features allow us to directly probe the nature of the bright cores and diffuse emission as well as the kinematics of the gas in VV 114E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Evidence for AGN Activity Previous multiwavelength observations of VV 114 hinted at an AGN contribution in the X-ray, mid-IR, and submm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Le Floc’h et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Grimes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Iono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' As the brightest compact sources from the mid-IR to radio, the NE and SW cores are the most likely to harbor an AGN, with Iono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2013) identifying the NE core as a potential AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We do not detect coronal lines in either the NE or SW core, but this does not rule out an AGN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' the ULIRG Mrk 231 is a well-known optically classified Seyfert 1 with no ob- served coronal lines in the mid-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Instead, very low 8 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' NIRSpec spectra and PAH diagnostic plots used (a): NIRSpec Spectra of the NE and SW cores in VV 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The SW core shows a strongly rising continuum characteristic of hot dust associated with an AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 µm PAH feature is detected in both spectra, but is significantly weaker in the SW core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (b) & (c): expanded region and fit to the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 µm PAH feature in the NE (b) and SW (c) cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (d): Equivalent width of the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm PAH feature plotted vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Silicate strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Measurements for our MIRI apertures plotted as colored circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Values measured with Spitzer/IRS for ULIRGs and LIRGs from the GOALS sample are plotted as “+” symbols, with VV 114E and three comparative ULIRGs denoted with red diamonds (Stierwalt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Starburst galaxy M82 is shown as a blue square (Spoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (e): Equivalent widths of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm PAH features for the SW and NE cores (Inami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (f): Flux Density ratio and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm PAH EQW for the SW and NE cores (Inami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The SW core lies near AGN-dominated ULIRGs in both diagnostic plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' PAH equivalent width and silicate absorption indicate the presence of the AGN in Mrk 231 (Armus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Imanishi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Inami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Stierwalt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' To compare with other AGN and starburst dominated LIRGs we plot our measured values on several PAH diagnostic diagrams (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We first plot s9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' EQW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 4d): the apertures extracted from dif- fuse emission (with the exception of d) as well as the star forming clump in aperture f have values consistent with starburst galaxies and lie near the literature val- ues for M82 (Spoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The SW core shows contribution from an AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Spoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Mar- shall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2018) and lies near Mrk 273, a ULIRG with a Seyfert 2 nucleus (Armus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Stierwalt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' U et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Aperture d is directly adjacent to the SW core, in fact partially overlapping, and is likely showing some contribution from the AGN that is more clearly seen in the SW core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The NE core has a corre- spondingly higher obscuration and lies near the values observed for the ULIRG Arp 220, a deeply embedded starburst that also has C2H2 and HCN absorption crys- talline silicate features in its mid-IR spectrum (Spoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Lahuis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 10- (a) a(NE) (d) NE Core .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 b c(SW) NE Core SW Core d S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7 SW Core Mrk 231 0 100 EQW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2μm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='8- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7 PAH) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 10-3 u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='4 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0 Wavelength [um] NE Core 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2 Q NE Core SW Core asw Core le-12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Spline Fit Spline Fit (b) (c) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='06 Local Continuum Local Continuum EQW(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3μm PAH) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='8- 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='5 _un 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='16- (f) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='4- AH) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='14- cm- cm NE Core 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='12 P 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2 - μm I-s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='10- s 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='081 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 3 Starburst dominated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' W AGN dominated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='02 SW Core 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='00- 100 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='4 F(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3μm)/F(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='8μm) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='5 Wavelength [um] Wavelength [um]9 The spectrum of the SW core at shorter IR wave- lengths is consistent with the heating and processing of dust by an AGN, which reduces the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm PAH EQW (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 4a-c) as well as the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm EQW (EQW3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='04µm and EQW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='20µm, Imanishi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Petric et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm PAH feature is directly adjacent to a broad absorption feature caused by H2O ice that may impact our continuum measurement and in turn the measured EQW3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm (Imanishi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Inami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Moreover, the attenuation of the con- tinuum and the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm feature may be different depend- ing on the geometry of the dust and PAH emisison (Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We make a simple estimate of the impact by performing a power law fit to the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='0µm spectrum and divide our integrated flux by the continuum flux density at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' This decreases the EQW3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm slightly from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='12µm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='017µm to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='11µm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='015µm for the NE and SW core, but does not affect our conclusions regarding the nature the NE and SW cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Inami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2018) used AKARI to propose revised AGN diagnostics including the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 µm PAH feature as well as the Fν(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3)/Fν(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='8) flux density ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' When placed on a plot of EQW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' EQW3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 4e, following Inami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2018), the SW core lies in a region populated by known AGN while the high EQW3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm of the NE core is more consistent with ULIRGs dominated by star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We also place the NE and SW core on the EQW3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' flux density ratio diagnostic pro- posed by Inami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2018) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 4f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The SW core again clearly lies in the AGN-dominated portion of the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' To estimate the contribution of the AGN in the SW core to the total luminosity of VV 114 we take the es- timation of LIR ∼ 5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='5 × 1010L⊙ by Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2022), about 12% of the total LIR, as an upper limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The presence of PAH features in the SW core indicates a blended contribution to the IR of star formation and AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Sources with similar EQW3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 and colors have a bolometric AGN contribution of ∼ 30 − 50% (D´ıaz- Santos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Inami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2018), which means the contribution of the AGN to the total luminosity of the entire VV 114 system is ∼ 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Although Iono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2013) identified the NE core as potentially harboring an AGN due to the enhanced HCN/HCO+ ratio, an analysis of X-ray and millime- ter data by Privon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2020) showed that in fact the HCN/HCO+ is not a robust indicator of total AGN lu- minosity or its fractional contribution to infrared lumi- nosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Our findings regarding the nature of the two bright cores in VV 114E are not in agreement with the analysis of JWST MIRI data by Donnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2022) who propose the NE core as a potential AGN host based on its compact nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Interestingly, the NE core does have a somewhat low EQW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm despite a strong EQW3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3µm, placing it in a region of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 4e with few other galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The galaxies with the most similar values to the NE core are II Zw 96 and IRAS F19297-0406, which also have similar s9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7µm when comparing the values measured using Spitzer for all three galaxies (Stierwalt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' II Zw 96 is an unusually compact and powerful starburst (Inami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2010, 2022) and IRAS F19297-0406 has a powerful star- burst driven outflow (Soto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Veilleux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2013), physical conditions which may warm dust in a way that lowers EQW6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The curious nature of both the NE and SW cores warrants a more thorough follow- up analysis decomposing the relative contribution of the stellar, PAH, dust, and AGN components of the SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Shocks and Tidal Features Evidence for extended shocks in VV 114 has pre- viously been suggested at visible wavelengths via en- hanced [O I] and [S II] emission and broadened line pro- files across VV 114 (Rich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2011, 2015) and in the submm from the presence of methanol (CH3OH) in the “overlap” region between VV 114W & E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Apertures f, g, h, and i lie closest to the overlap region and both emis- sion line ratios and molecular and atomic line widths show evidence of shocked gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The atomic lines in aperture f all have narrow line widths (∼100-150 km/s) and ratios typical of buried, young star-forming regions as this source was revealed to be in (Linden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Aperture i encompasses a fainter star forming knot seen in both MIRI and NIR- CAM imaging, and several emission line profiles show a double peaked profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The broader H2 lines in aper- tures f and i (∼200-300 km/s) are similar to the values extracted just to the north in apertures g and h, which show elevated [Fe II]/Pfα values indicative of shocks, also seen in MRS observations of NGC 7469 (U et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' These regions are consistent with ALMA observations of VV 114 where the elevated FWHM of 150-300 km/s in the molecular gas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' CO(1-0), CH3OH) is found in the “overlap” region and is suggested to be the prod- uct of both shocks, and overlapping kinematic compo- nents due to the merger (Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2017, 2018) and may be similar in nature to the shocked bridge seen in Stephan’s Quintet (Guillard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2012b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Appleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' However, the clusters in this region of VV 114 are 1-2 orders of magnitude more massive and ∼2-3 times dustier than those seen in the bridge of Stephan’s Quintet (Fedotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Linden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 10 Resolved emission line profiles in aperture j show a strong indication of both blue and red-shifted wings, po- tentially due to projection effects of the tidal arm that extends from VV 114W across the IR bright cores south- ward beyond the FOV of our aperture extractions (Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' These values are supported by examining the [Ne II] emission line profiles which fall in Channel 3 and has a wider FOV than Channel 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Looking sev- eral arcseconds southeast of the SW core, the [Ne II] line profiles show significant broadening and wings in several spaxels (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' We also see enhanced H2/[Ne II] in this region and at the furthest SE region of our spaxel-by-spaxel maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The value of [Fe II]/Pfα ∼ 9 in aperture j also indicates the presence of shocked gas which likely extends beyond the Channel 1 FOV and fol- lows the elevated emission line ratios caused by shocks seen in visible light IFU observations (Rich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Aperture b appears to be dominated by diffuse emis- sion when examining the MIRI and NIRCAM images, but does not display the same characteristics of shock excitation as the other apertures that trace diffuse gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Previous observations of shock excitation in VV 114 have suggested both galactic winds and tidally driven gas flows as sources of shock excitation (Rich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The JWST data show some shocked gas coincident with tidal features previously observed in the submm and radio as well as kinematic profiles potentially associated with galactic winds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Follow-up work mapping the two dimensional kinematics of the molecular and atomic gas, as well as the temperature and distribution of H2 gas using these data will provide a more complete picture of the shock excitation in VV 114, especially when combined with wide field optical IFU observations from MUSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Star Formation Rates If we assume that the atomic and fine structure line emission is dominated by star formation in the aper- tures with bright star-forming clumps (a, e, f), we can calculate star formation rates using a hydrogen recom- bination line flux or Neon emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' If we scale either Pfund or Humphreys α to Hα assuming Case B recombi- nation (Hummer & Storey 1987) and use equation (2) in Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2011) the estimated SFR for each spec- trum is ∼ 1M⊙/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Using the [Ne II] and [Ne III] fluxes and equation (3) in Ho & Keto (2007) yields an SFR from ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='5 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='5M⊙/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' These values are consistent with those found by Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' (2022) using the radio emission of bright knots measured with the VLA and amount to ∼ 2 − 3% of the total SFR per aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' CONCLUSIONS Our analysis of MIRI MRS and NIRSpec IFU spec- troscopy of VV 114E shows emission and absorption fea- tures that allow us to resolve variations in the proper- ties of the IR bright nuclear cores, unresolved clumps, and diffuse gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The integrated properties of the two bright cores and the diffuse gas agree with past multi-wavelength observations of VV 114E that show widespread star formation and diffuse emission from shocks, and reveal spectroscopic evidence of an AGN in the SW core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' More specifically: The SW core harbors an AGN as indicated by its extremely low 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='2 µm PAH equiva- lent widths and strong 3-5µm continuum, consis- tent with AGN-dominated LIRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The SW core is also surrounded by star forming knots and diffuse emission, which is evident in the atomic line ratios at longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The AGN in the SW core likely accounts for ∼ 5% of the total luminosity of VV 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The NE core is deeply embedded, its mid-IR spectrum displays strong 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='7µm silicate absorp- tion, crystalline silicate features, aliphatic hydro- carbons, and HCN absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Using the 14µm HCN absorption line we calculate a temperature and column density of 300-500 K and N(HCN) of 1 − 5 × 1016 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Our data show no evidence of an AGN impacting the atomic or molecular gas at mid-IR wavelengths and we conclude that this source is a deeply buried star forming region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The diffuse gas NW of the nuclear region shows el- evated [Fe II]/Pfα and higher H2 line widths along with double peaked profiles caused by shocked gas in the overlap region previously observed by ALMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' A fit to [Ne II] across the MIRI FOV reveals broader emission profiles to the south of the nucleus consistent with the extended tidal fea- ture observed by ALMA, as well as shocked gas ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='5′′ SE of the SW Core that likely extends be- yond the Channel 1 FOV, consistent with previous visible light IFU observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' These early 3D spectral data highlight the power of combining the NIRSpec and MIRI data to elucidate the nature of complex, obscured star formation and AGN in the local Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Taken together the spectroscopic datasets from both JWST instruments are extremely rich and will facilitate detailed and thorough analysis in future papers in this series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' ACKNOWLEDGEMENTS 11 We thank the referee for their helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' This work is based on observations made with the NASA/ESA/CSA JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The research was supported by NASA grant JWST-ERS-01328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The data were ob- tained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is op- erated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', under NASA contract NAS 5-03127 for JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The specific observations analyzed can be accessed via 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='17909/yqk1-jr92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' VU acknowledges funding support from NASA Astrophysics Data Anal- ysis Program (ADAP) grant 80NSSC20K0450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The Flatiron Institute is supported by the Simons Founda- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' AMM acknowledges support from the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2009416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' ASE and SL acknowledge support from NASA grants HST- GO15472 and HST-GO16914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' YS was funded in part by the NSF through the Grote Reber Fellowship Program administered by Associated Universities, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='/National Radio Astronomy Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agree- ment by Associated Universities, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='M-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' ac- knowledges support from NASA through ADAP award 80NSSC19K1096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' SA gratefully acknowledges support from an ERC Advanced Grant 789410, from the Swedish Research Council and from the Knut and Alice Wallen- berg (KAW) Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' SA gratefully acknowledges John Black for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' KI acknowledges support by the Spanish MCIN under grant PID2019- 105510GB-C33/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='13039/501100011033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' HI and TB acknowledge support from JSPS KAKENHI grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' JP21H01129 and the Ito Foundation for Pro- motion of Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' This work was also partly sup- ported by the Spanish program Unidad de Excelen- cia Mara de Maeztu CEX2020-001058-M, financed by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='13039/501100011033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' The computa- tions presented here were conducted through Carnegie’s partnership in the Resnick High Performance Comput- ing Center, a facility supported by Resnick Sustainabil- ity Institute at the California Institute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Finally, this research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Tech- nology, under contract with the National Aeronautics and Space Administration.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Rigopoulou, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='04647 Evans, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Frayer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Charmandaris, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content='14507 Falstad, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Aalto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', K¨onig, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2021, A&A, 649, A105 Fedotov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Gallagher, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Konstantopoulos, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2011, AJ, 142, 42 Garofali, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Lehmer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Basu-Zych, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2020, ApJ, 903, 79 Genzel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Lutz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Sturm, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 1998, ApJ, 498, 579 Grimes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Heckman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Hoopes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2006, ApJ, 648, 310 Guillard, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Ogle, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Emonts, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', & Herbst, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2013, ApJ, 765, 108 Ho, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', & Keto, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2007, ApJ, 658, 314 Howell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Ohyama, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2008, PASJ, 60, S489 12 Imanishi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Nakagawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Shirahata, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Ohyama, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', & Onaka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2010, ApJ, 721, 1233 Inami, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Armus, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Surace, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2010, AJ, 140, 63 Inami, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Armus, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Charmandaris, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2013, ApJ, 777, 156 Inami, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Armus, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Matsuhara, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2000, A&A, 355, 699 Lahuis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Spoon, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Tielens, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2007, ApJ, 659, 296 Lai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Smith, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Baba, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Spoon, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', & Imanishi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2020, ApJ, 905, 55 Laurent, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Mirabel, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Charmandaris, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2000, A&A, 359, 887 Le Floc’h, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Charmandaris, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Laurent, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2002, A&A, 391, 417 Linden, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Evans, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2000, ApJ, 536, 697 Marshall, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Elitzur, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Armus, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Diaz-Santos, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2011, ApJ, 737, 67 Newville, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Stensitzki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', Allen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', & Ingargiola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2014, LMFIT: Non-Linear Least-Square Minimization and Curve-Fitting for Python, Zenodo, doi:10.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=', & Dopita, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2011, ApJ, 734, 87 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' 2015, ApJS, 221, 28 Rieke, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE0T4oBgHgl3EQfawAZ/content/2301.02338v1.pdf'} 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/dev/null +++ b/sdFKT4oBgHgl3EQf1i6S/content/tmp_files/2301.11920v1.pdf.txt @@ -0,0 +1,2555 @@ +Magnomechanical backaction corrections due to coupling to higher order Walker +modes and Kerr nonlinearities +V.A.S.V. Bittencourt,1, 2, ∗ C.A. Potts,3, 4 Y. Huang,4 J.P. Davis,4 and S. Viola Kusminskiy5, 2, † +1ISIS (UMR 7006), Universit´e de Strasbourg, 67000 Strasbourg, France +2Max Planck Institute for the Science of Light, Staudtstr. 2, PLZ 91058 Erlangen, Germany +3Kavli Institute of NanoScience, Delft University of Technology, PO Box 5046, 2600 GA Delft, Netherlands +4Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E9, Canada +5Institute for Theoretical Solid State Physics, RWTH Aachen University, 52074 Aachen, Germany +(Dated: January 30, 2023) +The radiation pressure-like coupling between magnons and phonons in magnets can modify the +phonon frequency (magnomechanical spring effect) and decay rate (magnomechanical decay) via +dynamical backaction. Such effects have been recently observed by coupling the uniform magnon +mode of a magnetic sphere (the Kittel mode) to a microwave cavity. In particular, the ability to +evade backaction effects was demonstrated [C.A. Potts et al., arXiv:2211.13766 [quant-ph] (2022)], +a requisite for applications such as magnomechanical based thermometry. However, deviations were +observed from the predicted magnomechanical decay rate within the standard theoretical model. +In this work, we account for these deviations by considering corrections due to (i) magnetic Kerr +nonlinearities and (ii) the coupling of phonons to additional magnon modes. Provided that such +additional modes couple weakly to the driven cavity, our model yields a correction proportional to +the average Kittel magnon mode occupation. We focus our results on magnetic spheres, where we +show that the magnetostatic Walker modes couple to the relevant mechanical modes as efficiently +as the Kittel mode. Our model yields excellent agreement with the experimental data. +The dipolar interaction between the magnetization and +microwaves confined in a cavity can yield strong coupling +between magnons (quanta of spin waves) and microwave +photons. +After the first theoretical predictions of the +strong magnon-microwave coupling [1], cavity magnonic +systems consisting of a magnetic element loaded in a mi- +crowave cavity were realized in different architectures [2– +8]. The unique tunability of magnons combined with the +ability to drive and read out the microwave cavity makes +such systems a promising platform for several applica- +tions [9–13], such as the generation of squeezed and en- +tangled states [14, 15], the indirect coupling to qubits to +detect and manipulate magnons [16–20], and sensing of +magnetic fields [21–24]. +Magnons can also couple to other degrees of free- +dom, opening the opportunity of probing and manipu- +lating these via their coupling to the hybridized magnon- +microwave polaritons [13]. In particular, magnetoelastic +effects [25–28] couple the magnetization and the mechani- +cal vibrations of a magnetic material, yielding an interac- +tion between magnons and phonons [29]. Such magnome- +chanical coupling can be either resonant or parametric +[30]. The resonant coupling is relevant for specific geome- +tries where certain magnon modes are resonant with the +elastic vibrations of the medium, for example, for mag- +netic spheres with radii ranging from ∼ 10 nm to ∼ 10 +µm [30] and in magnetic films [31–33]. The second type of +coupling, parametric coupling, is relevant for geometries +in which the magnon frequency is far detuned from the +phonon, such as for micrometer-sized magnetic spheres +∗ sant@unistra.fr +† kusminskiy@physik.rwth-aachen.de +[29, 34]. The interaction Hamiltonian resembles the ra- +diation pressure coupling between phonons and photons +commonly found in optomechanical systems [35]. When +magnons are driven, the magnomechanical interaction is +enhanced and the driven-dissipative dynamics of the cou- +pled system result in dynamical backaction on the vibra- +tional modes. Specifically, the phonon frequency and de- +cay rate are modified, referred to as magnomechanical +spring effect and magnomechanical decay, respectively +[29, 34, 36]. +Dynamical backaction is the basis of several proposed +applications of magnomechanical systems, from state +preparation and generation of entangled states [15, 37– +40], to effects that are closely related to the optomechani- +cal counterpart, such as magnomechanical sideband cool- +ing and amplification of phonons [41, 42], albeit operating +in the microwave regime. +Moreover, cavity magnome- +chanical systems provide a unique tunability of dynami- +cal backaction due to the hybridization between magnons +and microwaves, which can be used to fulfill a triple reso- +nance condition by tuning an external bias magnetic field. +Dynamical backaction was first probed in magnomechan- +ics in a system consisting of a magnetic sphere of yttrium +iron garnet (YIG) loaded into a 3D microwave cavity [29]. +More recent experiments have demonstrated the full ar- +ray of dynamical backaction effects in these systems [34] +and demonstrated the capability of avoiding the induced +magnomechanical decay [43]. Dynamical backaction eva- +sion can enable the application of cavity magnomechanics +in thermometry [36], where, similar to proposal and ex- +periments in optomechanical systems [44, 45], the phonon +mode should be neither cooled nor heated by the drive. +Experiments and proposals for cavity magnomechan- +ical systems have so far focused on the coupling to a +arXiv:2301.11920v1 [cond-mat.mes-hall] 27 Jan 2023 + +2 +single magnon mode, the uniform precession of the mag- +netization called the Kittel mode. Nevertheless, a mag- +netic sphere supports a whole set of magnon modes called +Walker modes [46, 47] which can also couple to a given +vibration mode, in principle even stronger than the Kittel +mode. Weak inhomogeneities in the microwave field can +drive such higher-order magnon modes, modifying the +backaction effects. Furthermore, magnon nonlinearities +due to crystalline anisotropy [48] can also affect dynami- +cal backaction beyond the static frequency shift reported +in Ref. [49], akin to recently measured effects of the cav- +ity Kerr nonlinearity in an electromechanical system [50] +under sideband cooling. +In this work, we extend the theory of dynamical back- +action in cavity magnomechanical systems to include +Kerr nonlinearities and the coupling of a phonon mode to +several magnon modes. We consider the framework de- +picted in Fig. 1, where a microwave cavity mode couples +strongly to a magnon mode and weakly to a set of addi- +tional magnon modes. Those in turn exhibit nonlineari- +ties and interact via a radiation pressure-like coupling to +a single phonon mode. We derive the phonon self-energy, +describing the frequency shift and the magnomechanical +decay rate, generalizing previous results [34, 36]. +The +overall effect of the coupling to the additional magnon +modes is a correction proportional to the average number +of Kittel magnons. We evaluate our model for the case of +a magnetic sphere, computing numerically the coupling +rates between the (magnetic) Walker modes and the me- +chanical mode probed in Ref. [43]. At low driving powers +our model introduces corrections that agree well with the +measured data in Ref. [43], explaining the observed shift +in the magnomechanical decay rate. At higher driving +powers, there are further deviations which are not cap- +tured by our model. These are however only relevant for +driving frequencies detuned from the backaction evasion +point. +ˆa +ˆm +{ +} +ˆb +g0 +mb ˆm† ˆm +⇣ +ˆb† + ˆb +⌘ +{K(i) +cr,m ˆm† ˆm ˆm† +i ˆmi} +gam +� +ˆa ˆm† + ˆa† ˆm +� +{gamj +⇣ +ˆa ˆm† +j + ˆa† ˆmj +⌘ +} +ˆmj +{g0 +mjb ˆm† +j ˆmjˆb + H.c.} +Drive +FIG. 1. Schematics of the model describing the cavity mag- +nomechanical system with several magnon modes coupled to +the same phonon mode, see the Hamiltonian (7). The red ar- +row indicates the microwave drive, and we have omitted the +self-Kerr nonlinear terms and the magnon-phonon cross Kerr +nonlinearity. +This paper is organized as follows. In Sec. I we present +a brief review of the description of dynamical backac- +tion in cavity magnomechanics for the cases described in +the literature, e.g. [29, 34]. +In Sec. II we include Kerr +nonlinearities and the coupling to weakly driven addi- +tional magnon modes, and the phonon self-energy. Since +those corrections depend on how strongly the additional +magnon modes couple to the phonon mode, we special- +ize further our model to a magnetic sphere geometry as +probed in Ref. [43]. +In Sec. +III, we briefly review the +derivation of the magnomechanical coupling following the +literature [30], and in subsection III A we use the model to +numerically evaluate the coupling between Walker modes +with frequencies in a small range around the Kittel mode +frequency and a relevant mechanical mode of a magnetic +sphere. +In Sec. IV, we compare our model to the ex- +perimental results presented in Ref. [43] and show that +our generalized theory quantitatively accounts for the ob- +served magnomechanical decay for a large range of pa- +rameters. Finally, in Sec. V we present our conclusions. +I. +PHONON SELF-ENERGY AND DYNAMICAL +BACKACTION EVASION +The dynamics of a cavity magnomechanical system +consisting of a microwave mode (ˆa with frequency ωa), +a magnon mode ( ˆm with frequency ωm) coupled para- +metrically to a phonon mode (ˆb with frequency Ωb) is +described by the Hamiltonian [29] +ˆH +ℏ = ωaˆa†ˆa + ωm ˆm† ˆm + Ωbˆb†ˆb ++ gam +� +ˆa ˆm† + ˆa† ˆm +� ++ g0 +mb ˆm† ˆm +� +ˆb† + ˆb +� ++ i√κeϵd +� +ˆaeiωd − ˆa†e−iωd� +. +(1) +The magnon-microwave coupling rate gam is due to a +magnetic dipole interaction between the ferromagnetic +resonance of the material and the microwave cavity. +The parametric magnon-phonon coupling, with the sin- +gle magnon coupling rate g0 +mb, is due to magnetoelastic +effects. The last term in Eq. (1) describes the coherent +drive of the microwave cavity at a frequency ωd with an +amplitude ϵd = +� +P/ℏωd, where P is the drive power and +κe is the decay rate of the cavity to the external drive +port. +In the weak magnomechanical coupling limit, both the +phonon frequency Ωb and the decay rate Γb are modified +by the coupling to the driven magnons. The respective +shifts are given by +δΩb = −Re[Σ[Ωb]], +Γmag = 2Im[Σ[Ωb]], +(2) +where Σ[ω] is the phonon self-energy, obtained by ana- +lyzing the linearized dynamics of the system [34, 36] and +reads +Σ[ω] = i|gmb|2(Ξ[ω] − Ξ∗[−ω]). +(3) + +3 +From now on we refer to δΩb as the magnomechanical +frequency shift and to Γmag as the magnomechanical de- +cay rate. +Here, gmb = g0 +mb⟨ ˆm⟩ is the enhanced mag- +nomechanical coupling rate, with |⟨ ˆm⟩|2 the steady-state +magnon population. The function Ξ[ω] is a modified Kit- +tel mode susceptibility given by +Ξ−1[ω] = χ−1 +m [ω] + g2 +amχa[ω] +(4) +which depends on the magnon susceptibility χm[ω] = +[−i(∆m + ω) + γm/2], and the microwave susceptibil- +ity χa[ω] = [−i(∆a + ω) + κ/2]. +The detuning be- +tween the microwave (magnon) mode and the drive is +∆a(m) = ωd − ωa(m). γm is the magnon decay rate and κ +the total microwave decay. +The value and the sign of the magnomechanical de- +cay rate depend on the drive frequency, which can tune +scattering processes that upconvert or downconvert exci- +tations in the system. Depending on the drive, it is pos- +sible to make one of such processes more efficient than +the other, yielding a positive (cooling) or negative (am- +plification) magnomechanical decay rate, in a situation +akin to what is found in standard optomechanical sys- +tems [35, 51, 52]. Different from optomechanics, in cavity +magnomechanical systems magnons and microwaves hy- +bridize, yielding the unique situation where the different +scattering processes that contribute to dynamical back- +action are associated with mechanical sidebands of hy- +bridized modes [29, 43]. The hybrid magnon-microwave +modes have frequencies ω± that are separated by +ω+ − ω− = +� +4g2am + (ωa − ωm)2. +(5) +We will refer to the mode with frequency ω+ as the up- +per hybrid mode, and the mode with frequency ω− as +the lower hybrid mode. When the microwave drive is set +at a frequency between ω+ and ω−, the scattering from +the blue sideband of the lower hybrid mode can be bal- +anced by the scattering to the red sideband of the upper +hybrid mode yielding dynamical backaction evasion: the +magnomechanical decay rate vanishes. Consequently, the +drive at which dynamical backaction evasion happens can +be obtained from the condition +Γmag = 0. +(6) +In a system satisfying the two-phonon triple resonance +condition ω+ − ω− = 2Ωb, and for resonant magnons +and microwaves, such drive frequency is exactly at (ω+ + +ω−)/2 [43]. The ability to tune a magnomechanical sys- +tem in the dynamical backaction evasion regime was re- +cently demonstrated in Ref. [43], and is a requirement for +implementing a magnomechanical-based primary ther- +mometer [36]. +Equation (3) only takes into account the interaction +of a phonon mode with a single magnon mode. +How- +ever, multiple magnetostatic modes can couple to a +given phonon mode [30], modifying the magnomechan- +ical decay rate. The different scattering processes to and +from the additional magnomechanical sidebands can thus +change the frequency at which dynamical backaction is +evaded. +For instance, in the experimental data shown +in Ref. [43], the measured magnomechanical decay rate +exhibits a shift with respect to the theoretical prediction +obtained from the Hamiltonian given in Eq. +(1). This +shift was taken into account by adding to the magnome- +chanical decay rate a phenomenological correction pro- +portional to |⟨ ˆm⟩|2, which depends on the average num- +ber of magnons driven by the microwave tone. Such cor- +rection can be attributed to the coupling to additional +magnon modes, which are weakly driven by their cou- +pling to the microwave cavity, and to magnon nonlinear- +ities. +While nonlinearities in magnetic spheres are generally +weak, the microwave drive combined with the strong +magnon-microwave coupling can make nonlinear effects +prominent. This is the case provided that the power of +the drive is strong enough to induce an average number +of magnons above a certain threshold [53], with implica- +tions for magnetoelastic effects [54]. Even for drive pow- +ers below the nonlinear threshold, magnon nonlinearities +can affect the hybrid system dynamics. +For instance, +in a cavity-magnonic system, the Kittel mode self-Kerr +nonlinearity was shown to yield considerable cavity and +magnon frequency shifts under moderate driving powers +[55]. Experimentally, a phonon frequency shift, as well as +mechanical bistability, was reported recently [49], which +points to the importance of considering such nonlineari- +ties in the description of dynamical backaction effects. +In what follows, we include in the description of +dynamical backaction both the coupling to additional +magnon modes as well as magnon nonlinearities. +II. +INCLUSION OF KERR NONLINEARITY +AND COUPLING TO ADDITIONAL MAGNON +MODES IN THE PHONON SELF ENERGY +To derive the correction term to the self-energy, we +consider adding to the Hamiltonian in Eq. (1) self- and +cross- Kerr nonlinearities, and coupling to N additional +magnon modes, each with annihilation operators { ˆmj} +and frequencies ωj (j = 1, · · · , N). The total Hamilto- + +4 +nian is thus: +ˆH +ℏ = ωaˆa†ˆa + ωm ˆm† ˆm + Ωbˆb†ˆb + +N +� +j=1 +ωj ˆm† +j ˆmj ++ gam +� +ˆa ˆm† + ˆa† ˆm +� ++ +N +� +j=1 +gamj +� +ˆa ˆm† +j + ˆa† ˆmj +� ++ g0 +mb ˆm† ˆm +� +ˆb† + ˆb +� ++ +N +� +j=1 +ˆm† +j ˆmj +� +g0 +mjbˆb + (g0 +mjb)∗ˆb†� ++ Km( ˆm† ˆm)2 + Kcr ˆm† ˆmˆb†ˆb + +N +� +j=1 +K(j) +cr,m ˆm† ˆm ˆm† +j ˆmj ++ i√κeϵd +� +ˆaeiωd − ˆa†e−iωd� +. +(7) +From now on, we will refer to the magnon mode ˆm as the +Kittel mode, since this is typically the magnon mode that +has the strongest coupling to the cavity, while we refer +to the magnon modes ˆmj as additional magnon modes. +We specify such additional magnon modes for the case of +a magnetic sphere in Sec. III A. +Compared with the Hamiltonian in Eq. (1), the above +equation includes the following terms: +the additional +magnon modes; the coupling between these and (i) the +microwave mode, each with a coupling rate gamj and +(ii) the phonon mode, each with a coupling rate g0 +mjb; +the self-Kerr term for the Kittel mode; the cross-Kerr +term between the Kittel and the phonon mode; and the +cross-Kerr term between the Kittel mode and the other +magnon modes. Provided that the magnetic anisotropy +axis of the YIG sphere is aligned with the external bias +field, the Kittel mode self-Kerr nonlinear coefficient is +given by K0 +m = 13ℏKanγ2/(16M 2 +s V ), where Kan = −610 +J/m2 at room temperature and V is the sphere vol- +ume [48]. +We assume a rotating wave approximation +for the magnon-microwave coupling, as is done to obtain +Eq. (1), which eliminates any term of the form ˆm(j)ˆa +and ˆm† +(j)ˆa†. The rotating wave approximation is also as- +sumed for the magnomechanical coupling, as explained +in Section III. For the experiment in Ref. [43] this cor- +responds to K0 +m/2π = −5.15 nHz. The magnon-phonon +and magnon-magnon cross Kerr nonlinear coefficients de- +pend on the overlap between these modes and the Kittel +mode. In general, the magnon-magnon cross-Kerr coeffi- +cient is around the same order of magnitude as K0 +m [56], +while the magnon-phonon cross-Kerr coefficient is ∼ −5 +pHz [49]. We should notice that in the experiment shown +in [43], the magnetic anisotropy axis was not aligned with +the bias magnetic field, which effectively causes Km to be +smaller than the value K0 +m. Figure 1 shows a schematic +of the model, including the different coupling terms. +The values of the magnomechanical couplings depend +on the geometry of the magnet, which defines the magnon +and phonon mode profiles. In the system under study, +the coupling between the Kittel mode and a relevant me- +chanical mode of a sphere, discussed in Section III A, is +g0 +mb/2π = 4.56 mHz. In Section III we discuss in detail +the values of g0 +mjb for a magnetic sphere. It is impor- +tant to point out that, due to better mode overlap, in +principle, g0 +mjb can be comparable to or larger than g0 +mb +for some modes. +The coupling between magnons and +microwaves depends on the microwave field at the mag- +net position. For homogeneous fields, only the coupling +to the Kittel mode does not vanish. Nevertheless, small +inhomogeneities could yield a small microwave-magnon +coupling gamj which would, in turn, drive weakly such +magnon modes. For different cavity geometry, such cou- +plings can be strong [7, 8, 57, 58], a framework which we +do not consider here. +In correspondence with the experiment [43], we assume +that the additional magnon modes are weakly driven via +their coupling to the cavity mode, such that we expect a +small steady-state amplitude for those modes. Thus we +can safely disregard any self- and cross-Kerr nonlinear- +ity of the form ˆm† +k ˆmk ˆm† +j ˆmj. The Heisenberg-Langevin +equations describing the dynamics of the coupled modes +in the rotating frame with the drive frequency are +˙ˆa = +� +i∆a − κ +2 +� +ˆa − igam ˆm − i +N +� +j=1 +gamj ˆmj +− √κi ˆξI(t) − √κeϵd, +˙ˆm = +� +i∆m − γ +2 +� +ˆm − igamˆa − ig0 +mb ˆm +� +ˆb† + ˆb +� +− iKm ˆm +� +1 + 2 ˆm† ˆm +� +− iKcr ˆmˆb†ˆb +− i +N +� +j=1 +K(j) +cr,m ˆm ˆm† +j ˆmj + √γm ˆξm(t), +˙ˆmj = +� +i∆mj − γj +2 +� +ˆmj − igamjˆa − i ˆmj +� +g0 +mjbˆb + g0,∗ +mjbˆb†� +− iK(j) +cr,m ˆmj ˆm† ˆm + √γj ˆξmj(t), +˙b = − +� +iΩb − Γb +2 +� +ˆb − ig0 +mb ˆm† ˆm − iKcrˆb ˆm† ˆm +− i +N +� +j=1 +g0 +mjb ˆm† +j ˆmj + +� +Γb ˆξb(t). +(8) +In the above equations, κ = κi + κe denotes the total +microwave cavity decay rate, which is composed of the +intrinsic cavity decay κi and the decay into the exter- +nal port κe. The additional magnon modes decay rates +are indicated by γmj which, for magnetostatic modes of +a sphere, have the same value of the Kittel mode de- +cay [59]. All parameters appearing in Eq. (8) are sum- +marized in Table I, with the values that will be used +throughout this paper. The noise terms denoted by ˆξη(t) +(η = I, e, m, mj, b) describe thermal noises with correla- +tions +⟨ˆξη(t)ˆξ† +η′(t′)⟩ = (nTh,η + 1)δηη′δ(t − t′), +⟨ˆξ† +η(t)ˆξη′(t′)⟩ = nTh,ηδηη′δ(t − t′), +(9) +with nTh,η = [exp(ℏωη/kBT) − 1]−1 the number of ther- + +5 +mal excitations of mode η at a temperature T. +The steady state in a mean-field approximation is ob- +tained by taking the expectation values of the oper- +ators in Eqs. (8) and ignoring any quantum correla- +tions, i.e. ⟨ ˆmˆb⟩ ≈ ⟨ ˆm⟩⟨ˆb⟩. Since we are assuming that +the magnon modes { ˆmj} are weakly coupled to the mi- +crowaves, gamjgamk ≪ gamjgam, we discard any other in- +direct coupling between the additional magnon modes via +the cavity. These approximations yield +⟨ˆb⟩ = +ig0 +mb|⟨ ˆm⟩|2 +Fb − iKcr|⟨ ˆm⟩|2 + +i �N +j=1 g0 +mjb|⟨ ˆmj⟩|2 +Fb − iKcr|⟨ ˆm⟩|2 , +⟨ ˆmj⟩ = +igamj +√κeϵd +FmjFa + g2amj − iK(j) +cr,m|⟨ ˆm⟩|2 +− +gamjgam⟨ ˆm⟩ +FmjFa + g2amj − iK(j) +cr,m|⟨ ˆm⟩|2 , +(10) +where we have defined +Fb = −iΩb − Γb +2 , +Fmj(m) = i∆mj(m) − γmj(m) +2 +, +Fa = i∆a − γa +2 . +(11) +For ⟨ ˆmj⟩, we have also discarded the term ∝ g0 +mjb. The +steady-state of the Kittel mode reads +A⟨ ˆm⟩ = igam +√κeϵdB +(12) +where +A = FmFa + g2 +am − iKm +� +1 + 2|⟨ ˆm⟩|2� +− 2iFag0 +mbRe +� +⟨ˆb⟩ +� ++ g2 +amB − iKcr|⟨ˆb⟩|2 +B = 1 − +N +� +j=1 +g2 +amj +FmjFa + g2amj − iK(j) +cr,m|⟨ ˆm⟩|2 . +(13) +Equation (12) is solved numerically. Depending on the +drive power and the detuning, the equation can have +two bistable solutions. +We will focus our analysis on +a detuning range lying in between the hybridized Kit- +tel magnon-microwave modes. In the considered range +the magnomechanical decay rate of Eq. (2) changes its +sign, and in such region, the nonlinear equation for ⟨ ˆm⟩ +has only one solution. Furthermore, we can discard the +terms proportional to K(j) +cr,m, Kcr and g0 +mb to obtain the +solutions of Eq. (12). +A. +Linearized dynamics +We can now consider the fluctuations around the +steady-state values. We write ˆo = δˆo + ⟨ˆo⟩, and discard +any terms involving more than two fluctuations. +The +quadratic Hamiltonian describing the dynamics of the +fluctuations is given by +ˆHLin +ℏ += −∆aδˆa†δˆa + ˜Ωbδˆb†δˆb − ˜∆mδ ˆm†δ ˆm +− +N +� +j=1 +˜∆mjδ ˆm† +jδ ˆmj + +ˆHInt +ℏ +, +(14) +with the coupling terms included in ˆHInt given by +ˆHInt +ℏ += gamδˆa†δ ˆm + GRδ ˆm†δˆb + GBδ ˆm†δˆb† ++ gms +� +δ ˆm†�2 + +� +j=1 +gamjδˆa†δ ˆmj ++ +N +� +j=1 +� +GR,jδ ˆm† +jδˆb + GB,jδ ˆmjδˆb +� ++ +N +� +j=1 +� +gR,jδ ˆm†δ ˆmj + gB,jδ ˆm†δ ˆm† +j +� ++ H.c. +(15) +The interacting terms appearing in the Hamiltonian of +Eq. (7) induce frequency shifts for the fluctuations, which +are given by +˜∆m(mj) = ωd − ˜ωm(mj), +˜ωm = ωm + 2g0 +mbRe +� +⟨ˆb⟩ +� ++ 4Km|⟨ ˆm⟩|2 + Kcr|⟨ˆb⟩|2 ++ +N +� +j=1 +K(j) +cr,m|⟨ ˆmj⟩|2, +˜ωmj = ωmj + 2Re +� +g0 +mjb⟨ˆb⟩ +� ++ K(j) +cr,m|⟨ ˆm⟩|2, +˜Ωb = Ωb + Kcr|⟨ ˆm⟩|2. +(16) +The coupling rates between the fluctuations are enhanced +and modified with respect to the bare ones. Their expres- +sions are shown in Table II. +TABLE II. Enhanced couplings appearing in the linearized +Hamiltonian in Eq. (15) +gmb +g0 +mb⟨ ˆm⟩ +GR gmb + Kcr⟨ ˆm⟩⟨ˆb⟩∗ +GB +gmb + Kcr⟨ ˆm⟩⟨ˆb⟩ +GR,j +g0 +mjb⟨ ˆmj⟩ +GB,j +g0 +mjb⟨ ˆmj⟩∗ +gms +Km⟨ ˆm⟩2 +gR,j +K(j) +cr,m⟨ ˆm⟩⟨ ˆmj⟩∗ +gB,j +K(j) +cr,m⟨ ˆm⟩⟨ ˆmj⟩ +B. +Calculation of the phonon self-energy +The phonon self-energy is obtained by solving the lin- +ear Heisenberg-Langevin equations describing the cou- + +6 +TABLE I. Parameters of the magnomechanical system appearing in Eq. (8). The values correspond to the experiment in [43]. +The magnon self-Kerr term value corresponds to the case where the bias magnetic field is aligned with the magnetic anisotropy +axis. +Parameter +Symbol +Value +Microwave mode frequency +ωa +2π × 7.11 GHz +Kittel mode frequency +ωm +2π × 7.09 GHz +Additional magnon modes frequencies +ωmj +See section II A +Phonon mode frequency +Ωb +2π × 12.45 MHz +Drive frequency +ωd +2π × [7.096, 7.099] GHz +Microwave intrinsic decay rate +κI +2π × 2.91 MHz +Microwave external decay rate +κe +2π × 3.17 MHz +Kittel mode decay rate +γm +2π × 2.55 MHz +Additional magnon modes decay rate +γmj +2π × 2.55 MHz +Phonon intrinsic decay rate +Γb +2π × 3.74 kHz +Kittel mode - Microwave coupling rate +gam +2π × 9.34 MHz +Additional magnon modes - microwave coupling rate +gamj +See Section IV +Magnomechanical coupling to the Kittel mode +g0 +mb +2π × 4.56 mHz +Magnomechanical couplings to the j-th additional magnon mode +g0 +mjb +See section III A +Kittel mode self-Kerr nonlinearity +K0 +m +−2π × 5.15 nHz +Magnon cross-Kerr nonlinearity +K(j) +cr,m +−2π × 5.15 nHz +Magnon-phonon cross-Kerr nonlinearity +Kcr +−2π × 5.4 pHz +pled dynamics of the fluctuations for the phonon opera- +tor. To compute the effects of backaction in the response +of the phonon mode to noise, we consider the Fourier +transformed operators defined by +ˆo(t) = +� +dωe−iωtˆo[ω], +(17) +where ˆo = δˆa(†), δ ˆm(†), δ ˆm(†) +j , δˆb(†). We skip the algebraic +steps, but outline the main differences with respect to the +results in Refs. [34, 36]. After writing the cavity operator +in terms of the magnon operators, we obtain the following +equation for the additional magnon modes +Ξj[ω]−1δ ˆmj[ω] = −i +� +g∗ +R,j − igamjgamχa[ω] +� +δ ˆm[ω] +− igB,jδ ˆm†[ω] − iGR,jδˆb[ω] − iG∗ +B,jδˆb†[ω] ++ gamjχa[ω] +� +k̸=j +gamkδ ˆmk[ω] + ˆ˜ξmj[ω], +(18) +where ˆ˜ξmj represents the noise term modified by the inter- +action with the cavity, and we have defined the effective +magnon susceptibility in correspondence with the previ- +ous case in Eq. (4) +Ξj[ω]−1 = χ−1 +mj [ω] + g2 +amjχa[ω]. +(19) +We notice that the first term on the right-hand side of +Eq. (18) includes the indirect coupling between the j-th +magnon mode and the Kittel mode via the cavity. A sim- +ilar term related to the coupling between the additional +magnon modes appears in the last line of Eq. (18), and +since gamj ≪ gam, we discard these contributions. +After using Eq. (18) to eliminate the additional +magnon modes in favor of the Kittel mode and the +phonon fluctuations, we obtain the following set of cou- +pled equations +Ξ−1 +m [ω]δ ˆm[ω] = ηm[ω] ˆm[ω] − iΛm[ω] ˆm†[ω] +− iGm,R[ω]δˆb[ω] +− iGm,B[ω]δˆb†[ω] + ˆ˜ξm[ω], +� +�χ−1 +b [ω] − i +N +� +j=1 +σj[ω] +� +� δˆb[ω] = −i ˜G∗ +b,R[−ω]δ ˆm[ω] +− i ˜Gb,B[ω]δ ˆm†[ω] + ˆ˜ξb[ω] ++ i +� +� +N +� +j=1 +λj[ω] +� +� δˆb†[ω]. +(20) +We have included all the noise terms in ˆ˜ξm,b[ω]; all other +functions appearing in the equations below are defined +in the following. The coupling to the additional magnon +modes has the following effects: the introduction of a self- +energy term on the phonon mode; the modification of the +coupling between the Kittel mode and the phonon mode; +and a modification of the Kittel mode susceptibility and +squeezing. We briefly comment on each of these effects. +At this intermediate step, the phonon susceptibility is +modified by the self-energy term +N +� +j=1 +σj[ω] = i +N +� +j=1 +|g0 +mjb⟨ ˆmj⟩|2 � +Ξj[ω] − Ξ∗ +j[ω] +� +, +(21) +which is defined in analogy with the self-energy term de- +rived in Refs. +[34, 36] and given in Eq. (3). +Such a +term represents the direct dynamical backaction of the + +7 +coupling between the phonon mode and the additional +magnon modes. +The additional magnon modes modify the couplings +between the Kittel mode and the phonon mode. In fact, +the effective coupling constants appearing in Eqs. (20) +are given by +˜Gb,R[ω] = GR − gamχ∗ +a[−ω] +� +j +gamjGR,jΞ∗ +j[−ω] − i +� +j +� +GB,jgb,jΞj[ω] − GR,jgR,jΞ∗ +j[−ω] +� +, +˜Gb,B[ω] = GB − gamχ∗ +a[−ω] +� +j +gamjG∗ +B,jΞ∗ +j[−ω] − i +� +j +� +G∗ +R,jgb,jΞj[ω] − G∗ +B,jgR,jΞ∗ +j[−ω] +� +, +Gm,R[ω] = GR − gamχa[ω] +� +j +gamjGR,jΞj[ω] − i +� +j +� +gR,jGR,jΞj[ω] − gB,jGB,jΞ∗ +j[−ω] +� +, +Gm,B[ω] = GB − gamχa[ω] +� +j +gamjG∗ +B,jΞj[ω] − i +� +j +� +gR,jG∗ +B,jΞj[ω] − gB,jG∗ +R,jΞ∗ +j[−ω] +� +. +(22) +From these expressions, we notice two types of modifica- +tions. The second terms in Eqs. (22) represent the effect +of the indirect coupling between the additional magnon +modes and the Kittel mode via the cavity. +The third +terms are proportional to gB(R),j, which in turn (see Ta- +ble II) are due to the magnon cross-Kerr nonlinearity, +ˆm† +j ˆmj ˆm† ˆm in the Hamiltonian of Eq. (7). The relevance +of these corrections for a given drive frequency is deter- +mined by the susceptibilities of the additional magnon +modes. +The Kittel mode squeezing term Λm[ω] reads +Λm[ω] = 2gms +− gam(χa[ω] + χ∗ +a[−ω]) +� +j +gB,jgamjΞj[ω], (23) +where the first term is due to the self-Kerr nonlinearity, +while the second term is a combination of the magnon +cross-Kerr nonlinearity with the indirect coupling be- +tween the magnon modes via the microwave cavity. The +Kittel mode susceptibility is also modified by the term +ηm[ω], which is given by +ηm[ω] = g2 +amχ2 +a[ω] +� +j +g2 +amjΞj[ω] ++ 2igamχa[ω] +� +j +Re[gR,j]gamjΞj[ω] +− +� +j +� +|gR,j|2Ξj[ω] − |gB,j|2Ξ∗ +j[−ω] +� +. +(24) +The three terms in Eq. (24) describe the effects of a two- +mode squeezing between the Kittel mode and each of +the additional magnon modes. The first term is related +to the indirect coupling of the modes via the microwave +cavity while the last term is the direct two mode squeez- +ing induced by the magnon cross-Kerr nonlinearity. The +second term is a combination of both effects. +From Eqs. (20), we eliminate the Kittel mode operator +and obtain an equation for the phonon mode operator +� +χ−1 +b [ω] − iΣTot[ω] +� +δˆb[ω] = iΛb[ω]δˆb†[ω] + ˆΥb[ω], (25) +where ˆΥb[ω] includes all the noise terms driving the +phonon fluctuations, Λb[ω] describes phonon squeezing, +and ΣTot[ω] is the total self-energy. We focus only on the +self-energy term which is given by +ΣTot[ω] = Σm[ω] + +N +� +j=1 +σj[ω], +(26) +where the contribution of Kittel mode to the self-energy +is given by +Σm[ω] = i +� +˜G∗ +b,R[−ω] ˜Gm,R[ω]˜Ξm[ω] +− ˜Gb,B[ω] ˜G∗ +m,B[−ω]˜Ξ∗ +m[−ω] +� +. +(27) +In this equation, we have defined +˜Gm,R[ω] = Gm,R[ω] + i +Λm[ω]G∗ +m,B[−ω] +Ξ∗,−1 +m +[−ω] − η∗m[−ω] +, +˜Gm,B[ω] = Gm,B[ω] + i +Λm[ω]G∗ +m,R[−ω] +Ξ∗,−1 +m +[−ω] − η∗m[−ω] +, +(28) +which are the magnomechanical couplings modified by +the Kittel mode squeezing term Λm[ω], and the Kittel +mode susceptibility +˜Ξ−1 +m [ω] = Ξ−1 +m [ω] − ηm[ω] − +Λm[ω]Λ∗ +m[−ω] +Ξ∗,−1 +m +[−ω] − η∗m[−ω] +, (29) +which includes both the Kittel magnon squeezing in +Λm[ω] and the effects of two-mode squeezing interactions +with the additional magnon modes in ηm[ω]. +C. +Magnomechanical decay rate corrections +We turn our attention now to the effects on the mag- +nomechanical decay rate, in connection with the observed + +8 +shift reported in Ref. [43]. The total change in the phonon +linewidth is given by +Γmag[ω] = 2Im[ΣTot[ω]] = 2Im[Σm[ω]] + +N +� +j=1 +Γj[ω] +Γj[ω] = 2Im[σj[ω]]. +(30) +The contribution of self and cross-Kerr nonlinear terms +are included in the above self-energy. The corresponding +frequency shift is different than the static one, observed +in Ref. [49], which has already been included in the mod- +ified phonon frequency ˜Ωb defined in Eqs. (16). The self- +Kerr nonlinearity changes the behavior of the magnome- +chanical decay rate with the detuning. This is due to +three main factors: the modification of the steady-state +number of magnons |⟨ ˆm⟩|2, the induced static magnon +frequency shift, given in Eq. (16), and the generation of +squeezing in the magnon fluctuations. This has a conse- +quence for both dynamical backaction evasion, as we will +show, and for backaction cooling. The latter has been +recently reported in a system where a mechanical oscil- +lator is parametrically coupled to a nonlinear cavity [50]. +The magnomechanical frequency shift is given by +δΩb[ω] = −Re [ΣTot[ω]] , +(31) +which has a decomposition similar to the one shown in +Eq. (30). +The relevance of the different corrections due to the +additional magnon modes depend on their frequencies +with respect to the drive and their coupling rates to +the cavity. Recalling that Gj = g0 +mjb⟨ ˆmj⟩, we use the +steady state from Eq. (10). +For the experimental pa- +rameters in consideration, given in table I, due to the +strong coupling between the Kittel mode and the cavity, +√κeϵd ≪ gamRe[⟨ ˆm⟩] and the steady state of the weakly +driven Walker modes can be approximately written as +|⟨ ˆmj⟩|2 = +g2 +amjg2 +am|⟨ ˆm⟩|2 +|FmjFa + g2amj − iK(j) +cr,m|⟨ ˆm⟩|2|2 . +(32) +Within these approximations, the contribution of such +Walker modes to the magnomechanical linewidth is given +by +Γj[ω] = 2g2 +am|⟨ ˆm⟩|2 +|g0 +mjb|2g2 +amj +|FmjFa + g2amj − iK(j) +cr,m|⟨ ˆm⟩|2|2 +× Re +� +Ξj[ω] − Ξ∗ +j [ω] +� +, +(33) +which for small K(j) +cr,m gives a contribution to the mag- +nomechanical decay rate which is proportional to |⟨ ˆm⟩|2. +The contribution to the magnomechanical decay rate +also depends on the detuning between the drive and the +magnon frequency. +While Γj[ω] quantifies the direct effect of the coupling +between the phonon mode and the additional magnon +modes, there are also indirect effects due to the cou- +pling between the additional magnon modes and the +Kittel mode via the microwave cavity. +Those are in- +cluded in the term 2Im[Σm[ω]] via the modified coupling +rates ˜Gm(b),R(B)[ω] and the modified Kittel mode sus- +ceptibility ˜Ξm[ω]. In general, the corrections included in +those terms are proportional to |gms|2, to |GR(B),j|2 or to +|gR(B),j|2. Following the same procedure outlined above, +it is possible to show that those are all proportional to the +steady-state Kittel mode occupation. Their frequency- +dependent coefficients have a more complicated form due +to the explicit dependence of the modified couplings and +susceptibilities on frequency. We can describe all the cor- +rections by the expression +Γmag[ω] = Γ0 +mag[ω] + α[ω]|⟨ ˆm⟩|2, +(34) +where Γ0 +mag[ω] is given in Eq. (3), without including ad- +ditional magnon modes and nonlinearities. Such a cor- +rection was used phenomenologically in Ref. [43] to ex- +plain the observed discrepancies of the experimental re- +sults with Γ0 +mag[ω]. +III. +MAGNOMECHANICAL COUPLING +In the last section, we obtained the modifications of +the phonon self-energy due to the coupling between the +phonon mode and additional magnon modes. Such con- +tributions depend on the relative strength of the mag- +nomechanical coupling to the additional magnon modes +with respect to the coupling to the Walker mode, which +in turn depends on the geometry of the magnet. Before +specifying the latter, we briefly review the main points of +the derivation of the magnomechanical coupling Hamil- +tonian, done in detail in the literature [29, 30, 60, 61], +starting from the magnetoelastic energy [25, 26] +EME = B1 +M 2 +S +� +d3r +� +M 2 +xεxx + M 2 +y εyy + M 2 +z εzz +� ++ 2B2 +MS +� +d3r +� +MxMyεxy + MyMzεyz ++ MxMzεxz +� +, +(35) +where Mx,y,z are the magnetization components, MS is +the saturation magnetization, and +εij = (∂iuj + ∂jui) /2 +(36) +is the linear strain tensor with u(r, t) the displacement. +The magnetoelastic coefficients B1 and B2 are material +and temperature-dependent constants. For YIG at room +temperature, MS = 140 kA/m [48], B1 = 3.48 × 105 +J/m3 and B1 = 6.4 × 105 J/m3 [29]. +The magnome- +chanical Hamiltonian is then obtained by quantizing the +magnetization and displacement fields. +We first quantize the magnetization. We consider that +the magnetization displays small fluctuations around a + +9 +uniform saturation value MS [48] (such assumption can +be generalized to non-uniform magnetic ground states +[62]). In this framework Mz/MS ≫ Mx,y/MS. The x and +y components of the magnetization can then be written +as a superposition of modes, each labelled with a general +index j, such that the quantized magnetization field is +given by [63] +ˆ +Mx,y(r, t) = +� +j +Mj +� +δmx,y;j(r) ˆmj + δm∗ +x,y;j(r) ˆm† +j +� +, +(37) +where { ˆmj} is a set of bosonic operators satisfying +[ ˆmj, ˆm† +j] = 1. +The quantization procedure is valid in +the so called spin-wave limit for magnetic excitations. +The mode functions δmj(r) are obtained by solving the +Landau-Lifshitz equation for the magnetization fluctua- +tions [48], plus the appropriate boundary conditions. The +quantities Mj are the zero-point fluctuations of the mode +j given by +Mj = +� +ℏ|γ|MS +2Vj +, +(38) +where the mode volume is given by [63] +Vj = 2Im +�� +d3r δmy;j(r)δm∗ +x;j(r) +� +. +(39) +Such mode decomposition ensures that the magnetic en- +ergy density yields the Hamiltonian for a set of uncoupled +harmonic oscillators of the form ˆHm = � +j ℏωj ˆm† +j ˆmj, +with frequencies ωj obtained from the imposed bound- +ary conditions. +The elastic vibrations are quantized in terms of phonon +modes [64]. The displacement field is given by the super- +position of modes +ˆu = +� +α +Xα +� +fα(r)ˆbα + f ∗ +α(r)ˆb† +α +� +. +(40) +The mode functions fα(r) are dimensionless, and given +as the solution of the elastic boundary problem [65]. The +zero-point fluctuations are given by +Xα = +� +ℏ +2ρΩαNα +, +(41) +where +Nα = +� +d3r |fα(r)|2, +(42) +is the mode normalization. Such a mode decomposition +yields, for the non-interacting phonons, the Hamiltonian +ˆHb = � +α Ωαˆb† +αˆbα. +Substituting Eqs. (37) and (40) in the magnetoelas- +tic energy given by Eq. (35), we obtain an interaction +Hamiltonian describing the coupling between magnons +and phonons. +Such a Hamiltonian includes the fol- +lowing terms: +(i) linear magnon-phonon coupling ∝ +g(L) +mjbα ˆm† +jˆbα+H.c., relevant only for resonant magnon and +phonon modes, for example, for small magnetic parti- +cles [30] and for magnetic films [31–33]; (ii) spontaneous +parametric conversion terms ∝ ˆmj ˆmkˆb† +α and ∝ ˆm† +j ˆm† +kˆbα, +relevant when the phonon mode frequency matches the +sum of the frequency of the magnon modes j and k. +Such an interaction describes the creation of a pair of +magnons via the annihilation of a phonon; (iii) para- +metric phonon-magnon coupling ∝ g0 +mjmkbα ˆm† +j ˆmkˆb+H.c. +The off-resonant terms ˆmj ˆmkˆbα and ˆm† +j ˆm† +kˆb† +α can be +eliminated via a rotating wave approximation. +Here, we focus on the interaction (iii), the parametric +phonon-magnon coupling, which is given by the Hamil- +tonian +ˆHmb/ℏ = +� +{j̸=k},α +� +g0 +mkmjbα ˆm† +k ˆmjˆbα + ˜g0 +mkmjbα ˆm† +k ˆmjˆb† +α +� ++ +� +j,α +g0 +mjbα ˆm† +j ˆmjˆbα + H.c., +(43) +where {j ̸= k} indicates that the sum is over all j’s not +equal to k and without repeating combinations. In the +above equation we have separated the coupling terms +between one magnon mode and one phonon mode and +the terms involving two different magnon modes and a +phonon mode. The coupling rates are given explicitly by +g0 +mjbα +Nmjbα += B1 +� +d3r +� +|δmx;j(r)|2(∂xfx;α(r) − ∂zfz;α(r)) + |δmy;j(r)|2(∂xfx;α(r) − ∂zfz;α(r)) +� ++ B2 +� +d3r Re +� +δmx;j(r)δm∗ +y;j(r) +� +(∂yfx;α(r) + ∂xfy;α(r)) , +g0 +mkmjbα +Nmkmjbα += B1 +� +d3r +� +δm∗ +x;k(r)δmx;j(r)(∂xfx;α(r) − ∂zfz;α(r)) + δm∗ +y;k(r)δmy;j(r)(∂yfy;α(r) − ∂zfz;α(r)) +� ++ B2 +2 +� +d3r +� +δm∗ +y;k(r)δmx;j(r) + δm∗ +x;k(r)δmy;j(r) +� +(∂yfx;α(r) + ∂xfy;α(r)) . +(44) + +10 +where we have defined +Nmkmjbα = 2XαMkMj +ℏM 2 +S +, +(45) +and Nmjbα = Nmjmjbα. +The coupling ˜g0 +mkmjbα is ob- +tained from g0 +mkmjbα with the substitution ∂xifj;α(r) → +∂xif ∗ +j;α(r). +Focusing now on the coupling to a specific phonon +mode, the magnomechanical Hamiltonian is given by +ˆHmb +ℏ += ωm ˆm† ˆm + +� +j +ωj ˆm† +j ˆmj + Ωbˆb†ˆb ++ +ˆHmb,I +ℏ +, +(46) +where the coupling terms are +ˆHmb,I +ℏ += g0 +mb ˆm† ˆmˆb + +� +j +g0 +mjb ˆm† +j ˆmjˆb ++ ˆm† � +j +g0 +mmjb ˆmjˆb + +� +j̸=k +g0 +mkmjb ˆm† +k ˆmjˆb ++ H.c. +(47) +In the above equations, we have separated the terms of +the Kittel mode, which from now on we do not label, +while the other Walker modes are labelled by the index +j. The coupling rates are complex numbers, but we can +absorb the phase of one of such coupling rates into the +phonon field. +We write g0 +mb = |g0 +mb|eiφmb, and define +ˆ˜b = ˆbeiφmb, such that +ˆHm˜b +ℏ += ωm ˆm† ˆm + +� +j +ωj ˆm† +j ˆm + Ωbˆ˜b†ˆ˜b ++ g0 +m˜b ˆm† ˆm(ˆ˜b + ˆ˜b†) ++ +� +j +� +g0 +mj˜b ˆm† +j ˆmjˆ˜b + H.c. +� ++ ˆm† � +j +� +g0 +mmj˜b ˆmjˆ˜b + H.c. +� ++ +� +j̸=k +� +g0 +mkmj˜b ˆm† +k ˆmjˆ˜b + H.c. +� +, +(48) +where g0 +m˜b = |g0 +mb|, g0 +mj˜b = g0 +mjbe−iφmb and g0 +mmj˜b = +g0 +mmjbe−iφmb. Such a transformation corresponds to tak- +ing the phase of the coupling between the phonon mode +and the Kittel mode as a reference for the other cou- +plings. From now on, we take ˆ˜b → ˆb. This gauge trans- +formation of the phonon field does not change the Kerr +nonlinear terms, which are quadratic in the phonon field. +A. +Magnomechanical coupling rates for a sphere +The overlap integrals in Eq. (44) depend on the specific +geometry of the sample and the direction of the applied +magnetic field, which defines the mode functions. +We +consider now the case of a YIG sphere, corresponding to +the experimental configuration of [43]. +A sphere supports magnetostatic modes called Walker +modes [46, 47, 66], which have frequencies that can be +tuned by the value of the external bias field. To describe +such modes, it is convenient to introduce the following +characteristic frequencies +ωM = |γ|µ0MS, +ω0 = |γ|µ0 +� +H0 − MS +3 +� +, +(49) +where |γ|/2π = 28 GHz/T is the gyromagnetic ratio, +µ0 is vacuum permeability, and H0 is the applied bias +magnetic field. The Walker modes are conveniently given +in a nonorthogonal coordinate system {ξ, η, φ} defined by +the transformation [46] +x = R +� +−χP[ω] +� +1 − ξ2 sin η cos φ, +y = R +� +−χP[ω] +� +1 − ξ2 sin η sin φ, +z = R +� +χP[ω] +1 + χP +ξ cos η, +(50) +where +χP[ω] = +ωMω0 +ω2 +0 − ω2 . +(51) +At the the sphere’s surface η → θ and +ξ[ω] → ξ0[ω] = +� +1 + χP[ω] +χP[ω] +. +(52) +The frequencies of Walker modes are given by the non- +linear equation [46, 47] +ξ0[ω]∂ξP m +l (ξ[ω]) +P m +l (ξ[ω]) |ξ=ξ0 − mκP[ω] + n + 1 = 0, +(53) +where +κP[ω] = − ωMω +ω2 +0 − ω2 , +(54) +and P m +l +are the associated Legendre polynomials. The +Walker modes are labelled by three indices, {lmν}, with +l ≥ 1 and |m| ≤ l. For m > 0, Eq. (53) has (n − |m|)/2 +roots, while for m < 0 it has 1 + (n − |m|)/2 solutions +(both rounded down). The mode functions of the Walker +modes are given by +� +δmx;lmν +δmy;lmν +� += − +� +χP[ωlmν] +iκP[ωlmν] +−iκP[ωlmν] χP[ωlmν] +� � +∂xψlmν +∂yψlmν +� +(55) +where the magnetostatic potential inside the sphere is +ψlmν(r) = P m +l (ξ)Y m +l (η, φ). +(56) + +11 +For the phonon modes, we consider an unpinned sphere +and stress-free boundary conditions [65]. There are two +families of mechanical modes of a homogeneous sphere: +torsional (T) and spherical (S) modes. Torsional modes +are purely shear modes, while spherical modes involve +both shear and compression. Both families of modes are +labelled by three indexes {νlm}, where l and m are polar +and azimuthal indexes −l ≤ m ≤ l while ν is a radial +index. We focus here on S modes, whose frequencies are +given by [65] +T (a) +λν T (b) +λν − T (c) +λν T (d) +λν = 0, +(57) +where +T (a) +λν = +� +λ(λ − 1) − +˜β2[ω]R2 +2 +� +jλ(˜α[ω]R) ++ 2˜α[ω]Rjλ+1(˜α[ω]R) +T (b) +λν = +� +λ2 − 1 − +˜β2[ω]R2 +2 +� +jλ(˜β[ω]R) ++ ˜β[ω]Rjλ+1(˜β[ω]R) +T (c) +λν = λ(λ + 1) +� +(λ − 1)jλ(˜β[ω]R) +− ˜β[ω]Rjλ+1(˜β[ω]R) +� +T (d) +λν = (λ − 1)jλ(˜α[ω]R) − ˜α[ω]Rjλ+1(˜α[ω]R). +(58) +The parameters ˜α[ω] = ω/cL, and ˜β[ω] = ω/cT , are given +in terms of the longitudinal (L) and transverse (T) sound +velocities cL,T . jλ(x) denotes the spherical Bessel func- +tion. Since Eq. (57) does not depend on m, for given {νl} +there are 2l + 1 degenerate modes. The mode functions +for a S mode read, in spherical coordinates {er, eθ, eφ}, +fνλm = eiφm +� +� +Gνλ(r)P m +l (cos θ) +Fνλ(r)∂θP m +l (cos θ) +im +sin θFνλ(r)P m +l (cos θ) +� +� , +(59) +where +Gνλ(r) = R +r +� +λjλ(˜α[ω]r) − ˜α[ω]rjλ+1(˜α[ω]r) +− T (d) +λν +T (b) +λν +λ(λ + 1)jλ(˜β[ω]r) +� +, +Fνλ(r) = R +r +� +jλ(˜α[ω]r) + T (d) +λν +T (b) +λν +˜β[ω]rjλ+1(˜β[ω]r) +− T (d) +λν +T (b) +λν +(λ + 1)jλ(˜β[ω]r) +� +, +(60) +We focus our results for the mode probed in [43], the S122 +mode. +Even though the magnon and phonon modes functions +are given in terms of well-known special functions, the +coupling constant involves a non-trivial combination of +derivatives of those. Furthermore, the coordinate trans- +formation in Eqs. (50) is not easily invertible. While this +is not a problem when computing the coupling to the +Kittel mode, which is uniform, the exact expression for +the integrands of Eqs. (44) is not elucidating. Differently +from other parametrically coupled systems, it is hard to +infer from (44), e.g., selection rules. We, therefore, com- +pute the overlap integrals numerically and evaluate how +the couplings g0 +mjb compare with the coupling to the Kit- +tel mode g0 +mb. +It is also important to notice that the +magnomechanical couplings depend on both the inten- +sity of the bias magnetic field and its direction. In fact, +the coupling to the Kittel mode can even vanish for spe- +cific relative orientation of the magnetic field [29]. We +consider the case of a fixed bias field at a direction that +maximizes the coupling between the Kittel and the S122 +mode, as depicted in Fig. 2. +(a) +(b) +(c) +(d) +H0 k ez +R = 125 µm +� +���� +���� +���� +���� +|f|/max[|f|] +z/R +y/R +z/R +x/R +y/R +x/R +FIG. 2. Profile of the S122 mode of a sphere. (a) The bias field +H0 is parallel to the ez direction, and we consider a sphere +made of YIG with a radius R = 125 µm. (b-d) mode profile +|f(r)| for the spherical mode S122 in the (b)yz, (c) xz and +(d) zy planes. +Figure 3 shows the frequencies ωmj of the Walker +modes, the ratios |g0 +mjb|/|g0 +mb| between the magnome- +chanical coupling rate to the Walker mode (lmν) and to +the Kittel mode, and φj − φ, the relative phase between +g0 +mjb and g0 +mb. Results are shown for l up to 4 and for +Walker modes lying in a frequency range close to the Kit- +tel mode. Due to better mode overlap, some higher order +Walker modes, for example, the (200), couple strongly +with the phonon mode in comparison with the coupling +to the Kittel mode. In the theoretical analysis of section +I, we have not included in the magnomechanical Hamil- +tonian in Eq. (7) the last two terms of Eq. (48). Those +describe scattering processes between different magnon +modes via the phonon mode. For the considered case, +|g0 +mmjb|, |g0 +mkmjb| ≪ |g0 +mb|, |g0 +mjb|, and those processes can +be safely discarded. Nevertheless, it is possible that for + +0 +-1 +1 +0 +10 +-1 +-1 +0 +10 +-1 +-1 +0 +112 +some relative orientation between the magnon modes and +the phonon mode, set by the external bias field, those +processes can have a stronger coupling rate. +FIG. 3. +(a) Frequency of the Walker modes ωmjb in units +of the Kittel mode frequency ωmb. The labels by each point +indicate the radial magnon mode label ν; (b) Absolute value +of the magnomechanical coupling between the Walker modes +and the S122 mode in units of the coupling to the Kittel mode +g(0) +mb; (c) Phase of the magnomechanical coupling with respect +to the phase of the Kittel mode magnomechanical coupling +φmjb −φmb in radians. Results for a sphere of radius R = 125 +µm. +The dashed line is the reference value (for the Kittel +mode) for each quantity. +IV. +EVALUATION OF THE MODEL FOR THE +PHONON SELF-ENERGY ON DYNAMICAL +BACKACTION EVASION +The self-energy obtained in Eq. (30) includes contribu- +tions due to the Kerr nonlinearity and to the couplings to +higher order Walker modes. We focus our analysis now +on the effect of such contributions to the magnomechan- +ical decay for detunings in the vicinity of the backaction +evasion point. +In correspondence with the experiment [43], we con- +sider that the microwave drive frequency is varied be- +tween ωd,− and ωd,+ inside the frequency range {ω−, ω+} +between the hybrid modes frequencies, given by Eq. +(5). +The Walker modes contributing appreciably to +the phonon self-energy lie between (ωd,− − Ωb) and +(ωd,++Ωb). Since the system is in the resolved side-band +regime, any modes outside this frequency range would +not allow efficient scattering of phonons, and thus can be +neglected. The first condition corresponds to the lower +drive frequency corresponding to the blue side-band of a +magnon mode, while the second corresponds to driving +the red side-band of a magnon mode. For the parameters +summarized in Table I, only the mode (4, 3, 0) is in this +frequency range. In fact, the (4, 3, 0) mode is degener- +ate with the Kittel mode. The frequency configuration is +shown in Figs. 4(a,b), and the mode profile of the Walker +mode (4, 3, 0) is shown in Figs. 4 (c,d). +FIG. 4. Frequency configuration of the magnomechanical sys- +tem in consideration. (a) The microwave cavity frequency ωa +is higher than the Kittel mode frequency, which is degener- +ate with the (4, 3, 0) Walker mode. The red frequency range +corresponds to the microwave drive considered here. (b) Due +to strong coupling, the Kittel mode and the Microwave mode +hybridize, forming the two modes ω±. (c) Real part and (d) +imaginary parts of the transverse magnetization of the Walker +mode (4, 3, 0). The profiles were evaluated at z/R = cos π/4, +and for better visualization, the vectors were normalized to +max[ +� +Re[δmx]2 + Re[δmy]2]. +To address the effects of nonlinearities and coupling to +the higher order Walker mode, we define the dimension- +less parameters ηc = gmjc/gmc and ηK = Km/K0 +m. The +first parameter quantifies the strength of the coupling +between the Walker modes and the cavity compared to +the coupling between the Kittel mode and the cavity. +The second parameter quantifies the strength of the self- +Kerr nonlinearity compared to the value shown in Table I +K0 +m = −2π×5.15 nHz. We call here ηK the dimensionless +Kittel magnon self-Kerr nonlinearity. This parameter de- +pends on the alignment between the anisotropy axis of +the magnet with the external magnetic field, which has +not been taken into account in [43]. +Figure 5 shows (a) the magnomechanical decay for +ηc = 0 (the additional Walker mode is not driven by +the microwaves) and for several values of ηK, and (b) +the magnomechanical decay for several values of ηc at a + +13 +Magnomechanical +decay (Hz) +� +��� +��� +��� +��� +��� +⌘K +Drive detuning from the upper +hybrid mode (MHz) +� +���� +���� +���� +���� +���� +���� +⌘c +⌘c = 0 +⌘K = 0.2 +Magnomechanical +decay (Hz) +(a) +(b) +FIG. 5. Magnomechanical decay rate Γmag[Ωb] including the +contribution of the Walker mode (4, 3, 0) as a function of the +detuning from the upper hybrid mode for (a) ηc = 0 (without +microwave coupling to the additional Walker mode) and for +several values of ηK (dimensionless Kittel magnon self-Kerr +nonlinearity); and (b) for ηK = 0.2 and for several values +of ηc. The dashed line is the prediction from the self-energy +(3) derived in [36]. +The magnomechanical coupling to the +(4, 3, 0) Walker mode corresponds to that shown in Fig. 3. +The driving power is 15 mW. Parameters in correspondence +with the experiment [43], given in Table I. +fixed ηK = 0.2. The self-Kerr nonlinearity of the Kittel +mode changes the slope of the magnomechanical decay as +a function of the detuning. For a fixed Kerr nonlinearity, +the additional magnon mode shifts down the magnome- +chanical decay, that is, the weakly driven Walker mode +adds energy to the vibrational mode, yielding a negative +contribution to the decay rate. +The magnomechanical frequency shift is also modi- +fied by the Kittel mode nonlinearity and by the cou- +pling to the additional magnon mode. This is depicted in +Fig. 6, which shows the magnomechanical frequency shift +δΩ = −Re[ΣTot[Ωb]] for (a) several values of the self-Kerr +nonlinearity and (b) several values of the coupling to the +additional magnon mode. Whereas the Kerr nonlinearity +induced a tilt in the slope of the magnomechanical decay +rate, its effect on the the magnomechanical frequency +shift consists on an extra negative shift. +We also no- +tice that the magnomechanical frequency shift does not +vanish for a drive at the frequency where the magnome- +chanical decay vanishes. This is the case because for the +parameters considered, the Kittel mode frequency does +not match the microwave frequency. For perfectly match- +ing Kittel mode and microwave frequencies, and in the +absence of additional magnon modes, both the magnome- +chanical decay and the magnomechanical frequency shift +vanish at the same drive frequency [36]. +� +��� +��� +��� +��� +��� +⌘K +Drive detuning from the upper +hybrid mode (MHz) +� +���� +���� +���� +���� +���� +���� +⌘c +⌘c = 0 +⌘K = 0.2 +Magnomechanical +frequency shift (Hz) +(a) +(b) +Magnomechanical +frequency shift (Hz) +FIG. 6. Magnomechanical frequency shift δΩb including the +contribution of the Walker mode (4, 3, 0) as a function of the +detuning from the upper hybrid mode for (a) ηc = 0 (without +microwave coupling to the additional Walker mode) and for +several values of ηK (dimensionless magnon self-Kerr nonlin- +earity); and (b) for ηK = 0.2 and for several values of ηc. +The dashed line is the prediction from the self-energy Eq . (3) +derived in Ref. [36]. The magnomechanical coupling to the +(4, 3, 0) Walker mode corresponds to that shown in Fig. 3. +The driving power is 15 mW. Parameters in correspondence +with the experiment [43], given in Table I. +In Fig. 5 one notices that the drive frequency at which +the magnomechanical decay vanishes changes with both +ηK and ηc. For applications where evading backaction +is important, it is necessary that such modifications are +taken into account. We show in Fig. 7 the drive frequency +for backaction evasion (with respect to the upper hybrid +mode frequency) as a function of the drive power for (a) +several values of the Kittel self-Kerr nonlinearity and (b) +several values of the coupling to the additional magnon +mode at a fixed ηK. For the case without nonlinearities +and without coupling to the additional magnon mode, +the backaction evasion frequency has a weak dependency +on power (not perceptible in the plot). When the correc- +tions are included, a stronger linear dependency of the +backaction drive frequency with the power is induced. +For the parameters in consideration, the difference can +be of of the order of ∼ 0.1 MHz at moderate powers of +10 mW. + +100 +0 +-100 +-200 +-14. +-13. +12. +11100 +0 +-100 +-200 +-300 +14. +-13. +12. +11-180 +-200 +-220 +-240 +-14. +-13. +12. +11-180 +-200 +-220 +-240 +-14. +-13. +-12. +1114 +Backaction evasion +detuning (MHz) +Power (mW) +(a) +(b) +⌘K = 0.2 +⌘c = 0 +� +��� +��� +��� +��� +��� +⌘K = 0.2 +� +���� +���� +���� +���� +���� +���� +⌘c = 0 +FIG. 7. +Detuning between drive frequency and the upper +hybrid mode for backaction evasion as a function of power for +(a) no coupling to additional magnon modes and for several +values of the dimensionless Kittel self-Kerr nonlinearity ηK, +and (b) for several values of the coupling between the (4, 3, 0) +Walker mode and the microwave cavity at a fixed ηK = 0.2. +Parameters in correspondence with the experiment [43], given +in Table I. +In order to quantify the agreement between our model +and the measured data in Ref. [43], we study the dif- +ference between the theoretical magnomechanical decay +Γmag[Ωb] rate and the experimental data Γexp. In Fig. 8, +we show the absolute difference |Γmag[Ωb]−Γexp| between +theory and experiment as a function of the drive power +for different drive frequencies. +Our proposed model +agrees well with the experimental data, besides the differ- +ence at higher powers and drives farther from the upper +hybrid mode. In the worst case, the model proposed here +improves the discrepancy between data and theory from +∼ 120 Hz (red, dashed curve in Fig. 8), to a difference +of ∼ 50 Hz (red, solid curve in Fig. 8). Otherwise, we +notice good agreement between theory and experiment +for drive powers up to ∼ 14 mW. At such powers the co- +herent number of magnons generated by the microwave +drive |⟨ ˆm⟩|, see Eq. (12), is between ≈ 6.0 × 1013 at a de- +tuning from the upper hybrid mode ∆+ = −11 MHz and +≈ 7.4 × 1013 at ∆+ = −14 MHz. We should notice that +for the parameters considered here, the system is not in +a bistable regime. +Difference between +theory and data (Hz) +Power (mW) +(-13.4, -13.7) MHz +(-12.1, -12.0) MHz +-11 MHz +Detuning from the upper normal mode +� +�� +�� +� +�� +�� +��� +FIG. 8. Absolute difference between the theory for the mag- +nomechanical decay and the experimental data as a function +of power at different detunings. The dashed curves correspond +to the prediction of the previous theory using (4), while the +solid lines correspond to the theory developed in this paper. +Theory predictions use parameters in correspondence with the +experiment [43], given in Table I. +In Fig. 9 we show in (a-c) the magnomechanical decay +as a function of the drive frequency detuning from the +upper hybrid mode. As in the discussion above, we con- +sider the coupling only to the (4, 3, 0) Walker mode, and +we choose ηK = 0.2 and ηc = 0.3, which yields a good +agreement between theory and data. In the plots of Fig. +9 (d-f), we show the difference |Γmag[Ωb] − Γexp|. While +the correction due to the coupling to the (4, 3, 0) Walker +mode improves the agreement between theory and data +with respect to the previous theory framework [34], we +notice, as shown in Fig. 8, that at higher drive pow- +ers, there is a further discrepancy with the experiment +for drives away from the dynamical backaction evasion +points. Such drive frequencies are closer to frequencies +of the magnon modes, in particular to higher order modes +not considered here, and can thus induce a nonlinear be- +havior that was not taken into account. We also notice +that the errors of the data shown in Fig. 9 (a) do overlap +with both the present theory and the one used in [34], +the trend shown in Figs 9 (b-c) holds for the intermediate +drive powers not shown here. +V. +CONCLUSIONS +Dynamical backaction effects in magnomechanical sys- +tems are a consequence of the radiation pressure-like +coupling between magnons and phonons [29, 34] which +can be exploited for applications ranging from generat- +ing entangled states to noise-based therometry [36]. In +this paper, we have extended the description of dynami- +cal backaction in cavity magnomechanics by including in +the system’s dynamics self and cross-Kerr nonlinearities, +and the coupling between the phonon mode and addi- +tional magnon modes. While nonlinearities are intrinsic +to magnetic systems due to, e.g., magnetic anisotropy +[48], magnon modes other than the uniform Kittel mode +are always present and can couple to phonons as efficient +as (if not more than) the Kittel mode. A non-uniform mi- + +-11.75 +-11.85 +-11.95 +10 +¥15 20-11.75 +-11.85 +-11.95 +10 +15 +2015 +FIG. 9. Comparison between the magnomechanical decay rate Γmag[Ωb] predicted by Eq. (3) (Gray, dashed line), by Eq. (30) +(Blue line) and the experimental data measured in [43] (magenta points). In these plots we have used ηc = 0.3 and ηK = 0.2, +which yields a good agreement between our model and the experimental data specially close to the point of dynamical backaction +evasion. (a)-(c): Magnomechanical decay rate as a function for the detuning; (d)-(e) Absolute difference between the theory +and the experiment as a function of the detuning (we have omitted the error bars in these plots for a better visualization). +Theory curves with parameters in correspondence with the experiment [43], given in Table I. +crowave field can weakly drive such modes, which modi- +fies the backaction induced decay rate and frequency shift +of the phonon mode. Our framework considers a single +phonon mode, an assumption that can be readily gener- +alized. +We have obtained the phonon self-energy including the +aforementioned interactions and showed that, provided +that the additional magnon modes couple only weakly to +the microwave mode, the overall correction to the mag- +nomechanical decay rate is proportional to the average +number of Kittel magnons. We have then focused our +results on the case of a magnetic sphere, in connection +with the experiment performed in Ref. [43]. Our model +explains the observed shift in the magnomechanical de- +cay rate close to the dynamical backaction evasion drive +frequency. In this context, we have also evaluated the ef- +fects of the different corrections. Specifically, we showed +that the drive at which the dynamical backaction decay +is zero depends linearly on power. This is a consequence +of the corrections being proportional to the steady state +number of Kittel magnons, which scales linearly with the +drive power. +A small discrepancy with the experimental data is still +present at higher drive powers and for detunings far from +the upper hybrid mode. We attribute this difference to +higher order Walker modes that have not been taken into +account in the present model. Similar to the effects de- +scribed above for the mode included in our calculations, +even higher order Walker modes can lie in a frequency +range close to the microwave drive and, at higher powers, +can modify substantially the magnomechanical decay via +nonlinear effects. We should also point out that the ex- +perimental setup in [43] has particularities not included +here. +For instance, the magnetic sphere is glued on a +dielectric post, which modifies the photon, phonon and +magnon mode profiles. This in turn can change the mag- +nomechanical coupling constants as well as the frequency +of the Walker modes. A precise evaluation of such effects +requires a more refined numerical analysis, for example +using finite difference software and micromagnetic simu- +lations, which goes beyond the scope of our analysis. +While nonlinear effects in cavity magnomechanical sys- +tems have been previously computed for the nonlinear +dynamics of magnons [14, 49], the evaluation of such ef- +fects on the response of the mechanical degree-of-freedom +to noise, as computed by the self-energy, is a step forward +in the characterization of these systems as platforms for +quantum technologies. +Our analysis was restricted to +evaluate the effects of all the corrections included in the +model of Sec. II in the framework of dynamical backac- + +16 +tion evasion set by the experiment of Ref. [43]. Never- +theless, the model derived in Sec. II shows that several +phenomena play a role in the modification of dynami- +cal backaction, for example, magnon squeezing and two- +mode squeezing. It would be interesting to investigate +scenarios in which those terms can be harnessed to re- +duce noise for quantum metrology. Furthermore, the in- +clusion of the additional magnon modes opens new possi- +bilities for cavity magnomechanical systems, such as the +manipulation of the mechanics by driving different side- +bands of the different magnon modes in a Floquet-like +setup [67]. In this case, it would be interesting to go be- +yond the approximation used here, where only the Kit- +tel mode couples strongly to the microwave cavity. In +fact, several experiments have shown fingerprints of a +strong coupling between Walker modes of a sphere and +microwaves [7, 55, 58]. As we have numerically shown, +Walker modes other than the Kittel mode can couple bet- +ter to the phonons, which can be harnessed to applica- +tions, such as nonreciprocal transport between phonons +and microwaves [68]. +ACKNOWLEDGMENTS +The authors acknowledge helpful contributions from +S. Scharma and E. 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Zhang, +Floquet cavity electromagnonics, Phys. Rev. Lett. 125, +237201 (2020). +[68] L. Mercier de L´epinay, C. F. Ockeloen-Korppi, D. Malz, +and M. A. Sillanp¨a¨a, Nonreciprocal transport based on +cavity floquet modes in optomechanics, Phys. Rev. Lett. +125, 023603 (2020). + diff --git a/sdFKT4oBgHgl3EQf1i6S/content/tmp_files/load_file.txt b/sdFKT4oBgHgl3EQf1i6S/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2ee80e3dc2ea62d7e7bd130cb0f855bd43dcbe08 --- /dev/null +++ b/sdFKT4oBgHgl3EQf1i6S/content/tmp_files/load_file.txt @@ -0,0 +1,1324 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf,len=1323 +page_content='Magnomechanical backaction corrections due to coupling to higher order Walker modes and Kerr nonlinearities V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Bittencourt,1, 2, ∗ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Potts,3, 4 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Huang,4 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Davis,4 and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Viola Kusminskiy5, 2, † 1ISIS (UMR 7006), Universit´e de Strasbourg, 67000 Strasbourg, France 2Max Planck Institute for the Science of Light, Staudtstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' PLZ 91058 Erlangen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Germany 3Kavli Institute of NanoScience,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Delft University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' PO Box 5046,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 2600 GA Delft,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Netherlands 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' University of Alberta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Edmonton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Alberta T6G 2E9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Canada 5Institute for Theoretical Solid State Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' RWTH Aachen University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 52074 Aachen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Germany (Dated: January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 2023) The radiation pressure-like coupling between magnons and phonons in magnets can modify the phonon frequency (magnomechanical spring effect) and decay rate (magnomechanical decay) via dynamical backaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Such effects have been recently observed by coupling the uniform magnon mode of a magnetic sphere (the Kittel mode) to a microwave cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In particular, the ability to evade backaction effects was demonstrated [C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Potts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=', arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='13766 [quant-ph] (2022)], a requisite for applications such as magnomechanical based thermometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' However, deviations were observed from the predicted magnomechanical decay rate within the standard theoretical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In this work, we account for these deviations by considering corrections due to (i) magnetic Kerr nonlinearities and (ii) the coupling of phonons to additional magnon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Provided that such additional modes couple weakly to the driven cavity, our model yields a correction proportional to the average Kittel magnon mode occupation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We focus our results on magnetic spheres, where we show that the magnetostatic Walker modes couple to the relevant mechanical modes as efficiently as the Kittel mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Our model yields excellent agreement with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The dipolar interaction between the magnetization and microwaves confined in a cavity can yield strong coupling between magnons (quanta of spin waves) and microwave photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' After the first theoretical predictions of the strong magnon-microwave coupling [1], cavity magnonic systems consisting of a magnetic element loaded in a mi- crowave cavity were realized in different architectures [2– 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The unique tunability of magnons combined with the ability to drive and read out the microwave cavity makes such systems a promising platform for several applica- tions [9–13], such as the generation of squeezed and en- tangled states [14, 15], the indirect coupling to qubits to detect and manipulate magnons [16–20], and sensing of magnetic fields [21–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Magnons can also couple to other degrees of free- dom, opening the opportunity of probing and manipu- lating these via their coupling to the hybridized magnon- microwave polaritons [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In particular, magnetoelastic effects [25–28] couple the magnetization and the mechani- cal vibrations of a magnetic material, yielding an interac- tion between magnons and phonons [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Such magnome- chanical coupling can be either resonant or parametric [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The resonant coupling is relevant for specific geome- tries where certain magnon modes are resonant with the elastic vibrations of the medium, for example, for mag- netic spheres with radii ranging from ∼ 10 nm to ∼ 10 µm [30] and in magnetic films [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The second type of coupling, parametric coupling, is relevant for geometries in which the magnon frequency is far detuned from the phonon, such as for micrometer-sized magnetic spheres ∗ sant@unistra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='fr † kusminskiy@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='rwth-aachen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='de [29, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The interaction Hamiltonian resembles the ra- diation pressure coupling between phonons and photons commonly found in optomechanical systems [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' When magnons are driven, the magnomechanical interaction is enhanced and the driven-dissipative dynamics of the cou- pled system result in dynamical backaction on the vibra- tional modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Specifically, the phonon frequency and de- cay rate are modified, referred to as magnomechanical spring effect and magnomechanical decay, respectively [29, 34, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Dynamical backaction is the basis of several proposed applications of magnomechanical systems, from state preparation and generation of entangled states [15, 37– 40], to effects that are closely related to the optomechani- cal counterpart, such as magnomechanical sideband cool- ing and amplification of phonons [41, 42], albeit operating in the microwave regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Moreover, cavity magnome- chanical systems provide a unique tunability of dynami- cal backaction due to the hybridization between magnons and microwaves, which can be used to fulfill a triple reso- nance condition by tuning an external bias magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Dynamical backaction was first probed in magnomechan- ics in a system consisting of a magnetic sphere of yttrium iron garnet (YIG) loaded into a 3D microwave cavity [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' More recent experiments have demonstrated the full ar- ray of dynamical backaction effects in these systems [34] and demonstrated the capability of avoiding the induced magnomechanical decay [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Dynamical backaction eva- sion can enable the application of cavity magnomechanics in thermometry [36], where, similar to proposal and ex- periments in optomechanical systems [44, 45], the phonon mode should be neither cooled nor heated by the drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Experiments and proposals for cavity magnomechan- ical systems have so far focused on the coupling to a arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='11920v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='mes-hall] 27 Jan 2023 2 single magnon mode, the uniform precession of the mag- netization called the Kittel mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Nevertheless, a mag- netic sphere supports a whole set of magnon modes called Walker modes [46, 47] which can also couple to a given vibration mode, in principle even stronger than the Kittel mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Weak inhomogeneities in the microwave field can drive such higher-order magnon modes, modifying the backaction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Furthermore, magnon nonlinearities due to crystalline anisotropy [48] can also affect dynami- cal backaction beyond the static frequency shift reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [49], akin to recently measured effects of the cav- ity Kerr nonlinearity in an electromechanical system [50] under sideband cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In this work, we extend the theory of dynamical back- action in cavity magnomechanical systems to include Kerr nonlinearities and the coupling of a phonon mode to several magnon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We consider the framework de- picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 1, where a microwave cavity mode couples strongly to a magnon mode and weakly to a set of addi- tional magnon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Those in turn exhibit nonlineari- ties and interact via a radiation pressure-like coupling to a single phonon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We derive the phonon self-energy, describing the frequency shift and the magnomechanical decay rate, generalizing previous results [34, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The overall effect of the coupling to the additional magnon modes is a correction proportional to the average number of Kittel magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We evaluate our model for the case of a magnetic sphere, computing numerically the coupling rates between the (magnetic) Walker modes and the me- chanical mode probed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' At low driving powers our model introduces corrections that agree well with the measured data in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [43], explaining the observed shift in the magnomechanical decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' At higher driving powers, there are further deviations which are not cap- tured by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' These are however only relevant for driving frequencies detuned from the backaction evasion point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' ˆa ˆm { } ˆb g0 mb ˆm† ˆm ⇣ ˆb† + ˆb ⌘ {K(i) cr,m ˆm† ˆm ˆm† i ˆmi} gam � ˆa ˆm† + ˆa† ˆm � {gamj ⇣ ˆa ˆm† j + ˆa† ˆmj ⌘ } ˆmj {g0 mjb ˆm† j ˆmjˆb + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='} Drive FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Schematics of the model describing the cavity mag- nomechanical system with several magnon modes coupled to the same phonon mode, see the Hamiltonian (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The red ar- row indicates the microwave drive, and we have omitted the self-Kerr nonlinear terms and the magnon-phonon cross Kerr nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' I we present a brief review of the description of dynamical backac- tion in cavity magnomechanics for the cases described in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [29, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' II we include Kerr nonlinearities and the coupling to weakly driven addi- tional magnon modes, and the phonon self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Since those corrections depend on how strongly the additional magnon modes couple to the phonon mode, we special- ize further our model to a magnetic sphere geometry as probed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' III, we briefly review the derivation of the magnomechanical coupling following the literature [30], and in subsection III A we use the model to numerically evaluate the coupling between Walker modes with frequencies in a small range around the Kittel mode frequency and a relevant mechanical mode of a magnetic sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' IV, we compare our model to the ex- perimental results presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [43] and show that our generalized theory quantitatively accounts for the ob- served magnomechanical decay for a large range of pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' V we present our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' PHONON SELF-ENERGY AND DYNAMICAL BACKACTION EVASION The dynamics of a cavity magnomechanical system consisting of a microwave mode (ˆa with frequency ωa), a magnon mode ( ˆm with frequency ωm) coupled para- metrically to a phonon mode (ˆb with frequency Ωb) is described by the Hamiltonian [29] ˆH ℏ = ωaˆa†ˆa + ωm ˆm† ˆm + Ωbˆb†ˆb + gam � ˆa ˆm† + ˆa† ˆm � + g0 mb ˆm† ˆm � ˆb† + ˆb � + i√κeϵd � ˆaeiωd − ˆa†e−iωd� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (1) The magnon-microwave coupling rate gam is due to a magnetic dipole interaction between the ferromagnetic resonance of the material and the microwave cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The parametric magnon-phonon coupling, with the sin- gle magnon coupling rate g0 mb, is due to magnetoelastic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (1) describes the coherent drive of the microwave cavity at a frequency ωd with an amplitude ϵd = � P/ℏωd, where P is the drive power and κe is the decay rate of the cavity to the external drive port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In the weak magnomechanical coupling limit, both the phonon frequency Ωb and the decay rate Γb are modified by the coupling to the driven magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The respective shifts are given by δΩb = −Re[Σ[Ωb]], Γmag = 2Im[Σ[Ωb]], (2) where Σ[ω] is the phonon self-energy, obtained by ana- lyzing the linearized dynamics of the system [34, 36] and reads Σ[ω] = i|gmb|2(Ξ[ω] − Ξ∗[−ω]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (3) 3 From now on we refer to δΩb as the magnomechanical frequency shift and to Γmag as the magnomechanical de- cay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Here, gmb = g0 mb⟨ ˆm⟩ is the enhanced mag- nomechanical coupling rate, with |⟨ ˆm⟩|2 the steady-state magnon population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The function Ξ[ω] is a modified Kit- tel mode susceptibility given by Ξ−1[ω] = χ−1 m [ω] + g2 amχa[ω] (4) which depends on the magnon susceptibility χm[ω] = [−i(∆m + ω) + γm/2], and the microwave susceptibil- ity χa[ω] = [−i(∆a + ω) + κ/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The detuning be- tween the microwave (magnon) mode and the drive is ∆a(m) = ωd − ωa(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' γm is the magnon decay rate and κ the total microwave decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The value and the sign of the magnomechanical de- cay rate depend on the drive frequency, which can tune scattering processes that upconvert or downconvert exci- tations in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Depending on the drive, it is pos- sible to make one of such processes more efficient than the other, yielding a positive (cooling) or negative (am- plification) magnomechanical decay rate, in a situation akin to what is found in standard optomechanical sys- tems [35, 51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Different from optomechanics, in cavity magnomechanical systems magnons and microwaves hy- bridize, yielding the unique situation where the different scattering processes that contribute to dynamical back- action are associated with mechanical sidebands of hy- bridized modes [29, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The hybrid magnon-microwave modes have frequencies ω± that are separated by ω+ − ω− = � 4g2am + (ωa − ωm)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (5) We will refer to the mode with frequency ω+ as the up- per hybrid mode, and the mode with frequency ω− as the lower hybrid mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' When the microwave drive is set at a frequency between ω+ and ω−, the scattering from the blue sideband of the lower hybrid mode can be bal- anced by the scattering to the red sideband of the upper hybrid mode yielding dynamical backaction evasion: the magnomechanical decay rate vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Consequently, the drive at which dynamical backaction evasion happens can be obtained from the condition Γmag = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (6) In a system satisfying the two-phonon triple resonance condition ω+ − ω− = 2Ωb, and for resonant magnons and microwaves, such drive frequency is exactly at (ω+ + ω−)/2 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The ability to tune a magnomechanical sys- tem in the dynamical backaction evasion regime was re- cently demonstrated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [43], and is a requirement for implementing a magnomechanical-based primary ther- mometer [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Equation (3) only takes into account the interaction of a phonon mode with a single magnon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' How- ever, multiple magnetostatic modes can couple to a given phonon mode [30], modifying the magnomechan- ical decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The different scattering processes to and from the additional magnomechanical sidebands can thus change the frequency at which dynamical backaction is evaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For instance, in the experimental data shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [43], the measured magnomechanical decay rate exhibits a shift with respect to the theoretical prediction obtained from the Hamiltonian given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' This shift was taken into account by adding to the magnome- chanical decay rate a phenomenological correction pro- portional to |⟨ ˆm⟩|2, which depends on the average num- ber of magnons driven by the microwave tone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Such cor- rection can be attributed to the coupling to additional magnon modes, which are weakly driven by their cou- pling to the microwave cavity, and to magnon nonlinear- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' While nonlinearities in magnetic spheres are generally weak, the microwave drive combined with the strong magnon-microwave coupling can make nonlinear effects prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' This is the case provided that the power of the drive is strong enough to induce an average number of magnons above a certain threshold [53], with implica- tions for magnetoelastic effects [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Even for drive pow- ers below the nonlinear threshold, magnon nonlinearities can affect the hybrid system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For instance, in a cavity-magnonic system, the Kittel mode self-Kerr nonlinearity was shown to yield considerable cavity and magnon frequency shifts under moderate driving powers [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Experimentally, a phonon frequency shift, as well as mechanical bistability, was reported recently [49], which points to the importance of considering such nonlineari- ties in the description of dynamical backaction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In what follows, we include in the description of dynamical backaction both the coupling to additional magnon modes as well as magnon nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' INCLUSION OF KERR NONLINEARITY AND COUPLING TO ADDITIONAL MAGNON MODES IN THE PHONON SELF ENERGY To derive the correction term to the self-energy, we consider adding to the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (1) self- and cross- Kerr nonlinearities, and coupling to N additional magnon modes, each with annihilation operators { ˆmj} and frequencies ωj (j = 1, · · · , N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The total Hamilto- 4 nian is thus: ˆH ℏ = ωaˆa†ˆa + ωm ˆm† ˆm + Ωbˆb†ˆb + N � j=1 ωj ˆm† j ˆmj + gam � ˆa ˆm† + ˆa† ˆm � + N � j=1 gamj � ˆa ˆm† j + ˆa† ˆmj � + g0 mb ˆm† ˆm � ˆb† + ˆb � + N � j=1 ˆm† j ˆmj � g0 mjbˆb + (g0 mjb)∗ˆb†� + Km( ˆm† ˆm)2 + Kcr ˆm† ˆmˆb†ˆb + N � j=1 K(j) cr,m ˆm† ˆm ˆm† j ˆmj + i√κeϵd � ˆaeiωd − ˆa†e−iωd� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (7) From now on, we will refer to the magnon mode ˆm as the Kittel mode, since this is typically the magnon mode that has the strongest coupling to the cavity, while we refer to the magnon modes ˆmj as additional magnon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We specify such additional magnon modes for the case of a magnetic sphere in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Compared with the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (1), the above equation includes the following terms: the additional magnon modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' the coupling between these and (i) the microwave mode, each with a coupling rate gamj and (ii) the phonon mode, each with a coupling rate g0 mjb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' the self-Kerr term for the Kittel mode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' the cross-Kerr term between the Kittel and the phonon mode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' and the cross-Kerr term between the Kittel mode and the other magnon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Provided that the magnetic anisotropy axis of the YIG sphere is aligned with the external bias field, the Kittel mode self-Kerr nonlinear coefficient is given by K0 m = 13ℏKanγ2/(16M 2 s V ), where Kan = −610 J/m2 at room temperature and V is the sphere vol- ume [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We assume a rotating wave approximation for the magnon-microwave coupling, as is done to obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (1), which eliminates any term of the form ˆm(j)ˆa and ˆm† (j)ˆa†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The rotating wave approximation is also as- sumed for the magnomechanical coupling, as explained in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For the experiment in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [43] this cor- responds to K0 m/2π = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='15 nHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The magnon-phonon and magnon-magnon cross Kerr nonlinear coefficients de- pend on the overlap between these modes and the Kittel mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In general, the magnon-magnon cross-Kerr coeffi- cient is around the same order of magnitude as K0 m [56], while the magnon-phonon cross-Kerr coefficient is ∼ −5 pHz [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We should notice that in the experiment shown in [43], the magnetic anisotropy axis was not aligned with the bias magnetic field, which effectively causes Km to be smaller than the value K0 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Figure 1 shows a schematic of the model, including the different coupling terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The values of the magnomechanical couplings depend on the geometry of the magnet, which defines the magnon and phonon mode profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In the system under study, the coupling between the Kittel mode and a relevant me- chanical mode of a sphere, discussed in Section III A, is g0 mb/2π = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='56 mHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In Section III we discuss in detail the values of g0 mjb for a magnetic sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' It is impor- tant to point out that, due to better mode overlap, in principle, g0 mjb can be comparable to or larger than g0 mb for some modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The coupling between magnons and microwaves depends on the microwave field at the mag- net position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For homogeneous fields, only the coupling to the Kittel mode does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Nevertheless, small inhomogeneities could yield a small microwave-magnon coupling gamj which would, in turn, drive weakly such magnon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For different cavity geometry, such cou- plings can be strong [7, 8, 57, 58], a framework which we do not consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In correspondence with the experiment [43], we assume that the additional magnon modes are weakly driven via their coupling to the cavity mode, such that we expect a small steady-state amplitude for those modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Thus we can safely disregard any self- and cross-Kerr nonlinear- ity of the form ˆm† k ˆmk ˆm† j ˆmj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The Heisenberg-Langevin equations describing the dynamics of the coupled modes in the rotating frame with the drive frequency are ˙ˆa = � i∆a − κ 2 � ˆa − igam ˆm − i N � j=1 gamj ˆmj − √κi ˆξI(t) − √κeϵd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' ˙ˆm = � i∆m − γ 2 � ˆm − igamˆa − ig0 mb ˆm � ˆb† + ˆb � − iKm ˆm � 1 + 2 ˆm† ˆm � − iKcr ˆmˆb†ˆb − i N � j=1 K(j) cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='m ˆm ˆm† j ˆmj + √γm ˆξm(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' ˙ˆmj = � i∆mj − γj 2 � ˆmj − igamjˆa − i ˆmj � g0 mjbˆb + g0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='∗ mjbˆb†� − iK(j) cr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='m ˆmj ˆm† ˆm + √γj ˆξmj(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' ˙b = − � iΩb − Γb 2 � ˆb − ig0 mb ˆm† ˆm − iKcrˆb ˆm† ˆm − i N � j=1 g0 mjb ˆm† j ˆmj + � Γb ˆξb(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (8) In the above equations, κ = κi + κe denotes the total microwave cavity decay rate, which is composed of the intrinsic cavity decay κi and the decay into the exter- nal port κe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The additional magnon modes decay rates are indicated by γmj which, for magnetostatic modes of a sphere, have the same value of the Kittel mode de- cay [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' All parameters appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (8) are sum- marized in Table I, with the values that will be used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The noise terms denoted by ˆξη(t) (η = I, e, m, mj, b) describe thermal noises with correla- tions ⟨ˆξη(t)ˆξ† η′(t′)⟩ = (nTh,η + 1)δηη′δ(t − t′), ⟨ˆξ† η(t)ˆξη′(t′)⟩ = nTh,ηδηη′δ(t − t′), (9) with nTh,η = [exp(ℏωη/kBT) − 1]−1 the number of ther- 5 mal excitations of mode η at a temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The steady state in a mean-field approximation is ob- tained by taking the expectation values of the oper- ators in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (8) and ignoring any quantum correla- tions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' ⟨ ˆmˆb⟩ ≈ ⟨ ˆm⟩⟨ˆb⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Since we are assuming that the magnon modes { ˆmj} are weakly coupled to the mi- crowaves, gamjgamk ≪ gamjgam, we discard any other in- direct coupling between the additional magnon modes via the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' These approximations yield ⟨ˆb⟩ = ig0 mb|⟨ ˆm⟩|2 Fb − iKcr|⟨ ˆm⟩|2 + i �N j=1 g0 mjb|⟨ ˆmj⟩|2 Fb − iKcr|⟨ ˆm⟩|2 , ⟨ ˆmj⟩ = igamj √κeϵd FmjFa + g2amj − iK(j) cr,m|⟨ ˆm⟩|2 − gamjgam⟨ ˆm⟩ FmjFa + g2amj − iK(j) cr,m|⟨ ˆm⟩|2 , (10) where we have defined Fb = −iΩb − Γb 2 , Fmj(m) = i∆mj(m) − γmj(m) 2 , Fa = i∆a − γa 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (11) For ⟨ ˆmj⟩, we have also discarded the term ∝ g0 mjb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The steady-state of the Kittel mode reads A⟨ ˆm⟩ = igam √κeϵdB (12) where A = FmFa + g2 am − iKm � 1 + 2|⟨ ˆm⟩|2� − 2iFag0 mbRe � ⟨ˆb⟩ � + g2 amB − iKcr|⟨ˆb⟩|2 B = 1 − N � j=1 g2 amj FmjFa + g2amj − iK(j) cr,m|⟨ ˆm⟩|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (13) Equation (12) is solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Depending on the drive power and the detuning, the equation can have two bistable solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We will focus our analysis on a detuning range lying in between the hybridized Kit- tel magnon-microwave modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In the considered range the magnomechanical decay rate of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (2) changes its sign, and in such region, the nonlinear equation for ⟨ ˆm⟩ has only one solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Furthermore, we can discard the terms proportional to K(j) cr,m, Kcr and g0 mb to obtain the solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Linearized dynamics We can now consider the fluctuations around the steady-state values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We write ˆo = δˆo + ⟨ˆo⟩, and discard any terms involving more than two fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The quadratic Hamiltonian describing the dynamics of the fluctuations is given by ˆHLin ℏ = −∆aδˆa†δˆa + ˜Ωbδˆb†δˆb − ˜∆mδ ˆm†δ ˆm − N � j=1 ˜∆mjδ ˆm† jδ ˆmj + ˆHInt ℏ , (14) with the coupling terms included in ˆHInt given by ˆHInt ℏ = gamδˆa†δ ˆm + GRδ ˆm†δˆb + GBδ ˆm†δˆb† + gms � δ ˆm†�2 + � j=1 gamjδˆa†δ ˆmj + N � j=1 � GR,jδ ˆm† jδˆb + GB,jδ ˆmjδˆb � + N � j=1 � gR,jδ ˆm†δ ˆmj + gB,jδ ˆm†δ ˆm† j � + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (15) The interacting terms appearing in the Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (7) induce frequency shifts for the fluctuations, which are given by ˜∆m(mj) = ωd − ˜ωm(mj), ˜ωm = ωm + 2g0 mbRe � ⟨ˆb⟩ � + 4Km|⟨ ˆm⟩|2 + Kcr|⟨ˆb⟩|2 + N � j=1 K(j) cr,m|⟨ ˆmj⟩|2, ˜ωmj = ωmj + 2Re � g0 mjb⟨ˆb⟩ � + K(j) cr,m|⟨ ˆm⟩|2, ˜Ωb = Ωb + Kcr|⟨ ˆm⟩|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (16) The coupling rates between the fluctuations are enhanced and modified with respect to the bare ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Their expres- sions are shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Enhanced couplings appearing in the linearized Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (15) gmb g0 mb⟨ ˆm⟩ GR gmb + Kcr⟨ ˆm⟩⟨ˆb⟩∗ GB gmb + Kcr⟨ ˆm⟩⟨ˆb⟩ GR,j g0 mjb⟨ ˆmj⟩ GB,j g0 mjb⟨ ˆmj⟩∗ gms Km⟨ ˆm⟩2 gR,j K(j) cr,m⟨ ˆm⟩⟨ ˆmj⟩∗ gB,j K(j) cr,m⟨ ˆm⟩⟨ ˆmj⟩ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Calculation of the phonon self-energy The phonon self-energy is obtained by solving the lin- ear Heisenberg-Langevin equations describing the cou- 6 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Parameters of the magnomechanical system appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The values correspond to the experiment in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The magnon self-Kerr term value corresponds to the case where the bias magnetic field is aligned with the magnetic anisotropy axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Parameter Symbol Value Microwave mode frequency ωa 2π × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='11 GHz Kittel mode frequency ωm 2π × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='09 GHz Additional magnon modes frequencies ωmj See section II A Phonon mode frequency Ωb 2π × 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='45 MHz Drive frequency ωd 2π × [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='096, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='099] GHz Microwave intrinsic decay rate κI 2π × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='91 MHz Microwave external decay rate κe 2π × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='17 MHz Kittel mode decay rate γm 2π × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='55 MHz Additional magnon modes decay rate γmj 2π × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='55 MHz Phonon intrinsic decay rate Γb 2π × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='74 kHz Kittel mode - Microwave coupling rate gam 2π × 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='34 MHz Additional magnon modes - microwave coupling rate gamj See Section IV Magnomechanical coupling to the Kittel mode g0 mb 2π × 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='56 mHz Magnomechanical couplings to the j-th additional magnon mode g0 mjb See section III A Kittel mode self-Kerr nonlinearity K0 m −2π × 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='15 nHz Magnon cross-Kerr nonlinearity K(j) cr,m −2π × 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='15 nHz Magnon-phonon cross-Kerr nonlinearity Kcr −2π × 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='4 pHz pled dynamics of the fluctuations for the phonon opera- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' To compute the effects of backaction in the response of the phonon mode to noise, we consider the Fourier transformed operators defined by ˆo(t) = � dωe−iωtˆo[ω], (17) where ˆo = δˆa(†), δ ˆm(†), δ ˆm(†) j , δˆb(†).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We skip the algebraic steps, but outline the main differences with respect to the results in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [34, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' After writing the cavity operator in terms of the magnon operators, we obtain the following equation for the additional magnon modes Ξj[ω]−1δ ˆmj[ω] = −i � g∗ R,j − igamjgamχa[ω] � δ ˆm[ω] − igB,jδ ˆm†[ω] − iGR,jδˆb[ω] − iG∗ B,jδˆb†[ω] + gamjχa[ω] � k̸=j gamkδ ˆmk[ω] + ˆ˜ξmj[ω], (18) where ˆ˜ξmj represents the noise term modified by the inter- action with the cavity, and we have defined the effective magnon susceptibility in correspondence with the previ- ous case in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (4) Ξj[ω]−1 = χ−1 mj [ω] + g2 amjχa[ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (19) We notice that the first term on the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (18) includes the indirect coupling between the j-th magnon mode and the Kittel mode via the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' A sim- ilar term related to the coupling between the additional magnon modes appears in the last line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (18), and since gamj ≪ gam, we discard these contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' After using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (18) to eliminate the additional magnon modes in favor of the Kittel mode and the phonon fluctuations, we obtain the following set of cou- pled equations Ξ−1 m [ω]δ ˆm[ω] = ηm[ω] ˆm[ω] − iΛm[ω] ˆm†[ω] − iGm,R[ω]δˆb[ω] − iGm,B[ω]δˆb†[ω] + ˆ˜ξm[ω], � �χ−1 b [ω] − i N � j=1 σj[ω] � � δˆb[ω] = −i ˜G∗ b,R[−ω]δ ˆm[ω] − i ˜Gb,B[ω]δ ˆm†[ω] + ˆ˜ξb[ω] + i � � N � j=1 λj[ω] � � δˆb†[ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (20) We have included all the noise terms in ˆ˜ξm,b[ω];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' all other functions appearing in the equations below are defined in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The coupling to the additional magnon modes has the following effects: the introduction of a self- energy term on the phonon mode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' the modification of the coupling between the Kittel mode and the phonon mode;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' and a modification of the Kittel mode susceptibility and squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We briefly comment on each of these effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' At this intermediate step, the phonon susceptibility is modified by the self-energy term N � j=1 σj[ω] = i N � j=1 |g0 mjb⟨ ˆmj⟩|2 � Ξj[ω] − Ξ∗ j[ω] � , (21) which is defined in analogy with the self-energy term de- rived in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [34, 36] and given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Such a term represents the direct dynamical backaction of the 7 coupling between the phonon mode and the additional magnon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The additional magnon modes modify the couplings between the Kittel mode and the phonon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In fact, the effective coupling constants appearing in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (20) are given by ˜Gb,R[ω] = GR − gamχ∗ a[−ω] � j gamjGR,jΞ∗ j[−ω] − i � j � GB,jgb,jΞj[ω] − GR,jgR,jΞ∗ j[−ω] � , ˜Gb,B[ω] = GB − gamχ∗ a[−ω] � j gamjG∗ B,jΞ∗ j[−ω] − i � j � G∗ R,jgb,jΞj[ω] − G∗ B,jgR,jΞ∗ j[−ω] � , Gm,R[ω] = GR − gamχa[ω] � j gamjGR,jΞj[ω] − i � j � gR,jGR,jΞj[ω] − gB,jGB,jΞ∗ j[−ω] � , Gm,B[ω] = GB − gamχa[ω] � j gamjG∗ B,jΞj[ω] − i � j � gR,jG∗ B,jΞj[ω] − gB,jG∗ R,jΞ∗ j[−ω] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (22) From these expressions, we notice two types of modifica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The second terms in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (22) represent the effect of the indirect coupling between the additional magnon modes and the Kittel mode via the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The third terms are proportional to gB(R),j, which in turn (see Ta- ble II) are due to the magnon cross-Kerr nonlinearity, ˆm† j ˆmj ˆm† ˆm in the Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The relevance of these corrections for a given drive frequency is deter- mined by the susceptibilities of the additional magnon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The Kittel mode squeezing term Λm[ω] reads Λm[ω] = 2gms − gam(χa[ω] + χ∗ a[−ω]) � j gB,jgamjΞj[ω], (23) where the first term is due to the self-Kerr nonlinearity, while the second term is a combination of the magnon cross-Kerr nonlinearity with the indirect coupling be- tween the magnon modes via the microwave cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The Kittel mode susceptibility is also modified by the term ηm[ω], which is given by ηm[ω] = g2 amχ2 a[ω] � j g2 amjΞj[ω] + 2igamχa[ω] � j Re[gR,j]gamjΞj[ω] − � j � |gR,j|2Ξj[ω] − |gB,j|2Ξ∗ j[−ω] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (24) The three terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (24) describe the effects of a two- mode squeezing between the Kittel mode and each of the additional magnon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The first term is related to the indirect coupling of the modes via the microwave cavity while the last term is the direct two mode squeez- ing induced by the magnon cross-Kerr nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The second term is a combination of both effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (20), we eliminate the Kittel mode operator and obtain an equation for the phonon mode operator � χ−1 b [ω] − iΣTot[ω] � δˆb[ω] = iΛb[ω]δˆb†[ω] + ˆΥb[ω], (25) where ˆΥb[ω] includes all the noise terms driving the phonon fluctuations, Λb[ω] describes phonon squeezing, and ΣTot[ω] is the total self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We focus only on the self-energy term which is given by ΣTot[ω] = Σm[ω] + N � j=1 σj[ω], (26) where the contribution of Kittel mode to the self-energy is given by Σm[ω] = i � ˜G∗ b,R[−ω] ˜Gm,R[ω]˜Ξm[ω] − ˜Gb,B[ω] ˜G∗ m,B[−ω]˜Ξ∗ m[−ω] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (27) In this equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' we have defined ˜Gm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='R[ω] = Gm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='R[ω] + i Λm[ω]G∗ m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='B[−ω] Ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='−1 m [−ω] − η∗m[−ω] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' ˜Gm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='B[ω] = Gm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='B[ω] + i Λm[ω]G∗ m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='R[−ω] Ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='−1 m [−ω] − η∗m[−ω] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (28) which are the magnomechanical couplings modified by the Kittel mode squeezing term Λm[ω],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' and the Kittel mode susceptibility ˜Ξ−1 m [ω] = Ξ−1 m [ω] − ηm[ω] − Λm[ω]Λ∗ m[−ω] Ξ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='−1 m [−ω] − η∗m[−ω] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (29) which includes both the Kittel magnon squeezing in Λm[ω] and the effects of two-mode squeezing interactions with the additional magnon modes in ηm[ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Magnomechanical decay rate corrections We turn our attention now to the effects on the mag- nomechanical decay rate, in connection with the observed 8 shift reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The total change in the phonon linewidth is given by Γmag[ω] = 2Im[ΣTot[ω]] = 2Im[Σm[ω]] + N � j=1 Γj[ω] Γj[ω] = 2Im[σj[ω]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (30) The contribution of self and cross-Kerr nonlinear terms are included in the above self-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The corresponding frequency shift is different than the static one, observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [49], which has already been included in the mod- ified phonon frequency ˜Ωb defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The self- Kerr nonlinearity changes the behavior of the magnome- chanical decay rate with the detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' This is due to three main factors: the modification of the steady-state number of magnons |⟨ ˆm⟩|2, the induced static magnon frequency shift, given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (16), and the generation of squeezing in the magnon fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' This has a conse- quence for both dynamical backaction evasion, as we will show, and for backaction cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The latter has been recently reported in a system where a mechanical oscil- lator is parametrically coupled to a nonlinear cavity [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The magnomechanical frequency shift is given by δΩb[ω] = −Re [ΣTot[ω]] , (31) which has a decomposition similar to the one shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The relevance of the different corrections due to the additional magnon modes depend on their frequencies with respect to the drive and their coupling rates to the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Recalling that Gj = g0 mjb⟨ ˆmj⟩, we use the steady state from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For the experimental pa- rameters in consideration, given in table I, due to the strong coupling between the Kittel mode and the cavity, √κeϵd ≪ gamRe[⟨ ˆm⟩] and the steady state of the weakly driven Walker modes can be approximately written as |⟨ ˆmj⟩|2 = g2 amjg2 am|⟨ ˆm⟩|2 |FmjFa + g2amj − iK(j) cr,m|⟨ ˆm⟩|2|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (32) Within these approximations, the contribution of such Walker modes to the magnomechanical linewidth is given by Γj[ω] = 2g2 am|⟨ ˆm⟩|2 |g0 mjb|2g2 amj |FmjFa + g2amj − iK(j) cr,m|⟨ ˆm⟩|2|2 × Re � Ξj[ω] − Ξ∗ j [ω] � , (33) which for small K(j) cr,m gives a contribution to the mag- nomechanical decay rate which is proportional to |⟨ ˆm⟩|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The contribution to the magnomechanical decay rate also depends on the detuning between the drive and the magnon frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' While Γj[ω] quantifies the direct effect of the coupling between the phonon mode and the additional magnon modes, there are also indirect effects due to the cou- pling between the additional magnon modes and the Kittel mode via the microwave cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Those are in- cluded in the term 2Im[Σm[ω]] via the modified coupling rates ˜Gm(b),R(B)[ω] and the modified Kittel mode sus- ceptibility ˜Ξm[ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In general, the corrections included in those terms are proportional to |gms|2, to |GR(B),j|2 or to |gR(B),j|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Following the same procedure outlined above, it is possible to show that those are all proportional to the steady-state Kittel mode occupation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Their frequency- dependent coefficients have a more complicated form due to the explicit dependence of the modified couplings and susceptibilities on frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We can describe all the cor- rections by the expression Γmag[ω] = Γ0 mag[ω] + α[ω]|⟨ ˆm⟩|2, (34) where Γ0 mag[ω] is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (3), without including ad- ditional magnon modes and nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Such a cor- rection was used phenomenologically in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [43] to ex- plain the observed discrepancies of the experimental re- sults with Γ0 mag[ω].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' MAGNOMECHANICAL COUPLING In the last section, we obtained the modifications of the phonon self-energy due to the coupling between the phonon mode and additional magnon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Such con- tributions depend on the relative strength of the mag- nomechanical coupling to the additional magnon modes with respect to the coupling to the Walker mode, which in turn depends on the geometry of the magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Before specifying the latter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' we briefly review the main points of the derivation of the magnomechanical coupling Hamil- tonian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' done in detail in the literature [29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 60,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 61],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' starting from the magnetoelastic energy [25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 26] EME = B1 M 2 S � d3r � M 2 xεxx + M 2 y εyy + M 2 z εzz � + 2B2 MS � d3r � MxMyεxy + MyMzεyz + MxMzεxz � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (35) where Mx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='z are the magnetization components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' MS is the saturation magnetization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' and εij = (∂iuj + ∂jui) /2 (36) is the linear strain tensor with u(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' t) the displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The magnetoelastic coefficients B1 and B2 are material and temperature-dependent constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For YIG at room temperature, MS = 140 kA/m [48], B1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='48 × 105 J/m3 and B1 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='4 × 105 J/m3 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The magnome- chanical Hamiltonian is then obtained by quantizing the magnetization and displacement fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We first quantize the magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We consider that the magnetization displays small fluctuations around a 9 uniform saturation value MS [48] (such assumption can be generalized to non-uniform magnetic ground states [62]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In this framework Mz/MS ≫ Mx,y/MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The x and y components of the magnetization can then be written as a superposition of modes, each labelled with a general index j, such that the quantized magnetization field is given by [63] ˆ Mx,y(r, t) = � j Mj � δmx,y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='j(r) ˆmj + δm∗ x,y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='j(r) ˆm† j � , (37) where { ˆmj} is a set of bosonic operators satisfying [ ˆmj, ˆm† j] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The quantization procedure is valid in the so called spin-wave limit for magnetic excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The mode functions δmj(r) are obtained by solving the Landau-Lifshitz equation for the magnetization fluctua- tions [48], plus the appropriate boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The quantities Mj are the zero-point fluctuations of the mode j given by Mj = � ℏ|γ|MS 2Vj , (38) where the mode volume is given by [63] Vj = 2Im �� d3r δmy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='j(r)δm∗ x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='j(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (39) Such mode decomposition ensures that the magnetic en- ergy density yields the Hamiltonian for a set of uncoupled harmonic oscillators of the form ˆHm = � j ℏωj ˆm† j ˆmj, with frequencies ωj obtained from the imposed bound- ary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The elastic vibrations are quantized in terms of phonon modes [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The displacement field is given by the super- position of modes ˆu = � α Xα � fα(r)ˆbα + f ∗ α(r)ˆb† α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (40) The mode functions fα(r) are dimensionless, and given as the solution of the elastic boundary problem [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The zero-point fluctuations are given by Xα = � ℏ 2ρΩαNα , (41) where Nα = � d3r |fα(r)|2, (42) is the mode normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Such a mode decomposition yields, for the non-interacting phonons, the Hamiltonian ˆHb = � α Ωαˆb† αˆbα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (37) and (40) in the magnetoelas- tic energy given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (35), we obtain an interaction Hamiltonian describing the coupling between magnons and phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Such a Hamiltonian includes the fol- lowing terms: (i) linear magnon-phonon coupling ∝ g(L) mjbα ˆm† jˆbα+H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=', relevant only for resonant magnon and phonon modes, for example, for small magnetic parti- cles [30] and for magnetic films [31–33];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (ii) spontaneous parametric conversion terms ∝ ˆmj ˆmkˆb† α and ∝ ˆm† j ˆm† kˆbα, relevant when the phonon mode frequency matches the sum of the frequency of the magnon modes j and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Such an interaction describes the creation of a pair of magnons via the annihilation of a phonon;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (iii) para- metric phonon-magnon coupling ∝ g0 mjmkbα ˆm† j ˆmkˆb+H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The off-resonant terms ˆmj ˆmkˆbα and ˆm† j ˆm† kˆb† α can be eliminated via a rotating wave approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Here, we focus on the interaction (iii), the parametric phonon-magnon coupling, which is given by the Hamil- tonian ˆHmb/ℏ = � {j̸=k},α � g0 mkmjbα ˆm† k ˆmjˆbα + ˜g0 mkmjbα ˆm† k ˆmjˆb† α � + � j,α g0 mjbα ˆm† j ˆmjˆbα + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=', (43) where {j ̸= k} indicates that the sum is over all j’s not equal to k and without repeating combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In the above equation we have separated the coupling terms between one magnon mode and one phonon mode and the terms involving two different magnon modes and a phonon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The coupling rates are given explicitly by g0 mjbα Nmjbα = B1 � d3r � |δmx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='j(r)|2(∂xfx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='α(r) − ∂zfz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='α(r)) + |δmy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='j(r)|2(∂xfx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='α(r) − ∂zfz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='α(r)) � + B2 � d3r Re � δmx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='j(r)δm∗ y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='j(r) � (∂yfx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='α(r) + ∂xfy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='α(r)) , g0 mkmjbα Nmkmjbα = B1 � d3r � δm∗ x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='k(r)δmx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='j(r)(∂xfx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='α(r) − ∂zfz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='α(r)) + δm∗ y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='k(r)δmy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='j(r)(∂yfy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='α(r) − ∂zfz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='α(r)) � + B2 2 � d3r � δm∗ y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='k(r)δmx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='j(r) + δm∗ x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='k(r)δmy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='j(r) � (∂yfx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='α(r) + ∂xfy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='α(r)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (44) 10 where we have defined Nmkmjbα = 2XαMkMj ℏM 2 S , (45) and Nmjbα = Nmjmjbα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The coupling ˜g0 mkmjbα is ob- tained from g0 mkmjbα with the substitution ∂xifj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='α(r) → ∂xif ∗ j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='α(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Focusing now on the coupling to a specific phonon mode, the magnomechanical Hamiltonian is given by ˆHmb ℏ = ωm ˆm† ˆm + � j ωj ˆm† j ˆmj + Ωbˆb†ˆb + ˆHmb,I ℏ , (46) where the coupling terms are ˆHmb,I ℏ = g0 mb ˆm† ˆmˆb + � j g0 mjb ˆm† j ˆmjˆb + ˆm† � j g0 mmjb ˆmjˆb + � j̸=k g0 mkmjb ˆm† k ˆmjˆb + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (47) In the above equations, we have separated the terms of the Kittel mode, which from now on we do not label, while the other Walker modes are labelled by the index j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The coupling rates are complex numbers, but we can absorb the phase of one of such coupling rates into the phonon field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We write g0 mb = |g0 mb|eiφmb, and define ˆ˜b = ˆbeiφmb, such that ˆHm˜b ℏ = ωm ˆm† ˆm + � j ωj ˆm† j ˆm + Ωbˆ˜b†ˆ˜b + g0 m˜b ˆm† ˆm(ˆ˜b + ˆ˜b†) + � j � g0 mj˜b ˆm† j ˆmjˆ˜b + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' � + ˆm† � j � g0 mmj˜b ˆmjˆ˜b + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' � + � j̸=k � g0 mkmj˜b ˆm† k ˆmjˆ˜b + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' � , (48) where g0 m˜b = |g0 mb|, g0 mj˜b = g0 mjbe−iφmb and g0 mmj˜b = g0 mmjbe−iφmb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Such a transformation corresponds to tak- ing the phase of the coupling between the phonon mode and the Kittel mode as a reference for the other cou- plings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' From now on, we take ˆ˜b → ˆb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' This gauge trans- formation of the phonon field does not change the Kerr nonlinear terms, which are quadratic in the phonon field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Magnomechanical coupling rates for a sphere The overlap integrals in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (44) depend on the specific geometry of the sample and the direction of the applied magnetic field, which defines the mode functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We consider now the case of a YIG sphere, corresponding to the experimental configuration of [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' A sphere supports magnetostatic modes called Walker modes [46, 47, 66], which have frequencies that can be tuned by the value of the external bias field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' To describe such modes, it is convenient to introduce the following characteristic frequencies ωM = |γ|µ0MS, ω0 = |γ|µ0 � H0 − MS 3 � , (49) where |γ|/2π = 28 GHz/T is the gyromagnetic ratio, µ0 is vacuum permeability, and H0 is the applied bias magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The Walker modes are conveniently given in a nonorthogonal coordinate system {ξ, η, φ} defined by the transformation [46] x = R � −χP[ω] � 1 − ξ2 sin η cos φ, y = R � −χP[ω] � 1 − ξ2 sin η sin φ, z = R � χP[ω] 1 + χP ξ cos η, (50) where χP[ω] = ωMω0 ω2 0 − ω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (51) At the the sphere’s surface η → θ and ξ[ω] → ξ0[ω] = � 1 + χP[ω] χP[ω] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (52) The frequencies of Walker modes are given by the non- linear equation [46, 47] ξ0[ω]∂ξP m l (ξ[ω]) P m l (ξ[ω]) |ξ=ξ0 − mκP[ω] + n + 1 = 0, (53) where κP[ω] = − ωMω ω2 0 − ω2 , (54) and P m l are the associated Legendre polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The Walker modes are labelled by three indices, {lmν}, with l ≥ 1 and |m| ≤ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For m > 0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (53) has (n − |m|)/2 roots, while for m < 0 it has 1 + (n − |m|)/2 solutions (both rounded down).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The mode functions of the Walker modes are given by � δmx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='lmν δmy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='lmν � = − � χP[ωlmν] iκP[ωlmν] −iκP[ωlmν] χP[ωlmν] � � ∂xψlmν ∂yψlmν � (55) where the magnetostatic potential inside the sphere is ψlmν(r) = P m l (ξ)Y m l (η, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (56) 11 For the phonon modes, we consider an unpinned sphere and stress-free boundary conditions [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' There are two families of mechanical modes of a homogeneous sphere: torsional (T) and spherical (S) modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Torsional modes are purely shear modes, while spherical modes involve both shear and compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Both families of modes are labelled by three indexes {νlm}, where l and m are polar and azimuthal indexes −l ≤ m ≤ l while ν is a radial index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We focus here on S modes, whose frequencies are given by [65] T (a) λν T (b) λν − T (c) λν T (d) λν = 0, (57) where T (a) λν = � λ(λ − 1) − ˜β2[ω]R2 2 � jλ(˜α[ω]R) + 2˜α[ω]Rjλ+1(˜α[ω]R) T (b) λν = � λ2 − 1 − ˜β2[ω]R2 2 � jλ(˜β[ω]R) + ˜β[ω]Rjλ+1(˜β[ω]R) T (c) λν = λ(λ + 1) � (λ − 1)jλ(˜β[ω]R) − ˜β[ω]Rjλ+1(˜β[ω]R) � T (d) λν = (λ − 1)jλ(˜α[ω]R) − ˜α[ω]Rjλ+1(˜α[ω]R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (58) The parameters ˜α[ω] = ω/cL, and ˜β[ω] = ω/cT , are given in terms of the longitudinal (L) and transverse (T) sound velocities cL,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' jλ(x) denotes the spherical Bessel func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (57) does not depend on m, for given {νl} there are 2l + 1 degenerate modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The mode functions for a S mode read, in spherical coordinates {er, eθ, eφ}, fνλm = eiφm � � Gνλ(r)P m l (cos θ) Fνλ(r)∂θP m l (cos θ) im sin θFνλ(r)P m l (cos θ) � � , (59) where Gνλ(r) = R r � λjλ(˜α[ω]r) − ˜α[ω]rjλ+1(˜α[ω]r) − T (d) λν T (b) λν λ(λ + 1)jλ(˜β[ω]r) � , Fνλ(r) = R r � jλ(˜α[ω]r) + T (d) λν T (b) λν ˜β[ω]rjλ+1(˜β[ω]r) − T (d) λν T (b) λν (λ + 1)jλ(˜β[ω]r) � , (60) We focus our results for the mode probed in [43], the S122 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Even though the magnon and phonon modes functions are given in terms of well-known special functions, the coupling constant involves a non-trivial combination of derivatives of those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Furthermore, the coordinate trans- formation in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (50) is not easily invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' While this is not a problem when computing the coupling to the Kittel mode, which is uniform, the exact expression for the integrands of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (44) is not elucidating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Differently from other parametrically coupled systems, it is hard to infer from (44), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=', selection rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We, therefore, com- pute the overlap integrals numerically and evaluate how the couplings g0 mjb compare with the coupling to the Kit- tel mode g0 mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' It is also important to notice that the magnomechanical couplings depend on both the inten- sity of the bias magnetic field and its direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In fact, the coupling to the Kittel mode can even vanish for spe- cific relative orientation of the magnetic field [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We consider the case of a fixed bias field at a direction that maximizes the coupling between the Kittel and the S122 mode, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (a) (b) (c) (d) H0 k ez R = 125 µm � ���� ���� ���� ���� |f|/max[|f|] z/R y/R z/R x/R y/R x/R FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Profile of the S122 mode of a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (a) The bias field H0 is parallel to the ez direction, and we consider a sphere made of YIG with a radius R = 125 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (b-d) mode profile |f(r)| for the spherical mode S122 in the (b)yz, (c) xz and (d) zy planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Figure 3 shows the frequencies ωmj of the Walker modes, the ratios |g0 mjb|/|g0 mb| between the magnome- chanical coupling rate to the Walker mode (lmν) and to the Kittel mode, and φj − φ, the relative phase between g0 mjb and g0 mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Results are shown for l up to 4 and for Walker modes lying in a frequency range close to the Kit- tel mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Due to better mode overlap, some higher order Walker modes, for example, the (200), couple strongly with the phonon mode in comparison with the coupling to the Kittel mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In the theoretical analysis of section I, we have not included in the magnomechanical Hamil- tonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (7) the last two terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Those describe scattering processes between different magnon modes via the phonon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For the considered case, |g0 mmjb|, |g0 mkmjb| ≪ |g0 mb|, |g0 mjb|, and those processes can be safely discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Nevertheless, it is possible that for 0 1 1 0 10 1 1 0 10 1 1 0 112 some relative orientation between the magnon modes and the phonon mode, set by the external bias field, those processes can have a stronger coupling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (a) Frequency of the Walker modes ωmjb in units of the Kittel mode frequency ωmb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The labels by each point indicate the radial magnon mode label ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (b) Absolute value of the magnomechanical coupling between the Walker modes and the S122 mode in units of the coupling to the Kittel mode g(0) mb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (c) Phase of the magnomechanical coupling with respect to the phase of the Kittel mode magnomechanical coupling φmjb −φmb in radians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Results for a sphere of radius R = 125 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The dashed line is the reference value (for the Kittel mode) for each quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' EVALUATION OF THE MODEL FOR THE PHONON SELF-ENERGY ON DYNAMICAL BACKACTION EVASION The self-energy obtained in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (30) includes contribu- tions due to the Kerr nonlinearity and to the couplings to higher order Walker modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We focus our analysis now on the effect of such contributions to the magnomechan- ical decay for detunings in the vicinity of the backaction evasion point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In correspondence with the experiment [43], we con- sider that the microwave drive frequency is varied be- tween ωd,− and ωd,+ inside the frequency range {ω−, ω+} between the hybrid modes frequencies, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The Walker modes contributing appreciably to the phonon self-energy lie between (ωd,− − Ωb) and (ωd,++Ωb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Since the system is in the resolved side-band regime, any modes outside this frequency range would not allow efficient scattering of phonons, and thus can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The first condition corresponds to the lower drive frequency corresponding to the blue side-band of a magnon mode, while the second corresponds to driving the red side-band of a magnon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For the parameters summarized in Table I, only the mode (4, 3, 0) is in this frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In fact, the (4, 3, 0) mode is degener- ate with the Kittel mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The frequency configuration is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 4(a,b), and the mode profile of the Walker mode (4, 3, 0) is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 4 (c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Frequency configuration of the magnomechanical sys- tem in consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (a) The microwave cavity frequency ωa is higher than the Kittel mode frequency, which is degener- ate with the (4, 3, 0) Walker mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The red frequency range corresponds to the microwave drive considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (b) Due to strong coupling, the Kittel mode and the Microwave mode hybridize, forming the two modes ω±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (c) Real part and (d) imaginary parts of the transverse magnetization of the Walker mode (4, 3, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The profiles were evaluated at z/R = cos π/4, and for better visualization, the vectors were normalized to max[ � Re[δmx]2 + Re[δmy]2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' To address the effects of nonlinearities and coupling to the higher order Walker mode, we define the dimension- less parameters ηc = gmjc/gmc and ηK = Km/K0 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The first parameter quantifies the strength of the coupling between the Walker modes and the cavity compared to the coupling between the Kittel mode and the cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The second parameter quantifies the strength of the self- Kerr nonlinearity compared to the value shown in Table I K0 m = −2π×5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='15 nHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We call here ηK the dimensionless Kittel magnon self-Kerr nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' This parameter de- pends on the alignment between the anisotropy axis of the magnet with the external magnetic field, which has not been taken into account in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Figure 5 shows (a) the magnomechanical decay for ηc = 0 (the additional Walker mode is not driven by the microwaves) and for several values of ηK, and (b) the magnomechanical decay for several values of ηc at a 13 Magnomechanical decay (Hz) � ��� ��� ��� ��� ��� ⌘K Drive detuning from the upper hybrid mode (MHz) � ���� ���� ���� ���� ���� ���� ⌘c ⌘c = 0 ⌘K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='2 Magnomechanical decay (Hz) (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Magnomechanical decay rate Γmag[Ωb] including the contribution of the Walker mode (4, 3, 0) as a function of the detuning from the upper hybrid mode for (a) ηc = 0 (without microwave coupling to the additional Walker mode) and for several values of ηK (dimensionless Kittel magnon self-Kerr nonlinearity);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' and (b) for ηK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='2 and for several values of ηc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The dashed line is the prediction from the self-energy (3) derived in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The magnomechanical coupling to the (4, 3, 0) Walker mode corresponds to that shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The driving power is 15 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Parameters in correspondence with the experiment [43], given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' fixed ηK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The self-Kerr nonlinearity of the Kittel mode changes the slope of the magnomechanical decay as a function of the detuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For a fixed Kerr nonlinearity, the additional magnon mode shifts down the magnome- chanical decay, that is, the weakly driven Walker mode adds energy to the vibrational mode, yielding a negative contribution to the decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The magnomechanical frequency shift is also modi- fied by the Kittel mode nonlinearity and by the cou- pling to the additional magnon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' This is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 6, which shows the magnomechanical frequency shift δΩ = −Re[ΣTot[Ωb]] for (a) several values of the self-Kerr nonlinearity and (b) several values of the coupling to the additional magnon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Whereas the Kerr nonlinearity induced a tilt in the slope of the magnomechanical decay rate, its effect on the the magnomechanical frequency shift consists on an extra negative shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We also no- tice that the magnomechanical frequency shift does not vanish for a drive at the frequency where the magnome- chanical decay vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' This is the case because for the parameters considered, the Kittel mode frequency does not match the microwave frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For perfectly match- ing Kittel mode and microwave frequencies, and in the absence of additional magnon modes, both the magnome- chanical decay and the magnomechanical frequency shift vanish at the same drive frequency [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' � ��� ��� ��� ��� ��� ⌘K Drive detuning from the upper hybrid mode (MHz) � ���� ���� ���� ���� ���� ���� ⌘c ⌘c = 0 ⌘K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='2 Magnomechanical frequency shift (Hz) (a) (b) Magnomechanical frequency shift (Hz) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Magnomechanical frequency shift δΩb including the contribution of the Walker mode (4, 3, 0) as a function of the detuning from the upper hybrid mode for (a) ηc = 0 (without microwave coupling to the additional Walker mode) and for several values of ηK (dimensionless magnon self-Kerr nonlin- earity);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' and (b) for ηK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='2 and for several values of ηc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The dashed line is the prediction from the self-energy Eq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (3) derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The magnomechanical coupling to the (4, 3, 0) Walker mode corresponds to that shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The driving power is 15 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Parameters in correspondence with the experiment [43], given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 5 one notices that the drive frequency at which the magnomechanical decay vanishes changes with both ηK and ηc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For applications where evading backaction is important, it is necessary that such modifications are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 7 the drive frequency for backaction evasion (with respect to the upper hybrid mode frequency) as a function of the drive power for (a) several values of the Kittel self-Kerr nonlinearity and (b) several values of the coupling to the additional magnon mode at a fixed ηK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For the case without nonlinearities and without coupling to the additional magnon mode, the backaction evasion frequency has a weak dependency on power (not perceptible in the plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' When the correc- tions are included, a stronger linear dependency of the backaction drive frequency with the power is induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For the parameters in consideration, the difference can be of of the order of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='1 MHz at moderate powers of 10 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 100 0 100 200 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 11100 0 100 200 300 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 11-180 200 220 240 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 11-180 200 220 240 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 1114 Backaction evasion detuning (MHz) Power (mW) (a) (b) ⌘K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='2 ⌘c = 0 � ��� ��� ��� ��� ��� ⌘K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='2 � ���� ���� ���� ���� ���� ���� ⌘c = 0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Detuning between drive frequency and the upper hybrid mode for backaction evasion as a function of power for (a) no coupling to additional magnon modes and for several values of the dimensionless Kittel self-Kerr nonlinearity ηK, and (b) for several values of the coupling between the (4, 3, 0) Walker mode and the microwave cavity at a fixed ηK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Parameters in correspondence with the experiment [43], given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In order to quantify the agreement between our model and the measured data in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [43], we study the dif- ference between the theoretical magnomechanical decay Γmag[Ωb] rate and the experimental data Γexp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 8, we show the absolute difference |Γmag[Ωb]−Γexp| between theory and experiment as a function of the drive power for different drive frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Our proposed model agrees well with the experimental data, besides the differ- ence at higher powers and drives farther from the upper hybrid mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In the worst case, the model proposed here improves the discrepancy between data and theory from ∼ 120 Hz (red, dashed curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 8), to a difference of ∼ 50 Hz (red, solid curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Otherwise, we notice good agreement between theory and experiment for drive powers up to ∼ 14 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' At such powers the co- herent number of magnons generated by the microwave drive |⟨ ˆm⟩|, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (12), is between ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='0 × 1013 at a de- tuning from the upper hybrid mode ∆+ = −11 MHz and ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='4 × 1013 at ∆+ = −14 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We should notice that for the parameters considered here, the system is not in a bistable regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Difference between theory and data (Hz) Power (mW) (-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='4, -13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='7) MHz (-12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='1, -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='0) MHz 11 MHz Detuning from the upper normal mode � �� �� � �� �� ��� FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Absolute difference between the theory for the mag- nomechanical decay and the experimental data as a function of power at different detunings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' The dashed curves correspond to the prediction of the previous theory using (4), while the solid lines correspond to the theory developed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Theory predictions use parameters in correspondence with the experiment [43], given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 9 we show in (a-c) the magnomechanical decay as a function of the drive frequency detuning from the upper hybrid mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' As in the discussion above, we con- sider the coupling only to the (4, 3, 0) Walker mode, and we choose ηK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='2 and ηc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='3, which yields a good agreement between theory and data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In the plots of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 9 (d-f), we show the difference |Γmag[Ωb] − Γexp|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' While the correction due to the coupling to the (4, 3, 0) Walker mode improves the agreement between theory and data with respect to the previous theory framework [34], we notice, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 8, that at higher drive pow- ers, there is a further discrepancy with the experiment for drives away from the dynamical backaction evasion points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Such drive frequencies are closer to frequencies of the magnon modes, in particular to higher order modes not considered here, and can thus induce a nonlinear be- havior that was not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We also notice that the errors of the data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 9 (a) do overlap with both the present theory and the one used in [34], the trend shown in Figs 9 (b-c) holds for the intermediate drive powers not shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' CONCLUSIONS Dynamical backaction effects in magnomechanical sys- tems are a consequence of the radiation pressure-like coupling between magnons and phonons [29, 34] which can be exploited for applications ranging from generat- ing entangled states to noise-based therometry [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In this paper, we have extended the description of dynami- cal backaction in cavity magnomechanics by including in the system’s dynamics self and cross-Kerr nonlinearities, and the coupling between the phonon mode and addi- tional magnon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' While nonlinearities are intrinsic to magnetic systems due to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=', magnetic anisotropy [48], magnon modes other than the uniform Kittel mode are always present and can couple to phonons as efficient as (if not more than) the Kittel mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' A non-uniform mi- 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='75 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='85 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='95 10 ¥15 20-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='75 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='85 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='95 10 15 2015 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Comparison between the magnomechanical decay rate Γmag[Ωb] predicted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (3) (Gray, dashed line), by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (30) (Blue line) and the experimental data measured in [43] (magenta points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In these plots we have used ηc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='3 and ηK = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='2, which yields a good agreement between our model and the experimental data specially close to the point of dynamical backaction evasion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (a)-(c): Magnomechanical decay rate as a function for the detuning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' (d)-(e) Absolute difference between the theory and the experiment as a function of the detuning (we have omitted the error bars in these plots for a better visualization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Theory curves with parameters in correspondence with the experiment [43], given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' crowave field can weakly drive such modes, which modi- fies the backaction induced decay rate and frequency shift of the phonon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Our framework considers a single phonon mode, an assumption that can be readily gener- alized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We have obtained the phonon self-energy including the aforementioned interactions and showed that, provided that the additional magnon modes couple only weakly to the microwave mode, the overall correction to the mag- nomechanical decay rate is proportional to the average number of Kittel magnons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We have then focused our results on the case of a magnetic sphere, in connection with the experiment performed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Our model explains the observed shift in the magnomechanical de- cay rate close to the dynamical backaction evasion drive frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In this context, we have also evaluated the ef- fects of the different corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Specifically, we showed that the drive at which the dynamical backaction decay is zero depends linearly on power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' This is a consequence of the corrections being proportional to the steady state number of Kittel magnons, which scales linearly with the drive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' A small discrepancy with the experimental data is still present at higher drive powers and for detunings far from the upper hybrid mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We attribute this difference to higher order Walker modes that have not been taken into account in the present model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Similar to the effects de- scribed above for the mode included in our calculations, even higher order Walker modes can lie in a frequency range close to the microwave drive and, at higher powers, can modify substantially the magnomechanical decay via nonlinear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' We should also point out that the ex- perimental setup in [43] has particularities not included here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' For instance, the magnetic sphere is glued on a dielectric post, which modifies the photon, phonon and magnon mode profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' This in turn can change the mag- nomechanical coupling constants as well as the frequency of the Walker modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' A precise evaluation of such effects requires a more refined numerical analysis, for example using finite difference software and micromagnetic simu- lations, which goes beyond the scope of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' While nonlinear effects in cavity magnomechanical sys- tems have been previously computed for the nonlinear dynamics of magnons [14, 49], the evaluation of such ef- fects on the response of the mechanical degree-of-freedom to noise, as computed by the self-energy, is a step forward in the characterization of these systems as platforms for quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Our analysis was restricted to evaluate the effects of all the corrections included in the model of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' II in the framework of dynamical backac- 16 tion evasion set by the experiment of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Never- theless, the model derived in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' II shows that several phenomena play a role in the modification of dynami- cal backaction, for example, magnon squeezing and two- mode squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' It would be interesting to investigate scenarios in which those terms can be harnessed to re- duce noise for quantum metrology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Furthermore, the in- clusion of the additional magnon modes opens new possi- bilities for cavity magnomechanical systems, such as the manipulation of the mechanics by driving different side- bands of the different magnon modes in a Floquet-like setup [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In this case, it would be interesting to go be- yond the approximation used here, where only the Kit- tel mode couples strongly to the microwave cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' In fact, several experiments have shown fingerprints of a strong coupling between Walker modes of a sphere and microwaves [7, 55, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' As we have numerically shown, Walker modes other than the Kittel mode can couple bet- ter to the phonons, which can be harnessed to applica- tions, such as nonreciprocal transport between phonons and microwaves [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors acknowledge helpful contributions from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Scharma and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Varga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Bittencourt and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Viola Kusminskiy acknowledge financial sup- port from the Max Planck Society and from the Deutsche Forschungsgemeinschaft (DFG, German Re- search Foundation) through Project-ID 429529648–TRR 306 QuCoLiMa (“Quantum Cooperativity of Light and Matter”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Potts, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQf1i6S/content/2301.11920v1.pdf'} +page_content=' Huang, and J.' metadata={'source': 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+Michigan State University, MI 48824, USA. +2Department of Electrical and Computer Engineering, +Michigan State University, MI 48824, USA. +3 Department of Biochemistry and Molecular Biology, +Michigan State University, MI 48824, USA. +4 Research Center of Nonlinear Science, School of Mathematical and Physical Sciences, +Wuhan Textile University, Wuhan, 430200, P R. China +January 13, 2023 +Opioid use disorder (OUD) continuously poses major public health challenges and social implications +worldwide with dramatic rise of opioid dependence leading to potential abuse. Despite that a few phar- +macological agents have been approved for OUD treatment, the efficacy of said agents for OUD requires +further improvement in order to provide safer and more effective pharmacological and psychosocial treat- +ments. Preferable therapeutic treatments of OUD rely on the advances in understanding the neurobiological +mechanism of opioid dependence. Proteins including mu, delta, kappa, nociceptin, and zeta opioid recep- +tors are the direct targets of opioids. Each receptor has a large protein-protein interaction (PPI) network, +that behaves differently when subjected to various treatments, thus increasing the complexity in the drug +development process for an effective opioid addiction treatment. The report below analyzes the work by +presenting a PPI-network informed machine-learning study of OUD. We have examined more than 500 pro- +teins in the five opioid receptor networks and subsequently collected 74 inhibitor datasets. Machine learning +models were constructed by pairing gradient boosting decision tree (GBDT) algorithm with two advanced +natural language processing (NLP)-based molecular fingerprints. With these models, we systematically car- +ried out evaluations of screening and repurposing potential of drug candidates for four opioid receptors. +In addition, absorption, distribution, metabolism, excretion, and toxicity (ADMET) properties were also +considered in the screening of potential drug candidates. Our study can be a valuable and promising tool of +pharmacological development for OUD treatments. +Key words: Opioid use disorder, opioid receptor, machine-learning, cross-prediction, side effect, repur- +posing. +∗Corresponding author. Email: weig@msu.edu +i +arXiv:2301.04815v1 [q-bio.BM] 12 Jan 2023 + +Contents +1 +Introduction +1 +2 +Results +3 +2.1 +The Opioid receptors and addiction PPI networks. . . . . . . . . . . . . . . . . . . . . . . . . +3 +2.2 +Binding affinity predictions +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2.1 +Cross-target binding affinity predictions . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2.2 +Predictions of side effects and repurposing potentials . . . . . . . . . . . . . . . . . . . +6 +2.2.3 +Protein similarity inferred by cross-target BA correlations . . . . . . . . . . . . . . . . +6 +2.2.4 +Repurposing to opioid receptors and side effect on hERG +. . . . . . . . . . . . . . . . +7 +2.3 +Druggable property screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3 +Discussion +9 +3.1 +Side-effect evaluations of existing medications for OUD treatment . . . . . . . . . . . . . . . . +9 +3.2 +Nearly optimal lead compounds from screening and repurposing . . . . . . . . . . . . . . . . . +13 +4 +Methods +15 +4.1 +Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +4.2 +Molecular embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +4.2.1 +Sequence-to-sequence auto-encoder +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +4.2.2 +Bidirectional transformer +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +4.3 +Machine-learning models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +5 +Conclusion +17 +ii + +1 +Introduction +Over three million people in the United States are currently suffering or have previously suffered from opioid +use disorder (OUD). In 2020 alone, over 68,000 deaths were recorded from an overdose. Unfortunately, the +numbers are continuously rising, and have more than tripled throughout the past 10 years. The opioid +crisis or epidemic presents a substantial public health concern and costs the United States billions of dollars +annually. It takes many components, including increased public awareness, improved economic conditions, +and better therapies, to fully address the OUD crisis. Since the treatment of OUD with medications is +effective in reducing symptoms of drug withdrawal and cravings [1], there is a pressing need to further search +and develop more effective OUD treatments. +Opioid is a broad term for any natural or synthetic substance that binds to specific opioid receptors in the +human body. It is widely used as analgesics medications in modern pain management. The three main brain +receptors that opioids bind to are the mu opioid receptor (MOR), kappa opioid receptor (KOR), and delta- +opioid receptor (DOR) in the central nervous system (CNS) and peripheral organs [2], which are responsible +for a plethora of physiological functions, such as analgesia, respiration, and hormonal regulation. Synthetic +and exogenous opioids (i.e., morphine, heroin, oxycontin, etc.) act on the opioid receptors as endorphins, and +repeated exposure to escalating use of opioids causes gradual adaptations in the brain [1]. Tolerance develops +and leads to heightened uncontrolled intake of drugs. Consequently, physical dependence emerges with drug +craving to reduce withdrawal symptoms upon abstinence [3]. When opioids are improperly ingested, the +interaction with the opioid receptors can induce a harmful effect on the CNS impacting the respiratory system +and causing irreversible brain damage [4]. According to the Centers for Disease Control and Prevention +(CDC), methadone, oxycodone, and hydrocodone contribute to the highest number of fatalities, resulting +from opioid overdose. +MOR is critical in brain reward circuits. It has an important role in goal-directed behavior such as drug- +seeking behavior and represents a major factor in the initiation of addictive behaviors. In the development +of opioid addiction, poor decision-making, and cognition impairment, MOR translates the goal-directed +behaviors to habitual behaviors, promoting compulsive drug use [5]. Through experiments ran on animals, +with similar physiological functions, the results demonstrated that MOR is pivotal in mediating therapeutic +and adverse activities of opioids and is associated with the maintenance of drug use, drug craving, and +relapse [6]. KOR has anti-reward effects and can induce dysphoria. Upon long-term exposure to opioids, +KOR has impact on modifying the brain’s reward circuits, leading to a relapse [7]. The activation of KORs +suppresses unpleasant MOR/DOR-mediated side effects including the rewarding effect. KOR blockade may +beneficially alleviate stress responses, reduce drug cravings, and remediate depressive states. DORs reduce +levels of anxiety and attenuate depressive symptoms [8]. In addition, beneficial effects of DOR agonists were +found in treating chronic pain and psychiatric disorders [9]. +In growing efforts to combat the opioid epidemic, further research and development have been invested +in the treatments of OUD. Currently, there are three medications used to treat opioid dependency approved +by the Food and Drug Administration (FDA), methadone, buprenorphine, and naltrexone. Methadone is +a full agonist on MOR and is used to reduce withdrawal and craving symptoms in patients. Methadone +maintenance treatment (MMT) is useful in reducing the intensity of withdrawal symptoms and preventing +patients from ingesting more opioids to induce a euphoric effect. Since MMT can decrease the intensity of +cravings in patients, they are more willing to remain in treatment. However, methadone is associated with +the risk of causing respiratory depression when improperly administrated [10]. Buprenorphine, a partial +MOR agonist, is an alternative to methadone. It has a ceiling effect of stimulation on MOR than that +of methadone. Hence, buprenorphine provides a less euphoric effect and is less likely to cause respiratory +depression. MMT is more likely to keep patients in treatment than buprenorphine treatment [11]. Naltrexone +is an antagonist of MOR and is effective in attenuating drug cravings and reducing the risk of overdose. It +does not produce sedation, analgesia, euphoria, or potential for abuse or diversion [12]. However, it is not +1 + +as widely used as methadone or buprenorphine for several reasons including low rates of patient acceptance +and non-adherence. +Naloxone is a non-selective and competitive opioid receptor antagonist and is used +in treating opioid overdose or for opioid intoxication such as reversing respiratory depression. Take-home +naloxone programs were developed to prevent fatal overdoses [13]. Buprenorphine/naloxone formulations +are adopted. When injected, naloxone has higher bioavailability, thereby blocking the pain and craving- +reducing effects of buprenorphine [14]. However, Naloxone’s capabilities are limited when ingesting highly +potent opioids, such as fentanyl [15]. Psychosocial interventions were also combined with these medications +to improve the efficacy in treating opioid addictions [16]. Further studies on medication efficacy and the +mechanism of opioid addiction in the brain are needed to find better treatments to prevent relapse and +facilitate longer periods of abstinence. +The molecular mechanism underlying opioid tolerance, dependence, withdrawal, and addiction is com- +plicated, involving several systems located in different regions of the brain. Opioids target opioid receptors +in the brain and activate the mesolimbic (midbrain) reward system. Dopamine is produced in the ventral +tegmental area (VTA), and is then released into the nucleus accumbens (NAc) area, giving rise to the feeling +of pleasure. Opoid tolerance occurs because of the brain’s adaptation to repeated exposure to opioids. [1] +The withdrawal symptoms and opioid dependence are related to noradrenaline (NA) that is produced in +the locus ceruleus (LC) area [3]. Opioids impact brain areas with a fairly large number of proteins and +peptides that are responsible for a multitude of physiological and biological functions. It is challenging to +understand how so many proteins are simultaneously impacted by opioids, causing difficulty when designing +effective medications for OUD treatments. On one hand, medication compounds targeting opioid receptors +can potentially cause unintentional dependence or an overdose such as methadone, due to the possibility +of an agonist effect on MOR. On the other hand, blocking other proteins associated with the opioids can +interfere with the biological functions of these proteins and induces various side effects. It is necessary to +investigate the inhibition effects of compounds on the opioid receptors as well as the side effects of potential +medications by blocking other proteins. +The protein-protein interaction (PPI) network on the proteome scale forms a basis for systematically +studying potential treatment efficacy and side effects. A PPI network is constituted of proteins and cor- +responding direct and indirect interactions that contribute to certain biological activities. The String v11 +database (https://string-db.org/) [17] provides a large collection of protein-protein interactions for given +proteins or diseases. In the study of OUD, we can extract the PPI networks related to the major opioid re- +ceptors, based on which we can have systematic investigations of medication treatment and side effects. The +proteins in these PPI networks are the test targets of treatment or side effects but using traditional in vivo or +in vitro experiments is too time-consuming and expensive. Besides, large-scale experiments on animals raise +legal and ethical concerns. Machine learning/deep learning technology has recently gained wide popularity in +drug discovery and development, such as the generation of drug-like molecules [18], repositioning of existing +drugs for diseases [19], protein engineering [20] and predictions of chemical toxicity [21] in drug design. The +time and cost can be significantly reduced by machine learning as well as the erasure of ethical concerns. As +a result, machine-learning approaches were utilized in this study, to carry out large-scale predictions. +In this work, we developed a proteome-informed machine-learning (ML) platform for the discovery of +anti-opioid addiction compounds. From the String v11 database, we obtained PPI networks of the five ma- +jor opioid receptors with the associated proteins regarded as potential treatment and side effect targets. We +then collected inhibitor datasets with experimental binding affinity labels from ChEMBL database for these +protein targets and built machine learning models. The inhibitor compounds were represented by two forms +of latent-vector (LV) fingerprints generated by transformer and autoencoder learning models, respectively. +These latent vectors were paired with gradient boosting decision algorithm (GBDT) in building our bind- +ing affinity (BA) predictors. We then carried out cross-predictions to screen side effects and repurposing +potentials of more than 120,000 compounds. With these models, we had more side effect evaluations of +FDA-approved drugs or other existing medications. Another application using these cross-prediction mod- +2 + +els was to find promising lead compounds. In addition to the concern in potency and side effect, we also +considered evaluations of pharmacokinetic properties in compound filtering , i.e., absorption, distribution, +metabolism, excretion, and toxicity (ADMET) as well as synthesizability. Our platform is believed to be +useful in advancing the drug development in treating OUD. +2 +Results +2.1 +The Opioid receptors and addiction PPI networks. +Figure 1: The workflow for searching nearly optimal lead compounds. The inhibitor compounds were collected for the proteins +in protein-protein interaction networks of the five opioid receptors including mu, kappa, delta, nociceptin, and zeta opioid +receptors. Each receptor has a core and global PPI network. The proteins in one core network have direct interaction with +the opioid receptor, and those in a global network are related to opoid receptors through the proteins in the core network. +Abbreviations for the proteins in the core networks are shown. Full names of the proteins in the five core networks are provided +in the Supporting information. +Opioid receptors play critical roles in opioid dependence and are often the pharmacological targets of +medications. There are four major subtypes of opioid receptors, namely mu opioid receptor (MOR), delta +opioid receptor (DOR), kappa opioid receptor (KOR), and nociceptin opioid receptor (NOR). In addition, +zeta opioid receptor (ZOR) is also believed to be an important one. However, ZOR was recently discovered, +hence less studied and shares little sequence similarity with other opioid receptors. Opioid receptors are +crucial in various biological functions and have broad distributions in the brain, spinal cord, on peripheral +neurons, and digestive tract. MOR, KOR, and DOR are closely related to analgesia, opioid dependence and +the adverse effect of respiratory depression caused by opioids, but each of them is distributed in various re- +gions of the brain. NOR is distributed mainly in the cortex amygdala, hippocampus, septal nuclei, habenula, +and hypothalamus in the brain and spinal cord, and is linked to development of tolerance to MOR agonists. +ZORs widely exist in many parts of the body including the heart, liver, kidneys and brain. Its functions +are mainly on tissue growth. Several clinically useful medications for treating addiction target MOR, KOR, +and DOR [1], but the roles of NOR and ZOR in causing opioid dependence has not been much explored. +However, they cannot be neglected in the pharmaceutical treatment of opioid dependence as they are all +critical targets of opioids. +3 + +MOR PPI-network +KOR Ppl-network +ZOR PPI-network +HINT1 +POMC +MECP2 +SIN3A +SPATA4 +PENK +ZNE324 +FNIPI +GABARAPL1 +MOR +PDYN +POMC +MECP2 +CYB5D1 +MOR +WLS +STAT6 +CDK10 +PENK +GNAQ +PDYN PNOC +COG8 +KOR +ARRB1 +HDAC +ZOR +ADRBK1 +ARRB2 +SAG +SPC25 +FLNA +DOR +ARRB2 +DOK5 +SLC9A3R1 +ARRB1 +Treatment-target +Side-effect-target +Nearly Optimal Leads +Inhibitor Datasets +Inhibitor Datasets +ADMET +repurposing +Screening +Potency Predictors +Side-effect PredictorsOpioid receptors have wide distributions in the body, and the synergistic interactions between these +receptors and many other proteins upstream and downstream contribute to specific biological functions. As +discussed before, we carry out drug discovery in the PPI networks. We extracted five PPI networks centered +around each of the five opioid receptors by inputting receptor names, namely, mu-opioid receptor, delta- +opioid receptor, kappa opioid receptor, zeta opioid receptor, nociceptin opioid receptor, and OGFR into the +String database. In each network, there is a core subnetwork with proteins interacting directly with each +opioid receptor, while proteins with direct and indirect interactions jointly form the global network as shown +in Figure 1. We restrict the number of proteins in each global network to 101. Although more proteins +should be considered, we limit our efforts to critical ones. There are five global networks in which five core +networks exist. MOR, KOR, DOR, NOR, and ZOR are the most important proteins in the networks as +each core protein plays an essential role. The five networks are not independent of each other with a few +overlapping proteins found between the networks. +Compounds with agonist or antagonist effects on opioid receptors showed their pharmacological effects +in treating opioid dependence [4], hence encouraging us to look for more compounds that bind to the opioid +receptors. A desired drug must be specific to a target protein without causing adverse side effects to the +other proteins. To evaluate the binding effect of inhibitors to receptor proteins and other proteins in the +PPI networks, we collected inhibitor compounds from the ChEMBL database for each protein and built +machine-learning models. We then used them to systematically analyze the side effects and repurposing +potential of inhibitor compounds. We collected 74 datasets in total, with sufficient inhibitor data points for +the proteins in the five extracted PPI networks with a total of 129,515 inhibitor compounds. In addition, +we collected an inhibitor dataset for hERG protein and built an appropriate machine-learning model. The +hERG is a critical potassium channel that must be avoided in drug design and discovery, as the blockade +of the hERG channel is associated with prolongation of the long QT syndrome, eventually leading to fatal +arrhythmia, namely Torsade de Pointes (TdP) [22]. In total, we collected 75 protein targets and built 75 +machine-learning models. Since ZOR was recently discovered, not much inhibitor data was available to build +a model for the ZOR protein. However, we were able to build models for the four remaining receptors, i.e., +MOR, KOR, DOR, and NOR. Further, we used all the models to explore potential drugs that bind to the +four opioid receptors. The details about the collected datasets can be found in the Supporting information. +2.2 +Binding affinity predictions +The heatmap in Figure 2 shows the cross-target binding affinity (BA) predictions using the 75 machine- +learning models. The diagonal elements indicate the Pearson correlation coefficient (R) of five-fold cross- +validation for our machine-learning models. Two of the 75 models have R values greater than 0.9, and the +R values for fifty of them are greater than 0.8. The minimal R value of 0.604 is from the model built with +the FYN inhibitor dataset. Overall, these models show excellent prediction accuracy and are reliable for BA +predictions. +2.2.1 +Cross-target binding affinity predictions +Cross-target predictions can reveal side effects of drug candidates on other proteins. The off-diagonal elements +represent the maximal BA values (BA with the largest absolute value) of inhibitor compounds in one dataset +predicted by other models. The notations to the left of the heatmap indicate the 75 inhibitor datasets and +those on top of the heatmap represent all the 75 machine-learning models. Each column exhibits all the +predictions by one model. Specifically, the i-th element in the j-th column is the prediction result of i-th +dataset by the j-th model. These cross-target prediction results are indicators of side effects of one inhibitor +dataset on other proteins. The BA value of -9.54 kcal/mol (Ki= 0.1 µM) is widely accepted as an inhibition +threshold in the literature [23]. With this threshold, 5103 out of the 5625 cross-predictions were found to +have side effects, i.e., the predicted maximal BA less than -9.54 kcal/mol. On the other hand, the remaining +4 + +Figure 2: The heatmap of cross-target binding affinities (BAs) predictions revealing the inhibitor specificity of each dataset +on other protein targets. The notations above the heatmap shows the machine-learning models while the those on the left of +the heatmap denote all the inhibitor datasets. The diagonal elements in the heatmap indicate the Pearson correlation efficient +(R) of five-fold cross validations for all the predictive models. The off-diagonal elements in each row represent the highest BAs +values of inhibitors in one dataset predicted by 74 machine-learning models. +522 cross-prediction results with maximal BA greater than -9.54 kcal/mol suggest weak side effects. The +color of the off-diagonal elements indicates the strength of side effects. The lighter the color, the stronger +the side effects are. +Similar binding sites on off-target proteins is one of many reasons for side effects caused by drug candidates +for one designated protein. Proteins in the same family can have similar three-dimensional (3D) structures or +protein sequences, giving rise to the existence of similar binding sites. An inhibitor compound potent at one +protein likely binds to another protein in the same family. As observed in Figure 2, mutual side effects occur +among the four opioid receptors, i.e., MOR, DOR, KOR, and NOR. The 16 yellow square boxes on the upper +left corner of the heatmap showed the cross-prediction maximal BA value less than -9.54 kcal/mol. These +four proteins are all in the opioid receptor family and are highly similar in their 3D structure conformation +or 2D sequences. This is validated by the alignments of 3D structures and 2D sequences as shown in Figure +S2 of the Supporting information. The heatmap shows more examples of mutual side effects among proteins +in one family such as the family of tyrosine kinase protein (JAK1, JAK2, and JAK3), melanocortin receptor +(MCR1, MCR3, MCR4, and MCR5), and matrix metalloproteinases (MMP1, MMP2, MMP7, MMP8, and +MMP9). +5 + +5 +B +5 +RAAACCC +D4P +MOR +DOR +KOR +NOR +ACE +ACKR3 +ADAM17 +ADAMTS4 +ADAMTS5 +ADORA1 +ADRB1 +ADRB2 +ADRBK1 +AGTR1 +AKT1 +AR +ATG4B +AVPR2 +BDKRB1 +BDKRB2 +CASR +CCR5 +CDK1 +CNR1 +CREBBP +CRHR1 +CXCR1 +CXCR2 +CXCR4 +DNMT1 +EGFR +ERBB2 +ERBB4 +ESR1 +F11 +F2R +FYN +HDAC1 +HDAC2 +HGF +ITGB1 +JAKi +JAK2 +JAK3 +KLKB1 +KRAS +MAP3K5 +MAPK1 +MAPK10 +MC1R +MC3R +MC4R +MC5R +MDM2 +MMP1 +MMP2 +MMP? +MMP8 +MMP9 +NOS3 +NTRK1 +P2RY12 +PDE4A +PDGFRB +PI4KB +PPARG +PTPN2 +REN +S1PR1 +SMO +SRC +TBXA2R +TOP2A +TYK2 +hERG +0.65 +0.70 +0.75 +0.80 +0.85 +-15 +-14 +-13 +-12 +-10 +-g +0.90 +-8 +R of 10-fold CV +ML-BA (kcal/mol)2.2.2 +Predictions of side effects and repurposing potentials +The cross-target prediction is a useful tool to detect side effects and to evaluate repurposing potentials of +inhibitors. Side effects are caused when a drug candidate exhibits a strong binding affinity to the desired +target, but unintentionally acts as a potent inhibitor on other proteins. Drug candidates that exhibit a weak +binding affinity to their designated targets but an effective inhibitor to other proteins are deemed to have +repurposing potential. Figures 2a and 2b exemplify side effects and repurposing. Each panel involves one +target and two off-target proteins. The title, the x-axis and the y-axis of each panel stand for the target, +an off-target protein, and another off-target protein, respectively. The colors of the scattered points indicate +the experimental BAs of the inhibitors for the target protein. The red and green colors reveal high and low +binding affinities, respectively. The x-axis and y-axis indicate the predicted BAs from two machine learning +models built on inhibitor datasets for two off-target proteins. +The yellow frames in the nine panels of Figure 3a highlight the zone where no side effects are induced +on two off-target proteins according to our predictions. The three rows in Figure 3a show some examples +of inhibitors for one designated protein having side effects on zero, one, and two of the given two off-target +proteins, respectively. For instance, as shown on the second panel in the first row of Figure 3a, all inhibitors +for protein ADAM17 are predicted to be weak inhibitors, i.e., BA values greater than -9.54 kcal/mol, on two +off-target proteins. The first panel in the second row shows that around half of the inhibitors for the DOR +are predicted to be potent at the MDM2 protein, but all the inhibitors were predicted to not bind to the +MMP protein. In addition, the first panel in the third-row exhibits a significant amount of KOR inhibitors +that were predicted to be potent at JAK2 and JAK3, simultaneously. +The repurposing potentials of inhibitors can be revealed through cross-target predictions as well. Figure +3b provides a few prediction examples of repurposing using our models. The blue frames highlight the zone +where inhibitors for target proteins can bind strongly to one protein, i.e., predicted BAs less than -9.54 +kcal/mol, but are weaker binders to the other protein, i.e., predicted BAs greater than -9.54 kcal/mol. The +first panel in the first row in 3b shows that many inactive inhibitors for HDAC1 were predicted to have +repurposing potential for either MOR or DOR, but not bind to the other one. Since both MOR and DOR +are critical targets of medications in treating OUD [4], finding more drug candidates for these two proteins +is desirable. Buprenorphine is an FDA-approved drug that is a partial agonist of MOR and KOR, as well +as a weak DOR antagonist. As seen in the HDAC1-DOR-MOR panel on the first row of 3b, there are some +inactivate inhibitor compounds for HDAC1 that are effective inhibitors to MOR and DOR. Our models can +be used to find more inhibitors that can bind to both targets as Buprenorphine does. The second and third +rows in Figure 3b demonstrate additional examples of the inhibitors for one given protein having repurposing +potentials for two other proteins. +2.2.3 +Protein similarity inferred by cross-target BA correlations +As discussed above, cross-target BA prediction is useful in evaluating side effects and repurposing potentials. +Side effects can be caused when the drug candidate binds to proteins with similar 3D structures or sequences. +In such situation, the predicted BA values can be correlated. On the other hand, the correlated predicted BAs +can be an indicator of similar binding sites or 3D protein structures. As shown in Figure 4a, the predicted +BAs of inhibitors for JAK3 on HDAC1 and HDAC2 proteins have a nearly linear correlation. The Pearson +correlation coefficient (R) of the predicted BA is up to 0.838. This is due to the binding site similarity, which +is validated by the 3D protein structure and 2D sequence alignments as shown in 4a. The 3D structures +of the HDAC1 and HDAC2 proteins are found to be quite similar while the 2D sequence identity near the +binding site is around 85%. Two more examples of BA correlations revealing similar 3D protein structures +are seen in Figures 4b and 4c. For the case in Figure 4b, the Pearson correlation coefficient of the predicted +BAs for DOR inhibitors on MOR and KOR proteins is 0.569. For the case in Figure 4c, the R value of +predicted BAs of JAK3 inhibitors on JAK1 and JAK2 proteins is 0.561. The 3D protein structure and 2D +6 + +Figure 3: Examples of inhibitors’ predicted side effects and repurposing potentials. The three rows in panel a indicate example +inhibitor datasets have side effects on 0, 1, and 2 of the given two off-target proteins, respectively. The yellow frame outlines +therein. Yellow zones indicate where side effects are not found. The three rows in panel b reveal example inhibitor datasets +that show repurposing potentials on 0, 1, and 2 of the two given off-target proteins. The blue frames highlight the domains +where inhibitors have repurposing potential for one protein but have no side effect on the other proteins. +sequence alignments confirm the usefulness of cross-prediction in detecting protein similarities. In addition, +it was found that there is a bilinear correlation relationship among the predicted BAs and experimental BAs +in the case of Figures 4b and 4c. The target and two off-target proteins are in the same protein family +and share high 3D structure and 2D sequence similarities. A potent DOR inhibitor is likely to be a strong +binder on KOR and MOR proteins simultaneously. The high structural similarities form the basis of drug- +mediated trilinear target relationship. KOR, MOR, and DOR proteins are often pharmacological targets in +the treatment of opioid addiction [1]. The observed bilinear or trilinear relationship indicates the possibility +of developing inhibitors that simultaneously bind to multiple targets of the major opioid receptors, namely, +MOR, KOR, DOR, and NOR. Such binding effects on multiple opioid receptors have been observed on the +currently FDA-approved medications [4]. More examples of similar proteins with correlated predicted BAs +can be found in the Supporting information. +2.2.4 +Repurposing to opioid receptors and side effect on hERG +MOR, KOR, DOR, and NOR are the four major subtypes of opioid receptors and are the critical pharma- +cological targets in treating OUD. Inhibitors that bind to these receptors can be potential medications for +OUD treatments. We adopted our cross-target prediction strategy to evaluate the repurposing potential +of inhibitors on the four opioid receptors. The 75 collected inhibitor datasets contain more than 120,000 +compounds, providing a source of drug candidates in our repurposing study. +The side effect of hERG is a priority concern for novel medications, and hence we used our machine +learning model to predict the binding affinity of these inhibitors on hERG. A stricter side effect threshold +of −8.18 kcal/mol (Ki = 1 µM ) was adopted for the hERG. In this study, inhibitors are considered to have +7 + +a. +b. +MOR +ADAM17 +CRHR1 +HDAC1 +CRHR1 +-8- +MMP1 ML-BA +MMP1 ML-BA +CNR1 ML-BA +AR ML-BA +-9 +-8 +-8 +-9 +-10 +-10 +-9.8 +-9.8 +-9.8 +9 +-9.8 +-8 +-9.8 +-8 +-9.8 +-8 +-10 +-10 +6- +SRC ML-BA +ADORA1 ML-BA +HDAC2 ML-BA +DOR ML-BA +CCR5 ML-BA +DOR +CCR5 +HGF +MOR +DOR +-8 + ML-BA +-8 +1 ML-BA +2 ML-BA +MC4R ML-BA +MDM2 ML-BA +6- +-8 +-9 +PDGFRB I +MDM2 I +CRHR1 +-10 +-9.8 +-9.8 +-9 +-10 +-9.8 +-8 +-9.8 +-8 +-10 +.9 +-10 +-9 +10 +MMP1 ML-BA +PPARG ML-BA +MDM2 ML-BA +CNR1 ML-BA +MC4R ML-BA +KOR +REN +SRC +HDAC1 +JAK3 +-8 +-8- +-8. +CRHR1 ML-BA +ML-BA +3 ML-BA +-8 +EGFR ML-BA +-9 +ML-I +MAPK1 I +ERBB2 +JAK3 +-9.8 +-9.8 +9.8 +-10 +-11 +出 +-10 +-8 +-10 +-8 +-10 +-8 +-8 +-10 +-9.5 +-9 +JAK2 ML-BA +CNR1 ML-BA +JAK3 ML-BA +ESR1 ML-BA +CRHR1 ML-BA +-14 +-12 +-10 +8- +-6 +Experimental BA (kcal/mol)Figure 4: Three examples of predicted BA values being correlated. In each example, the chart shows the predicted BA values +on two other proteins. The 3D structure alignments are shown on the right of the chart, and the 2D sequence alignment is +exhibited on the second row. The 3D structures in the alignment are (PDB 4BKX and 4LY1 for HDAC1 and HDAC2), (PDB +5C1M, 4DJH, 4N6H for MOR, KOR, and DOR), and (PDB 6BBU, 2B7A, and 1YVJ for JAK1, JAK2, and JAK3). +Property +Optimal range +FDAMDD +Excellent: 0-0.3; medium: 0.3-0.7; poor: 0.7-1.0 +F20% +Excellent: 0-0.3; medium: 0.3-0.7; poor: 0.7-1.0 +Log P +The proper range: 0-3 log mol/L +Log S +The proper range: -4-0.5 log mol/L +T1/2 +Excellent: 0-0.3; medium: 0.3-0.7; poor: 0.7-1.0 +Caco-2 +The proper range: >-5.15 +SAS +The proper range: <6 +Table 1: The optimal ranges of six selected ADMET properties and synthesizability (SAS) used to screen nearly optimal +compounds. +no hERG side effect if the predicted BA value on hERG is greater than −8.18 kcal/mol. Figures S5 and +S6 provide the predicted BAs of the other 73 inhibitor datasets on MOR and hERG. The orange frames +highlight the zones where compounds can have repurposing potentials for MOR but do not cause hERG side +effects. Some of the 73 inhibitor datasets have almost no compounds in the orange frames such as CXCR2, +ERBB4, MAP3K5, PI4KB, and SMO, while other datasets still have a significant number of compounds +lying in these orange frames, such as HDAC1, MMP1, MMP2, DOR, KOR, and NOR. +2.3 +Druggable property screening +ADMET (absorption, distribution, metabolism, excretion, and toxicity) plays a key role in drug discovery +and development. It includes vast attributes associated with the pharmacokinetic studies of a compound. +A promising drug candidate should not only have sufficient efficacy on the therapeutic target but also +satisfies appropriate ADMET properties. Accurate predictions of ADMET are significant in drug design. +The successful ADMET screening at the design stage of new compounds is beneficial in reducing the risk of +late-stage attrition. To search for promising compounds in treating OUD, systematic screenings of ADMET +properties, synthetic accessibility (SAS), and the hERG risk of all inhibitor datasets are needed. We paid +attention to six indexes of ADMET, i.e., FDAMDD, T1/2 and F20%, log P, log S, and Caco-2, and SAS +as well as hERG risk assessment. +To evaluate the ADMET properties, we utilized the ADMETlab 2.0 +(https://admetmesh.scbdd.com/) solvers that provide machine-learning predictions [24,25]. Their documents +provide a set of optimal ranges of these ADMET properties. The SAS evaluation was obtained from Rdkit +packages [26]. The optimal ranges of ADMET properties and SAS are provided in Table 1, while the BA value +> -8.18 kcal/mol is applied as the required range for exempting hERG side effects. With the evaluations of +8 + +a. +b. +C. +JAK3 +JAK3 +DOR +6 +2 ML-BA +-8 +JAK2 ML-BA +ML-BA +-8 +8- +-10 +-9 +-10 +HDAC2 +KORI +-12 +-12 +-10 +12-10 -8 +-6 +-12-10-8 +-10 +6- +-8 +JAK1 ML-BA +HDAC1 ML-BA +MOR ML-BA +KOR +JAK1 +FGDVLCK +IVIsI +DYYNMFTSIF +FGDVLCK +(IVIS +DYYNMFTSIF +HDAC1 +D +IHHGDGVEEAFYTTDRVMTVS +JAK2 +FGELLCKAVLSIC +DYYNMFTSIF +DOR +DIHHGDGVEEAFYTTDRVMTVS +FGELLCKAVLSI +DYYNME +TS +HDAC2 +FGTILCKIVISIDYYNMFTSIF +MOR +FGTILCKIVISI +DYYNMFTSI +JAK3 +FHKYGEYFPGTGDLRDIGAGKG +TLTMMSVDRY +YIAVCHPVKALDF +FHKYGEYFPGTGDLRDIGAGKG +TLTMMSV +ALD +TLTMMSVDRYIAVCHPVKALDF +TLTMMSVDRYIAVCHPVKALDF +TLCTMSVDRYIAVCHPVKALDF +TLCTMSVI +IAVCFigure 5: Screening of example datasets on ADMET properties, synthesizability, and hERG side effects. The colors of scattered +points represent the experimental BA values of inhibitors in each dataset. The Orange frames highlight the optimal ranges of +the properties and side effects. +ADMET properties and SAS as well as cross-target prediction tools, we were able to systematically search for +promising compound leads. Figure 5 shows the ADMET screening of a few inhibitor datasets including MOR, +DOR, KOR, NOR, and MMP7. The four rows represent the eight property screenings of the five inhibitor +datasets. The colors of the scattered points indicate the experimental BA values of inhibitor compounds. +FDAMDD is the FDA maximum recommended daily dose, aimed at avoiding toxicity in the human +body. The half-life is the amount of time for a drug’s active substance to reduce by half in the human +body. The value of T1/2 stands for the probability of half-life less than 3 hours. F20% is the probability of +administered drug reaching systemic circulation with less than 20% of the initial dose. The values of property +log P and log S are the logarithm of the n-octanol/water distribution coefficient and aqueous solubility value, +respectively. Caco-2 is a measure used to estimate in vivo permeability of oral drugs. SAS quantifies the +synthesis difficulty of druglike molecules. As seen in Figure 5, the orange frames in the panels outline the +optimal ranges for a pair of screening properties denoted on the x- and y-axes. Each pair of screening forms +a screening filter for the inhibitors. T1/2 and F20% especially offer a stricter screening as only small portions +of inhibitors are covered in the orange frames. The SAS screening seems to be a loose filter, as a significant +portion of inhibitors remains in the orange frames after screening. Overall, these ADMET indexes and SAS +cause strict restrictions of finding inhibitors. +3 +Discussion +3.1 +Side-effect evaluations of existing medications for OUD treatment +Substantial pharmacological efforts have been dedicated to the treatment of OUD. Opioid replacement +therapy (ORT) is a popular method to treat people with opioid use disorder. It involves replacing an opioid +9 + +MOR +DOR +KOR +NOR +MMP7 +F9- + ML-BA + ML-BA +hERG ML-BA +ML-BA +ML-BA +.7 +-7 +-8 +.9 +-8 +hERG +hERG +hERG +hER +8. +-10 +9 +.9 +-11 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +FDAMDD +FDAMDD +FDAMDD +FDAMDD +FDAMDD +0.8 +0.8 +0.8 +0.8 +0.8 +20% +20% +20% +20% +20% +F +0.4- +0.4 +F +0.4- +F +0.4- +F +0.4 +0.0- +0.0 +0.0- +0.0 +0.0 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +0.0 +0.5 +1.0 +T1/2 +T1/2 +T1/2 +T1/2 +-1ti.. +-1 - +1 +1 +S +S +S +-4 +S +S +-7. +-7. +.7. +-7 +8- +-1 +4 +6 +1 +8 +15 +0 +3 +6 +0 +4 +8 +1 +4 +P +P +log P. +log P +-9 +6 +6- +5 +9 +SAS +SAS +4 +4 +S +3 +2 +2 +2. +2 +2 +.6 +-5 +-6 +-5 +-6 +-6 +-5 +-6 +5 +Caco-2 +Caco-2 +Caco-2 +Caco-2 +Caco-2 +13 +-9 +7 +ExperimentalBA(kcal/mol)with a longer-acting but less euphoric opioid. Three classes of medications acting directly on the opioid +receptors were found to be effective, namely full agonist, partial agonist, and antagonist. +Methadone, buprenorphine, and naltrexone are approved by the U.S. Food and Drug Administration +(FDA) for medication-assisted treatment (MAT). These medications are useful in reducing the risk of death +and preventing relapse. In addition, naloxone is a frequently used medication in reducing the risk of over- +dose, and take-home naloxone is crucial in stopping opioid overdose. It is necessary to evaluate the side +effects of these anti-OUD medications, including their actions on five opioid receptors and their multitude +of physiological functions affecting the human body. We used our machine learning models to predict the +BA values on the proteins in the five opioid networks as well as on the hERG channel. +Among the five important opioid receptors, MOR is typically the target of most clinically prescribed +medications. Methadone is a full MOR opioid agonist, while it has some agonist effect on KOR and possibly +DOR agonist [27]. The methadone maintenance treatment (MMT) is beneficial in reducing the intensity of +withdrawal symptoms including muscle aches and osteodynia in addicted individuals [27]. Methadone has a +long half-life that makes it more useful in reducing withdrawal symptoms in patients [28], and consequently +reducing patients’ compulsive drug-seeking and craving behavior. +Our BA predictions of methadone on +MOR, KOR, and DOR are -11.8 kcal/mol, -8.96 kcal/mol, and -8.52 kcal/mol, respectively, which agrees +with the methadone binding activity on opioid receptors, especially for MOR. It was reported that methadone +prolongs the QT interval in a dose-dependent manner, and high-dose methadone is associated with ventricular +tachycardia torsade de pointes [29]. The overall hERG side effect profile is safe. The predicted BA value +on hERG from our model is -7.73 kcal/mol, which is higher than the hERG side-effect threshold of -8.18 +kcal/mol, and confirms the safety profile of methadone on hERG. Our predictions indicate that the SMO +protein is the only target it can have side effects on. The predicted BA of methadone on SMO protein is +-9.67 kcal/mol, and the predicted BAs on all other targets are greater than -9.54 kcal/mol. SMO is targeted +and inhibited by small-molecule drugs for the treatment of advanced basal cell cancer. No serious side effects +were reported by inhibiting the SMO protein. Its low side effect profile might be one of the reasons that +methadone is the most used medication in MAT and the gold standard against which other medications are +compared [13]. +Buprenorphine is a partial agonist of MOR, the antagonist of KOR, and a weak antagonist of DOR [30]. +Unlike methadone and other full opioid receptor agonists, buprenorphine has a lower risk of respiratory +depression due to a low ceiling to the euphoric effect [31]. In treating opioid dependence, it is typically +administered sublingually as it has an extended half-life than that of intravenous buprenorphine [4], increas- +ing the potential of misuse or overdose. To avoid buprenorphine abuse, it is commonly used with opioid +antagonist naloxone via injection or insufflation without causing impairment when used appropriately [32]. +The predicted BA values of buprenorphine for MOR, KOR, DOR, and NOR are -12.5, -12.88, -11.64, and +-9.41 kcal/mol, which are consistent with the experimental BA values of -12.0, -11.2, -11.7, -9.69 kcal/mol +for MOR, KOR, DOR, and NOR, respectively [33,34]. Buprenorphine was predicted to have no side effects +on hERG with the predicted BA values of -7.31 kcal/mol. It was found to have a minimal impact on the +corrected QT interval [35], which is consistent with our hERG side effect prediction. However, it is predicted +to be a potent inhibitor on quite a few other proteins including REN, JAK1, BDKRB1, and NTRK1 with +BA values of -10.74, -10.67, -10.49, and -10.28 kcal/mol. JAK1 is a member of the Janus kinase family +and Janus kinase inhibitors are used in the treatment of cancer and inflammatory diseases. Clinical trials +indicated inhibitors for NTRK1 protein have shown efficacy as targeted therapies for extracranial tumors. +The inhibition of buprenorphine needs to be further investigated for its side effects on previously discussed +proteins and its usefulness in the treatment of alternative diseases. +LAAM, acting as a MOR agonist, can provide greater suppression of heroin use in comparison to +methadone. +Our predicted BAs of LAAM to MOR, DOR, KOR, and NOR are -9.34, -8.94, -8.83, and +-9.69 kcal/mol, respectively. The potency of LAAM on these receptors is not as strong as methadone and +10 + +buprenorphine. In addition, the safety profile of LAAM is low due to its potential for ventricular rhythm +disorders [36]. Such adverse effects are directly associated with the hERG blockade, which is verified by the +predicted relatively high BA of -7.94 kcal/mol on hERG, a value close to our hERG side effect threshold. +According to our models, the top potential targets with side effects imposed are SMO and JAK1 proteins +with predicted BA values of -9.82 and -9.75 kcal/mol. Similar to the predictions for buprenorphine, the +molecular binding on SMO and JAK1 might not cause a serious problem. +Naltrexone is an antagonist of MOR and a partial agonist of KOR. Long-acting injectable naltrexone +can block opioid receptors but does not activate them, reducing drug-seeking behavior and alleviating drug +craving [37]. Naltrexone is observed to have a continuous effect in reducing the frequency and dosage of +heroin use [38] and in decreasing the risk of opioid overdose. +Naltrexone and nalmefene have a longer +duration period, therefore drawing research and clinical interests to investigate their anti-overdose effect +against potent fentanyl analogs [39]. We predicted BA values for MOR, KOR, NOR, and DOR respectively +-12.54, -12.05, -9.88, and -10.49 kcal/mol. The predicted BA value on hERG was low with a value of -7.64 +kcal/mol, suggesting a low hERG side effect potential. Strong binding potency can occur on a few proteins +including JAK1, BDKRBA, and SMO with the predicted BA values of -10.30, -10.02, and -9.92 kcal/mol, +respectively. Analogous to naltrexone, nalmefene is also a MOR antagonist and a KOR partial agonist. +It has a prolonged duration of action and intravenous doses of nalmefene have been shown effective at +counteracting the respiratory depression produced by an opioid overdose [40]. It was predicted to be potent +at the four opioid receptors MOR, KOR, DOR, and NOR with BA values of -12.62, -12.11, -10.78, and +-10.05 kcal/mol. The proteins it can bind to, with a strong binding affinity, include JAK1, SMO, BDKRB1, +REN, and S1PR1 with predicted BA values of -10.25, -10.08, -10.06, -9.99, and -9.99 kcal/mol. Protein +BDKRB1 is a G-protein coupled receptor that mediates responses to pathophysiologic conditions such as +inflammation, trauma, burns, shock, and allergy. Antagonist inhibitors of this receptor were used to reverse +acute or persistent inflammatory pain in these pathophysiologic conditions. Naltrexone or nalmefene might +have some clinical significance as pain relief in these pathophysiologic conditions. Activation of receptor +protein S1PR1 is heavily involved in immune cell regulation and development. It is also responsible for +vascular growth and development, during embryogenesis. Inhibitions of protein S1PR1 by naltrexone may +interfere with normal growth and development. +Naloxone is a non-selective and competitive opioid antagonist that reverses opioid analgesic actions quite +effectively. Naloxone is commonly used for acute opioid intoxication, rescuing opioid-induced respiratory +depression [40]. However, it is of lower potency and shorter duration period, compared to other antagonists +such as naltrexone and nalmefene. +In addition, the capabilities of naloxone are limited when ingesting +highly potent opioids, such as fentanyl [15]. +The predicted BA value on MOR, KOR, and DOR are - +11.50, -10.93, and -9.79 kcal/mol, close to those experimental BA values of -12.21, -10,63, -10.75 kcal/mol, +respectively. The BA prediction for hERG is -7.58 kcal/mol, suggesting a safe hERG-blockade profile. The +potential side effect on other proteins included JAK1, JAK3, REN, BDKRB1, PDGFRB, and ERBB4 with +predicted BA values of -10.34, -9.92, -9.89, -9.85, -9.80, and -9.76 kcal/mol. Like naltrexone and nalmefene, +naloxone could also block protein BDKRB1, and may also be clinically useful in the pain relief of several +pathophysiologic conditions. Protein PDGFRB is essential for vascular development, and its inhibition may +compromise the integrity and/or functionality of the vasculature in multiple organs, including the brain, +heart, kidney, skin, and eyes [41]. HER4 is a receptor tyrosine kinase that is critical for normal body systems +such as the cardiovascular, nervous, and endocrine systems. Overexpression of HER4 kinase results in cancer +development [42], and activation of HER4 by ligand binding can potentially cause cancer, promoting drug +development to inhibit these HER4 [42]. The duality of naloxone’s ability to activate HER4, potentially +causing cancer or providing an effective HER4 inhibitor, needs to be investigated further. +Heroin (Diamorphine), a MOR agonist, was found to be useful in helping patients disengage from the use +of street heroin and reducing criminal involvement. It can be an effective adjunctive treatment for chronic, +relapsing opioid dependence [43]. Heroin assisted treatment is now available in Canada and some European +11 + +countries as a new treatment modality. It is administrated under direct medical or nurse supervision. Heroin +assisted treatment is intended for injection into patients suffering from OUD, who have not responded to +standard medications for OUD. However, compared to other medications for opioid use disorder, its safety +profile is low with major adverse effects, such as respiratory depression and seizures [43]. The predicted BA +of heroin on MOR by our model is -9.46 kcal/mol. In addition, it was predicted to be a potent inhibitor +at NOR, JAK1, JAK3, BDKRB1, SMO, REN, and ITGB1 with BA values of -10.67, -10.32, -10.15, -10.10, +-9.87, -9.77, and -9.72 kcal/mol. +The BA value on hERG was predicted to be -7.59 kcal/mol. +Protein +ITGB1 associates with integrin alpha 1 and integrin alpha 2 to form integrin complexes which function as +collagen receptors. Recent studies have shown that the inhibition of ITGB1 enhances the anti-tumor effect +of cetuximab in colorectal cancer cell [44]. Nonetheless, due to the severe addiction effect, heroin may not +be a good choice for cancer treatment. +Hydromorphone is also a MOR agonist. Injectable hydromorphone was also found to be as effective +as diacetylmorphine for patients who have not benefited from previous treatments, such as methadone or +suboxone [45]. Following studies showed that once-daily sustained-release oral hydromorphone was useful +in managing cravings without notable side effects. It has the advantage of no influence on the cardiac QTc +interval [46]. The treatment with hydromorphone requires supervised administration. In our prediction, +hydromorphone had strong binding affinities on all four opioid receptors, namely, MOR, KOR, NOR, and +DOR with the BA values of -12.9, -11.58, -10.35, and -10.08 kcal/mol, respectively. The predicted BA value +on hERG is -7.71 kcal/mol, which can be deemed as having a low potential of side effect on hERG. Other +targets that hydromorphone can possibly cause side effects on are JAK1, AR, PDGFRB proteins with the +predicted BA values of -10.20, -9.81, and -9.78 kcal/mol. +Dihydrocodeine is a semi-synthetic opioid analgesic and agonist. It is sometimes used for maintenance +treatment as an alternative to methadone or buprenorphine in some European countries. Our predicted BA +values of dihydrocodeine on MOR, DOR, KOR, and NOR are -9.41, -7.44, -7.91, and -11.06 kcal/mol. It +may be used as a second line treatment. Low quality evidence reported that dihydrocodeine may be no more +effective than other routinely used medication interventions in reducing illicit opiate use [47]. It can be a +potent inhibitor for several proteins such as JAK1, SMO, JAK3, BDKRB1, ITGB1, and AR, with predicted +BA values of -10.07, -9.89, -9.83, -9.76, -9.73, and -9.68 kcal/mol. The predicted BA value on hERG was +-8.08 kcal/mol, which shows a mild potential of side effects on hERG and proteins ITGB1 and AR. Androgen +receptor (AR) functions mainly as a DNA-binding transcription factor that regulates gene expression. High +expression in androgen receptor has been linked to aggression and sex drive [48]. AR also has roles in the +progression of prostate cancer and is an important therapeutic target in prostate cancer [49]. The inhibition +of AR by Dihydrocodeine may have an impact on male sexual phenotype. +Lofexidine is an α2-adrenergic receptor agonist but is not classified as an opioid. It is an alternative +for people with mild or uncertain opioid dependence in need of short-term detoxification and is effective in +reducing withdrawal symptoms of OUD. Its adverse side effects include QT prolongation. Our predicted +BA values for MOR, KOR, DOR, and NOR are -8.33, -8.53, -8.02, and -9.7 kcal/mol, which are consistent +with the fact that Lofexidine is not an opioid. Our BA prediction of Lofexidine on hERG is -7.30 kcal/mol, +showing a good side profile. The strongest predicted BAs were for FDE4A, SMO, and IITGB1 with the +values of -9.79, -9.76, and -9.56 kcal/mol. The side effects of these three proteins might be mild. +Currently, there are some drugs used in the treatment of opioid-induced constipation such as alvimopan, +methylnaltrexone, and naloxegol. The three drugs are all peripherally acting µ-opioid receptor antagonists +due to their limited ability to cross the blood–brain barrier and reach the MORs of the central nervous +system. Consequently, the effects of centrally-acting opioid antagonists are normally not notable. Alvimopan +is approved for the treatment of postoperative due to its intended blockade of MORs in the gastrointestinal +tract. We predicted BA values for the MOR, KOR, NOR, and DOR -12.40, -9.50, -10.18, and -11.02 kcal/mol, +respectively. Its potency on MOR is validated in our prediction. It can also be potent at several proteins +12 + +including JAK1, BDKRB1, S1PR1, ACE, and SMO with predicted BA values of -10.54, -10.36, -10.28, +-10.07, and -10.07 kcal/mol. The most common side effects of alvimopan include dyspepsia, hypokalemia, +back pain, and delayed micturition. Methylnaltrexone is used to treat opioid-induced constipation in chronic +non-cancer pain or when ordinary laxatives do not work well. It was also predicted to be a potent inhibitor +of MOR, KOR, and NOR with BA values of -11.04, -10.62, and -10.25 kcal/mol. Potential side effects can +be exhibited on proteins including JAK1, BDKRB1, and REN with BA values of -10.57, -10.05, and -10.04 +kcal/mol, respectively. Naloxegol is recommended for the treatment of opioid-induced constipation (OIC) in +patients with chronic non-cancer pain. Its predicted BA values for NOR, KOR, MOR, and DOR are -10.52, +-10.15, -9.96, and -9.26 kcal/mol, respectively. Based on our predictions, it can be potent for REN, JAK1, +BDKRB1, and ERBB4 with predicted BA values of -10.86, -10.59, -10.50, and -10.03 kcal/mol, respectively. +3.2 +Nearly optimal lead compounds from screening and repurposing +As previously discussed, opioid replacement therapy (ORT) replaces an opioid with a longer-acting but +less euphoric opioid, and is widely used in OUD treatment. +Drugs like methadone and buprenorphine +are commonly used for ORT, and act as agonist/antagonist depending on the different opioid receptors. +We dedicate our efforts to search for more potential inhibitors of the four opioid receptors, which can be +potential agonist/antagonist in OUD treatments. Screening and repurposing are the two avenues to find +more inhibitors. There were 74 models utilized to predict the cross-target bind affinity in the process of +screening and repurposing. In addition to the potency concern, optimal ranges for the ADMET properties, +synthetic accessibility in Table 1, and hERG side effect all needed to be satisfied. As we know, MOR, DOR, +KOR, and NOR are the four major opioid receptors, and critical pharmacological targets in OUD treatments. +In finding more promising potent compounds at these four receptors, the 74 inhibitor datasets are our source +of inhibitor compounds. In the screening process, we start with potent inhibitor compounds (experimental +BA values < -9.54 kcal/mol) in the inhibitor datasets of the four opioid receptors, and then evaluate a series +of other properties. It is important to note that if a designated inhibitor of a receptor exhibits notable +potency on any of the other three receptors, it is not considered as a side effect. It is very common that +one inhibitor can be effective on several of the four major opioid receptors simultaneously, as seen by the +currently approved drugs that act as agonists or antagonists on several receptors. However, the potential +for side effect concern must be evaluated on the other 70 protein targets including hERG. We require the +predicted BA values > -9.54 kcal/mol to exempt side effects except for hERG with a stricter BA > -8.18 +kcal/mol. In the repurposing process, we evaluate the binding potency of all weak inhibitors in the other 70 +datasets on the four opioid receptors. We start with inhibitors of experimental BA value > -9.54 kcal/mol, +and then find those with predicted BA values < -9.54 kcal/mol on the four opioid receptors. In search of +inhibitors with repurposing potential on the opioid receptors, their side effects need to be exempted on the +other 70 proteins. Following these, optimal range of ADMET properties and synthetic accessibility need to +be screened. +Two inhibitor compounds, CHEMBL466223 from the CNR1 dataset and CHEMBL355008 from the +CRHR1 dataset, were found to satisfy all the aforementioned criteria for repurposing. They were predicted to +be effective on NOR with predicted BA values of -9.543 kcal/mol and -9.88 kcal/mol while their experimental +BA values for the designated targets are -8.18 kcal/mol and -7.37 kcal/mol. They were not predicted to be +potent at the other three opioid receptors, MOR, DOR, and KOR, as seen by the following predicted BA +values of -8.26, -7.87, and -8.51 kcal/mol, respectively. Its predicted BA value on hERG was -7.34 kcal/mol. +The predicted BA values of compound CHEMBL355008 on MOR, DOR, and KOR were -8.46, -8.33, and +-8.25 kcal/mol, respectively, and -8.01 kcal/mol on hERG. These two compounds were predicted to have +no binding effects or side effects on all other 69 proteins. We carried out evaluations of more ADMET +properties regarding these two molecular compounds using the ADMETlab 2.0 predictive solver. As seen +in Figures 6a and 6b, the two compounds are still in the optimal ranges of these ADMET properties. The +meaning and optimal ranges of the 13 ADMET properties are provided in the Supporting information. We +13 + +are also interested in the molecular interaction between the two inhibitors and NOR protein structure and +utilized the software AutoDock Vina to perform protein-ligand docking to analyze the interactions. The 3D +docking structure and 2D interaction diagrams are shown in Figure 7. It can be seen in the figure below, +hydrogen bonds are formed between the inhibitors and the NOR protein. Compound CHEMBL355008 has +two hydrogen bonds formed with the Tyr131 (3.32 ˚A) and Tyr309 (3.17 ˚A), respectively. The compound +CHEMBL466223 has one strong hydrogen bond (2.82 ˚A) with Tyr309. The predicted binding energies by +CHEMBL355008 and CHEMBL466223 with NOR were -9.54 and -9.88 cal/mol, respectively. The single +strong hydrogen bond of CHEMBL466223 partially explains its predicted higher binding energy with NOR +even though CHEMBL466223 has two hydrogen bonds with NOR. It is observed that the side chains of +Tyr309 play roles in the formation of hydrogen bonds with the two compounds. No covalent bond is formed +by either of the two compounds with the side chains of the NOR protein, and hydrogen bonds play the +essential roles of binding energy. +Even though these two compounds were not predicted to be potent inhibitors on MOR, similar to the +approved drugs for OUD treatment, they are selective inhibitors for NOR. Selective antagonists are needed +in scientific research when one of the receptors needs to be blocked without affecting the others. These +two were predicted to be selective inhibitors of NOR with safe side effect profiles on all other 70 proteins. +Further studies can be carried out to test their physiological effect on NOR or their pharmacological effect +in treating OUD. +Figure 6: Evaluations of more ADMET properties for the found molecular compounds with repurposing potentials. +Panel +a and c represent the prediction results of ADMET properties and side effect evaluations for compound ChEMBL466223, +and panels b and d stand for these predictions for compound ChEMBL355008. The boundaries of yellow and red zones in +figure a and b highlight the upper and lower limits of the optimal ranges for the ADMET properties, respectively. The blue +curves represent values of the specified 13 ADMET properties. Figures a and b are the prediction results from ADMETlab +2.0 (https://admetmesh.scbdd.com/) website. Abbreviations: MW (Molecular Weight), logP (log of octanol/water partition +coefficient), logS (log of the aqueous solubility), logD (logP at physiological pH 7.4), nHA (Number of hydrogen bond acceptors), +nHD (Number of hydrogen bond donors), TPSA (Topological polar surface area), nRot (Number of rotatable bonds), nRing +(Number of rings), MaxRing (Number of atoms in the biggest ring), nHet (Number of heteroatoms), fChar (Formal charge), +and nRig (Number of rigid bonds). +14 + +b. +a. +Upper Limit + Lower Limit +Compound Properties +Upper Limit + Lower Limit +Compound Properties +MW +MW +LogP +LogP +nRig +nRig +fChar +fChar +LogS +LogS +nHet +LogD +LogD +nHet +MaxRing +MaxRing +nHA +nHA +nHD +nRing +nHD +nRing +TPSA +nRot +TPSA +nRot +C. +d. +H3 +CHEMBL466223 +CHEMBL355008 +CNR1-ex-BA: -8.18 kcal/mol +CRHR1-ex-BA: -7.37 kcal/mol +Predicted BA: +Predicted BA: +MOR: -8.26 kcal/mol +MOR: -8.46 kcal/mol +KOR: -8.51 kcal/mol +KOR: -8.25 kcal/mol +DOR: -7.87 kcal/mol +DOR: -8.33 kcal/mol +NOR: -9.54 kcal/mol +NOR: -9.88 kcal/mol +hERG: -7.35 kcal/mol +hERG: -8.01 kcal/molFigure 7: The docking structure of our nearly optimal lead compounds bound to nociceptin opioid receptor and their 2D +interaction diagrams. AutoDock Vina was used to performing the protein-ligand docking. Hydrogen bonds are playing essential +roles in binding energies. +4 +Methods +4.1 +Datasets +The inhibitor datasets were collected from the ChEMBL database for the proteins in the five investigated +opioid receptor networks. As machine-learning models rely on a training set of sufficient data points, we +require the size of the collected inhibitor dataset to be at least 250. A total of 74 datasets were then obtained. +The labels for the data points are IC50 or Ki. As suggested in [50], the IC50 values can be approximately +converted to Ki values with the relation Ki=IC50/2. These labels were used to compute the binding affinity +(BA) with the formula BA=1.3633× log10 Ki (kcal/mol), which were then used in building machine learning +models. Since the hERG is a critical target of side effects in drug design, an inhibitor dataset was also +collected for it from the ChEMBL database. The details regarding all these datasets are provided in the +Supporting information. +4.2 +Molecular embeddings +The molecular representation for the inhibitors in the collected 75 datasets is 2D SMILES strings. Two +forms of molecular fingerprints were used to build machine-learning models in this study. The molecular +fingerprints were generated by pre-trained models based on natural language processing (NLP) algorithms +including transformer [51] and sequence-to-sequence autoencoder [52]. The two pre-trained models encode +the 2D SMILES strings of inhibitor compounds in latent embedding vectors of lengths 512. We denote the +two types of fingerprints by the transformer and autoencoder models as TF-FP and AE-FP, respectively. +4.2.1 +Sequence-to-sequence auto-encoder +A data-driven unsupervised learning model was recently proposed to extract molecular information embed- +ded in the SMILES representation [52]. A sequence-to-sequence autoencoder was utilized to translate one +form of molecular representation to another, with a comprehensive description of the chemical structure +compressed in the latent representation between the encoder and decoder. The translation model extracts +the physicochemical information in the molecular representation when translated to another semantically +equivalent but syntactically different representation of the molecule. The translation model was trained on +a large set of chemical structures and allows for the molecular descriptor extraction for query compounds +15 + +Trp116(A) +Met134(A +a. +b. +TTT +Gln107(A +al283(A) +TT +Gln280(A +p110(A) +T +13 +Val126(AE +p276(A) +CD1 +Eu301(A) +01 +p130(A) +L +OH +Tyr309(A) +C19 +2.82 +CE2 +CD2 +3.32 +012 +Ouz +CG +C6C5 +/n286(A) +CEb1 +CD2 +CB +CA +E +Asp130(AE +Val279(A) +Gln107(A) +Met134(A) +C1 +TT +EL +E +Val283(AL +Tvr131(A) +Tp276(A) +Docking structure: +Docking structure: +NOR-ChEMBL466223 +NOR-ChEMBL355008 +EFL +Ie111(A) +PDB ID: 4EA3 +PDB ID: 4EA3 +Asp110(A) +Ile219(A)without retraining or using labels. +The translation model consists of the encoder and decoder networks. The information bottleneck in +between is used to compress the essential information of the input SMILES, and the embedded information +is then used as input in the translation through the decoder. In the encoder network, convolutional neural +network (CNN) and recurrent neural network (RNN) architectures were adopted. +Then fully connected +layers map the output of CNN or the concatenated cell states of the RNN networks to intermediate vector +representations between encoders and decoders. The decoder is comprised of RNN networks with latent vec- +tors as input. To embed more meaningful physicochemical information about molecules in the latent vectors, +a classification model was used to extend the translation model by predicting certain molecular properties +based on the latent vectors. +The output of the decoder’s RNN network is the probability distributions +over different characters in the translated molecular representations. In training the autoencoder model, +the loss function is the sum of cross-entropies between probability distributions and one-hot encoded cor- +rect characters as well as the mean squared errors for molecular property predictions from the classification +model. +The translation model was trained with approximately 72 million molecular compounds from ZINC +and PubChem databases. The preprocessing was carried out to filter compounds with a variety of criteria +including molecular weight, heavy atom numbers, partition coefficient, and other properties. After sufficient +training with the processed dataset, the resulting translation model yields the embedding vectors as molecular +fingerprints. +4.2.2 +Bidirectional transformer +A self-supervised learning (SSL)-based platform was recently developed to pre-train a deep learning network +from millions of unlabeled molecules. Predictive molecular fingerprints can be extracted from the pre-trained +models [51]. The self-supervised learning was achieved with the bidirectional encoder transformer (BET) +model that relies on the attention mechanism. The SSL has the advantage of avoiding the construction +of a complete encoder-decoder framework and solely using the decoder network to encode the SMILES of +molecules. +The SMILES strings of molecules were the input for the SSL pretraining platform. Pairs of real SMILES +and masked SMILES were constructed by hiding a certain percentage of some meaning symbols in the +strings. Then an SSL approach enables the model training with the data-mask pairs in a supervised way. In +the pretraining process, the symbols of masked symbols were referred to by learning the unprocessed ones +in SMILES, which then leads to the understanding of SMILES language. Data masking is preprocessed +before starting to train the model with SSL. A total of 51 symbols were considered as the components in the +SMILES strings. The SMILES were the input for training the model, and we required the maximal length +to be 256. Symbols ′⟨s⟩′ and ′⟨\s⟩′ were added to the beginning and the end of SMILES strings. If the +length is less than 256, the symbol ′⟨pad⟩′ was used to supplement a SMILES string. For the data masking, +a total of 15% of the symbols in all the SMILES were operated, among which 80% were masked, 10% were +unchanged and the remaining 10% were randomly changed. +The BET modules play critical roles in achieving SSL from a massive number of SMILES strings. The +attention mechanism in transformer modules captures the importance of each symbol in the inputted SMILES +sequences. The BET consists of eight bidirectional encoder layers, with each encoder layer composed of a +multi-head self-attention layer and a subsequent fully connected feed-forward neural network. The number +of heads in each self-attention layer is 8, and the embedding size of fully connected feed-forward layers is +1024. The Adam optimizer was used in the training process and weight decay of 0.1 was applied. The loss +function is defined to be the cross-entropy, measuring the difference between the real and predicted symbols +at masked positions. The maximum length of input SMILES is 256 including the added special symbols at +the two ends, while the embedding dimension of each symbol was 512. The resulting molecular embedding +16 + +matrix is comprised of 256 embedding vectors of dimension 512. The mean of embedding vectors for the +valid symbols in one SMILES string was used as molecular fingerprint of a given SMILES. +Due to the high parallelism capability and training efficiency from transformer modules, a massive number +of SMILES can be used to train deep learning models. In our implementations, SMILES strings from one or +the union of the ChEMBL, PubChem, and ZINC databases were employed, giving rise to three pre-trained +models [51]. In this study, transformer-based embeddings generated from the pre-trained model solely using +the ChEMBL database were used as molecular fingerprints. +4.3 +Machine-learning models +The gradient boosting decision tree (GBDT) algorithm was deployed to build our machine learning models. +The GBDT algorithm is a popular ensemble method and has the advantage of robustness against overfitting, +insensitiveness to hyperparameters, and ease of implementation. The methodology is to create many weak +learners (individual trees) by bootstrapping training samples and to make predictions by integrating the +outputs of weak learners. Weak learners are likely to make poor predictions, but through ensemble approach +the overall errors by combining all the weaker learners are reduced. GBDT is particularly useful when training +with small datasets and can deliver better prediction performance than deep neural network (DNN) and some +other machine learning algorithms. It gains wide popularity in a range of quantitative structure–activity +relationship (QSAR) prediction problems [53,54], and promotes the development of competitive predictive +ML models. The GBDT algorithm provided in the Scikit-learn (version 0.24.1) library was used in this work. +We collected a total of 75 inhibitor datasets with at least 250 data points in each dataset. It is preferable +to utilize GBDT in building models for these datasets. As aforementioned, two types of molecular fingerprints +including TF-FP and AE-FP were adopted to represent inhibitor compounds. Our machine-learning (ML) +models were built by integrating these molecular fingerprints with the GBDT algorithm. We built a total +of 75 ligand-based ML models with the 75 inhibitor datasets. For each dataset, two individual models were +built by pairing TF-FP and AE-FP with GBDT algorithm, and then the average of the predictions from +the two individual models was regarded as our final binding affinity prediction. Such average or consensus +results typically outperform those from individual models. To alleviate the effect of randomness, each of +the individual GBDT models were trained ten times with a different random seed. The average of the ten +predictions was regarded as the final outcome of each individual model. In the Supporting information, we +included the Pearson correlation coefficients of five-fold cross validations for modeling the 75 datasets. +5 +Conclusion +Opioid use disorder (OUD) is a chronic and complex disease with neurobiological, psychological, behavioral, +and medical implications. Each year in the United States and around the world, thousands of deaths are +caused by opioid abuse, and billions of dollars have been spent on OUD treatment. To combat the opioid +epidemic, efforts in novel treatment formulations and devices have been dedicated by pharmaceutical agencies +and scientists. Pharmacological or psychosocial interventions showed their efficacy for OUD treatment, but +many patients still drop out of treatment and return to opioid-dependent life because of the chronic and +relapsing nature of opioid addiction. The development of nonaddictive analgesics and anti-opioid vaccines +can be potentially effective in opioid abuse prevention and the OUD treatment, but the progresses seem very +slow. More options for treatment are needed to combat such destructive diseases. +Opioid receptors are the direct targets of opioids, and medications on them are found effective in opioid +addictions. OUD affects intricate molecular and biological activities in the brain involving significant protein- +protein interactions (PPI) in various brain areas. The development of anti-OUD medications cannot neglect +the impact of opioids or medications on the PPI networks of opioid receptors. In this work, we developed +proteome-informed machine learning protocol to study OUD and discover more drug candidates to treat +17 + +it. With molecular fingerprints generated by advanced NLP models based on transformer and autoencoder +algorithms, gradient boosting decision tree (GBDT) algorithm was used to build our predictive models. The +consensus predictions from two forms of molecular fingerprints could enhance the predictive performance. +We used these models to reevaluate the side effects of currently available medications for treating OUD. In +addition, these models were used to study the repurposing potentials of existing inhibitors on the major +opioid receptors and screened the possible side effects of these inhibitors. +The evaluations of ADMET +properties were then carried out with machine-learning predictions. +We identified a group of promising +compounds targeting the opioid receptors. Considering the therapeutic efficacy by antagonist or agonist +effect of currently approved drugs, further animal experiments with these compounds are needed to test the +antagonist/agonist properties. More tests in vitro or animal arrays are needed to scrutinize the toxicity and +blood-brain barrier permeability characteristics of these candidate compounds. Automated generation of +more drug candidates can be carried out using our generative network modules [18], and this study can be +employed for the screening of potential side effects. +Our machine-learning-based platform provides a novel approach for searching compound candidates to +treat OUD and can be generalized to the studies of other diseases with neurological implications. With +more advances in understanding the opioid addiction mechanism and more efforts from pharmacological +treatment, our platform can be assistive in combating the serious public health issues from OUD. +Data and code availability +The related datasets studied in this work are available at: https://weilab.math.msu.edu/DataLibrary/2D/. +Codes are available at https://github.com/WeilabMSU/OUD-PPI. +Acknowledgment +This work was supported in part by NIH grant GM126189, NSF Grants DMS-2052983, DMS-1761320, and +IIS-1900473, NASA 80NSSC21M0023, MSU Foundation, Michigan Economic Development Corporation, +George Mason University award PD45722, Bristol-Myers Squibb 65109, and Pfizer. +References +[1] Jennifer C Veilleux, Peter J Colvin, Jennifer Anderson, Catherine York, and Adrienne J Heinz. +A +review of opioid dependence treatment: pharmacological and psychosocial interventions to treat opioid +addiction. 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Journal of chemical information and +modeling, 60(3):1235–1244, 2020. +22 + diff --git a/stE3T4oBgHgl3EQf9Quo/content/tmp_files/load_file.txt b/stE3T4oBgHgl3EQf9Quo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9346c9392ba954f21572a5b5379b30d0055b472d --- /dev/null +++ b/stE3T4oBgHgl3EQf9Quo/content/tmp_files/load_file.txt @@ -0,0 +1,1557 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf,len=1556 +page_content='Machine-learning Analysis of Opioid Use Disorder Informed by MOR, DOR, KOR, NOR and ZOR-Based Interactome Networks Hongsong Feng1, Rana Elladki1, Jian Jiang4, and Guo-Wei Wei1,2,3∗ 1 Department of Mathematics, Michigan State University, MI 48824, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Michigan State University, MI 48824, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 2Department of Electrical and Computer Engineering, Michigan State University, MI 48824, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 3 Department of Biochemistry and Molecular Biology, Michigan State University, MI 48824, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 4 Research Center of Nonlinear Science, School of Mathematical and Physical Sciences, Wuhan Textile University, Wuhan, 430200, P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' China January 13, 2023 Opioid use disorder (OUD) continuously poses major public health challenges and social implications worldwide with dramatic rise of opioid dependence leading to potential abuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Despite that a few phar- macological agents have been approved for OUD treatment, the efficacy of said agents for OUD requires further improvement in order to provide safer and more effective pharmacological and psychosocial treat- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Preferable therapeutic treatments of OUD rely on the advances in understanding the neurobiological mechanism of opioid dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Proteins including mu, delta, kappa, nociceptin, and zeta opioid recep- tors are the direct targets of opioids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Each receptor has a large protein-protein interaction (PPI) network, that behaves differently when subjected to various treatments, thus increasing the complexity in the drug development process for an effective opioid addiction treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The report below analyzes the work by presenting a PPI-network informed machine-learning study of OUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We have examined more than 500 pro- teins in the five opioid receptor networks and subsequently collected 74 inhibitor datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Machine learning models were constructed by pairing gradient boosting decision tree (GBDT) algorithm with two advanced natural language processing (NLP)-based molecular fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' With these models, we systematically car- ried out evaluations of screening and repurposing potential of drug candidates for four opioid receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In addition, absorption, distribution, metabolism, excretion, and toxicity (ADMET) properties were also considered in the screening of potential drug candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Our study can be a valuable and promising tool of pharmacological development for OUD treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Key words: Opioid use disorder, opioid receptor, machine-learning, cross-prediction, side effect, repur- posing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Email: weig@msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='edu i arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='04815v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='BM] 12 Jan 2023 Contents 1 Introduction 1 2 Results 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='1 The Opioid receptors and addiction PPI networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2 Binding affinity predictions .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='1 Cross-target binding affinity predictions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2 Predictions of side effects and repurposing potentials .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='3 Protein similarity inferred by cross-target BA correlations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='4 Repurposing to opioid receptors and side effect on hERG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='3 Druggable property screening .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 8 3 Discussion 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='1 Side-effect evaluations of existing medications for OUD treatment .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 17 5 Conclusion 17 ii 1 Introduction Over three million people in the United States are currently suffering or have previously suffered from opioid use disorder (OUD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In 2020 alone, over 68,000 deaths were recorded from an overdose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Unfortunately, the numbers are continuously rising, and have more than tripled throughout the past 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The opioid crisis or epidemic presents a substantial public health concern and costs the United States billions of dollars annually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It takes many components, including increased public awareness, improved economic conditions, and better therapies, to fully address the OUD crisis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Since the treatment of OUD with medications is effective in reducing symptoms of drug withdrawal and cravings [1], there is a pressing need to further search and develop more effective OUD treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Opioid is a broad term for any natural or synthetic substance that binds to specific opioid receptors in the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It is widely used as analgesics medications in modern pain management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The three main brain receptors that opioids bind to are the mu opioid receptor (MOR), kappa opioid receptor (KOR), and delta- opioid receptor (DOR) in the central nervous system (CNS) and peripheral organs [2], which are responsible for a plethora of physiological functions, such as analgesia, respiration, and hormonal regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Synthetic and exogenous opioids (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=', morphine, heroin, oxycontin, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=') act on the opioid receptors as endorphins, and repeated exposure to escalating use of opioids causes gradual adaptations in the brain [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Tolerance develops and leads to heightened uncontrolled intake of drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Consequently, physical dependence emerges with drug craving to reduce withdrawal symptoms upon abstinence [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' When opioids are improperly ingested, the interaction with the opioid receptors can induce a harmful effect on the CNS impacting the respiratory system and causing irreversible brain damage [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' According to the Centers for Disease Control and Prevention (CDC), methadone, oxycodone, and hydrocodone contribute to the highest number of fatalities, resulting from opioid overdose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' MOR is critical in brain reward circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It has an important role in goal-directed behavior such as drug- seeking behavior and represents a major factor in the initiation of addictive behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In the development of opioid addiction, poor decision-making, and cognition impairment, MOR translates the goal-directed behaviors to habitual behaviors, promoting compulsive drug use [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Through experiments ran on animals, with similar physiological functions, the results demonstrated that MOR is pivotal in mediating therapeutic and adverse activities of opioids and is associated with the maintenance of drug use, drug craving, and relapse [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' KOR has anti-reward effects and can induce dysphoria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Upon long-term exposure to opioids, KOR has impact on modifying the brain’s reward circuits, leading to a relapse [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The activation of KORs suppresses unpleasant MOR/DOR-mediated side effects including the rewarding effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' KOR blockade may beneficially alleviate stress responses, reduce drug cravings, and remediate depressive states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' DORs reduce levels of anxiety and attenuate depressive symptoms [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In addition, beneficial effects of DOR agonists were found in treating chronic pain and psychiatric disorders [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In growing efforts to combat the opioid epidemic, further research and development have been invested in the treatments of OUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Currently, there are three medications used to treat opioid dependency approved by the Food and Drug Administration (FDA), methadone, buprenorphine, and naltrexone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Methadone is a full agonist on MOR and is used to reduce withdrawal and craving symptoms in patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Methadone maintenance treatment (MMT) is useful in reducing the intensity of withdrawal symptoms and preventing patients from ingesting more opioids to induce a euphoric effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Since MMT can decrease the intensity of cravings in patients, they are more willing to remain in treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' However, methadone is associated with the risk of causing respiratory depression when improperly administrated [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Buprenorphine, a partial MOR agonist, is an alternative to methadone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It has a ceiling effect of stimulation on MOR than that of methadone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Hence, buprenorphine provides a less euphoric effect and is less likely to cause respiratory depression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' MMT is more likely to keep patients in treatment than buprenorphine treatment [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Naltrexone is an antagonist of MOR and is effective in attenuating drug cravings and reducing the risk of overdose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It does not produce sedation, analgesia, euphoria, or potential for abuse or diversion [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' However, it is not 1 as widely used as methadone or buprenorphine for several reasons including low rates of patient acceptance and non-adherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Naloxone is a non-selective and competitive opioid receptor antagonist and is used in treating opioid overdose or for opioid intoxication such as reversing respiratory depression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Take-home naloxone programs were developed to prevent fatal overdoses [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Buprenorphine/naloxone formulations are adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' When injected, naloxone has higher bioavailability, thereby blocking the pain and craving- reducing effects of buprenorphine [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' However, Naloxone’s capabilities are limited when ingesting highly potent opioids, such as fentanyl [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Psychosocial interventions were also combined with these medications to improve the efficacy in treating opioid addictions [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Further studies on medication efficacy and the mechanism of opioid addiction in the brain are needed to find better treatments to prevent relapse and facilitate longer periods of abstinence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The molecular mechanism underlying opioid tolerance, dependence, withdrawal, and addiction is com- plicated, involving several systems located in different regions of the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Opioids target opioid receptors in the brain and activate the mesolimbic (midbrain) reward system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Dopamine is produced in the ventral tegmental area (VTA), and is then released into the nucleus accumbens (NAc) area, giving rise to the feeling of pleasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Opoid tolerance occurs because of the brain’s adaptation to repeated exposure to opioids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' [1] The withdrawal symptoms and opioid dependence are related to noradrenaline (NA) that is produced in the locus ceruleus (LC) area [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Opioids impact brain areas with a fairly large number of proteins and peptides that are responsible for a multitude of physiological and biological functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It is challenging to understand how so many proteins are simultaneously impacted by opioids, causing difficulty when designing effective medications for OUD treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' On one hand, medication compounds targeting opioid receptors can potentially cause unintentional dependence or an overdose such as methadone, due to the possibility of an agonist effect on MOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' On the other hand, blocking other proteins associated with the opioids can interfere with the biological functions of these proteins and induces various side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It is necessary to investigate the inhibition effects of compounds on the opioid receptors as well as the side effects of potential medications by blocking other proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The protein-protein interaction (PPI) network on the proteome scale forms a basis for systematically studying potential treatment efficacy and side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' A PPI network is constituted of proteins and cor- responding direct and indirect interactions that contribute to certain biological activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The String v11 database (https://string-db.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='org/) [17] provides a large collection of protein-protein interactions for given proteins or diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In the study of OUD, we can extract the PPI networks related to the major opioid re- ceptors, based on which we can have systematic investigations of medication treatment and side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The proteins in these PPI networks are the test targets of treatment or side effects but using traditional in vivo or in vitro experiments is too time-consuming and expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Besides, large-scale experiments on animals raise legal and ethical concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Machine learning/deep learning technology has recently gained wide popularity in drug discovery and development, such as the generation of drug-like molecules [18], repositioning of existing drugs for diseases [19], protein engineering [20] and predictions of chemical toxicity [21] in drug design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The time and cost can be significantly reduced by machine learning as well as the erasure of ethical concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' As a result, machine-learning approaches were utilized in this study, to carry out large-scale predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In this work, we developed a proteome-informed machine-learning (ML) platform for the discovery of anti-opioid addiction compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' From the String v11 database, we obtained PPI networks of the five ma- jor opioid receptors with the associated proteins regarded as potential treatment and side effect targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We then collected inhibitor datasets with experimental binding affinity labels from ChEMBL database for these protein targets and built machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The inhibitor compounds were represented by two forms of latent-vector (LV) fingerprints generated by transformer and autoencoder learning models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' These latent vectors were paired with gradient boosting decision algorithm (GBDT) in building our bind- ing affinity (BA) predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We then carried out cross-predictions to screen side effects and repurposing potentials of more than 120,000 compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' With these models, we had more side effect evaluations of FDA-approved drugs or other existing medications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Another application using these cross-prediction mod- 2 els was to find promising lead compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In addition to the concern in potency and side effect, we also considered evaluations of pharmacokinetic properties in compound filtering , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=', absorption, distribution, metabolism, excretion, and toxicity (ADMET) as well as synthesizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Our platform is believed to be useful in advancing the drug development in treating OUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 2 Results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='1 The Opioid receptors and addiction PPI networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Figure 1: The workflow for searching nearly optimal lead compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The inhibitor compounds were collected for the proteins in protein-protein interaction networks of the five opioid receptors including mu, kappa, delta, nociceptin, and zeta opioid receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Each receptor has a core and global PPI network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The proteins in one core network have direct interaction with the opioid receptor, and those in a global network are related to opoid receptors through the proteins in the core network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Abbreviations for the proteins in the core networks are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Full names of the proteins in the five core networks are provided in the Supporting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Opioid receptors play critical roles in opioid dependence and are often the pharmacological targets of medications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' There are four major subtypes of opioid receptors, namely mu opioid receptor (MOR), delta opioid receptor (DOR), kappa opioid receptor (KOR), and nociceptin opioid receptor (NOR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In addition, zeta opioid receptor (ZOR) is also believed to be an important one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' However, ZOR was recently discovered, hence less studied and shares little sequence similarity with other opioid receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Opioid receptors are crucial in various biological functions and have broad distributions in the brain, spinal cord, on peripheral neurons, and digestive tract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' MOR, KOR, and DOR are closely related to analgesia, opioid dependence and the adverse effect of respiratory depression caused by opioids, but each of them is distributed in various re- gions of the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' NOR is distributed mainly in the cortex amygdala, hippocampus, septal nuclei, habenula, and hypothalamus in the brain and spinal cord, and is linked to development of tolerance to MOR agonists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' ZORs widely exist in many parts of the body including the heart, liver, kidneys and brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Its functions are mainly on tissue growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Several clinically useful medications for treating addiction target MOR, KOR, and DOR [1], but the roles of NOR and ZOR in causing opioid dependence has not been much explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' However, they cannot be neglected in the pharmaceutical treatment of opioid dependence as they are all critical targets of opioids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='MOR PPI-network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='KOR Ppl-network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='ZOR PPI-network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='HINT1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='POMC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='MECP2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='SIN3A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='SPATA4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='PENK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='ZNE324 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='FNIPI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='GABARAPL1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='MOR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='PDYN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='POMC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='MECP2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='CYB5D1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='MOR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='WLS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='STAT6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='CDK10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='PENK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='GNAQ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='PDYN PNOC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='COG8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='KOR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='ARRB1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='HDAC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='ZOR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='ADRBK1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='ARRB2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='SAG ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='SPC25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='FLNA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='DOR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='ARRB2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='DOK5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='SLC9A3R1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='ARRB1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='Treatment-target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='Side-effect-target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='Nearly Optimal Leads ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='Inhibitor Datasets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='Inhibitor Datasets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='ADMET ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='repurposing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='Screening ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='Potency Predictors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='Side-effect PredictorsOpioid receptors have wide distributions in the body,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' and the synergistic interactions between these receptors and many other proteins upstream and downstream contribute to specific biological functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' As discussed before, we carry out drug discovery in the PPI networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We extracted five PPI networks centered around each of the five opioid receptors by inputting receptor names, namely, mu-opioid receptor, delta- opioid receptor, kappa opioid receptor, zeta opioid receptor, nociceptin opioid receptor, and OGFR into the String database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In each network, there is a core subnetwork with proteins interacting directly with each opioid receptor, while proteins with direct and indirect interactions jointly form the global network as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We restrict the number of proteins in each global network to 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Although more proteins should be considered, we limit our efforts to critical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' There are five global networks in which five core networks exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' MOR, KOR, DOR, NOR, and ZOR are the most important proteins in the networks as each core protein plays an essential role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The five networks are not independent of each other with a few overlapping proteins found between the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Compounds with agonist or antagonist effects on opioid receptors showed their pharmacological effects in treating opioid dependence [4], hence encouraging us to look for more compounds that bind to the opioid receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' A desired drug must be specific to a target protein without causing adverse side effects to the other proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' To evaluate the binding effect of inhibitors to receptor proteins and other proteins in the PPI networks, we collected inhibitor compounds from the ChEMBL database for each protein and built machine-learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We then used them to systematically analyze the side effects and repurposing potential of inhibitor compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We collected 74 datasets in total, with sufficient inhibitor data points for the proteins in the five extracted PPI networks with a total of 129,515 inhibitor compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In addition, we collected an inhibitor dataset for hERG protein and built an appropriate machine-learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The hERG is a critical potassium channel that must be avoided in drug design and discovery, as the blockade of the hERG channel is associated with prolongation of the long QT syndrome, eventually leading to fatal arrhythmia, namely Torsade de Pointes (TdP) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In total, we collected 75 protein targets and built 75 machine-learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Since ZOR was recently discovered, not much inhibitor data was available to build a model for the ZOR protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' However, we were able to build models for the four remaining receptors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=', MOR, KOR, DOR, and NOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Further, we used all the models to explore potential drugs that bind to the four opioid receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The details about the collected datasets can be found in the Supporting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2 Binding affinity predictions The heatmap in Figure 2 shows the cross-target binding affinity (BA) predictions using the 75 machine- learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The diagonal elements indicate the Pearson correlation coefficient (R) of five-fold cross- validation for our machine-learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Two of the 75 models have R values greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='9, and the R values for fifty of them are greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The minimal R value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='604 is from the model built with the FYN inhibitor dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Overall, these models show excellent prediction accuracy and are reliable for BA predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='1 Cross-target binding affinity predictions Cross-target predictions can reveal side effects of drug candidates on other proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The off-diagonal elements represent the maximal BA values (BA with the largest absolute value) of inhibitor compounds in one dataset predicted by other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The notations to the left of the heatmap indicate the 75 inhibitor datasets and those on top of the heatmap represent all the 75 machine-learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Each column exhibits all the predictions by one model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Specifically, the i-th element in the j-th column is the prediction result of i-th dataset by the j-th model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' These cross-target prediction results are indicators of side effects of one inhibitor dataset on other proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The BA value of -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54 kcal/mol (Ki= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='1 µM) is widely accepted as an inhibition threshold in the literature [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' With this threshold, 5103 out of the 5625 cross-predictions were found to have side effects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=', the predicted maximal BA less than -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' On the other hand, the remaining 4 Figure 2: The heatmap of cross-target binding affinities (BAs) predictions revealing the inhibitor specificity of each dataset on other protein targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The notations above the heatmap shows the machine-learning models while the those on the left of the heatmap denote all the inhibitor datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The diagonal elements in the heatmap indicate the Pearson correlation efficient (R) of five-fold cross validations for all the predictive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The off-diagonal elements in each row represent the highest BAs values of inhibitors in one dataset predicted by 74 machine-learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 522 cross-prediction results with maximal BA greater than -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54 kcal/mol suggest weak side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The color of the off-diagonal elements indicates the strength of side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The lighter the color, the stronger the side effects are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Similar binding sites on off-target proteins is one of many reasons for side effects caused by drug candidates for one designated protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Proteins in the same family can have similar three-dimensional (3D) structures or protein sequences, giving rise to the existence of similar binding sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' An inhibitor compound potent at one protein likely binds to another protein in the same family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' As observed in Figure 2, mutual side effects occur among the four opioid receptors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=', MOR, DOR, KOR, and NOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The 16 yellow square boxes on the upper left corner of the heatmap showed the cross-prediction maximal BA value less than -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' These four proteins are all in the opioid receptor family and are highly similar in their 3D structure conformation or 2D sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' This is validated by the alignments of 3D structures and 2D sequences as shown in Figure S2 of the Supporting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The heatmap shows more examples of mutual side effects among proteins in one family such as the family of tyrosine kinase protein (JAK1, JAK2, and JAK3), melanocortin receptor (MCR1, MCR3, MCR4, and MCR5), and matrix metalloproteinases (MMP1, MMP2, MMP7, MMP8, and MMP9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 5 5 B 5 RAAACCC D4P MOR DOR KOR NOR ACE ACKR3 ADAM17 ADAMTS4 ADAMTS5 ADORA1 ADRB1 ADRB2 ADRBK1 AGTR1 AKT1 AR ATG4B AVPR2 BDKRB1 BDKRB2 CASR CCR5 CDK1 CNR1 CREBBP CRHR1 CXCR1 CXCR2 CXCR4 DNMT1 EGFR ERBB2 ERBB4 ESR1 F11 F2R FYN HDAC1 HDAC2 HGF ITGB1 JAKi JAK2 JAK3 KLKB1 KRAS MAP3K5 MAPK1 MAPK10 MC1R MC3R MC4R MC5R MDM2 MMP1 MMP2 MMP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' MMP8 MMP9 NOS3 NTRK1 P2RY12 PDE4A PDGFRB PI4KB PPARG PTPN2 REN S1PR1 SMO SRC TBXA2R TOP2A TYK2 hERG 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='85 15 14 13 12 10 g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='90 8 R of 10-fold CV ML-BA (kcal/mol)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2 Predictions of side effects and repurposing potentials The cross-target prediction is a useful tool to detect side effects and to evaluate repurposing potentials of inhibitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Side effects are caused when a drug candidate exhibits a strong binding affinity to the desired target, but unintentionally acts as a potent inhibitor on other proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Drug candidates that exhibit a weak binding affinity to their designated targets but an effective inhibitor to other proteins are deemed to have repurposing potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Figures 2a and 2b exemplify side effects and repurposing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Each panel involves one target and two off-target proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The title, the x-axis and the y-axis of each panel stand for the target, an off-target protein, and another off-target protein, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The colors of the scattered points indicate the experimental BAs of the inhibitors for the target protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The red and green colors reveal high and low binding affinities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The x-axis and y-axis indicate the predicted BAs from two machine learning models built on inhibitor datasets for two off-target proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The yellow frames in the nine panels of Figure 3a highlight the zone where no side effects are induced on two off-target proteins according to our predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The three rows in Figure 3a show some examples of inhibitors for one designated protein having side effects on zero, one, and two of the given two off-target proteins, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' For instance, as shown on the second panel in the first row of Figure 3a, all inhibitors for protein ADAM17 are predicted to be weak inhibitors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=', BA values greater than -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54 kcal/mol, on two off-target proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The first panel in the second row shows that around half of the inhibitors for the DOR are predicted to be potent at the MDM2 protein, but all the inhibitors were predicted to not bind to the MMP protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In addition, the first panel in the third-row exhibits a significant amount of KOR inhibitors that were predicted to be potent at JAK2 and JAK3, simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The repurposing potentials of inhibitors can be revealed through cross-target predictions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Figure 3b provides a few prediction examples of repurposing using our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The blue frames highlight the zone where inhibitors for target proteins can bind strongly to one protein, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=', predicted BAs less than -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54 kcal/mol, but are weaker binders to the other protein, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=', predicted BAs greater than -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The first panel in the first row in 3b shows that many inactive inhibitors for HDAC1 were predicted to have repurposing potential for either MOR or DOR, but not bind to the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Since both MOR and DOR are critical targets of medications in treating OUD [4], finding more drug candidates for these two proteins is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Buprenorphine is an FDA-approved drug that is a partial agonist of MOR and KOR, as well as a weak DOR antagonist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' As seen in the HDAC1-DOR-MOR panel on the first row of 3b, there are some inactivate inhibitor compounds for HDAC1 that are effective inhibitors to MOR and DOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Our models can be used to find more inhibitors that can bind to both targets as Buprenorphine does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The second and third rows in Figure 3b demonstrate additional examples of the inhibitors for one given protein having repurposing potentials for two other proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='3 Protein similarity inferred by cross-target BA correlations As discussed above, cross-target BA prediction is useful in evaluating side effects and repurposing potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Side effects can be caused when the drug candidate binds to proteins with similar 3D structures or sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In such situation, the predicted BA values can be correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' On the other hand, the correlated predicted BAs can be an indicator of similar binding sites or 3D protein structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' As shown in Figure 4a, the predicted BAs of inhibitors for JAK3 on HDAC1 and HDAC2 proteins have a nearly linear correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The Pearson correlation coefficient (R) of the predicted BA is up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='838.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' This is due to the binding site similarity, which is validated by the 3D protein structure and 2D sequence alignments as shown in 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The 3D structures of the HDAC1 and HDAC2 proteins are found to be quite similar while the 2D sequence identity near the binding site is around 85%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Two more examples of BA correlations revealing similar 3D protein structures are seen in Figures 4b and 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' For the case in Figure 4b, the Pearson correlation coefficient of the predicted BAs for DOR inhibitors on MOR and KOR proteins is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' For the case in Figure 4c, the R value of predicted BAs of JAK3 inhibitors on JAK1 and JAK2 proteins is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The 3D protein structure and 2D 6 Figure 3: Examples of inhibitors’ predicted side effects and repurposing potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The three rows in panel a indicate example inhibitor datasets have side effects on 0, 1, and 2 of the given two off-target proteins, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The yellow frame outlines therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Yellow zones indicate where side effects are not found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The three rows in panel b reveal example inhibitor datasets that show repurposing potentials on 0, 1, and 2 of the two given off-target proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The blue frames highlight the domains where inhibitors have repurposing potential for one protein but have no side effect on the other proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' sequence alignments confirm the usefulness of cross-prediction in detecting protein similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In addition, it was found that there is a bilinear correlation relationship among the predicted BAs and experimental BAs in the case of Figures 4b and 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The target and two off-target proteins are in the same protein family and share high 3D structure and 2D sequence similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' A potent DOR inhibitor is likely to be a strong binder on KOR and MOR proteins simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The high structural similarities form the basis of drug- mediated trilinear target relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' KOR, MOR, and DOR proteins are often pharmacological targets in the treatment of opioid addiction [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The observed bilinear or trilinear relationship indicates the possibility of developing inhibitors that simultaneously bind to multiple targets of the major opioid receptors, namely, MOR, KOR, DOR, and NOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Such binding effects on multiple opioid receptors have been observed on the currently FDA-approved medications [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' More examples of similar proteins with correlated predicted BAs can be found in the Supporting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='4 Repurposing to opioid receptors and side effect on hERG MOR, KOR, DOR, and NOR are the four major subtypes of opioid receptors and are the critical pharma- cological targets in treating OUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Inhibitors that bind to these receptors can be potential medications for OUD treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We adopted our cross-target prediction strategy to evaluate the repurposing potential of inhibitors on the four opioid receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The 75 collected inhibitor datasets contain more than 120,000 compounds, providing a source of drug candidates in our repurposing study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The side effect of hERG is a priority concern for novel medications, and hence we used our machine learning model to predict the binding affinity of these inhibitors on hERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' A stricter side effect threshold of −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='18 kcal/mol (Ki = 1 µM ) was adopted for the hERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In this study, inhibitors are considered to have 7 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' MOR ADAM17 CRHR1 HDAC1 CRHR1 8- MMP1 ML-BA MMP1 ML-BA CNR1 ML-BA AR ML-BA 9 8 8 9 10 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 8 10 10 6- SRC ML-BA ADORA1 ML-BA HDAC2 ML-BA DOR ML-BA CCR5 ML-BA DOR CCR5 HGF MOR DOR 8 ML-BA 8 1 ML-BA 2 ML-BA MC4R ML-BA MDM2 ML-BA 6- 8 9 PDGFRB I MDM2 I CRHR1 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 9 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 8 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='9 10 9 10 MMP1 ML-BA PPARG ML-BA MDM2 ML-BA CNR1 ML-BA MC4R ML-BA KOR REN SRC HDAC1 JAK3 8 8- 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' CRHR1 ML-BA ML-BA 3 ML-BA 8 EGFR ML-BA 9 ML-I MAPK1 I ERBB2 JAK3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 10 11 出 10 8 10 8 10 8 8 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='5 9 JAK2 ML-BA CNR1 ML-BA JAK3 ML-BA ESR1 ML-BA CRHR1 ML-BA 14 12 10 8- 6 Experimental BA (kcal/mol)Figure 4: Three examples of predicted BA values being correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In each example, the chart shows the predicted BA values on two other proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The 3D structure alignments are shown on the right of the chart, and the 2D sequence alignment is exhibited on the second row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The 3D structures in the alignment are (PDB 4BKX and 4LY1 for HDAC1 and HDAC2), (PDB 5C1M, 4DJH, 4N6H for MOR, KOR, and DOR), and (PDB 6BBU, 2B7A, and 1YVJ for JAK1, JAK2, and JAK3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Property Optimal range FDAMDD Excellent: 0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' medium: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='3-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' poor: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='0 F20% Excellent: 0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' medium: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='3-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' poor: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='0 Log P The proper range: 0-3 log mol/L Log S The proper range: -4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='5 log mol/L T1/2 Excellent: 0-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' medium: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='3-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' poor: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='7-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='0 Caco-2 The proper range: >-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='15 SAS The proper range: <6 Table 1: The optimal ranges of six selected ADMET properties and synthesizability (SAS) used to screen nearly optimal compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' no hERG side effect if the predicted BA value on hERG is greater than −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='18 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Figures S5 and S6 provide the predicted BAs of the other 73 inhibitor datasets on MOR and hERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The orange frames highlight the zones where compounds can have repurposing potentials for MOR but do not cause hERG side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Some of the 73 inhibitor datasets have almost no compounds in the orange frames such as CXCR2, ERBB4, MAP3K5, PI4KB, and SMO, while other datasets still have a significant number of compounds lying in these orange frames, such as HDAC1, MMP1, MMP2, DOR, KOR, and NOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='3 Druggable property screening ADMET (absorption, distribution, metabolism, excretion, and toxicity) plays a key role in drug discovery and development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It includes vast attributes associated with the pharmacokinetic studies of a compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' A promising drug candidate should not only have sufficient efficacy on the therapeutic target but also satisfies appropriate ADMET properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Accurate predictions of ADMET are significant in drug design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The successful ADMET screening at the design stage of new compounds is beneficial in reducing the risk of late-stage attrition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' To search for promising compounds in treating OUD, systematic screenings of ADMET properties, synthetic accessibility (SAS), and the hERG risk of all inhibitor datasets are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We paid attention to six indexes of ADMET, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=', FDAMDD, T1/2 and F20%, log P, log S, and Caco-2, and SAS as well as hERG risk assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' To evaluate the ADMET properties, we utilized the ADMETlab 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='0 (https://admetmesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='scbdd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='com/) solvers that provide machine-learning predictions [24,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Their documents provide a set of optimal ranges of these ADMET properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The SAS evaluation was obtained from Rdkit packages [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The optimal ranges of ADMET properties and SAS are provided in Table 1, while the BA value > -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='18 kcal/mol is applied as the required range for exempting hERG side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' With the evaluations of 8 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='JAK3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='JAK3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='DOR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2 ML-BA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='JAK2 ML-BA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='ML-BA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='HDAC2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='KORI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='12-10 -8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='12-10-8 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='TLCTMSVI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='IAVCFigure 5: Screening of example datasets on ADMET properties,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' synthesizability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' and hERG side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The colors of scattered points represent the experimental BA values of inhibitors in each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The Orange frames highlight the optimal ranges of the properties and side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' ADMET properties and SAS as well as cross-target prediction tools, we were able to systematically search for promising compound leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Figure 5 shows the ADMET screening of a few inhibitor datasets including MOR, DOR, KOR, NOR, and MMP7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The four rows represent the eight property screenings of the five inhibitor datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The colors of the scattered points indicate the experimental BA values of inhibitor compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' FDAMDD is the FDA maximum recommended daily dose, aimed at avoiding toxicity in the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The half-life is the amount of time for a drug’s active substance to reduce by half in the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The value of T1/2 stands for the probability of half-life less than 3 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' F20% is the probability of administered drug reaching systemic circulation with less than 20% of the initial dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The values of property log P and log S are the logarithm of the n-octanol/water distribution coefficient and aqueous solubility value, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Caco-2 is a measure used to estimate in vivo permeability of oral drugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' SAS quantifies the synthesis difficulty of druglike molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' As seen in Figure 5, the orange frames in the panels outline the optimal ranges for a pair of screening properties denoted on the x- and y-axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Each pair of screening forms a screening filter for the inhibitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' T1/2 and F20% especially offer a stricter screening as only small portions of inhibitors are covered in the orange frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The SAS screening seems to be a loose filter, as a significant portion of inhibitors remains in the orange frames after screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Overall, these ADMET indexes and SAS cause strict restrictions of finding inhibitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 3 Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='1 Side-effect evaluations of existing medications for OUD treatment Substantial pharmacological efforts have been dedicated to the treatment of OUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Opioid replacement therapy (ORT) is a popular method to treat people with opioid use disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It involves replacing an opioid 9 MOR DOR KOR NOR MMP7 F9- ML-BA ML-BA hERG ML-BA ML-BA ML-BA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='7 7 8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='9 8 hERG hERG hERG hER 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 10 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='9 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 7 8- 1 4 6 1 8 15 0 3 6 0 4 8 1 4 P P log P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' log P 9 6 6- 5 9 SAS SAS 4 4 S 3 2 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='6 5 6 5 6 6 5 6 5 Caco-2 Caco-2 Caco-2 Caco-2 Caco-2 13 9 7 ExperimentalBA(kcal/mol)with a longer-acting but less euphoric opioid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Three classes of medications acting directly on the opioid receptors were found to be effective, namely full agonist, partial agonist, and antagonist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Methadone, buprenorphine, and naltrexone are approved by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Food and Drug Administration (FDA) for medication-assisted treatment (MAT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' These medications are useful in reducing the risk of death and preventing relapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In addition, naloxone is a frequently used medication in reducing the risk of over- dose, and take-home naloxone is crucial in stopping opioid overdose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It is necessary to evaluate the side effects of these anti-OUD medications, including their actions on five opioid receptors and their multitude of physiological functions affecting the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We used our machine learning models to predict the BA values on the proteins in the five opioid networks as well as on the hERG channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Among the five important opioid receptors, MOR is typically the target of most clinically prescribed medications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Methadone is a full MOR opioid agonist, while it has some agonist effect on KOR and possibly DOR agonist [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The methadone maintenance treatment (MMT) is beneficial in reducing the intensity of withdrawal symptoms including muscle aches and osteodynia in addicted individuals [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Methadone has a long half-life that makes it more useful in reducing withdrawal symptoms in patients [28], and consequently reducing patients’ compulsive drug-seeking and craving behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Our BA predictions of methadone on MOR, KOR, and DOR are -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='8 kcal/mol, -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='96 kcal/mol, and -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='52 kcal/mol, respectively, which agrees with the methadone binding activity on opioid receptors, especially for MOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It was reported that methadone prolongs the QT interval in a dose-dependent manner, and high-dose methadone is associated with ventricular tachycardia torsade de pointes [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The overall hERG side effect profile is safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The predicted BA value on hERG from our model is -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='73 kcal/mol, which is higher than the hERG side-effect threshold of -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='18 kcal/mol, and confirms the safety profile of methadone on hERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Our predictions indicate that the SMO protein is the only target it can have side effects on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The predicted BA of methadone on SMO protein is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='67 kcal/mol, and the predicted BAs on all other targets are greater than -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' SMO is targeted and inhibited by small-molecule drugs for the treatment of advanced basal cell cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' No serious side effects were reported by inhibiting the SMO protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Its low side effect profile might be one of the reasons that methadone is the most used medication in MAT and the gold standard against which other medications are compared [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Buprenorphine is a partial agonist of MOR, the antagonist of KOR, and a weak antagonist of DOR [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Unlike methadone and other full opioid receptor agonists, buprenorphine has a lower risk of respiratory depression due to a low ceiling to the euphoric effect [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In treating opioid dependence, it is typically administered sublingually as it has an extended half-life than that of intravenous buprenorphine [4], increas- ing the potential of misuse or overdose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' To avoid buprenorphine abuse, it is commonly used with opioid antagonist naloxone via injection or insufflation without causing impairment when used appropriately [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The predicted BA values of buprenorphine for MOR, KOR, DOR, and NOR are -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='5, -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='88, -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='64, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='41 kcal/mol, which are consistent with the experimental BA values of -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='0, -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2, -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='7, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='69 kcal/mol for MOR, KOR, DOR, and NOR, respectively [33,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Buprenorphine was predicted to have no side effects on hERG with the predicted BA values of -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='31 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It was found to have a minimal impact on the corrected QT interval [35], which is consistent with our hERG side effect prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' However, it is predicted to be a potent inhibitor on quite a few other proteins including REN, JAK1, BDKRB1, and NTRK1 with BA values of -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='74, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='67, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='49, and -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='28 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' JAK1 is a member of the Janus kinase family and Janus kinase inhibitors are used in the treatment of cancer and inflammatory diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Clinical trials indicated inhibitors for NTRK1 protein have shown efficacy as targeted therapies for extracranial tumors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The inhibition of buprenorphine needs to be further investigated for its side effects on previously discussed proteins and its usefulness in the treatment of alternative diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' LAAM, acting as a MOR agonist, can provide greater suppression of heroin use in comparison to methadone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Our predicted BAs of LAAM to MOR, DOR, KOR, and NOR are -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='34, -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='94, -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='83, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='69 kcal/mol, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The potency of LAAM on these receptors is not as strong as methadone and 10 buprenorphine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In addition, the safety profile of LAAM is low due to its potential for ventricular rhythm disorders [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Such adverse effects are directly associated with the hERG blockade, which is verified by the predicted relatively high BA of -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='94 kcal/mol on hERG, a value close to our hERG side effect threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' According to our models, the top potential targets with side effects imposed are SMO and JAK1 proteins with predicted BA values of -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='82 and -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='75 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Similar to the predictions for buprenorphine, the molecular binding on SMO and JAK1 might not cause a serious problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Naltrexone is an antagonist of MOR and a partial agonist of KOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Long-acting injectable naltrexone can block opioid receptors but does not activate them, reducing drug-seeking behavior and alleviating drug craving [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Naltrexone is observed to have a continuous effect in reducing the frequency and dosage of heroin use [38] and in decreasing the risk of opioid overdose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Naltrexone and nalmefene have a longer duration period, therefore drawing research and clinical interests to investigate their anti-overdose effect against potent fentanyl analogs [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We predicted BA values for MOR, KOR, NOR, and DOR respectively 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54, -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='05, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='88, and -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='49 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The predicted BA value on hERG was low with a value of -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='64 kcal/mol, suggesting a low hERG side effect potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Strong binding potency can occur on a few proteins including JAK1, BDKRBA, and SMO with the predicted BA values of -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='30, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='02, and -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='92 kcal/mol, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Analogous to naltrexone, nalmefene is also a MOR antagonist and a KOR partial agonist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It has a prolonged duration of action and intravenous doses of nalmefene have been shown effective at counteracting the respiratory depression produced by an opioid overdose [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It was predicted to be potent at the four opioid receptors MOR, KOR, DOR, and NOR with BA values of -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='62, -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='11, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='78, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='05 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The proteins it can bind to, with a strong binding affinity, include JAK1, SMO, BDKRB1, REN, and S1PR1 with predicted BA values of -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='25, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='08, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='06, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='99, and -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='99 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Protein BDKRB1 is a G-protein coupled receptor that mediates responses to pathophysiologic conditions such as inflammation, trauma, burns, shock, and allergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Antagonist inhibitors of this receptor were used to reverse acute or persistent inflammatory pain in these pathophysiologic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Naltrexone or nalmefene might have some clinical significance as pain relief in these pathophysiologic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Activation of receptor protein S1PR1 is heavily involved in immune cell regulation and development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It is also responsible for vascular growth and development, during embryogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Inhibitions of protein S1PR1 by naltrexone may interfere with normal growth and development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Naloxone is a non-selective and competitive opioid antagonist that reverses opioid analgesic actions quite effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Naloxone is commonly used for acute opioid intoxication, rescuing opioid-induced respiratory depression [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' However, it is of lower potency and shorter duration period, compared to other antagonists such as naltrexone and nalmefene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In addition, the capabilities of naloxone are limited when ingesting highly potent opioids, such as fentanyl [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The predicted BA value on MOR, KOR, and DOR are - 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='50, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='93, and -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='79 kcal/mol, close to those experimental BA values of -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='21, -10,63, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='75 kcal/mol, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The BA prediction for hERG is -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='58 kcal/mol, suggesting a safe hERG-blockade profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The potential side effect on other proteins included JAK1, JAK3, REN, BDKRB1, PDGFRB, and ERBB4 with predicted BA values of -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='34, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='92, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='89, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='85, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='80, and -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='76 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Like naltrexone and nalmefene, naloxone could also block protein BDKRB1, and may also be clinically useful in the pain relief of several pathophysiologic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Protein PDGFRB is essential for vascular development, and its inhibition may compromise the integrity and/or functionality of the vasculature in multiple organs, including the brain, heart, kidney, skin, and eyes [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' HER4 is a receptor tyrosine kinase that is critical for normal body systems such as the cardiovascular, nervous, and endocrine systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Overexpression of HER4 kinase results in cancer development [42], and activation of HER4 by ligand binding can potentially cause cancer, promoting drug development to inhibit these HER4 [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The duality of naloxone’s ability to activate HER4, potentially causing cancer or providing an effective HER4 inhibitor, needs to be investigated further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Heroin (Diamorphine), a MOR agonist, was found to be useful in helping patients disengage from the use of street heroin and reducing criminal involvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It can be an effective adjunctive treatment for chronic, relapsing opioid dependence [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Heroin assisted treatment is now available in Canada and some European 11 countries as a new treatment modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It is administrated under direct medical or nurse supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Heroin assisted treatment is intended for injection into patients suffering from OUD, who have not responded to standard medications for OUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' However, compared to other medications for opioid use disorder, its safety profile is low with major adverse effects, such as respiratory depression and seizures [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The predicted BA of heroin on MOR by our model is -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='46 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In addition, it was predicted to be a potent inhibitor at NOR, JAK1, JAK3, BDKRB1, SMO, REN, and ITGB1 with BA values of -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='67, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='32, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='15, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='10, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='87, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='77, and -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='72 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The BA value on hERG was predicted to be -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='59 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Protein ITGB1 associates with integrin alpha 1 and integrin alpha 2 to form integrin complexes which function as collagen receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Recent studies have shown that the inhibition of ITGB1 enhances the anti-tumor effect of cetuximab in colorectal cancer cell [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Nonetheless, due to the severe addiction effect, heroin may not be a good choice for cancer treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Hydromorphone is also a MOR agonist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Injectable hydromorphone was also found to be as effective as diacetylmorphine for patients who have not benefited from previous treatments, such as methadone or suboxone [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Following studies showed that once-daily sustained-release oral hydromorphone was useful in managing cravings without notable side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It has the advantage of no influence on the cardiac QTc interval [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The treatment with hydromorphone requires supervised administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In our prediction, hydromorphone had strong binding affinities on all four opioid receptors, namely, MOR, KOR, NOR, and DOR with the BA values of -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='9, -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='58, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='35, and -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='08 kcal/mol, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The predicted BA value on hERG is -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='71 kcal/mol, which can be deemed as having a low potential of side effect on hERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Other targets that hydromorphone can possibly cause side effects on are JAK1, AR, PDGFRB proteins with the predicted BA values of -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='20, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='81, and -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='78 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Dihydrocodeine is a semi-synthetic opioid analgesic and agonist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It is sometimes used for maintenance treatment as an alternative to methadone or buprenorphine in some European countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Our predicted BA values of dihydrocodeine on MOR, DOR, KOR, and NOR are -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='41, -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='44, -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='91, and -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='06 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It may be used as a second line treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Low quality evidence reported that dihydrocodeine may be no more effective than other routinely used medication interventions in reducing illicit opiate use [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It can be a potent inhibitor for several proteins such as JAK1, SMO, JAK3, BDKRB1, ITGB1, and AR, with predicted BA values of -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='07, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='89, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='83, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='76, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='73, and -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='68 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The predicted BA value on hERG was 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='08 kcal/mol, which shows a mild potential of side effects on hERG and proteins ITGB1 and AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Androgen receptor (AR) functions mainly as a DNA-binding transcription factor that regulates gene expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' High expression in androgen receptor has been linked to aggression and sex drive [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' AR also has roles in the progression of prostate cancer and is an important therapeutic target in prostate cancer [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The inhibition of AR by Dihydrocodeine may have an impact on male sexual phenotype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Lofexidine is an α2-adrenergic receptor agonist but is not classified as an opioid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It is an alternative for people with mild or uncertain opioid dependence in need of short-term detoxification and is effective in reducing withdrawal symptoms of OUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Its adverse side effects include QT prolongation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Our predicted BA values for MOR, KOR, DOR, and NOR are -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='33, -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='53, -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='02, and -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='7 kcal/mol, which are consistent with the fact that Lofexidine is not an opioid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Our BA prediction of Lofexidine on hERG is -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='30 kcal/mol, showing a good side profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The strongest predicted BAs were for FDE4A, SMO, and IITGB1 with the values of -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='79, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='76, and -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='56 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The side effects of these three proteins might be mild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Currently, there are some drugs used in the treatment of opioid-induced constipation such as alvimopan, methylnaltrexone, and naloxegol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The three drugs are all peripherally acting µ-opioid receptor antagonists due to their limited ability to cross the blood–brain barrier and reach the MORs of the central nervous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Consequently, the effects of centrally-acting opioid antagonists are normally not notable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Alvimopan is approved for the treatment of postoperative due to its intended blockade of MORs in the gastrointestinal tract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We predicted BA values for the MOR, KOR, NOR, and DOR -12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='40, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='50, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='18, and -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='02 kcal/mol, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Its potency on MOR is validated in our prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It can also be potent at several proteins 12 including JAK1, BDKRB1, S1PR1, ACE, and SMO with predicted BA values of -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='36, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='28, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='07, and -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='07 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The most common side effects of alvimopan include dyspepsia, hypokalemia, back pain, and delayed micturition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Methylnaltrexone is used to treat opioid-induced constipation in chronic non-cancer pain or when ordinary laxatives do not work well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It was also predicted to be a potent inhibitor of MOR, KOR, and NOR with BA values of -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='04, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='62, and -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='25 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Potential side effects can be exhibited on proteins including JAK1, BDKRB1, and REN with BA values of -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='57, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='05, and -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='04 kcal/mol, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Naloxegol is recommended for the treatment of opioid-induced constipation (OIC) in patients with chronic non-cancer pain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Its predicted BA values for NOR, KOR, MOR, and DOR are -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='52, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='15, -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='96, and -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='26 kcal/mol, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Based on our predictions, it can be potent for REN, JAK1, BDKRB1, and ERBB4 with predicted BA values of -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='86, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='59, -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='50, and -10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='03 kcal/mol, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2 Nearly optimal lead compounds from screening and repurposing As previously discussed, opioid replacement therapy (ORT) replaces an opioid with a longer-acting but less euphoric opioid, and is widely used in OUD treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Drugs like methadone and buprenorphine are commonly used for ORT, and act as agonist/antagonist depending on the different opioid receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We dedicate our efforts to search for more potential inhibitors of the four opioid receptors, which can be potential agonist/antagonist in OUD treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Screening and repurposing are the two avenues to find more inhibitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' There were 74 models utilized to predict the cross-target bind affinity in the process of screening and repurposing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In addition to the potency concern, optimal ranges for the ADMET properties, synthetic accessibility in Table 1, and hERG side effect all needed to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' As we know, MOR, DOR, KOR, and NOR are the four major opioid receptors, and critical pharmacological targets in OUD treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In finding more promising potent compounds at these four receptors, the 74 inhibitor datasets are our source of inhibitor compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In the screening process, we start with potent inhibitor compounds (experimental BA values < -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54 kcal/mol) in the inhibitor datasets of the four opioid receptors, and then evaluate a series of other properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It is important to note that if a designated inhibitor of a receptor exhibits notable potency on any of the other three receptors, it is not considered as a side effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It is very common that one inhibitor can be effective on several of the four major opioid receptors simultaneously, as seen by the currently approved drugs that act as agonists or antagonists on several receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' However, the potential for side effect concern must be evaluated on the other 70 protein targets including hERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We require the predicted BA values > -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54 kcal/mol to exempt side effects except for hERG with a stricter BA > -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='18 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In the repurposing process, we evaluate the binding potency of all weak inhibitors in the other 70 datasets on the four opioid receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We start with inhibitors of experimental BA value > -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54 kcal/mol, and then find those with predicted BA values < -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54 kcal/mol on the four opioid receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In search of inhibitors with repurposing potential on the opioid receptors, their side effects need to be exempted on the other 70 proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Following these, optimal range of ADMET properties and synthetic accessibility need to be screened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Two inhibitor compounds, CHEMBL466223 from the CNR1 dataset and CHEMBL355008 from the CRHR1 dataset, were found to satisfy all the aforementioned criteria for repurposing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' They were predicted to be effective on NOR with predicted BA values of -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='543 kcal/mol and -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='88 kcal/mol while their experimental BA values for the designated targets are -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='18 kcal/mol and -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='37 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' They were not predicted to be potent at the other three opioid receptors, MOR, DOR, and KOR, as seen by the following predicted BA values of -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='26, -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='87, and -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='51 kcal/mol, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Its predicted BA value on hERG was -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='34 kcal/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The predicted BA values of compound CHEMBL355008 on MOR, DOR, and KOR were -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='46, -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='33, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='25 kcal/mol, respectively, and -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='01 kcal/mol on hERG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' These two compounds were predicted to have no binding effects or side effects on all other 69 proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We carried out evaluations of more ADMET properties regarding these two molecular compounds using the ADMETlab 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='0 predictive solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' As seen in Figures 6a and 6b, the two compounds are still in the optimal ranges of these ADMET properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The meaning and optimal ranges of the 13 ADMET properties are provided in the Supporting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We 13 are also interested in the molecular interaction between the two inhibitors and NOR protein structure and utilized the software AutoDock Vina to perform protein-ligand docking to analyze the interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The 3D docking structure and 2D interaction diagrams are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It can be seen in the figure below, hydrogen bonds are formed between the inhibitors and the NOR protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Compound CHEMBL355008 has two hydrogen bonds formed with the Tyr131 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='32 ˚A) and Tyr309 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='17 ˚A), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The compound CHEMBL466223 has one strong hydrogen bond (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='82 ˚A) with Tyr309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The predicted binding energies by CHEMBL355008 and CHEMBL466223 with NOR were -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54 and -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='88 cal/mol, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The single strong hydrogen bond of CHEMBL466223 partially explains its predicted higher binding energy with NOR even though CHEMBL466223 has two hydrogen bonds with NOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It is observed that the side chains of Tyr309 play roles in the formation of hydrogen bonds with the two compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' No covalent bond is formed by either of the two compounds with the side chains of the NOR protein, and hydrogen bonds play the essential roles of binding energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Even though these two compounds were not predicted to be potent inhibitors on MOR, similar to the approved drugs for OUD treatment, they are selective inhibitors for NOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Selective antagonists are needed in scientific research when one of the receptors needs to be blocked without affecting the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' These two were predicted to be selective inhibitors of NOR with safe side effect profiles on all other 70 proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Further studies can be carried out to test their physiological effect on NOR or their pharmacological effect in treating OUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Figure 6: Evaluations of more ADMET properties for the found molecular compounds with repurposing potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Panel a and c represent the prediction results of ADMET properties and side effect evaluations for compound ChEMBL466223, and panels b and d stand for these predictions for compound ChEMBL355008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The boundaries of yellow and red zones in figure a and b highlight the upper and lower limits of the optimal ranges for the ADMET properties, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The blue curves represent values of the specified 13 ADMET properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Figures a and b are the prediction results from ADMETlab 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='0 (https://admetmesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='scbdd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='com/) website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Abbreviations: MW (Molecular Weight), logP (log of octanol/water partition coefficient), logS (log of the aqueous solubility), logD (logP at physiological pH 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='4), nHA (Number of hydrogen bond acceptors), nHD (Number of hydrogen bond donors), TPSA (Topological polar surface area), nRot (Number of rotatable bonds), nRing (Number of rings), MaxRing (Number of atoms in the biggest ring), nHet (Number of heteroatoms), fChar (Formal charge), and nRig (Number of rigid bonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 14 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Upper Limit Lower Limit Compound Properties Upper Limit Lower Limit Compound Properties MW MW LogP LogP nRig nRig fChar fChar LogS LogS nHet LogD LogD nHet MaxRing MaxRing nHA nHA nHD nRing nHD nRing TPSA nRot TPSA nRot C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' H3 CHEMBL466223 CHEMBL355008 CNR1-ex-BA: -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='18 kcal/mol CRHR1-ex-BA: -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='37 kcal/mol Predicted BA: Predicted BA: MOR: -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='26 kcal/mol MOR: -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='46 kcal/mol KOR: -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='51 kcal/mol KOR: -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='25 kcal/mol DOR: -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='87 kcal/mol DOR: -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='33 kcal/mol NOR: -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='54 kcal/mol NOR: -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='88 kcal/mol hERG: -7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='35 kcal/mol hERG: -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='01 kcal/molFigure 7: The docking structure of our nearly optimal lead compounds bound to nociceptin opioid receptor and their 2D interaction diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' AutoDock Vina was used to performing the protein-ligand docking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Hydrogen bonds are playing essential roles in binding energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 4 Methods 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='1 Datasets The inhibitor datasets were collected from the ChEMBL database for the proteins in the five investigated opioid receptor networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' As machine-learning models rely on a training set of sufficient data points, we require the size of the collected inhibitor dataset to be at least 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' A total of 74 datasets were then obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The labels for the data points are IC50 or Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' As suggested in [50], the IC50 values can be approximately converted to Ki values with the relation Ki=IC50/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' These labels were used to compute the binding affinity (BA) with the formula BA=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='3633× log10 Ki (kcal/mol), which were then used in building machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Since the hERG is a critical target of side effects in drug design, an inhibitor dataset was also collected for it from the ChEMBL database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The details regarding all these datasets are provided in the Supporting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2 Molecular embeddings The molecular representation for the inhibitors in the collected 75 datasets is 2D SMILES strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Two forms of molecular fingerprints were used to build machine-learning models in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The molecular fingerprints were generated by pre-trained models based on natural language processing (NLP) algorithms including transformer [51] and sequence-to-sequence autoencoder [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The two pre-trained models encode the 2D SMILES strings of inhibitor compounds in latent embedding vectors of lengths 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We denote the two types of fingerprints by the transformer and autoencoder models as TF-FP and AE-FP, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='1 Sequence-to-sequence auto-encoder A data-driven unsupervised learning model was recently proposed to extract molecular information embed- ded in the SMILES representation [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' A sequence-to-sequence autoencoder was utilized to translate one form of molecular representation to another, with a comprehensive description of the chemical structure compressed in the latent representation between the encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The translation model extracts the physicochemical information in the molecular representation when translated to another semantically equivalent but syntactically different representation of the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The translation model was trained on a large set of chemical structures and allows for the molecular descriptor extraction for query compounds 15 Trp116(A) Met134(A a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' TTT Gln107(A al283(A) TT Gln280(A p110(A) T 13 Val126(AE p276(A) CD1 Eu301(A) 01 p130(A) L OH Tyr309(A) C19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='82 CE2 CD2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='32 012 Ouz CG C6C5 /n286(A) CEb1 CD2 CB CA E Asp130(AE Val279(A) Gln107(A) Met134(A) C1 TT EL E Val283(AL Tvr131(A) Tp276(A) Docking structure: Docking structure: NOR-ChEMBL466223 NOR-ChEMBL355008 EFL Ie111(A) PDB ID: 4EA3 PDB ID: 4EA3 Asp110(A) Ile219(A)without retraining or using labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The translation model consists of the encoder and decoder networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The information bottleneck in between is used to compress the essential information of the input SMILES, and the embedded information is then used as input in the translation through the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In the encoder network, convolutional neural network (CNN) and recurrent neural network (RNN) architectures were adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Then fully connected layers map the output of CNN or the concatenated cell states of the RNN networks to intermediate vector representations between encoders and decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The decoder is comprised of RNN networks with latent vec- tors as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' To embed more meaningful physicochemical information about molecules in the latent vectors, a classification model was used to extend the translation model by predicting certain molecular properties based on the latent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The output of the decoder’s RNN network is the probability distributions over different characters in the translated molecular representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In training the autoencoder model, the loss function is the sum of cross-entropies between probability distributions and one-hot encoded cor- rect characters as well as the mean squared errors for molecular property predictions from the classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The translation model was trained with approximately 72 million molecular compounds from ZINC and PubChem databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The preprocessing was carried out to filter compounds with a variety of criteria including molecular weight, heavy atom numbers, partition coefficient, and other properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' After sufficient training with the processed dataset, the resulting translation model yields the embedding vectors as molecular fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='2 Bidirectional transformer A self-supervised learning (SSL)-based platform was recently developed to pre-train a deep learning network from millions of unlabeled molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Predictive molecular fingerprints can be extracted from the pre-trained models [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The self-supervised learning was achieved with the bidirectional encoder transformer (BET) model that relies on the attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The SSL has the advantage of avoiding the construction of a complete encoder-decoder framework and solely using the decoder network to encode the SMILES of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The SMILES strings of molecules were the input for the SSL pretraining platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Pairs of real SMILES and masked SMILES were constructed by hiding a certain percentage of some meaning symbols in the strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Then an SSL approach enables the model training with the data-mask pairs in a supervised way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In the pretraining process, the symbols of masked symbols were referred to by learning the unprocessed ones in SMILES, which then leads to the understanding of SMILES language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Data masking is preprocessed before starting to train the model with SSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' A total of 51 symbols were considered as the components in the SMILES strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The SMILES were the input for training the model, and we required the maximal length to be 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Symbols ′⟨s⟩′ and ′⟨\\s⟩′ were added to the beginning and the end of SMILES strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' If the length is less than 256, the symbol ′⟨pad⟩′ was used to supplement a SMILES string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' For the data masking, a total of 15% of the symbols in all the SMILES were operated, among which 80% were masked, 10% were unchanged and the remaining 10% were randomly changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The BET modules play critical roles in achieving SSL from a massive number of SMILES strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The attention mechanism in transformer modules captures the importance of each symbol in the inputted SMILES sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The BET consists of eight bidirectional encoder layers, with each encoder layer composed of a multi-head self-attention layer and a subsequent fully connected feed-forward neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The number of heads in each self-attention layer is 8, and the embedding size of fully connected feed-forward layers is 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The Adam optimizer was used in the training process and weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='1 was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The loss function is defined to be the cross-entropy, measuring the difference between the real and predicted symbols at masked positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The maximum length of input SMILES is 256 including the added special symbols at the two ends, while the embedding dimension of each symbol was 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The resulting molecular embedding 16 matrix is comprised of 256 embedding vectors of dimension 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The mean of embedding vectors for the valid symbols in one SMILES string was used as molecular fingerprint of a given SMILES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Due to the high parallelism capability and training efficiency from transformer modules, a massive number of SMILES can be used to train deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In our implementations, SMILES strings from one or the union of the ChEMBL, PubChem, and ZINC databases were employed, giving rise to three pre-trained models [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In this study, transformer-based embeddings generated from the pre-trained model solely using the ChEMBL database were used as molecular fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='3 Machine-learning models The gradient boosting decision tree (GBDT) algorithm was deployed to build our machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The GBDT algorithm is a popular ensemble method and has the advantage of robustness against overfitting, insensitiveness to hyperparameters, and ease of implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The methodology is to create many weak learners (individual trees) by bootstrapping training samples and to make predictions by integrating the outputs of weak learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Weak learners are likely to make poor predictions, but through ensemble approach the overall errors by combining all the weaker learners are reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' GBDT is particularly useful when training with small datasets and can deliver better prediction performance than deep neural network (DNN) and some other machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It gains wide popularity in a range of quantitative structure–activity relationship (QSAR) prediction problems [53,54], and promotes the development of competitive predictive ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The GBDT algorithm provided in the Scikit-learn (version 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='1) library was used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We collected a total of 75 inhibitor datasets with at least 250 data points in each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' It is preferable to utilize GBDT in building models for these datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' As aforementioned, two types of molecular fingerprints including TF-FP and AE-FP were adopted to represent inhibitor compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Our machine-learning (ML) models were built by integrating these molecular fingerprints with the GBDT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We built a total of 75 ligand-based ML models with the 75 inhibitor datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' For each dataset, two individual models were built by pairing TF-FP and AE-FP with GBDT algorithm, and then the average of the predictions from the two individual models was regarded as our final binding affinity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Such average or consensus results typically outperform those from individual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' To alleviate the effect of randomness, each of the individual GBDT models were trained ten times with a different random seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The average of the ten predictions was regarded as the final outcome of each individual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In the Supporting information, we included the Pearson correlation coefficients of five-fold cross validations for modeling the 75 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' 5 Conclusion Opioid use disorder (OUD) is a chronic and complex disease with neurobiological, psychological, behavioral, and medical implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Each year in the United States and around the world, thousands of deaths are caused by opioid abuse, and billions of dollars have been spent on OUD treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' To combat the opioid epidemic, efforts in novel treatment formulations and devices have been dedicated by pharmaceutical agencies and scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Pharmacological or psychosocial interventions showed their efficacy for OUD treatment, but many patients still drop out of treatment and return to opioid-dependent life because of the chronic and relapsing nature of opioid addiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The development of nonaddictive analgesics and anti-opioid vaccines can be potentially effective in opioid abuse prevention and the OUD treatment, but the progresses seem very slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' More options for treatment are needed to combat such destructive diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Opioid receptors are the direct targets of opioids, and medications on them are found effective in opioid addictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' OUD affects intricate molecular and biological activities in the brain involving significant protein- protein interactions (PPI) in various brain areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The development of anti-OUD medications cannot neglect the impact of opioids or medications on the PPI networks of opioid receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In this work, we developed proteome-informed machine learning protocol to study OUD and discover more drug candidates to treat 17 it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' With molecular fingerprints generated by advanced NLP models based on transformer and autoencoder algorithms, gradient boosting decision tree (GBDT) algorithm was used to build our predictive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The consensus predictions from two forms of molecular fingerprints could enhance the predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We used these models to reevaluate the side effects of currently available medications for treating OUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' In addition, these models were used to study the repurposing potentials of existing inhibitors on the major opioid receptors and screened the possible side effects of these inhibitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' The evaluations of ADMET properties were then carried out with machine-learning predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' We identified a group of promising compounds targeting the opioid receptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Considering the therapeutic efficacy by antagonist or agonist effect of currently approved drugs, further animal experiments with these compounds are needed to test the antagonist/agonist properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' More tests in vitro or animal arrays are needed to scrutinize the toxicity and blood-brain barrier permeability characteristics of these candidate compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Automated generation of more drug candidates can be carried out using our generative network modules [18], and this study can be employed for the screening of potential side effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Our machine-learning-based platform provides a novel approach for searching compound candidates to treat OUD and can be generalized to the studies of other diseases with neurological implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' With more advances in understanding the opioid addiction mechanism and more efforts from pharmacological treatment, our platform can be assistive in combating the serious public health issues from OUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Data and code availability The related datasets studied in this work are available at: https://weilab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='msu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='edu/DataLibrary/2D/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Codes are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content='com/WeilabMSU/OUD-PPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE3T4oBgHgl3EQf9Quo/content/2301.04815v1.pdf'} +page_content=' Acknowledgment This work was supported in part by NIH grant GM126189, NSF Grants DMS-2052983, DMS-1761320, and IIS-1900473, NASA 80NSSC21M0023, MSU Foundation, Michigan Economic Development Corporation, George Mason University award PD45722, Bristol-Myers Squibb 65109, and Pfizer.' metadata={'source': 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a/ttFAT4oBgHgl3EQfhR0m/content/tmp_files/2301.08592v1.pdf.txt b/ttFAT4oBgHgl3EQfhR0m/content/tmp_files/2301.08592v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..30950f0d172deeaea681b362af45fddac01c005d --- /dev/null +++ b/ttFAT4oBgHgl3EQfhR0m/content/tmp_files/2301.08592v1.pdf.txt @@ -0,0 +1,1441 @@ +Search for the dipole portal of heavy neutral leptons at future colliders +Maksym Ovchynnikov1, 2, ∗ and Jing-Yu Zhu1, 3, † +1Institut f¨ur Astroteilchen Physik, Karlsruher Institut f¨ur Technologie (KIT), +Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany +2Instituut-Lorentz for Theoretical Physics, Universiteit Leiden, +Niels Bohrweg 2, 2333 CA Leiden, The Netherlands +3School of Physics and Astronomy and Tsung-Dao Lee Institute, +Shanghai Jiao Tong University, 800 Dongchuan Rd, Shanghai 200240, China +In this paper, we study the potential of future colliders to explore the parameter space of heavy +neutral leptons (HNLs) through the dipole portal. We consider hadron colliders such as the LHC in +the high luminosity phase and FCC-hh, and lepton colliders, such as FCC-ee. We consider various +signatures for the HNLs, including the missing energy signature and displaced decays, and discuss +the complementarity between the hadron and lepton colliders. In particular, we find that thanks to +a much clearer environment, FCC-ee may search for the HNLs with masses up to ≃ 30 GeV and +proper lifetimes cτN ≳ 1 cm, which is well beyond the reach of the experiments to be launched in +the next decade. +CONTENTS +I. Introduction +1 +II. HNLs with dipole coupling at colliders +2 +A. Phenomenology +2 +B. Signatures +3 +1. Hadron colliders +3 +2. Lepton colliders +4 +III. Hadron colliders +4 +A. Background +4 +B. Sensitivity +5 +IV. Lepton colliders +6 +A. Backgrounds +6 +B. Sensitivity +7 +1. Selection efficiencies +7 +2. Number of events and sensitivity curves +8 +V. Conclusions +8 +References +10 +A. Events selection at FCC-ee +12 +I. +INTRODUCTION +The neutrino dipole portal of the heavy neutral lepton +(HNL) originates from a six-dimension operator [1]: +Ldipole = ¯L(dWWa +µντa + dBBµν) ˜HσµνN + h.c. , +(1) +where L denotes the SM lepton doublet, Wa +µν and Bµν +stand for the SU(2)L field strength tensor, τa = σa/2 +∗ maksym.ovchynnikov@kit.edu +† zhujingyu@sjtu.edu.cn +with σa being Pauli matrix, ˜H = iσ2H∗ is the conjugate +Higgs field, and N is the HNL gauge singlet. After the +spontaneous symmetry breaking, we have +Leff +dipole =¯ναL(dαFµν − dZZµν)σµνN ++ dW ¯lLWµνσµνN + h.c., +(2) +where the couplings dZ, dW vary within the range +|dZ/dα| ∈ (0, cot θW ), +|dW /dα| ∈ +� +0, +√ +2 +sin θW +� +(3) +and dα stands for the dipole coupling with α = e, µ, τ. +The first works discussing this model were motivated +by the MiniBooNE and LSND anomalies [2, 3]. +Since +then, many studies have been made on the constraints +and sensitivities of future experiments to the dipole por- +tal, especially very recently [4–24]. +The constraints on the HNLs can be briefly summarized +as follows [1, 8]: +– From the viewpoint of outer space, HNL may in- +fluence the stellar cooling process as well as the ex- +pansion and cooling of the universe. The latter can +be scrutinized by analyzing Big Bang Nucleosyn- +thesis and CMB constraints. The above processes +are mainly sensitive to mN with masses below 300 +MeV [25]. +– One of the efficient production mechanisms of +HNLs is the neutrinos up-scattering off electrons +or nucleons[1, 8]. Therefore, the dipole portal may +be constrained in neutrino or dark matter exper- +iments. +Examples include CHARM-II, DONUT, +NOMAD, +LSND, +MiniBooNE, +SBND, +Micro- +BooNE, CHARM-II, DONUT, NOMAD, Borex- +ino, Super-Kamiokande, IceCube, SHiP, DUNE, +COHERENT, NUCLEUS, Xenon1T, SuperCDMS, +etc., see [1, 8, 12]. Because of the limited energy of +neutrino sources, these constraints or sensitivities +to HNL masses can be at most at a few GeV. +arXiv:2301.08592v1 [hep-ph] 20 Jan 2023 + +2 +– At the energy frontier, it is feasible to probe the +existence of much heavier HNL through the dipole +portal. A discussion of the constraints coming from +the LEP and the LHC has been made in [1]. +The currently unexplored parameter space may be +probed in various future experiments. +Relatively light +HNLs with masses mN ≲ O(1 GeV) may be produced +by decays of light mesons such as π, K, and in neu- +trino up-scatterings [1, 8] at neutrino factories. Exam- +ples are currently running SND@LHC [26], FASERν [27] +as well as their future upgrades – advSND [28] and +FASER2/FASERν2 [6], SHiP [29], and DUNE [30]. The +plot showing sensitivities of DUNE and FASER2 to HNLs +coupled to τ neutrinos is shown in Fig. 1. Another poten- +tial probe of the HNL parameter space may come from +Belle II [16]. +DUNE FD +DUNE ND +FASER2 +0.5 +1 +5 +10 +50 +1. × 10-7 +5. × 10-7 +1. × 10-6 +5. × 10-6 +1. × 10-5 +5. × 10-5 +mN [GeV] +|dτ| [GeV-1] +|dZ/dτ| = cot(θW), |dW/dτ| = +2 /sin(θW) +Figure +1: +Sensitivity +of +neutrino +factories +(such +as +DUNE [22] and FASER2 [6]) to HNLs with the dipole coupling +to τ neutrinos. The bounds from past experiments (shown as +gray region) have been obtained from [8]. +However, the sensitivity of neutrino factories quickly +diminishes with the increase of the HNL mass mN. This +is because of the quick decrease of the up-scattering pro- +duction cross-section with mN. Therefore, we need other +experiments to explore heavier HNLs with small cou- +plings. Future ultrahigh energy neutrino telescope may +contribute to the sensitivities of HNL masses as large +as 30 TeV [17]. +To probe masses above 1 GeV and +small couplings, one may consider various signatures with +HNLs at colliders: lepton colliders, such as FCC-ee [31], +CEPC [32, 33], the muon collider [34, 35], and hadron +colliders – the LHC in its high luminosity phase, FCC- +eh [31] and FCC-hh [36]. +The sensitivity of CEPC to the HNLs for the partic- +ular signature of a photon and missing energy has been +obtained in [24]. However, to the best of our knowledge, +a systematic study of the potential of the colliders to +explore the dipole portal is lacking. +This work analyzes how different signatures with HNLs +may be used to search for them at the lepton and hadron +colliders. The paper is organized as follows: in Sec. II, we +study the phenomenology of HNLs at colliders, including +their production and decay, and consider different signa- +tures used to search for these HNLs. We also discuss how +the events originating from the dipole coupling may be +distinguished from those originating from the HNL mix- +ing with neutrinos. Subsequently, the sensitivities to the +dipole portal at hadron and lepton colliders are discussed +in Sec. III and Sec. IV, respectively. Finally, in Sec. V, +we make conclusions. +II. +HNLS WITH DIPOLE COUPLING AT +COLLIDERS +In this section, we discuss details on the production +and decay of HNLs, and how to search for them at col- +liders. +Below, we marginalize over dZ, dW by choos- +ing their maximal possible values (see Eq. (3)), namely, +dZ = cot(θW )dα and dW = +√ +2/ sin(θW )dα. We also as- +sume the Dirac nature of HNLs. +The analysis for the +Majorana HNLs would be completely similar, except for +the twice larger decay length for the fixed coupling and +mass (see a discussion in [8]). +A. +Phenomenology +Experiment +NW +NZ +HL-LHC +6 · 1011 +1.5 · 1011 +FCC-hh +1.2 · 1013 +9 · 1012 +FCC-ee +5 · 108 +5 · 1012 +Table I: The numbers of bosons used in the simplified esti- +mates for the HNL number events in Sec. II, which are taken +from Refs. [31, 37] (for HL-LHC and FCC-ee), or obtained +using MadGraph tree-level simulation [38] (for FCC-hh). +a. +Production. +At colliders, there are two mecha- +nisms of the production of HNLs with masses mN ≳ +1 GeV. The first mechanism is direct production: +l+ + l− → N + ν +(4) +at the lepton colliders, and +p + p → N + ν + X, +p + p → N + l + X +(5) +at the hadron colliders. +The processes with neutri- +nos may go via all the possible couplings dα, dZ, dW in +Eq. (2), while the process with the charged lepton is via +dW . The second production mechanism is via decays of +W and Z bosons, +W → N + l, +Z → N + ν, +(6) +controlled by dW and dZ coupling, respectively. +The +amounts of these bosons at the LHC, FCC-hh, and the +Z-pole mode of FCC-ee are reported in Table I. In the + +3 +low-mass limit mN ≪ mW/Z, the branching ratios of +these decay processes behave as +Br(W → N +l) ≈ d2 +W m3 +W +12πΓW +≈ 6.5·103 +� +dW +GeV−1 +�2 +, (7) +Br(Z → N + ν) ≈ d2 +Zm3 +Z +12πΓZ +≈ 8 · 103 +� +dZ +GeV−1 +�2 +. +(8) +For the HNLs with masses below the W/Z mass, the +prompt production is suppressed compared to the decay +of the heavy bosons. This in particular means that the +muon collider, operating at energies much above the Z +boson mass, is not as efficient in probing the parameter +space of such HNLs as the electron colliders. +b. +Decays. +Decays of HNLs in the mass range mN ≪ +mW,Z occur mainly via the coupling dα. This is because +the decay widths for the processes mediated by dW,Z are +suppressed by m4 +NG2 +F ≪ 1. The main decay channel is +the 2-body decay N → ν + γ. The corresponding decay +width is [1] +ΓN→νγ = d2 +αm3 +N +4π +≈ ΓN,tot +(9) +However, this process does not suit the searches at col- +liders that require observing a displaced decay vertex. +Instead, sub-dominant decay channels should be consid- +ered – the 3-body decays N → f + ¯f + ν, which occur +via the virtual photon. +In the limit mN ≫ 2mf, the +corresponding decay width behaves as [22] +ΓN→νf ¯ +f ≈ +αEM|dα|2m3 +NQ2 +fNf +12π2 +� +log +� +m2 +N +m2 +f +� +− 3 +� +, +(10) +where Qf is the electric charge of the fermion f, while +Nf = 1 for leptons or Nf = 3 for quarks. The branching +ratios of these processes are shown in Fig. 2. +B. +Signatures +Having discussed the phenomenology of HNLs, we may +now consider the signatures at lepton and hadron collid- +ers. +1. +Hadron colliders +a. +Monophoton and missing energy. +One of the pos- +sible signatures is the event with a monophoton and miss- +ing energy, e.g. +q + ¯q → N + ¯ν, +N → ν + γ, +(11) +with the missing energy carried away by neutrinos. This +signature has been analyzed in [1], where the authors uti- +lized ATLAS search [39] performed for the dataset corre- +sponding to the integrated luminosity L = 36.1 fb−1. It +N → ν+ee +N → ν+μμ +N → ν+hadrons +0.001 +0.010 +0.100 +1 +10 +10-6 +10-5 +10-4 +0.001 +0.010 +mN [GeV] +Brprocess, d = 1 GeV-1 +Figure 2: The branching ratios of sub-dominant decays of +HNLs: the di-lepton decays N → ν + l+ + l−, where l = e, µ, +and the hadronic decays N → ν + hadrons, approximated by +the decay N → ν + q + ¯q, where q = u, d, s, c, b. Note that be- +low mN ≃ ΛQCD ∼ 1 GeV, perturbative QCD breaks down, +and the corresponding prediction for the hadronic branching +ratio becomes invalid. +was shown that with this type of search, it might be possi- +ble to probe the couplings higher than d ≳ 10−5 GeV−1. +It is simple to estimate the sensitivity of the LHC in +the high luminosity phase. In the accessible parameter +space, the HNLs have microscopic proper lifetimes cτN ≪ +1 mm, and the probability for the HNL to decay is ≈ 1. +Therefore, the number of events scales as +Nev ∼ NN,prod × Pdecay × ϵsel ∝ L × d2 +Z, +(12) +where NN,prod ∝ L × d2 +Z is the total number of the pro- +duced HNLs, and ϵsel is the selection efficiency, which +we assume will not affect the scaling of the number of +events with the HNL model parameters. Given that the +background also scales with L, the lower bound of the +sensitivity in the plane mN–d changes as L−1/4. For HL- +LHC, LHL = 3000 fb−1, and therefore it should probe a +factor of (3000/38)1/4 ≈ 3 smaller couplings than it was +derived in [1] for the old dataset. We show the corre- +sponding bounds in Figs. 5 and 9. +b. +Displaced +vertices. +HNLs +with +decay +lengths +lN,decay ≳ O(1 mm) and masses mN < mW may be +searched using displaced vertex (DV) techniques. An ex- +ample of such a DV scheme is the scheme [40] used at +CMS to look for HNLs with the mixing coupling. The +scheme utilizes the process chain +p+p → W +X, +W → N +l, +N → l +′++l +′′−+ν, (13) +where l, l′, l′′ are electrons or muons. +To discriminate +HNLs from backgrounds, it is required to detect the final +state leptons l′, l′′, and the prompt lepton l. These par- +ticles must have kinematic properties that satisfy some +selection criteria. Examples of such properties are large +enough transverse momentum and transverse impact pa- +rameters. + +4 +2. +Lepton colliders +a. +Monophoton and missing energy. +Similarly to the +hadron colliders, it may be possible to search for the +HNLs via the missing energy signature at the lepton col- +liders. The process of interest is [1] +l+ + l− → N + ¯ν, +N → γ + ν +(14) +Similarly to the case of the analogical search at the LHC, +it may be possible to derive the sensitivity of FCC-ee +from the result of the older searches at DELPHI. The +upper bound from LEP on the cross-section of the process +e+e− → γ + inv obtained at the Z pole mode is [41–43] +σDELPHI +mono-γ += 0.1 pb, where for the energy of the photon +and its polar angle it was required Eγ > 0.7 GeV and +| cos(θγ)| < 0.7. Given the similar background at FCC- +ee and LEP (and in particular that both LEP and FCC- +ee are free from pileup), and assuming conservatively the +same detector properties of FCC-ee as for LEP, the lower +bound of the sensitivity of FCC-ee would be +σFCC-ee +mono-γ +σFCC-ee +mono-γ +≃ +� +LLEP +Z-pole +LFCC-ee +Z-pole +� 1 +4 += 3.3 · 10−2 +(15) +where LLEP +Z-pole = 0.2 fb−1 and LFCC-ee +Z-pole = 150 · 103 fb−1. +We show the corresponding sensitivity in Figs. 8 and 9. +Note that this simple estimate roughly agrees with the +sensitivity of CEPC that has been recently computed +in [24]. +Let us now discuss how to distinguish leptonic decays +of the HNLs that have either the mixing or the dipole +couplings. The simplest way would be to check the pres- +ence of the leptons of different flavors in the lepton pair: +such type of decays is common for the HNLs with the +mixing coupling (it occurs via the charged current) [44] +but is highly suppressed for the HNLs with the dipole +coupling. Another way would be to compare the distri- +bution of the lepton pair in invariant mass. For the dipole +coupling, the leptons appear via a virtual photon. There- +fore, the distribution has the maximum at minv = 2me +and quickly drops with the increase of minv. In contrast, +for the mixing coupling, the mediator is a heavy W/Z, +the corresponding propagator is a constant, and the dis- +tribution is rather flat in the range 0 < minv < mN, see +Fig. 3. +b. +Displaced decays +Another way to search for HNLs +may be to look for the displaced decays. +Unlike the +LHC, lepton colliders have a much cleaner background; +in particular, no pile-up events [45, 46]. +Therefore, it +may be much simpler to distinguish a hypothetical SM +background and the signal from decaying HNLs. In par- +ticular, instead of searching for the events with prompt +leptons, one may consider only the events with the dis- +placed vertex – the Z boson decays +e+ + e− → Z, +Z → N + ν, +N → l+ + l− + ν +(16) +To summarize, we conclude that there is a comple- +mentarity between the mentioned signatures at colliders +Mixing coupling +Dipole coupling +0 +5 +10 +15 +20 +10-5 +10-4 +0.001 +0.010 +0.100 +1 +10 +mee [GeV] +Probability density +N → νe+e-, mN = 20 GeV +Figure 3: The distribution of the electron-positron pair from +the HNL decay N → νe+e− in the invariant mass mee = +� +(pe+ + pe−)2, assuming the mixing (the red histogram) or +the dipole (the blue histogram) coupling of the HNL to the +SM particles. +and the non-collider experiments. While the latter would +probe mainly dα, the former may explore the couplings +dZ, dW . The displaced vertex signatures may contribute +to the sensitivity only if dZ,W ̸= 0, since these couplings +determine the production of the HNLs. In contrast, the +missing energy signature may still provide the sensitiv- +ity, given that the HNL production, in this case, is also +controlled by dα. +In addition, we should stress another complementarity +– between the lepton colliders may mainly probe the dZ +coupling, the hadronic colliders suit better for probing +dW . +III. +HADRON COLLIDERS +A. +Background +In [40], the search for HNLs with the mixing coupling +has been performed using the statistics accumulated dur- +ing 2016-2018, corresponding to the integrated luminos- +ity 138 fb−1 at CMS. The results of this search may be +extrapolated to the high-luminosity LHC, with the cor- +responding scaling of the SM background. +In order to reduce backgrounds, the following selection +cuts have been imposed: +– One prompt lepton l1 and two displaced leptons l2,3 +within the pseudorapidity range |η| < 2.5. +– Prompt electron (muon): pT > 30 − 32 (25) GeV, +transverse impact parameter |d0| < 0.05 cm and +longitudinal impact parameter |dz| < 0.1 cm. +– Displaced electrons (muons): +pT > 7 (5) GeV, +|d0| > 0.01 cm, |dz| < 10 cm. The total transverse +momentum of the two displaced leptons should be +pT,23 > 15 GeV. + +5 +– The invariant mass of 3 leptons should be within +50 GeV < √s123 < 80 GeV; the invariant mass of +the displaced leptons √s23 should not be close to +the invariant mass of the SM resonances (such as +ω, φ, J/ψ,. . . ). +– Angular constraints: the angle between the HNL +direction (assumed to be given by the vector from +the primary vertex to the secondary vertex) and +the direction given by the total momentum of l2, p3 +is cos(θSV,23) > 0.99; the azimuthal separation +between the prompt and each of the displaced +leptons should be |∆φ(l1, l2/3)| > 1; the angu- +lar separation between l2,3 should be ∆R(l2, l3) = +� +∆η2 +23 + ∆φ2 +23 < 1. +– Maximal displacement constraints: displaced ver- +tex within the tracker, i.e. +the transverse dis- +tance ∆2D < 0.5 m and the longitudinal distance +∆|| < 3 m. +The reconstruction efficiency for the prompt leptons is ≃ +90%. The reconstruction efficiency for displaced leptons +depends on the lepton type, its relative isolation, and the +displacement. In particular, for the displacement 10 (25) +cm, depending on the relative isolation, the efficiency for +the electron reconstruction varies in the limits from 20%- +40% to 60%-80% (15%-20% to50%-60%), while for the +muons with the displacement 10 (50) cm the numbers +change to from 85%-90% to 95% (40%-50% to 80%). +Backgrounds for this selection set may come from the +events with misidentified hadrons, muons from pion or +kaon decays, and leptons coming from decays of heavy +flavor hadrons. For the luminosity corresponding to the +data set collected at CMS in 2016-2018, the total number +of predicted background events is ≃ 100 − 200. The col- +lected data was in agreement with the theoretical back- +ground prediction, which was used to impose the exclu- +sion bound on the parameter space of the HNLs with the +mixing coupling. +B. +Sensitivity +Let us estimate the sensitivity of this scheme to the +HNLs with the dipole coupling. +We will consider the +LHC in its high luminosity phase (HL-LHC) and FCC- +hh, assuming for the latter the same search scheme as +for the LHC. The parameters of these two detectors are +summarized in Table II. Due to larger energies, the back- +Detector +|η| +R × L +CMS@LHC < 2.5 0.5 m × 3 m +FCC-hh +< 4 +1.6 m × 5 m +Table II: Parameters of the trackers at CMS@LHC and +the FCC-hh reference design detector: pseudorapidity cov- +erage, transverse and longitudinal size. The values are taken +from [47] and [36]. +ground at the FCC-hh may qualitatively change. There- +fore, we will present the sensitivity of the FCC-hh in the +form of iso-contours. +We start with evaluating the selection efficiency for the +signal. We define it as +ϵselection ≡ +� +l=e,µ Br(N → νl+l−) × ϵll +sel +� +l=e,µ Br(N → νl+l−) +, +(17) +where ϵll +sel is the selection efficiency for the decay into +a lepton pair l+l−. +For simplicity, we perform a pure +MC simulation, where the kinematics reconstruction ef- +fects are not considered. For the LHC, we approximate +the displaced leptons reconstruction efficiency by a linear +function of the transverse displacement, adopting con- +servatively the lowest values reported in [40] for the in- +terpolation points. As a cross-check of the calculations, +we have reproduced the sensitivity to HNLs with the +mixing coupling reported in [40] within a factor of 1.5, +which is appropriate given the simplicity of the simula- +tion. For the FCC-hh, we assume unit displaced leptons +reconstruction efficiency, motivated by a possible devel- +opment of technologies at the time of the construction +of FCC-hh. Compared to the CMS@LHC case, we also +change the pseudorapidity/displacement cuts due to the +changed tracker size (see Table II), leaving the other cuts +unchanged. +The mass and lifetime dependence of ϵselection for the +HNLs with the dipole coupling case is shown in Fig. 4. +From the figure, we see that for HNLs with mass mN ≲ +10 GeV, ϵselection does not exceed ≃ 10−2 at the LHC. +The corresponding values at the FCC-hh are at least one +order of magnitude larger. This is a combined effect of +the larger tracker volume and the unit displaced leptons +reconstruction efficiency. For the fixed decay length, the +efficiency increases with the HNL mass. The reason is an +increase of the pT of the produced leptons relative to the +direction of the incoming HNL, and hence the transverse +impact parameter. +The number of events is given by +Nevents = NW × Br(W → N + l)× +× +� +l=e,µ +Br(N → νl+l−) × ϵselection +(18) +The behavior of the number of events with the coupling +for the fixed mass is shown in Fig. 5. The number of +events at the FCC-hh is a factor of a few hundred larger +than at the LHC. This increase is due to the larger selec- +tion efficiency and a gain in the luminosity and W boson +production cross-section. +The sensitivities of the searches for the displaced ver- +tices at the HL-LHC and FCC-hh assuming the coupling +of the HNLs to the electron flavor are shown in Fig. 5. +Although the sensitivity of the LHC is completely within +the sensitivity of DUNE, it may still be a useful probe +of the dipole portal, since it probes not only the dα cou- +pling, but also the coupling to W bosons, and hence the +LHC is complementary to other probes. + +6 +LHC, mN = 1 GeV +LHC, mN = 5 GeV +FCC-hh, mN = 1 GeV +FCC-hh, mN = 5 GeV +10-4 +0.001 +0.010 +0.100 +1 +10 +10-5 +10-4 +0.001 +0.010 +0.100 +cτN [m] +ϵselection +LHC, cτN = 0.1 m +LHC, cτN = 1 mm +FCC-hh, cτN = 0.1 m +FCC-hh, cτN = 1 mm +0.5 +1 +2 +5 +10-5 +10-4 +0.001 +0.010 +0.100 +mN [GeV] +ϵselection +Figure 4: Selection efficiencies for the events with decaying HNLs at CMS@LHC and at FCC-hh, assuming the same experi- +mental setup as at CMS@LHC. Left panel: as a function of the HNL decay length for several choices of its mass. Right panel: +as a function of the HNL mass. +FCC-hh, mN = 3 GeV +FCC-hh, mN = 5 GeV +LHC, mN = 1 GeV +LHC, mN = 2 GeV +1. × 10-7 +5. × 10-7 1. × 10-6 +5. × 10-6 1. × 10-5 +0.1 +1 +10 +100 +1000 +dα [GeV-1] +Nevents +WN+e, Nνe+e-, |dW/dα| = +2 /sin(θW) +LHCdispl +LHCγ+Emiss +FCC-hh50 events +FCC-hh100 events +DUNE ND +0.5 +1 +2 +5 +1. × 10-7 +5. × 10-7 +1. × 10-6 +5. × 10-6 +1. × 10-5 +5. × 10-5 +mN [GeV] +|de|, GeV-1 +|dW/de| = +2 /sin(θW) +Figure 5: Top panel: the behavior of the number of events +for HNLs with different masses as a function of the dipole +coupling dα. Bottom panel: the potential of the hadron col- +liders – high luminosity LHC and FCC-hh – to probe the HNL +parameter space. For the LHC case, we show the sensitivities +coming from two signatures (Sec. II B 1): the dilepton dis- +placed vertex searches, for which we report the 90% CL limit, +as well as the projected sensitivity from the searches for the +events with mono γ and missing energy at ATLAS. In the +case of the FCC-hh, we show the iso-contours corresponding +to 50 and 100 events. +In the same figure, we also show the projected limits +for the parameter space that may be probed by the mono +γ searches (remind Sec. II B 1). Because of a huge back- +ground, this search cannot explore unconstrained HNL +couplings. +IV. +LEPTON COLLIDERS +A. +Backgrounds +Lepton colliders are free from pileup and have a low +beam-induced background. Therefore, for the given pro- +cess with an HNL, +e+ + e− → Z → N + ν → Y + ¯Y + ν, +(19) +where Y, ¯Y denote visible HNL decay products, the only +possible background comes from single events of e+, e− +collisions. The latter includes Z boson decays +e+ + e− → Z → f + ¯f → Y + ¯Y + inv, +(20) +where f = l = e, µ, τ or q = u, d, s, c, b, and prompt +4-fermion production +e+ + e− → f + ¯f ′ + f ′′ + ¯f ′′′ → Y + ¯Y + inv, +(21) +see Fig. 6. +By “inv”, we denote the particles that leave the detec- +tor invisibly; examples include neutrinos or the particles +that have not been detected due to the inefficiency of the +detector. +In [48], a preliminary background analysis for FCC-ee +has been performed for the minimal HNL model with +the mixing coupling. +For the particular decay process +N → e+ + e− + ν, backgrounds from the decays of Z +bosons have been considered. +The simulation started +by generating events in MadGraph [38], followed by +Pythia8 [49] for the hadronization and DELPHES [50] for +the simulation of the detector response. The background +reduction has been studied using pre-selection cuts, i.e. +without requiring the candidates Y , ¯Y to form a good +vertex. +The selection started from the requirement to +have the visible final state consisting solely of a pair of + +7 +l− +l+ +X +¯X +(c) +Z +¯ν +l+ +l− +ν +(a) +¯X +X +inv +(b) +l− +l+ +e− +e+ +ν +¯ν +(d) +e+ +e− +N +Figure 6: Events at lepton colliders. An event (19) with an HNL decaying into a pair of charged leptons l+, l− (the diagram +(a)), and possible background processes to it: decays Z → X ¯ +X → l+ + l− + inv (the diagrams (b), (c)), as well as 4-fermion +process e+e− → l+l− + ν + ¯ν. +e+, e− particles. Then, the event was required to have +non-zero missing momentum /p = |pe+ + pe−| > 10 GeV, +to account for finite momentum reconstruction resolution +and remove a huge fraction of background from decays +Z → ee. +Then, the cut on the transverse impact pa- +rameter, the minimal distance |d0| > 0.5 mm from the +track helical trajectory to the beam line, has been ap- +plied to both e+ and e−. This selection allowed reducing +backgrounds from promptly produced e+, e−. +In total, the pre-selection reduced backgrounds down +to ∼ 105 - mostly coming from +Z → τ + ¯τ → e+ + e− + inv +(22) +Therefore, an additional selection is needed to remove +the background. In addition, the impact parameter cut +harms the sensitivity to short-lived HNLs, being in par- +ticular much more restrictive than the requirement for +the vertex displacement rdispl > 400 µm used in [48] to +demonstrate the potential of FCC-ee to explore the pa- +rameter space of the HNLs with the mixing coupling (see +also Fig. 7). +An examination of the kinematics for the process (22) +and the signal (remind Sec. II A) suggests that the +amount of the remaining background events may be sig- +nificantly reduced if imposing the cut on the angle be- +tween two electrons from above and their energies from +below, see Appendix A. In particular, performing the +pure MC simulation of the decays (22), we have found +that the cut +cos(θee) > −0.5, +Ee+ > 2 GeV, +Ee− > 2 GeV (23) +leaves no events even before imposing the |d0| cut while +keeping a large signal selection efficiency independent +of the lifetime of the HNL. Apart from this selection, +the background may also be reduced by requiring the +electron-positron pair to form a good vertex (with e.g. +a small distance of the closest approach between their +tracks). +It may suggest that the |d0| cut may be re- +laxed to allow for probing short-lived HNLs, as the se- +lection (23) should also work properly for the other Z +decays. +Realistic simulations are required to examine +this question further, which is left for future work. +However, the cuts (23) are not efficient in the case of +the 4-fermion production processes (21). To examine this +question, we have simulated the purely leptonic process +e+ + e− → e+ + e− + ν + ¯ν +(24) +in MadGraph. +The total cross section of this process +requiring pT,l,ν > 0.1 GeV has been found at the level of +σee→eeνν ≈ 1.7 pb, which results in NZ · σee→eeνν +σee→Z +≈ 2·108 +of such events during the Z-pole mode timeline. +The +e+, e− pair typically originates from the same vertex and +hence may be as collimated as the signal, while neutrinos +carry away missing momentum. +However, the 4-fermion process is prompt. Unlike the +background coming from the decays of Z, the produced +e+, e− pair has zero displacement from the collision point. +To reduce this background to zero, one may additionally +require non-zero displacement of the vertex formed by +the e+, e− pair. The exact cut depends on the spatial +resolution of the tracker. We will exploit two different +choices for the displacement cut: +rdispl > 0.4 mm, +or rdispl > 0.1 mm +(25) +The cuts considered in [48] and the pre-selection we pro- +pose in this work are summarized in Table III. +B. +Sensitivity +1. +Selection efficiencies +The signal efficiency for the selection criteria from Ta- +ble III for various HNL masses and decay lengths, con- + +8 +Selection cuts +Ref. [48] +Only e+, e− in an event, /p > 10 GeV +|d0| > 0.5 mm +This work +Only l+, l− in an event, /p > 10 GeV +cos(θll) > −0.5, El+, El− > 2 GeV +rdispl > 0.4 mm, or rdispl > 1 µm +Table III: Summary of the selection cuts required to remove +the background for different HNL decay processes, as imposed +in [48] and considered in this work. +Here, d0 denotes the +transverse impact parameter of any of the two tracks, /p = +| � preconstructed| corresponds to the missing momentum in +an event, θab is the angle between the two particles a, b, and +rdispl is the vertex displacement from the collision point. +sidering both the mixing and dipole couplings, is shown +in Fig. 7. +Let us first consider the cuts set from [48]. We repro- +duce the values of the efficiencies reported for particular +masses and lifetimes of HNLs with the mixing coupling +in Table 3 of this paper. The figures show that the cuts’ +impact depends significantly on the HNL mass and life- +time. +The efficiency, being ≈ 1 for ldecay ≫ 0.5 mm +independently on the HNL mass, starts dropping at +ldecay ≃ 1 cm. For the given decay length, the decrease +of ϵ is larger for smaller HNL masses. The reason is that +the impact parameter (and hence the efficiency) of the +decay products is higher if they gain large pT relatively +to the direction of the HNL, and the magnitude of pT +is controlled by the HNL mass. The efficiency for the +dipole coupling case has similar behavior. However, the +impact of efficiency however is less severe. Indeed, be- +cause of the kinematics of the decay process N → l+l−ν +(remind Sec. II A), in the dipole case, the leptons typi- +cally gain smaller energies and than in the mixing case. +Due to this feature, their deflection relative to the HNL +is larger, which results in a larger IP on average. +For the cuts set proposed in this paper, the situation +is different. The decrease at small lifetimes is obviously +less significant. As for the mN behavior, the efficiency +slightly drops once mass increases because of an increase +of the mean angle between leptons with mN. In particu- +lar, for heavy HNLs with mN ≃ mZ a sizable fraction of +events may have cos(θ) < −0.5. This effect is more sig- +nificant for HNLs with mixing because of the process’s +kinematics. On the other hand, since leptons produced +via the dipole coupling are less energetic, the efficiency +is lower at low HNL masses because of the El cut. +2. +Number of events and sensitivity curves +Let us now estimate the sensitivity of FCC-ee to HNLs. +We will consider the reference Innovative Detector for +Electron–positron Accelerators (IDEA), which is a cylin- +der having the radius r = 4.5 m and longitudinal size +L = 11 m [48]. The other reference detector, CLD, has +very similar specifications, and therefore the sensitivity +would be completely similar. +The expected number of events with decays of HNLs +at IDEA@FCC-ee is +Nevents = 2·NZ ·BrZ→N+ν × +� +l=e,µ +BrN→l+l−ν ×ϵ(l) +sel, (26) +where ϵsel = ϵsel(mN, dα) is the fraction of events with +HNLs decaying inside the decay volume and that sat- +isfy the selection cuts from Table III. In the limit when +ldecay,N ≫ O(1 mm), the displacement selection has unit +efficiency, and ϵsel becomes decay length-independent: +ϵsel ≈ ϵ(mN) +π +π +� +0 +dθ +� +exp +� +− +lmin +ldecay,N +� +− exp +� +−lmax(θ) +ldecay,N +�� +, +(27) +where the integration is performed over all directions of +the cylindrical decay volume of IDEA. +The sensitivity of the FCC-ee to the HNLs with the +dipole coupling is shown in Fig. 8, where we also in- +clude the sensitivity of the missing energy search (remind +Sec. II B 2). To fix the excluded parameter space, we as- +sume dα = dµ. We stress however that the sensitivity of +the FCC-ee is flavor-universal since both the production +and decay of the HNL are flavor-agnostic. +From the figure, we conclude that depending on the dis- +placement cut, with the displaced decay searches FCC- +ee may probe the HNLs with masses up to mN = 30 +GeV. The upper bound of the sensitivity is caused by +the HNL decay vertex displacement selection. The shape +of the lower bound is changing: below mN ≃ 3 GeV it +gets smoothly improved, while at larger masses it be- +comes plateau. The reason is that at small masses, the +HNL decay length at the lower bound is lN,decay ≫ 1 m, +and therefore the decay probability scales as Pdecay ≈ +lN,decay/lfid ∝ m−4 +N , where the scaling comes from the +behavior of the HNL decay width (9) and the γ factor. +At large masses, the HNL decay length becomes small +enough such that HNLs have a unit probability of decay- +ing inside the detector. The lower bound in this case is +determined by the condition NN,prod × ϵsel > 2.3, which +is almost mass-independent in the mass range of interest. +The missing energy search is complementary compared +to the displaced decays. Namely, it cannot probe as small +couplings as probed by the displaced decay search be- +cause of significant background. However, it may explore +higher HNL masses, since there is no displacement cut. +V. +CONCLUSIONS +In this paper, we have analyzed the potential of hadron +and lepton colliders to probe the parameter space of +HNLs with the dipole coupling. +We have first discussed the phenomenology of HNLs – +including their production, decays, and possible signa- +tures – at the LHC, FCC-hh, and FCC-ee (Sec. II A). +We have also commented on how to distinguish decays of + +9 +FCC-ee reach for rdispl > 0.4 mm +Cuts from [2205.05502], mixing +Cuts from this work, mixing +Cuts from [2205.05502], dipole +Cuts from this work, dipole +10-5 +0.001 +0.100 +10 +10-5 +10-4 +0.001 +0.010 +0.100 +1 +ldecay = cτNpN/mN [m] +ϵselection +N→νe+e-, mN = 20 GeV +Cuts from [2205.05502], mixing +Cuts from this work, mixing +Cuts from [2205.05502], dipole +Cuts from this work, dipole +5 +10 +20 +50 +10-5 +10-4 +0.001 +0.010 +0.100 +1 +mN [GeV] +ϵselection +N→νe+e-, ldecay = 0.0005 mm +Figure 7: Selection efficiency for the process N → e+e−ν (for both the mixing and dipole couplings) based on the cuts from +Table III: the ones considered in [48] (the blue lines), and the ones discussed in this work (the red lines), assuming the minimal +displacement ldispl > 0.4 mm. The left panel: as a function of the HNL decay length lN,decay = cτNpN/mN for the fixed HNL +mass mN = 30 GeV. The vertical dashed gray line denotes the minimal decay length of the HNL with the mixing coupling to +which FCC-ee may be sensitive if requiring only the displacement rdispl > 0.4 mm (from [48]). The right panel: as a function +of the HNL mass for the fixed HNL decay length lN,decay = 0.5 mm. +Displ0.4 mm +Displ0.1 mm +γ+Emiss +0.5 +1 +5 +10 +10-8 +10-7 +10-6 +10-5 +mN [GeV] +|dμ| [GeV-1] +|dZ/dμ| = cot(θW) +Figure 8: The potential of FCC-ee to probe the parameter +space of the HNLs with the dipole coupling, see Sec. II B 2. +The solid and short-dashed dark blue lines show the 90% +CL sensitivity corresponding to the displaced decay signa- +ture, assuming the event selection considered in this paper +(see Sec. IV A and Table III). The long-dashed lighter blue +line denotes the sensitivity corresponding to the γ+missing +energy signature. +HNLs with mixing and dipole couplings. Thanks to the +different working modes of the lepton and hadron collid- +ers, they complement each other in exploring the param- +eter space of HNLs: the hadron colliders may probe the +coupling of HNLs to W bosons, while the lepton colliders +are more efficient in probing the coupling to Z bosons. +In addition, because of the production channels of HNLs, +from decays of W, Z bosons, as well as due to the small +distance from the production point to the decay volume, +the colliders may probe the parameter space in the mass +range inaccessible to neutrino factories such as DUNE +and FASER2. +Then, +we +have +considered +the +hadron +colliders +(Sec. III), utilizing the search for displaced vertices with +dileptons at CMS as well as the missing energy searches +at ATLAS. We have derived the sensitivity of the LHC +in the high luminosity phase and estimated the potential +of FCC-hh (Fig. 5). A detailed background study for the +FCC-hh case is required, which however goes beyond the +scope of this paper. +Next, we have considered the lepton colliders, see +Sec. IV, concentrating on FCC-ee. We have first made +a simplified background analysis demonstrating that the +HNL decay signal may be clearly distinguished from the +background using kinematic properties (Sec. IV A). De- +pending on the model parameters, it may be possible to +probe the HNL masses up to mN ≃ 30 GeV, see Fig. 8. +The final plot combining the sensitivities of lepton and +hadron colliders is shown in Fig. 9, where we marginalize +over the couplings to Z, W assuming their maximal pos- +sible values. From the figures, we conclude that FCC-ee +may explore the HNL masses up to mN ≃ 30 GeV, while +the exploration potential of the hadron colliders is lim- +ited by mN ≃ 3 GeV. This is due to the different back- +ground environments of these colliders: FCC-ee is free +from pileup events, and therefore background is much +cleaner, which allows for softer selection which keeps high +efficiency for events with HNLs and simultaneously effi- +ciently reduces the yield of the pure SM events. +ACKNOWLEDGEMENTS +We thank Juliette Alimena, Suchita Kulkarni, and Re- +beca Gonzalez Suarez for discussing the background esti- +mates at FCC-ee performed in [48], and Lesya Shchutska +for discussing the backgrounds at the LHC. This project +has received support from the European Union’s Horizon +2020 research and innovation program under the Marie +Sklodowska-Curie grant agreement No. 860881-HIDDeN. +Jing-yu Zhu is grateful for the support from the China +and Germany Postdoctoral Exchange Program from the + +10 +DUNE +FCC-ee +LHC +FCC-hh50 events +FCC-eeγ+Emiss +0.5 +1 +5 +10 +10-8 +10-7 +10-6 +10-5 +mN [GeV] +|de|, GeV-1 +|dZ/de| = cot(θW), |dW/de| = +2 /sin(θW) +DUNE +FCC-ee +LHC +FCC-hh50 events +FCC-eeγ+Emiss +0.5 +1 +5 +10 +10-8 +10-7 +10-6 +10-5 +mN [GeV] +|dμ|, GeV-1 +|dZ/dμ| = cot(θW), |dW/dμ| = +2 /sin(θW) +DUNE ND +DUNE FD +FCC-ee +FASER2 +FCC-eeγ+Emiss +0.5 +1 +5 +10 +10-8 +10-7 +10-6 +10-5 +mN [GeV] +|dτ|, GeV-1 +|dZ/dτ| = cot(θW), |dW/dτ| = +2 /sin(θW) +Figure 9: Potential of colliders – FCC-ee, LHC in the high luminosity phase, and FCC-hh – to explore the parameter space of +HNLs with the dipole coupling. 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A small +fraction of events with a small angle between the momenta of e+, e− have the following pattern: one of the particles +from the pair has very small energy, Ee± ≪ mZ/2. +Background (ττ) +Signal (HNLs mixing) +Signal (HNLs dipole) +-1.0 +-0.5 +0.0 +0.5 +1.0 +10-4 +0.001 +0.010 +0.100 +1 +10 +cos(θee) +Fraction +mN = 20 GeV, N → νe+e- +Figure 10: The distribution of the e+e− pair in cosine of the angle between the e+, e− at FCC-ee in the Z-pole operating +mode. Three processes are considered: the background process Z → τ ¯τ → e+e−¯νeνe¯ντντ, and the HNL decays N → e+e−ν, +assuming the dipole and the mixing couplings (mixing with νe is considered). +The detector reconstruction effects are not +included. +The situation with the signal is different: the angle distribution between the e+, e− originated from the HNL decay +is peaked at θ = 0, and the situation remains the same even for heavy HNLs mN ≃ mZ. Therefore, the background +yield may be reduced without a significant impact on the signal if one requires a cut on cos(θee) and Ee+, Ee− from +below. +To estimate the effect of such a cut on the background and signal, we have simulated ≃ 5 · 109 decays Z → +τ ¯τ → e+e−νe¯νeντ ¯ντ, which corresponds to the full statistics expected during the full timeline of FCC-ee in the Z +pole mode [48]. +In the simulation, we included neither finite detector reconstruction resolution1 nor the particle +identification efficiency. Therefore, its predictions should be validated with full-scale simulations. +The distribution in cos(θee) for the e+e− pair from the background and the decays N → e+e−ν, considering both +the models of the dipole portal and the minimal HNL model, is shown in Fig. 10. With the simulated sample, we +have reproduced the selection efficiencies reported in Table 2 of [48] for the process Z → ττ → ee. Next, we found +that the cut +cos(θee) > −0.5, +Ee+ > 2 GeV, +Ee− > 2 GeV +(A2) +reduces the number of backgrounds to zero even before imposing the d0 and /p cuts. +The same conclusion may hold for other decays Z → f ¯f → e+e− + inv. The e+, e− pair originates either from +the single process f → · · · → e+e− + inv2 such that ¯f → · · · → inv (the diagram (b) in Fig. 6), or from the two +independent processes f → · · · → e− + inv, ¯f → · · · → e+ + inv (the diagram (c)). By inspecting the decay modes +of possible products of f in [51] and assuming a perfect detector efficiency in detecting charged particles and neutral +long-lived mesons such as K0 +L (via deposition in HCAL), we have not found the combination f ¯f which may lead to +the diagram (b). Therefore, we conclude that this category of events may be a subject of the detector inefficiency +only. The level of this inefficiency is to be determined by the full-scale simulations. +Nevertheless, we believe that the combination of the presented cuts in addition to the vertex criteria (such as the +small distance of the closest approach between the tracks) would allow reducing the background from Z boson SM +1 Nevertheless, as is demonstrated in [48], FCC-ee has perfect re- +construction capabilities of both the lepton energies and momen- +tum (and hence cos(θee)). +2 Here, if f is a quark, by . . . we mean a hadronization. + +13 +decays to zero. Further, we will assume zero background from the processes of the type Z → f ¯f → Y ¯Y + inv, +constituting the background for various decay modes of the HNL N → Y ¯Y + ν. This may be especially the case +for the decays e.g. N → q + ¯q + ν, for which the background process, Z → q + ¯q, would have even more marginal +kinematics. + diff --git a/ttFAT4oBgHgl3EQfhR0m/content/tmp_files/load_file.txt b/ttFAT4oBgHgl3EQfhR0m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..46fa838f73ac653bf0c1e7d04438446ec7f9c0d2 --- /dev/null +++ b/ttFAT4oBgHgl3EQfhR0m/content/tmp_files/load_file.txt @@ -0,0 +1,740 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf,len=739 +page_content='Search for the dipole portal of heavy neutral leptons at future colliders Maksym Ovchynnikov1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' ∗ and Jing-Yu Zhu1,' 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Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Universiteit Leiden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Niels Bohrweg 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 2333 CA Leiden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The Netherlands 3School of Physics and Astronomy and Tsung-Dao Lee Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Shanghai Jiao Tong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 800 Dongchuan Rd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Shanghai 200240,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' China In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' we study the potential of future colliders to explore the parameter space of heavy neutral leptons (HNLs) through the dipole portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We consider hadron colliders such as the LHC in the high luminosity phase and FCC-hh, and lepton colliders, such as FCC-ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We consider various signatures for the HNLs, including the missing energy signature and displaced decays, and discuss the complementarity between the hadron and lepton colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In particular, we find that thanks to a much clearer environment, FCC-ee may search for the HNLs with masses up to ≃ 30 GeV and proper lifetimes cτN ≳ 1 cm, which is well beyond the reach of the experiments to be launched in the next decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' CONTENTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Introduction 1 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' HNLs with dipole coupling at colliders 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Phenomenology 2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Signatures 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Hadron colliders 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Lepton colliders 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Hadron colliders 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Background 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Sensitivity 5 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Lepton colliders 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Backgrounds 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Sensitivity 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Selection efficiencies 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Number of events and sensitivity curves 8 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Conclusions 8 References 10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Events selection at FCC-ee 12 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' INTRODUCTION The neutrino dipole portal of the heavy neutral lepton (HNL) originates from a six-dimension operator [1]: Ldipole = ¯L(dWWa µντa + dBBµν) ˜HσµνN + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' , (1) where L denotes the SM lepton doublet, Wa µν and Bµν stand for the SU(2)L field strength tensor, τa = σa/2 ∗ maksym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='ovchynnikov@kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='edu † zhujingyu@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='cn with σa being Pauli matrix, ˜H = iσ2H∗ is the conjugate Higgs field, and N is the HNL gauge singlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' After the spontaneous symmetry breaking, we have Leff dipole =¯ναL(dαFµν − dZZµν)σµνN + dW ¯lLWµνσµνN + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=', (2) where the couplings dZ, dW vary within the range |dZ/dα| ∈ (0, cot θW ), |dW /dα| ∈ � 0, √ 2 sin θW � (3) and dα stands for the dipole coupling with α = e, µ, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The first works discussing this model were motivated by the MiniBooNE and LSND anomalies [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Since then, many studies have been made on the constraints and sensitivities of future experiments to the dipole por- tal, especially very recently [4–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The constraints on the HNLs can be briefly summarized as follows [1, 8]: – From the viewpoint of outer space, HNL may in- fluence the stellar cooling process as well as the ex- pansion and cooling of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The latter can be scrutinized by analyzing Big Bang Nucleosyn- thesis and CMB constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The above processes are mainly sensitive to mN with masses below 300 MeV [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' – One of the efficient production mechanisms of HNLs is the neutrinos up-scattering off electrons or nucleons[1, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Therefore, the dipole portal may be constrained in neutrino or dark matter exper- iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Examples include CHARM-II, DONUT, NOMAD, LSND, MiniBooNE, SBND, Micro- BooNE, CHARM-II, DONUT, NOMAD, Borex- ino, Super-Kamiokande, IceCube, SHiP, DUNE, COHERENT, NUCLEUS, Xenon1T, SuperCDMS, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=', see [1, 8, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Because of the limited energy of neutrino sources, these constraints or sensitivities to HNL masses can be at most at a few GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='08592v1 [hep-ph] 20 Jan 2023 2 – At the energy frontier, it is feasible to probe the existence of much heavier HNL through the dipole portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' A discussion of the constraints coming from the LEP and the LHC has been made in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The currently unexplored parameter space may be probed in various future experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Relatively light HNLs with masses mN ≲ O(1 GeV) may be produced by decays of light mesons such as π, K, and in neu- trino up-scatterings [1, 8] at neutrino factories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Exam- ples are currently running SND@LHC [26], FASERν [27] as well as their future upgrades – advSND [28] and FASER2/FASERν2 [6], SHiP [29], and DUNE [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The plot showing sensitivities of DUNE and FASER2 to HNLs coupled to τ neutrinos is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Another poten- tial probe of the HNL parameter space may come from Belle II [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' DUNE FD DUNE ND FASER2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 1 5 10 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-5 mN [GeV] |dτ| [GeV-1] |dZ/dτ| = cot(θW), |dW/dτ| = 2 /sin(θW) Figure 1: Sensitivity of neutrino factories (such as DUNE [22] and FASER2 [6]) to HNLs with the dipole coupling to τ neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The bounds from past experiments (shown as gray region) have been obtained from [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' However, the sensitivity of neutrino factories quickly diminishes with the increase of the HNL mass mN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' This is because of the quick decrease of the up-scattering pro- duction cross-section with mN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Therefore, we need other experiments to explore heavier HNLs with small cou- plings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Future ultrahigh energy neutrino telescope may contribute to the sensitivities of HNL masses as large as 30 TeV [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' To probe masses above 1 GeV and small couplings, one may consider various signatures with HNLs at colliders: lepton colliders, such as FCC-ee [31], CEPC [32, 33], the muon collider [34, 35], and hadron colliders – the LHC in its high luminosity phase, FCC- eh [31] and FCC-hh [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The sensitivity of CEPC to the HNLs for the partic- ular signature of a photon and missing energy has been obtained in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' However, to the best of our knowledge, a systematic study of the potential of the colliders to explore the dipole portal is lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' This work analyzes how different signatures with HNLs may be used to search for them at the lepton and hadron colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The paper is organized as follows: in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' II, we study the phenomenology of HNLs at colliders, including their production and decay, and consider different signa- tures used to search for these HNLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We also discuss how the events originating from the dipole coupling may be distinguished from those originating from the HNL mix- ing with neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Subsequently, the sensitivities to the dipole portal at hadron and lepton colliders are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' III and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' IV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' V, we make conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' HNLS WITH DIPOLE COUPLING AT COLLIDERS In this section, we discuss details on the production and decay of HNLs, and how to search for them at col- liders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Below, we marginalize over dZ, dW by choos- ing their maximal possible values (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' (3)), namely, dZ = cot(θW )dα and dW = √ 2/ sin(θW )dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We also as- sume the Dirac nature of HNLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The analysis for the Majorana HNLs would be completely similar, except for the twice larger decay length for the fixed coupling and mass (see a discussion in [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Phenomenology Experiment NW NZ HL-LHC 6 · 1011 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 · 1011 FCC-hh 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='2 · 1013 9 · 1012 FCC-ee 5 · 108 5 · 1012 Table I: The numbers of bosons used in the simplified esti- mates for the HNL number events in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' II, which are taken from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' [31, 37] (for HL-LHC and FCC-ee), or obtained using MadGraph tree-level simulation [38] (for FCC-hh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' At colliders, there are two mecha- nisms of the production of HNLs with masses mN ≳ 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The first mechanism is direct production: l+ + l− → N + ν (4) at the lepton colliders, and p + p → N + ν + X, p + p → N + l + X (5) at the hadron colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The processes with neutri- nos may go via all the possible couplings dα, dZ, dW in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' (2), while the process with the charged lepton is via dW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The second production mechanism is via decays of W and Z bosons, W → N + l, Z → N + ν, (6) controlled by dW and dZ coupling, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The amounts of these bosons at the LHC, FCC-hh, and the Z-pole mode of FCC-ee are reported in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In the 3 low-mass limit mN ≪ mW/Z, the branching ratios of these decay processes behave as Br(W → N +l) ≈ d2 W m3 W 12πΓW ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5·103 � dW GeV−1 �2 , (7) Br(Z → N + ν) ≈ d2 Zm3 Z 12πΓZ ≈ 8 · 103 � dZ GeV−1 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' (8) For the HNLs with masses below the W/Z mass, the prompt production is suppressed compared to the decay of the heavy bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' This in particular means that the muon collider, operating at energies much above the Z boson mass, is not as efficient in probing the parameter space of such HNLs as the electron colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Decays of HNLs in the mass range mN ≪ mW,Z occur mainly via the coupling dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' This is because the decay widths for the processes mediated by dW,Z are suppressed by m4 NG2 F ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The main decay channel is the 2-body decay N → ν + γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The corresponding decay width is [1] ΓN→νγ = d2 αm3 N 4π ≈ ΓN,tot (9) However, this process does not suit the searches at col- liders that require observing a displaced decay vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Instead, sub-dominant decay channels should be consid- ered – the 3-body decays N → f + ¯f + ν, which occur via the virtual photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In the limit mN ≫ 2mf, the corresponding decay width behaves as [22] ΓN→νf ¯ f ≈ αEM|dα|2m3 NQ2 fNf 12π2 � log � m2 N m2 f � − 3 � , (10) where Qf is the electric charge of the fermion f, while Nf = 1 for leptons or Nf = 3 for quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The branching ratios of these processes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Signatures Having discussed the phenomenology of HNLs, we may now consider the signatures at lepton and hadron collid- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Hadron colliders a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Monophoton and missing energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' One of the pos- sible signatures is the event with a monophoton and miss- ing energy, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' q + ¯q → N + ¯ν, N → ν + γ, (11) with the missing energy carried away by neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' This signature has been analyzed in [1], where the authors uti- lized ATLAS search [39] performed for the dataset corre- sponding to the integrated luminosity L = 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='1 fb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' It N → ν+ee N → ν+μμ N → ν+hadrons 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='100 1 10 10-6 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='010 mN [GeV] Brprocess, d = 1 GeV-1 Figure 2: The branching ratios of sub-dominant decays of HNLs: the di-lepton decays N → ν + l+ + l−, where l = e, µ, and the hadronic decays N → ν + hadrons, approximated by the decay N → ν + q + ¯q, where q = u, d, s, c, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Note that be- low mN ≃ ΛQCD ∼ 1 GeV, perturbative QCD breaks down, and the corresponding prediction for the hadronic branching ratio becomes invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' was shown that with this type of search, it might be possi- ble to probe the couplings higher than d ≳ 10−5 GeV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' It is simple to estimate the sensitivity of the LHC in the high luminosity phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In the accessible parameter space, the HNLs have microscopic proper lifetimes cτN ≪ 1 mm, and the probability for the HNL to decay is ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Therefore, the number of events scales as Nev ∼ NN,prod × Pdecay × ϵsel ∝ L × d2 Z, (12) where NN,prod ∝ L × d2 Z is the total number of the pro- duced HNLs, and ϵsel is the selection efficiency, which we assume will not affect the scaling of the number of events with the HNL model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Given that the background also scales with L, the lower bound of the sensitivity in the plane mN–d changes as L−1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' For HL- LHC, LHL = 3000 fb−1, and therefore it should probe a factor of (3000/38)1/4 ≈ 3 smaller couplings than it was derived in [1] for the old dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We show the corre- sponding bounds in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 5 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Displaced vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' HNLs with decay lengths lN,decay ≳ O(1 mm) and masses mN < mW may be searched using displaced vertex (DV) techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' An ex- ample of such a DV scheme is the scheme [40] used at CMS to look for HNLs with the mixing coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The scheme utilizes the process chain p+p → W +X, W → N +l, N → l ′++l ′′−+ν, (13) where l, l′, l′′ are electrons or muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' To discriminate HNLs from backgrounds, it is required to detect the final state leptons l′, l′′, and the prompt lepton l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' These par- ticles must have kinematic properties that satisfy some selection criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Examples of such properties are large enough transverse momentum and transverse impact pa- rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Lepton colliders a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Monophoton and missing energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Similarly to the hadron colliders, it may be possible to search for the HNLs via the missing energy signature at the lepton col- liders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The process of interest is [1] l+ + l− → N + ¯ν, N → γ + ν (14) Similarly to the case of the analogical search at the LHC, it may be possible to derive the sensitivity of FCC-ee from the result of the older searches at DELPHI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The upper bound from LEP on the cross-section of the process e+e− → γ + inv obtained at the Z pole mode is [41–43] σDELPHI mono-γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='1 pb, where for the energy of the photon and its polar angle it was required Eγ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='7 GeV and | cos(θγ)| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Given the similar background at FCC- ee and LEP (and in particular that both LEP and FCC- ee are free from pileup), and assuming conservatively the same detector properties of FCC-ee as for LEP, the lower bound of the sensitivity of FCC-ee would be σFCC-ee mono-γ σFCC-ee mono-γ ≃ � LLEP Z-pole LFCC-ee Z-pole � 1 4 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='3 · 10−2 (15) where LLEP Z-pole = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='2 fb−1 and LFCC-ee Z-pole = 150 · 103 fb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We show the corresponding sensitivity in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 8 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Note that this simple estimate roughly agrees with the sensitivity of CEPC that has been recently computed in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Let us now discuss how to distinguish leptonic decays of the HNLs that have either the mixing or the dipole couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The simplest way would be to check the pres- ence of the leptons of different flavors in the lepton pair: such type of decays is common for the HNLs with the mixing coupling (it occurs via the charged current) [44] but is highly suppressed for the HNLs with the dipole coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Another way would be to compare the distri- bution of the lepton pair in invariant mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' For the dipole coupling, the leptons appear via a virtual photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' There- fore, the distribution has the maximum at minv = 2me and quickly drops with the increase of minv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In contrast, for the mixing coupling, the mediator is a heavy W/Z, the corresponding propagator is a constant, and the dis- tribution is rather flat in the range 0 < minv < mN, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Displaced decays Another way to search for HNLs may be to look for the displaced decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Unlike the LHC, lepton colliders have a much cleaner background;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' in particular, no pile-up events [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Therefore, it may be much simpler to distinguish a hypothetical SM background and the signal from decaying HNLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In par- ticular, instead of searching for the events with prompt leptons, one may consider only the events with the dis- placed vertex – the Z boson decays e+ + e− → Z, Z → N + ν, N → l+ + l− + ν (16) To summarize, we conclude that there is a comple- mentarity between the mentioned signatures at colliders Mixing coupling Dipole coupling 0 5 10 15 20 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='100 1 10 mee [GeV] Probability density N → νe+e-, mN = 20 GeV Figure 3: The distribution of the electron-positron pair from the HNL decay N → νe+e− in the invariant mass mee = � (pe+ + pe−)2, assuming the mixing (the red histogram) or the dipole (the blue histogram) coupling of the HNL to the SM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' and the non-collider experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' While the latter would probe mainly dα, the former may explore the couplings dZ, dW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The displaced vertex signatures may contribute to the sensitivity only if dZ,W ̸= 0, since these couplings determine the production of the HNLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In contrast, the missing energy signature may still provide the sensitiv- ity, given that the HNL production, in this case, is also controlled by dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In addition, we should stress another complementarity – between the lepton colliders may mainly probe the dZ coupling, the hadronic colliders suit better for probing dW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' HADRON COLLIDERS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Background In [40], the search for HNLs with the mixing coupling has been performed using the statistics accumulated dur- ing 2016-2018, corresponding to the integrated luminos- ity 138 fb−1 at CMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The results of this search may be extrapolated to the high-luminosity LHC, with the cor- responding scaling of the SM background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In order to reduce backgrounds, the following selection cuts have been imposed: – One prompt lepton l1 and two displaced leptons l2,3 within the pseudorapidity range |η| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' – Prompt electron (muon): pT > 30 − 32 (25) GeV, transverse impact parameter |d0| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='05 cm and longitudinal impact parameter |dz| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='1 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' – Displaced electrons (muons): pT > 7 (5) GeV, |d0| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='01 cm, |dz| < 10 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The total transverse momentum of the two displaced leptons should be pT,23 > 15 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 5 – The invariant mass of 3 leptons should be within 50 GeV < √s123 < 80 GeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' the invariant mass of the displaced leptons √s23 should not be close to the invariant mass of the SM resonances (such as ω, φ, J/ψ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' – Angular constraints: the angle between the HNL direction (assumed to be given by the vector from the primary vertex to the secondary vertex) and the direction given by the total momentum of l2, p3 is cos(θSV,23) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='99;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' the azimuthal separation between the prompt and each of the displaced leptons should be |∆φ(l1, l2/3)| > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' the angu- lar separation between l2,3 should be ∆R(l2, l3) = � ∆η2 23 + ∆φ2 23 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' – Maximal displacement constraints: displaced ver- tex within the tracker, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' the transverse dis- tance ∆2D < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 m and the longitudinal distance ∆|| < 3 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The reconstruction efficiency for the prompt leptons is ≃ 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The reconstruction efficiency for displaced leptons depends on the lepton type, its relative isolation, and the displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In particular, for the displacement 10 (25) cm, depending on the relative isolation, the efficiency for the electron reconstruction varies in the limits from 20%- 40% to 60%-80% (15%-20% to50%-60%), while for the muons with the displacement 10 (50) cm the numbers change to from 85%-90% to 95% (40%-50% to 80%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Backgrounds for this selection set may come from the events with misidentified hadrons, muons from pion or kaon decays, and leptons coming from decays of heavy flavor hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' For the luminosity corresponding to the data set collected at CMS in 2016-2018, the total number of predicted background events is ≃ 100 − 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The col- lected data was in agreement with the theoretical back- ground prediction, which was used to impose the exclu- sion bound on the parameter space of the HNLs with the mixing coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Sensitivity Let us estimate the sensitivity of this scheme to the HNLs with the dipole coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We will consider the LHC in its high luminosity phase (HL-LHC) and FCC- hh, assuming for the latter the same search scheme as for the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The parameters of these two detectors are summarized in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Due to larger energies, the back- Detector |η| R × L CMS@LHC < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 m × 3 m FCC-hh < 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='6 m × 5 m Table II: Parameters of the trackers at CMS@LHC and the FCC-hh reference design detector: pseudorapidity cov- erage, transverse and longitudinal size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The values are taken from [47] and [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' ground at the FCC-hh may qualitatively change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' There- fore, we will present the sensitivity of the FCC-hh in the form of iso-contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We start with evaluating the selection efficiency for the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We define it as ϵselection ≡ � l=e,µ Br(N → νl+l−) × ϵll sel � l=e,µ Br(N → νl+l−) , (17) where ϵll sel is the selection efficiency for the decay into a lepton pair l+l−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' For simplicity, we perform a pure MC simulation, where the kinematics reconstruction ef- fects are not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' For the LHC, we approximate the displaced leptons reconstruction efficiency by a linear function of the transverse displacement, adopting con- servatively the lowest values reported in [40] for the in- terpolation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' As a cross-check of the calculations, we have reproduced the sensitivity to HNLs with the mixing coupling reported in [40] within a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5, which is appropriate given the simplicity of the simula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' For the FCC-hh, we assume unit displaced leptons reconstruction efficiency, motivated by a possible devel- opment of technologies at the time of the construction of FCC-hh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Compared to the CMS@LHC case, we also change the pseudorapidity/displacement cuts due to the changed tracker size (see Table II), leaving the other cuts unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The mass and lifetime dependence of ϵselection for the HNLs with the dipole coupling case is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' From the figure, we see that for HNLs with mass mN ≲ 10 GeV, ϵselection does not exceed ≃ 10−2 at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The corresponding values at the FCC-hh are at least one order of magnitude larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' This is a combined effect of the larger tracker volume and the unit displaced leptons reconstruction efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' For the fixed decay length, the efficiency increases with the HNL mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The reason is an increase of the pT of the produced leptons relative to the direction of the incoming HNL, and hence the transverse impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The number of events is given by Nevents = NW × Br(W → N + l)× × � l=e,µ Br(N → νl+l−) × ϵselection (18) The behavior of the number of events with the coupling for the fixed mass is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The number of events at the FCC-hh is a factor of a few hundred larger than at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' This increase is due to the larger selec- tion efficiency and a gain in the luminosity and W boson production cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The sensitivities of the searches for the displaced ver- tices at the HL-LHC and FCC-hh assuming the coupling of the HNLs to the electron flavor are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Although the sensitivity of the LHC is completely within the sensitivity of DUNE, it may still be a useful probe of the dipole portal, since it probes not only the dα cou- pling, but also the coupling to W bosons, and hence the LHC is complementary to other probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 6 LHC, mN = 1 GeV LHC, mN = 5 GeV FCC-hh, mN = 1 GeV FCC-hh, mN = 5 GeV 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='100 1 10 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='100 cτN [m] ϵselection LHC, cτN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='1 m LHC, cτN = 1 mm FCC-hh, cτN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='1 m FCC-hh, cτN = 1 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 1 2 5 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='100 mN [GeV] ϵselection Figure 4: Selection efficiencies for the events with decaying HNLs at CMS@LHC and at FCC-hh, assuming the same experi- mental setup as at CMS@LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Left panel: as a function of the HNL decay length for several choices of its mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Right panel: as a function of the HNL mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' FCC-hh, mN = 3 GeV FCC-hh, mN = 5 GeV LHC, mN = 1 GeV LHC, mN = 2 GeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='1 1 10 100 1000 dα [GeV-1] Nevents W\uf522N+e, N\uf522νe+e-, |dW/dα| = 2 /sin(θW) LHCdispl LHCγ+Emiss FCC-hh50 events FCC-hh100 events DUNE ND 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 1 2 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' × 10-5 mN [GeV] |de|, GeV-1 |dW/de| = 2 /sin(θW) Figure 5: Top panel: the behavior of the number of events for HNLs with different masses as a function of the dipole coupling dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Bottom panel: the potential of the hadron col- liders – high luminosity LHC and FCC-hh – to probe the HNL parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' For the LHC case, we show the sensitivities coming from two signatures (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' II B 1): the dilepton dis- placed vertex searches, for which we report the 90% CL limit, as well as the projected sensitivity from the searches for the events with mono γ and missing energy at ATLAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In the case of the FCC-hh, we show the iso-contours corresponding to 50 and 100 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In the same figure, we also show the projected limits for the parameter space that may be probed by the mono γ searches (remind Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' II B 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Because of a huge back- ground, this search cannot explore unconstrained HNL couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' LEPTON COLLIDERS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Backgrounds Lepton colliders are free from pileup and have a low beam-induced background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Therefore, for the given pro- cess with an HNL, e+ + e− → Z → N + ν → Y + ¯Y + ν, (19) where Y, ¯Y denote visible HNL decay products, the only possible background comes from single events of e+, e− collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The latter includes Z boson decays e+ + e− → Z → f + ¯f → Y + ¯Y + inv, (20) where f = l = e, µ, τ or q = u, d, s, c, b, and prompt 4-fermion production e+ + e− → f + ¯f ′ + f ′′ + ¯f ′′′ → Y + ¯Y + inv, (21) see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' By “inv”, we denote the particles that leave the detec- tor invisibly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' examples include neutrinos or the particles that have not been detected due to the inefficiency of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In [48], a preliminary background analysis for FCC-ee has been performed for the minimal HNL model with the mixing coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' For the particular decay process N → e+ + e− + ν, backgrounds from the decays of Z bosons have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The simulation started by generating events in MadGraph [38], followed by Pythia8 [49] for the hadronization and DELPHES [50] for the simulation of the detector response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The background reduction has been studied using pre-selection cuts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' without requiring the candidates Y , ¯Y to form a good vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The selection started from the requirement to have the visible final state consisting solely of a pair of 7 l− l+ X ¯X (c) Z ¯ν l+ l− ν (a) ¯X X inv (b) l− l+ e− e+ ν ¯ν (d) e+ e− N Figure 6: Events at lepton colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' An event (19) with an HNL decaying into a pair of charged leptons l+, l− (the diagram (a)), and possible background processes to it: decays Z → X ¯ X → l+ + l− + inv (the diagrams (b), (c)), as well as 4-fermion process e+e− → l+l− + ν + ¯ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' e+, e− particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Then, the event was required to have non-zero missing momentum /p = |pe+ + pe−| > 10 GeV, to account for finite momentum reconstruction resolution and remove a huge fraction of background from decays Z → ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Then, the cut on the transverse impact pa- rameter, the minimal distance |d0| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 mm from the track helical trajectory to the beam line, has been ap- plied to both e+ and e−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' This selection allowed reducing backgrounds from promptly produced e+, e−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In total, the pre-selection reduced backgrounds down to ∼ 105 - mostly coming from Z → τ + ¯τ → e+ + e− + inv (22) Therefore, an additional selection is needed to remove the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In addition, the impact parameter cut harms the sensitivity to short-lived HNLs, being in par- ticular much more restrictive than the requirement for the vertex displacement rdispl > 400 µm used in [48] to demonstrate the potential of FCC-ee to explore the pa- rameter space of the HNLs with the mixing coupling (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' An examination of the kinematics for the process (22) and the signal (remind Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' II A) suggests that the amount of the remaining background events may be sig- nificantly reduced if imposing the cut on the angle be- tween two electrons from above and their energies from below, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In particular, performing the pure MC simulation of the decays (22), we have found that the cut cos(θee) > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5, Ee+ > 2 GeV, Ee− > 2 GeV (23) leaves no events even before imposing the |d0| cut while keeping a large signal selection efficiency independent of the lifetime of the HNL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Apart from this selection, the background may also be reduced by requiring the electron-positron pair to form a good vertex (with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' a small distance of the closest approach between their tracks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' It may suggest that the |d0| cut may be re- laxed to allow for probing short-lived HNLs, as the se- lection (23) should also work properly for the other Z decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Realistic simulations are required to examine this question further, which is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' However, the cuts (23) are not efficient in the case of the 4-fermion production processes (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' To examine this question, we have simulated the purely leptonic process e+ + e− → e+ + e− + ν + ¯ν (24) in MadGraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The total cross section of this process requiring pT,l,ν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='1 GeV has been found at the level of σee→eeνν ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='7 pb, which results in NZ · σee→eeνν σee→Z ≈ 2·108 of such events during the Z-pole mode timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The e+, e− pair typically originates from the same vertex and hence may be as collimated as the signal, while neutrinos carry away missing momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' However, the 4-fermion process is prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Unlike the background coming from the decays of Z, the produced e+, e− pair has zero displacement from the collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' To reduce this background to zero, one may additionally require non-zero displacement of the vertex formed by the e+, e− pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The exact cut depends on the spatial resolution of the tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We will exploit two different choices for the displacement cut: rdispl > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='4 mm, or rdispl > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='1 mm (25) The cuts considered in [48] and the pre-selection we pro- pose in this work are summarized in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Sensitivity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Selection efficiencies The signal efficiency for the selection criteria from Ta- ble III for various HNL masses and decay lengths, con- 8 Selection cuts Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' [48] Only e+, e− in an event, /p > 10 GeV |d0| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 mm This work Only l+, l− in an event, /p > 10 GeV cos(θll) > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5, El+, El− > 2 GeV rdispl > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='4 mm, or rdispl > 1 µm Table III: Summary of the selection cuts required to remove the background for different HNL decay processes, as imposed in [48] and considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Here, d0 denotes the transverse impact parameter of any of the two tracks, /p = | � preconstructed| corresponds to the missing momentum in an event, θab is the angle between the two particles a, b, and rdispl is the vertex displacement from the collision point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' sidering both the mixing and dipole couplings, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Let us first consider the cuts set from [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We repro- duce the values of the efficiencies reported for particular masses and lifetimes of HNLs with the mixing coupling in Table 3 of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The figures show that the cuts’ impact depends significantly on the HNL mass and life- time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The efficiency, being ≈ 1 for ldecay ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 mm independently on the HNL mass, starts dropping at ldecay ≃ 1 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' For the given decay length, the decrease of ϵ is larger for smaller HNL masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The reason is that the impact parameter (and hence the efficiency) of the decay products is higher if they gain large pT relatively to the direction of the HNL, and the magnitude of pT is controlled by the HNL mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The efficiency for the dipole coupling case has similar behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' However, the impact of efficiency however is less severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Indeed, be- cause of the kinematics of the decay process N → l+l−ν (remind Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' II A), in the dipole case, the leptons typi- cally gain smaller energies and than in the mixing case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Due to this feature, their deflection relative to the HNL is larger, which results in a larger IP on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' For the cuts set proposed in this paper, the situation is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The decrease at small lifetimes is obviously less significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' As for the mN behavior, the efficiency slightly drops once mass increases because of an increase of the mean angle between leptons with mN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In particu- lar, for heavy HNLs with mN ≃ mZ a sizable fraction of events may have cos(θ) < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' This effect is more sig- nificant for HNLs with mixing because of the process’s kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' On the other hand, since leptons produced via the dipole coupling are less energetic, the efficiency is lower at low HNL masses because of the El cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Number of events and sensitivity curves Let us now estimate the sensitivity of FCC-ee to HNLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We will consider the reference Innovative Detector for Electron–positron Accelerators (IDEA), which is a cylin- der having the radius r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 m and longitudinal size L = 11 m [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The other reference detector, CLD, has very similar specifications, and therefore the sensitivity would be completely similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The expected number of events with decays of HNLs at IDEA@FCC-ee is Nevents = 2·NZ ·BrZ→N+ν × � l=e,µ BrN→l+l−ν ×ϵ(l) sel, (26) where ϵsel = ϵsel(mN, dα) is the fraction of events with HNLs decaying inside the decay volume and that sat- isfy the selection cuts from Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In the limit when ldecay,N ≫ O(1 mm), the displacement selection has unit efficiency, and ϵsel becomes decay length-independent: ϵsel ≈ ϵ(mN) π π � 0 dθ � exp � − lmin ldecay,N � − exp � −lmax(θ) ldecay,N �� , (27) where the integration is performed over all directions of the cylindrical decay volume of IDEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The sensitivity of the FCC-ee to the HNLs with the dipole coupling is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 8, where we also in- clude the sensitivity of the missing energy search (remind Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' II B 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' To fix the excluded parameter space, we as- sume dα = dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We stress however that the sensitivity of the FCC-ee is flavor-universal since both the production and decay of the HNL are flavor-agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' From the figure, we conclude that depending on the dis- placement cut, with the displaced decay searches FCC- ee may probe the HNLs with masses up to mN = 30 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The upper bound of the sensitivity is caused by the HNL decay vertex displacement selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The shape of the lower bound is changing: below mN ≃ 3 GeV it gets smoothly improved, while at larger masses it be- comes plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The reason is that at small masses, the HNL decay length at the lower bound is lN,decay ≫ 1 m, and therefore the decay probability scales as Pdecay ≈ lN,decay/lfid ∝ m−4 N , where the scaling comes from the behavior of the HNL decay width (9) and the γ factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' At large masses, the HNL decay length becomes small enough such that HNLs have a unit probability of decay- ing inside the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The lower bound in this case is determined by the condition NN,prod × ϵsel > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='3, which is almost mass-independent in the mass range of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The missing energy search is complementary compared to the displaced decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Namely, it cannot probe as small couplings as probed by the displaced decay search be- cause of significant background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' However, it may explore higher HNL masses, since there is no displacement cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' CONCLUSIONS In this paper, we have analyzed the potential of hadron and lepton colliders to probe the parameter space of HNLs with the dipole coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We have first discussed the phenomenology of HNLs – including their production, decays, and possible signa- tures – at the LHC, FCC-hh, and FCC-ee (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' II A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We have also commented on how to distinguish decays of 9 FCC-ee reach for rdispl > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='4 mm Cuts from [2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='05502], mixing Cuts from this work, mixing Cuts from [2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='05502], dipole Cuts from this work, dipole 10-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='100 10 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='100 1 ldecay = cτNpN/mN [m] ϵselection N→νe+e-, mN = 20 GeV Cuts from [2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='05502], mixing Cuts from this work, mixing Cuts from [2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='05502], dipole Cuts from this work, dipole 5 10 20 50 10-5 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='100 1 mN [GeV] ϵselection N→νe+e-, ldecay = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='0005 mm Figure 7: Selection efficiency for the process N → e+e−ν (for both the mixing and dipole couplings) based on the cuts from Table III: the ones considered in [48] (the blue lines), and the ones discussed in this work (the red lines), assuming the minimal displacement ldispl > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='4 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The left panel: as a function of the HNL decay length lN,decay = cτNpN/mN for the fixed HNL mass mN = 30 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The vertical dashed gray line denotes the minimal decay length of the HNL with the mixing coupling to which FCC-ee may be sensitive if requiring only the displacement rdispl > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='4 mm (from [48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The right panel: as a function of the HNL mass for the fixed HNL decay length lN,decay = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Displ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='4 mm Displ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='1 mm γ+Emiss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 1 5 10 10-8 10-7 10-6 10-5 mN [GeV] |dμ| [GeV-1] |dZ/dμ| = cot(θW) Figure 8: The potential of FCC-ee to probe the parameter space of the HNLs with the dipole coupling, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' II B 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The solid and short-dashed dark blue lines show the 90% CL sensitivity corresponding to the displaced decay signa- ture, assuming the event selection considered in this paper (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' IV A and Table III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The long-dashed lighter blue line denotes the sensitivity corresponding to the γ+missing energy signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' HNLs with mixing and dipole couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Thanks to the different working modes of the lepton and hadron collid- ers, they complement each other in exploring the param- eter space of HNLs: the hadron colliders may probe the coupling of HNLs to W bosons, while the lepton colliders are more efficient in probing the coupling to Z bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In addition, because of the production channels of HNLs, from decays of W, Z bosons, as well as due to the small distance from the production point to the decay volume, the colliders may probe the parameter space in the mass range inaccessible to neutrino factories such as DUNE and FASER2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Then, we have considered the hadron colliders (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' III), utilizing the search for displaced vertices with dileptons at CMS as well as the missing energy searches at ATLAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We have derived the sensitivity of the LHC in the high luminosity phase and estimated the potential of FCC-hh (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' A detailed background study for the FCC-hh case is required, which however goes beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Next, we have considered the lepton colliders, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' IV, concentrating on FCC-ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' We have first made a simplified background analysis demonstrating that the HNL decay signal may be clearly distinguished from the background using kinematic properties (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' IV A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' De- pending on the model parameters, it may be possible to probe the HNL masses up to mN ≃ 30 GeV, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The final plot combining the sensitivities of lepton and hadron colliders is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 9, where we marginalize over the couplings to Z, W assuming their maximal pos- sible values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' From the figures, we conclude that FCC-ee may explore the HNL masses up to mN ≃ 30 GeV, while the exploration potential of the hadron colliders is lim- ited by mN ≃ 3 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' This is due to the different back- ground environments of these colliders: FCC-ee is free from pileup events, and therefore background is much cleaner, which allows for softer selection which keeps high efficiency for events with HNLs and simultaneously effi- ciently reduces the yield of the pure SM events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank Juliette Alimena, Suchita Kulkarni, and Re- beca Gonzalez Suarez for discussing the background esti- mates at FCC-ee performed in [48], and Lesya Shchutska for discussing the backgrounds at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' This project has received support from the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 860881-HIDDeN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Jing-yu Zhu is grateful for the support from the China and Germany Postdoctoral Exchange Program from the 10 DUNE FCC-ee LHC FCC-hh50 events FCC-eeγ+Emiss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 1 5 10 10-8 10-7 10-6 10-5 mN [GeV] |de|, GeV-1 |dZ/de| = cot(θW), |dW/de| = 2 /sin(θW) DUNE FCC-ee LHC FCC-hh50 events FCC-eeγ+Emiss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 1 5 10 10-8 10-7 10-6 10-5 mN [GeV] |dμ|, GeV-1 |dZ/dμ| = cot(θW), |dW/dμ| = 2 /sin(θW) DUNE ND DUNE FD FCC-ee FASER2 FCC-eeγ+Emiss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 1 5 10 10-8 10-7 10-6 10-5 mN [GeV] |dτ|, GeV-1 |dZ/dτ| = cot(θW), |dW/dτ| = 2 /sin(θW) Figure 9: Potential of colliders – FCC-ee, LHC in the high luminosity phase, and FCC-hh – to explore the parameter space of HNLs with the dipole coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' For the LHC, we report the 90% CL sensitivity based on the search scheme and backgrounds from [40] (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' III B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' For FCC-hh, we assume the same search scheme as for the LHC and show the iso-contour corresponding to 50 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' For FCC-ee, we report the 90% CL sensitivity assuming that the background is absent (see the corresponding discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' IV A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Office of China Postdoctoral Council and the Helmholtz Centre under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 2020031 and by the National Natural Science Foundation of China under Grant No.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' [51] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Workman and Others (Particle Data Group), “Review of Particle Physics,” PTEP 2022, 083C01 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 12 Appendix A: Events selection at FCC-ee Let us first analyze the kinematics of Z boson decays into two τ leptons at FCC-ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Since Zs are at rest, their decay products τ, ¯τ fly in exactly opposite directions and have the same energy Eτ = E¯τ = mZ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The e+, e− pair without any other visible particle can originate only from the two decays (the diagram (c) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 6) τ → e− + ¯νe + ντ, ¯τ → e+ + νe + ¯ντ , (A1) where the distribution of e+, e− in the angle θee between their directions of motion is peaked around θee = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' A small fraction of events with a small angle between the momenta of e+, e− have the following pattern: one of the particles from the pair has very small energy, Ee± ≪ mZ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Background (ττ) Signal (HNLs mixing) Signal (HNLs dipole) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='0 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='100 1 10 cos(θee) Fraction mN = 20 GeV, N → νe+e- Figure 10: The distribution of the e+e− pair in cosine of the angle between the e+, e− at FCC-ee in the Z-pole operating mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Three processes are considered: the background process Z → τ ¯τ → e+e−¯νeνe¯ντντ, and the HNL decays N → e+e−ν, assuming the dipole and the mixing couplings (mixing with νe is considered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The detector reconstruction effects are not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The situation with the signal is different: the angle distribution between the e+, e− originated from the HNL decay is peaked at θ = 0, and the situation remains the same even for heavy HNLs mN ≃ mZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Therefore, the background yield may be reduced without a significant impact on the signal if one requires a cut on cos(θee) and Ee+, Ee− from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' To estimate the effect of such a cut on the background and signal, we have simulated ≃ 5 · 109 decays Z → τ ¯τ → e+e−νe¯νeντ ¯ντ, which corresponds to the full statistics expected during the full timeline of FCC-ee in the Z pole mode [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' In the simulation, we included neither finite detector reconstruction resolution1 nor the particle identification efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Therefore, its predictions should be validated with full-scale simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The distribution in cos(θee) for the e+e− pair from the background and the decays N → e+e−ν, considering both the models of the dipole portal and the minimal HNL model, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' With the simulated sample, we have reproduced the selection efficiencies reported in Table 2 of [48] for the process Z → ττ → ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Next, we found that the cut cos(θee) > −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='5, Ee+ > 2 GeV, Ee− > 2 GeV (A2) reduces the number of backgrounds to zero even before imposing the d0 and /p cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The same conclusion may hold for other decays Z → f ¯f → e+e− + inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The e+, e− pair originates either from the single process f → · · · → e+e− + inv2 such that ¯f → · · · → inv (the diagram (b) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 6), or from the two independent processes f → · · · → e− + inv, ¯f → · · · → e+ + inv (the diagram (c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' By inspecting the decay modes of possible products of f in [51] and assuming a perfect detector efficiency in detecting charged particles and neutral long-lived mesons such as K0 L (via deposition in HCAL), we have not found the combination f ¯f which may lead to the diagram (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Therefore, we conclude that this category of events may be a subject of the detector inefficiency only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' The level of this inefficiency is to be determined by the full-scale simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Nevertheless, we believe that the combination of the presented cuts in addition to the vertex criteria (such as the small distance of the closest approach between the tracks) would allow reducing the background from Z boson SM 1 Nevertheless, as is demonstrated in [48], FCC-ee has perfect re- construction capabilities of both the lepton energies and momen- tum (and hence cos(θee)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 2 Here, if f is a quark, by .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' we mean a hadronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' 13 decays to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' Further, we will assume zero background from the processes of the type Z → f ¯f → Y ¯Y + inv, constituting the background for various decay modes of the HNL N → Y ¯Y + ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' This may be especially the case for the decays e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} +page_content=' N → q + ¯q + ν, for which the background process, Z → q + ¯q, would have even more marginal kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttFAT4oBgHgl3EQfhR0m/content/2301.08592v1.pdf'} diff --git a/x9E4T4oBgHgl3EQfYQwo/content/tmp_files/2301.05047v1.pdf.txt b/x9E4T4oBgHgl3EQfYQwo/content/tmp_files/2301.05047v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f6a3571ed0d2460e8be4b073de46315895572a91 --- /dev/null +++ b/x9E4T4oBgHgl3EQfYQwo/content/tmp_files/2301.05047v1.pdf.txt @@ -0,0 +1,1504 @@ +arXiv:2301.05047v1 [math.DG] 12 Jan 2023 +THE SCALAR CURVATURE IN WEDGE SPACES: EXISTENCE AND +OBSTRUCTIONS +LEVI LOPES DE LIMA +ABSTRACT. We study the scalar curvature of incomplete wedge metrics in certain +stratified spaces with a single singular stratum (wedge spaces). Building upon sev- +eral well established technical tools for this category of spaces (the corresponding +Yamabe, elliptic and index theories) we provide existence and obstruction results +for such metrics under suitable positivity assumptions on the underlying geom- +etry. This is meant to be a follow-up to a previous paper of ours (AGAG, 2022), +where the case of spaces with an isolated conical singularity was considered. +CONTENTS +1. +Introduction +1 +2. +Wedge spaces and statements of the main results +3 +3. +The Yamabe problem in the wedge setting +7 +4. +The wedge elliptic theory for the scalar Laplacian +10 +5. +An existence result: the proof of Theorem 2.7 +19 +6. +A topological obstruction: the proof of Theorem 2.10 +20 +References +23 +1. INTRODUCTION +The general problem of prescribing the scalar curvature function in a given +smooth closed manifold is a central theme in Riemannian Geometry. Since in prin- +ciple this metric invariant only affects the underlying geometry at a local level with +no direct influence on its large scale behavior, it is expected that a huge amount of +functions might be realized as the scalar curvature. In a sense this has been con- +firmed by Kazdan and Warner [KW75], who showed that any function on a closed +manifold of dimension n ≥ 3 which is negative somewhere is the scalar curva- +ture of some metric. Versions of this result in the setting of compact manifolds +with boundary have been established in [CV19]. These contributions should be +contrasted with the well-known topological obstructions to the existence of met- +rics with positive scalar curvature in the spin setting stemming from the works +of Lichnerowicz [Lic63], Hitchin [Hit74] and Gromov-Lawson [GL80b, Gro96] and +L.L. de Lima has been supported by CNPq 312485/2018-2 and FUNCAP/CNPq/PRONEX +00068.01.00/15. +1 + +2 +LEVI LOPES DE LIMA +relying upon the remarkable properties of the Dirac operator acting on spinors; +see also [BH20] for similar obstructions in the presence of a boundary. +The analysis in [dL22b] has suitably extended the aforementioned results to +the category of spaces with an isolated conical singularity. The purpose of this +note is to investigate further extensions of these contributions to the more general +category of (incomplete) edge spaces with a single singular stratum, also named +wedge spaces. The main results here are Theorems 2.7 and 2.10 dealing respectively +with existence and obstructions of wedge metrics with prescribed scalar curvature +on such spaces under certain positivity assumptions on the underlying geometry. +In particular, we provide a unified treatment that retrieves the conical case studied +in [dL22b]. Also, as explained in Remark 2.12, the obstruction in Theorem 2.10 in a +sense complements those obtained in [AGR16, BPR21] by using similar tools (the +wedge index theory in [AGR16]). +In order to carry out the arguments needed to establish Theorem 2.7, certain as- +pects of the solution of the Yamabe problem in this wedge category are reviewed in +Section 3, following [ACM14] closely. Another key ingredient here is the mapping +theory of geometric elliptic operators on such spaces (specifically, the Laplacian +acting on functions and the Dirac operator acting on spinors), which is described +in Sections 4 and 6; see, among others, [Maz91, Mel93, Les97, Sch98, Gri01, ES12] +for the general theory and also [dL22a] for an informal account of this topic in the +conical setting. This theory is used in Section 4 to establish good mapping proper- +ties for the scalar Laplacian. More precisely, we check here that in all cases of in- +terest this operator is essentially self-adjoint when viewed as an unbounded and +symmetric operator acting on a certain weighted Sobolev space, with the corre- +sponding self-adjoint extension being Fredholm (Theorem 4.11). This justifies the +integration by parts needed to carry out the perturbative scheme leading to Theo- +rem 2.7. We remark that surjectivity would suffice in this part of the argument, but +we prefer to establish the finer mapping properties for at least two reasons: they +provide a local description of the space of solutions of an associated non-linear +problem (see Remark 5.4) and, more fundamentally, the corresponding analysis +may be easily transplanted to the Dirac setting, where surjectivity does not suffice +and both self-adjointness and Fredholmness are crucial to the proof of Theorem +2.10. +The mapping properties mentioned above should routinely follow from results +in the existing literature (as in the accounts for the Hodge Laplacian in [MV12, +ARS22]) but since we were unable to locate a specific source carrying out the anal- +ysis for the scalar Laplacian, we supply a fairly detailed guide to the quite involved +proof, as this also will allow us to keep track of the rather subtle role (the dimen- +sion of) the link manifold plays in achieving self-adjointness; see Remarks 2.9 and +4.15. Among the various routes available, we have chosen to adopt as a key input +here an argument employed in [ALMP12, ALMP18, AGR16] for Dirac-type oper- +ators and in [ARS22] for the Hodge Laplacian, both relying on certain regularity +results in [GM03, GKM13]. This same reasoning is used in Section 6 to probe the +mapping properties of the Dirac operator needed in the proof of Theorem 2.10. +Besides its relevance to the applications mentioned above, we believe that the ap- +proach we take here has an independent interest as it might be implemented in + +THE SCALAR CURVATURE IN WEDGE SPACES +3 +other non-linear geometric problems where Laplace-type operators show up at an +infinitesimal level. +We now briefly describe the strategy behind the proofs of our results. Regard- +ing Theorem 2.7, we adapt the argument laid down in [KW75] in the smooth set- +ting. The first step is to find a wedge metric with constant negative (say −1) scalar +curvature (Theorem 3.9). In the smooth category, this corresponds to the “easy” +case of the solution of the classical Yamabe problem [LP87], hence always solv- +able. Unfortunately, finding such a metric in the wedge category with the avail- +able technology (the singular Yamabe theory in [ACM14]) turns out to be consid- +erably much harder and seems to require some kind of positivity condition on +the geometry of the link manifold. The simplest possibility, which suffices for the +applications we have in mind, is to assume that the link metric has constant posi- +tive scalar curvature (the more general case treated in Theorem 2.7, which merely +assumes that the link metric is Yamabe positive, may be reduced to this simpler +case in view of the well-known solution of the Yamabe problem for closed mani- +folds [Sch84, LP87]; see Remark 2.6). In any case, with this wedge metric (say g) +at hand, the next step consists in showing that any function “close” enough to −1 +may be realized as the scalar function of some conformal wedge metric (if this is +the case, it is straightforward to check that the natural action of the group of diffeo- +morphisms of bounded distortion on the space of wedge metrics takes care of the +general case, in which the prescribed scalar curvature function is negative some- +where; see Proposition 5.1). This step involves a perturbative argument relying on +the invertibility of the Laplace-type operator −∆g + λ, λ > 0, arising as the liner- +ization at g of the associated non-linear problem. Again, in the smooth category +the invertibility of this operator in the standard Sobolev scale is well established. +In our case, however, one is led to work in a Sobolev scale that takes into account +the structure of the singular stratum (the wedge) and it is precisely here that the +mapping theory mentioned above is needed. The outcome of applying this the- +ory to our problem appears in Theorem 4.11, confirming that in most cases the +Laplacian has good mapping properties, which leads to Theorem 2.7. Finally, we +mention that this same mapping theory is also needed to establish self-adjointness +and Fredholmness for the Dirac operator (Theorem 6.1), which enables the use of +the index theory from [AGR16] employed in the proof of Theorem 2.10. +Acknowledgments. The author would like to thank S. Almaraz for helpful dis- +cussions and comments. +2. WEDGE SPACES AND STATEMENTS OF THE MAIN RESULTS +We start by describing the class of spaces we are interested in. For simplicity, +all smooth manifolds appearing below are assumed to be oriented. +Definition 2.1. A stratified space with a single singular stratum is a topological +space X satisfying: +(1) X is a (compact and connected) metric space with distance function d; +(2) X = Y ⊔Xs, where Xs is a (open and dense) smooth manifold of dimension +n ≥ 3 (the smooth stratum of X); + +4 +LEVI LOPES DE LIMA +(3) Y is a (connected and boundaryless) smooth manifold of dimension b ≥ 0 +(the singular stratum); +(4) there exists a (connected) neighborhood U of Y (the wedge region) such +that: +• X\U is a smooth manifold with boundary Y •; +• there exists a retraction πY : U → Y with πY |U\Y being smooth; +(5) there exists a “radial function” x = xY : U → [0, 1) such that x−1(0) = Y +and with x|U\Y being smooth; +(6) πY is a locally trivial fibration whose typical fiber is the (truncated) cone +CZ = [0, 1) × Z/ ∼ over a closed connected manifold Z (here, ∼ means +that {0} × Z has been collapsed into a point), with atlas (φ, V), where each +φ : π−1 +Y (V) → V × CZ is a trivialization and the corresponding transition +functions preserve x (so that, restricted to each fiber, x ◦ φ−1 corresponds +to the natural projection CZ → [0, 1)). +In particular, for each x ∈ (0, 1) one has a submersion x−1 +Y (x) → Y whose +typical fiber is Z, so if we send x → 1 we obtain a submersion Y • → Y again with +typical fiber Z. Hence, the manifold X\U has a “fibred boundary” Y • which may +be viewed as the “resolution” of the singular locus Y . +We assume that x has been smoothly extended to Xs so that x|Xs\U ≡ 1. Notice +that +n = b + f + 1, +where f is the dimension of Z. Due to our use of the singular Yamabe theory in +[ACM14], in the following we will make the key assumption +(2.1) +b ≤ n − 2 ⇐⇒ f ≥ 1 +We now introduce the appropriate geometric data in X (more precisely, in the +smooth stratum Xs = X\Y ). For simplicity we assume that πY is trivial and we +fix a trivializing chart providing an identification +U ≃ Y × CZ ≃ Y × ([0, 1) × Z/∼) , +with x corresponding to the projection onto the [0, 1)-factor. Locally around each +cone fiber over a fixed fiber of the submersion Y • → X we may introduce coor- +dinates (x, y, z), where (y, z) are (local) coordinates on Y × Z. This allows us to +consider a class of adapted metrics on X. +Definition 2.2. An (incomplete) wedge metric in a stratified space X as above is a +Riemannian metric g on its smooth stratum Xs such that: +(1) there holds +(2.2) +g|U(x, y, z) = dx2 + x2gZ(z) + gY (y) + o(x), +x → 0, +where gY and gZ are fixed metrics in Y and Z, respectively; +(2) the distance induced by g coincides with d|Xs. +Remark 2.3. Under suitable, but still quite restrictive, assumptions we may also +treat the case in which the metric gZ varies with y ∈ Y in a smooth way. For in- +stance, we may assume more generally that the family of varying metrics gZ(y, ·), +is isospectral. For simplicity, however, we prefer to avoid this complication. + +THE SCALAR CURVATURE IN WEDGE SPACES +5 +Definition 2.4. A pair (X, g) as above is a wedge space and the closed Riemannian +manifold (Z, gZ) is the link of the singular stratum Y . In the cone-edge case f = 1, +the length of the linking circle is the cone angle. +We denote by dvolg the volume element associated to g. We may define Sobolev +spaces Hk(X, dvolg), k ≥ 0 an integer, in the usual way (just take the closure of +Lipschitz functions under the usual Sobolev norm induced by dvolg). If f ≥ 3 we +recall that the conformal Laplacian of (Z, gZ) is the elliptic operator +Lf +gZ = −∆gZ + +f − 2 +4(f − 1)RgZ. +Here and in the following, ∆ stands for the Laplacian and R for the scalar curva- +ture. +Definition 2.5. We say that (Z, gZ) is Yamabe positive if Lf +gZ is positive definite +(viewed as a self-adjoint operator acting on L2(Z, dvolgZ)). +Remark 2.6. It is known that Yamabe positivity is a conformal property of gZ: +it is equivalent to the conformal class [gZ] of gZ carrying a metric with positive +scalar curvature [LP87]. If this is the case then the solution of the Yamabe prob- +lem for closed manifolds [Sch84, LP87] allows us to replace gZ by a Yamabe metric +g′ +Z ∈ [gZ], the conformal class of gZ, without affecting the wedge character of the +underlying metric (in fact, the corresponding wedge metrics are easily seen to be +quasi-isometric to each other, with the quasi-isometry bounds depending only on +bounds on the conformal factor). In particular, after a further rescaling of the link +we may assume that Rg′ +Z = f(f − 1). This quasi-isometric replacement of wedge +metrics, induced by conformal deformations on the link, is crucial here since only +for g′ +Z as a link metric we may use Theorem 3.9, which relies on Theorem 3.7, to +find a wedge metric with constant negative scalar curvature to which the pertur- +bative scheme leading to the proof of Theorem 2.7 below may be applied. +We may now state our first result, which provides an existence theorem for +wedge metrics with prescribed scalar curvature under appropriate assumptions +on the link. +Theorem 2.7. Let (X, g) be a wedge space whose link (Z, gZ) satisfies either f = 1 or +f ≥ 3 and it is Yamabe positive. Then any smooth and bounded function which is negative +somewhere in Xs\U is the scalar curvature of some wedge metric on X. +Remark 2.8. The Yamabe positivity of (Z, gZ) in case f ≥ 3 relates to the already +mentioned difficulty in using the singular Yamabe theory in [ACM14] to find a +background wedge metric with constant negative scalar curvature to which the +perturbative scheme in Section 5 could be applied (see Theorem 3.9 (3)). As men- +tioned in Remark 2.6, after possibly passing to a Yamabe metric g′ +Z in the confor- +mal class [gZ] of gZ satisfying Rg′ +Z = f(f − 1), Yamabe positivity suffices to carry +out the proof of Theorem 2.7. In any case, the fact that the class of closed mani- +folds of dimension f ≥ 3 carrying metrics with positive scalar curvature is stable +under surgeries of co-dimension at least 3 [GL80a, SY79] provides many examples +of wedge spaces to which Theorem 2.7 applies. On the other hand, if f = 2 then +Yamabe positivity morally corresponds to asking that the link surface Z is (topo- +logically) a sphere, precisely the case covered by Theorem 3.9 (2); see also Remark + +6 +LEVI LOPES DE LIMA +3.8. But notice that now (Z, g′ +Z) is a round sphere, which means that the original +wedge manifold is quasi-isometric (in the “conformal” sense of Remark 2.6) to a +smooth manifold, in which case the conclusion of Theorem 2.7 already follows from +[KW75]. Of course, this justifies the omission of this case in Theorem 2.7. Finally, +as it is apparent from Theorem 3.9 (1), if f = 1 the existence of a Yamabe wedge +metric with negative scalar curvature can always be taken for granted. +Remark 2.9. As far as essential self-adjointness of the Laplacian operator is con- +cerned, the analytic machinery employed in Section 4 works fine universally (that +is, with no restriction on the link) if f ≥ 3, but it does not seem to deliver this +specific mapping property if f = 2; see Remarks 4.13 and 4.15. Note that mere +surjectivity would suffice for our purpose of extending Theorem 2.7 to this case +but this is not needed here anyway due to the fact that, as explained in Remark +2.8, the existence of a Yamabe wedge metric forces the surface link to be a sphere, +in which case the result follows from [KW75]. On the other hand, the cone-edge +case (f = 1) is treated here and, as expected, self-adjointness is fulfilled if the cone +angle is at most 2π. Again, this may always be achieved by a quasi-isometry of +the underlying wedge space induced by a rescaling the circle link (as in Remark +2.6), which suffices for our purposes. For other instances of the usage of this “at +most 2π” condition in the cone-edge setting, with relevant applications to rigidity +phenomena in Geometry, we refer to [HK98, MM11, Don12, LM19, CLT21]. +The existence results in Theorem 2.7 should be compared with our next contri- +bution, which provides topological obstructions to the existence of wedge metrics +with (strictly) positive scalar curvature on certain spin wedge spaces. For the no- +tion of infinite K-area, see Definition 6.3. +Theorem 2.10. Let (Xn, g), n = 2k, be a spin wedge manifold with infinite K-area +and whose link satisfies either 1 < f ≤ n − 1 or f = 1 and the cone angle is at most +2π. Then X carries no wedge metric with (strictly) positive scalar curvature (in the given +quasi-isometry class of wedge metrics). Also, the same conclusion holds true for n odd if +X × S1 has infinite K-area. +Remark 2.11. The arguments presented in [dL22b, Sections 2 and 5.2] may be +easily adapted to provide versions of the theorems above in case the wedge space +carries a boundary ∂X disjoint from the wedge region U: one gains minimality of +∂X in the analogue of Theorem 2.7 and should require mean convexity of ∂X for +the analogue of Theorem 2.10. +Remark 2.12. The obstruction in Theorem 2.10 may be thought of as being “com- +plementary” to [AGR16, Theorem 1.3] and [BPR21, Theorem 2.3] in a sense that +we now discuss. There exist two fundamental lines of inquiry concerning obstruc- +tions to the existence of metrics of positive scalar curvature in the spin category, +which in a sense reflect the quite diverse topological contributions to the Atiyah- +Singer index formula for Dirac operators. In the untwisted case, the only contri- +bution comes from the tangent bundle, which has been explored by Lichnerow- +icz [Lic63] to check that the �A-genus obstructs such metrics; this line of thought +has been refined by Hitchin [Hit74], with the corresponding obstruction coming +from KO-theory. On the other hand, it has been shown by Gromov and Law- +son [GL80a, GL80b, GL83] that twisting the Dirac operator with almost flat vector +bundles leads to a “complementary” obstruction for enlargeable manifolds (for + +THE SCALAR CURVATURE IN WEDGE SPACES +7 +example, this works for tori, whose �A-genus vanish). Later on, Gromov [Gro96] +was able to somehow quantify this latter proposal by means of the notion of “in- +finite K-area” (compare with Definition 6.3). Hence, whereas the obstructions in +[AGR16, BPR21] referred to above provide versions of the Lichnerowicz-Hitchin +approach in the wedge category, Theorem 2.10 aligns with Gromov’s philosophy. +3. THE YAMABE PROBLEM IN THE WEDGE SETTING +A first step towards the proof of Theorem 2.7 involves constructing a wedge +metric with constant negative scalar curvature on the given wedge space. Here we +explain how this result (Theorem 3.9 below) follows from the singular Yamabe +theory developed in [ACM14]. +Given a wedge space (X, g) as in Definition 2.4, we denote by [g]w the space +of wedge metrics �g in X which are conformal to g (in the sense that there exists +u : Xs → R smooth and positive such that �g = u +4 +n−2 g). The corresponding Yamabe +problem asks: there exists �g ∈ [g]w with the property that its scalar curvature R�g is +constant? +As in the smooth case, this admits a variational formulation. We fix a back- +ground wedge metric g in the given conformal class and consider the quadratic +form Q : H1(X, dvolg) → R, +(3.3) +Q(u) = +ˆ +X +� +|du|2 + cnRgu2� +dvolg, +cn = +n − 2 +4(n − 1), +and the constraint sphere +Bn∗ = {u ∈ H1(X, dvolg); ∥u∥n∗ = 1}, +n∗ = +2n +n − 2. +We note that C∞ +cpt(Xs) ⊂ H1(X, dvolg) densely, which is a consequence of the di- +mensional assumption (2.1); see [ACM14, Section 2.2]. This justifies the integration +by parts leading to the following result. +Proposition 3.1. Critical points of Q|Bn∗ precisely correspond to (weak) solutions of +(3.4) +Lgu = µu +n+2 +n−2 , +µ ∈ R, +where +Lg := −∆g + cnRg +is the conformal Laplacian (here, ∆g is the Laplacian of the background metric g). In +particular, if u is smooth and further satisfies 0 < c−1 ≤ u(x, y, z) ≤ c for some c > 0 +and (x, y, z) ∈ U then �g := u +4 +n−2 g is a solution of the corresponding Yamabe problem (in +the conformal class [g]w). +The preferred way to produce critical points for Q|Bn∗ is by minimization. Un- +fortunately, the Direct Method in the Calculus of Variations can not be applied +here due to the fact that the continuous embedding +H1(X, dvolg) ⊂ Lp(X, dvolg) +is only compact for p < n∗ [ACM14, Proposition 1.6]. Thus, a minimizer, if it +exists, must be located by alternative methods. Also, from past experience with + +8 +LEVI LOPES DE LIMA +the smooth case fully discussed in [LP87], we expect that a minimizer should exist +only in case the “total energy” of [g]w, as measured by the global Yamabe invariant +(3.5) +Yglo(X, [g]w) := +inf +u∈Bn∗ Q(u), +which is a conformal invariant of (X, g), lies below a certain threshold value (this +is just a manifestation of the ubiquitous bubbling off phenomenon characteristic +of conformally invariant problems). A major contribution in [ACM14] is precisely +to identify this critical threshold. +To explain this latter point we consider, for each V ⊂ X open, +Y (V ) = inf +�ˆ +V +� +|du|2 + cnRgu2� +dvolg; u ∈ H1 +0(V ∩ Xs); ∥u∥n∗ = 1 +� +. +In particular, by (3.5), +Y (X) = Yglo(X, [g]w). +Notice that in principle we might have Y (X) = −∞ (in other words, Q|Bn∗ might +not be bounded from below). +Definition 3.2. The local Yamabe invariant of (X, [g]w) is given by +Yloc(X, [g]w) = inf +x∈X lim +r→0 Y (Br(x)). +Remark 3.3. There always holds Yloc(X, [g]w) ≤ Yn, where Yn is the Yamabe in- +variant of the round metric in the unit sphere Sn, which follows from the fact that +(3.6) +lim +r→0 Y (Br(x)) = Yn, +x ∈ Xs. +Although some progress has been made in this regard [Mon17, AM22], the ac- +tual computation of the local Yamabe invariant is notoriously hard, the reason +being that, amazingly enough, it depends globally on the link (Z, gZ). Precisely, +Yloc(X, [g]w) += +Y(Rb × CZ, [dy2 + dx2 + x2gZ]) += +Y(Hb+1 × Z, [ghyp + gZ]), +where (Hb+1, ghyp) is hyperbolic space and Y denotes the standard Yamabe in- +variant. Nonetheless, some of its qualitative properties may be established as a +consequence of certain integrability conditions on the scalar curvature of the back- +ground wedge metric, which also imply that Q|Bn∗ is bounded from below. +Theorem 3.4. [ACM14] Assume that either +(3.7) +Rg ∈ Lq(Xs, dvolg), +for some q > n/2, +or +(3.8) +sup +r>0 +rq−n +ˆ +Br(x) +|Rg|qdvolg ≤ C, +for some q > 1 and all x ∈ X. +Then Yloc(X, [g]w) > 0 and Yglo(X, [g]w) > −∞. +The main result in [ACM14], as applied to wedge spaces, yields the following +criterion for the existence of minimizers for the Yamabe functional in (3.3). +Theorem 3.5. [ACM14] Assume that either (3.7) or (3.8) holds and that +(3.9) +Yglo(X, [g]w) < Yloc(X, [g]w). +Then there exists a minimizer for Q|Bn∗. + +THE SCALAR CURVATURE IN WEDGE SPACES +9 +Remark 3.6. We stress that the theory in [ACM14] applies to a much larger class of +singular spaces than that considered here. Also, in the smooth case, it follows from +(3.6) that Theorem 3.5 reproduces the classical Aubin’s criterion for the existence +of minimizers [LP87, Theorem 4.5]. +In order to use Theorem 3.5 to solve the Yamabe problem for a given wedge +conformal class, we must first come to grips with the integrability conditions (3.7)- +(3.8) on the scalar curvature of the background metric g. As already observed in +[ACM14, Lemma 2.4], it follows from the asymptotic expansion +(3.10) +Rg ∼ (RgZ − f(f − 1)) x−2 + O(x−1), +x → 0, +that either (3.7) or (3.8) hold true if the link (Z, gZ) satisfies +(3.11) +RgZ = f(f − 1). +As a consequence, we obtain the following criterion for the existence of Yamabe +wedge metrics. +Theorem 3.7. [ACM14] Let (X, g) be a wedge space whose link (Z, gZ) satisfies one of +the following conditions: +(1) f = 1; +(2) f = 2 and (Z, gZ) is the round unit sphere; +(3) RgZ = f(f − 1) if 3 ≤ f ≤ n − 1. +Then there exists a minimizer for Q|Bn∗ as long as (3.9) is satisfied. +Remark 3.8. Under the conditions of Theorem 3.7 (2), the wedge metric g actually +extends smoothly to Y and the result reduces to the classical criterion due to Aubin +[LP87, Theorem 4.5]; see Remark 3.6. +Hence, under the assumptions on the link displayed in the previous theorem, +ensuring that the scalar curvature meets the appropriate integrability condition, +the general solution of the Yamabe problem gets reduced to checking the validity +of the Aubin-type condition (3.9), a task that, as already observed, lies beyond the +current technology due to the fact that the local Yamabe invariant is mostly inac- +cessible. Nonetheless, since here we are interested in merely constructing a wedge +metric with constant negative scalar curvature, we may take advantage of the free- +dom to suitably modify the background metric to which Theorem 3.7 should be +applied. +Theorem 3.9. Let (X, g) be a wedge space whose link (Z, gZ) satisfies one of the following +conditions: +(1) f = 1; +(2) f = 2 and Z is (topologically) a sphere; +(3) (Z, gZ) is Yamabe positive if 3 ≤ f ≤ n − 1. +Then Xs carries a wedge metric with constant negative scalar curvature. +Proof. We first observe that, as explained in Remark 2.6, if item (3) above is satis- +fied then we can replace gZ by a Yamabe metric whose constant scalar curvature is +f(f − 1), so as to ensure that item (3) in Theorem 3.7 is met. Also, if f = 2 we may +also reduce to Theorem 3.7 (2) by replacing the link by the unit sphere (but recall + +10 +LEVI LOPES DE LIMA +that here the new wedge metric turns out to be smooth). After these preliminaries, +we proceed by using (3.3) with u a suitable constant to get +Yglo(X, [g]w) ≤ C +ˆ +X +Rgdvolg. +As in [Bes07, Subsection 4.32] we can inject a sufficiently large amount of neg- +ative scalar curvature around some point in Xs\U (without further altering the +wedge region) so as to make the new background metric, still denoted by g, to +satisfy ´ +X Rgdvolg < 0. Thus, Yglo(X, [g]w) < 0 and Theorem 3.7 applies since +Yloc(X, [g]w) > 0 and hence (3.9) is satisfied. Leading the minimizer to (3.4) we +immediately see that the (constant) scalar curvature of the corresponding Yamabe +wedge metric is negative, as desired. +□ +Remark 3.10. The conformal factor, say u, obtained in Theorem 3.9 is strictly posi- +tive and remains uniformly bounded from above as x → 0. Thus, the Yamabe met- +ric so obtained is quasi-isometric to the original background wedge metric. In fact, +the regularity theory in [ACM14, Section 3] guarantees that u(x, y, z) = u0(y)+o(1) +as x → 0, where u0 : Y → R is smooth and strictly positive. Hence, after possibly +implementing a conformal deformation on gY and a (y-dependent) change in the +radial coordinate, the new metric still satisfies (2.2), so that the corresponding link +metric remains independent of y ∈ Y . +Remark 3.11. It follows from the analysis in [ACM14, Section 2.3] that Theorem +3.7 (3) holds true under the weaker assumption that the first eigenvalue λ0(Ln +gZ) +of the operator +Ln +gZ = −∆gZ + cnRgZ, +cn = +n − 2 +4(n − 1), +is exactly equal to cnf(f −1); indeed, it suffices to assume λ0(Ln +gZ) > 0 if f = n−1. +Since in this case there exist positive constants A e B depending only on f and n +such that +Lf +gZ = ALn +gZ − B∆gZ, +any of these assumptions on Ln +gZ actually imply that (Z, gZ) is Yamabe positive, +which is the requirement in Theorem 3.9 (3). Hence, as far as the conclusion of +Theorem 2.7 is concerned, nothing is gained if we use this finer result based on +λ0(Ln +gZ). +4. THE WEDGE ELLIPTIC THEORY FOR THE SCALAR LAPLACIAN +We describe how the wedge elliptic theory applies to Laplace-type operators +on a wedge space (X, g) as above. As hinted at in the Introduction, this involves +considering ∆g as acting on a suitable weighted Sobolev scale and then employ- +ing the powerful micro-local methods in [Maz91, Les97, Sch98, ES12], with further +developments in [GM03, GKM13, ALMP12, AGR16, ALMP18, ARS22], to check +that ∆g, viewed as an unbounded, symmetric operator, has good mapping prop- +erties (essential self-adjointness and Fredholmness) for a carefully chosen value of +the weight. We emphasize that surjectivity would suffice for the application we +have in mind (Theorem 2.7), but we have chosen to establish finer mapping prop- +erties not only because, as explained in Remark 5.4, they provide a precise (local) +description of the space of solutions of certain semi-linear elliptic equations on +(X, g), but also because we intend to transplant the analysis to the Dirac operator + +THE SCALAR CURVATURE IN WEDGE SPACES +11 +treated in Section 6, where self-adjointness and Fredholmness are crucial in appli- +cations. Also, as the statement of our ultimate goal, Theorem 4.11, makes it clear, +in this section we pose no restriction on the geometry of the link (Z, gZ) if f ≥ 3. +We next consider the b-density +dvolb = x−1dxdvolgY dvolgZ +in the wedge region U and extend it to Xs in the obvious manner. +Definition 4.1. Given an integer k ≥ 0 and a cutoff function ϕ with ϕ ≡ 1 near +Y and ϕ ≡ 0 outside U, we define Hk +b(X) to be the space of all distributions u ∈ +D′(Xs) such that: +• (1 − ϕ)u lies in the standard Sobolev space Hk(Xs, dvolg); +• there holds +(x∂x)j(x∂y)µ∂ν +z (ϕu)(x, y, z) ∈ L2(X, dvolb), +j + |µ| + |ν| ≤ k. +Definition 4.2. (Weighted Sobolev scale) If β ∈ R we set +xβHk +b(X) = +� +v; x−βv ∈ Hk +b(X) +� +. +Remark 4.3. Using interpolation and duality we may define xβHσ +b(X) for any +σ ∈ R. This turns out to be a Sobolev scale of Hilbert spaces. For instance, +(4.12) +⟨u, v⟩xβH0 +b(X) = +ˆ +X +x−2βuv dvolb. +In particular, one has the continuous inclusion +xβ′Hσ′ +b (X) ⊂ xβHσ +b(X), +β′ ≥ β, +σ′ ≥ σ, +which is compact if strict inequalities hold. Also, if σ > n/2 then any u ∈ xβHσ +b(X) +is continuous in Xs and satisfies u(x) = O(xβ) as x → 0. +We are interested here not only in the bounded, weighted Laplacian +(4.13) +Mg,β := −x2∆g : xβHσ +b(X) → xβHσ−2 +b +(X), +induced by the natural action of Mg := −x2∆g on the weighted Sobolev scale, but +also in the unbounded Laplacian +(4.14) +Cg,β := −∆g : C∞ +cpt(X) ⊂ xβH0 +b(X) → xβH0 +b(X), +viewed as a densely defined operator. Their analysis will eventually hinge on the +pair of f-dependent constants +βf = −f + 1 +2 +, +γf = 1 − f +2 +. +Proposition 4.4. The Laplacian Cg,β is symmetric if β = βf. +Proof. Since in the wedge region dvolb relates to the Riemannian volume element +dvolg of the underlying wedge metric g by +(4.15) +dvolb = x2βf dvolg, +it follows from (4.12) that +⟨∆gu, v⟩xβH0 +b(X) = +ˆ +X +x−2(β−βf)v∆gu dvolg. + +12 +LEVI LOPES DE LIMA +The result follows if we take u, v ∈ C∞ +cpt(X). +□ +Remark 4.5. The assertion in Proposition 4.4 also holds true if we replace ∆g by +any operator which is formally self-adjoint with respect to L2(X, dvolg) (the Dirac +operator, for instance). +At least in the conical case (f = n − 1) the mapping properties of Mg,β are +completely determined by the spectral resolution of ∆gZ in a way that we now +explain. We first note that for a general wedge space, +Mg,β|U = −D2 +x − (f − 1) Dx − ∆gZ − x2∆gY + o(1), +where Dx = x∂x and of course the term x2∆gY should be omitted in the conical +case (Y collapses into a point). In any case, after discarding the last two terms with +an explicit dependence on x we obtain the indicial operator +(4.16) +IMg,β = −D2 +x − (f − 1) Dx − ∆gZ. +Also, if we replace Dx by ζ ∈ C we get the indicial symbol of Mg,β, +(4.17) +IMg,β(ζ) = −ζ2 − (f − 1)ζ − ∆gZ, +a one-parameter family of elliptic operators on (Z, gZ). +Let us consider the spectrum of ∆gZ, +Spec(∆gZ ) = {µ ∈ R; ∃u ̸= 0 satisfying − ∆gZu = µu} ⊂ [0, +∞). +For each µ ∈ Spec(∆gZ), let {ζ± +µ,f} be the (real) roots of the indicial equation +ζ2 + (f − 1)ζ − µ = 0, +which is obtained by equating to zero the restriction of the right-hand side of (4.17) +to each eigenspace of ∆gZ. Explicitly, the indicial roots are +(4.18) +ζ± +µ,f = γf ± +� +γ2 +f + µ. +The corresponding indicial set is +If +Mg = +� +µ∈ Spec(∆gZ ) +{ζ± +µ,f}, +which is a discrete subset of R (because Z is closed). Finally, the non-indicial set is +I/f +Mg = R\If +Mg, +a countable union of bounded, open intervals. In the conical case, we have the +following well-known result. +Theorem 4.6. The map (4.13) is Fredholm if and only if β ∈ I/n−1 +Mg , with the corresponding +Fredholm index remaining the same as long as β varies in a given connected component of +I/n−1 +Mg . +This follows from [Maz91, Theorem 4.4]. An outline of its proof, along the lines +of the theory developed in [Sch98, Les97], may be found in [dL22b, dL22a], albeit +the weight numerics there is slightly different from ours. Also, see [SS01] for the +specific information regarding the dependence of the Fredholm index on β. +Although its proof requires a somewhat delicate analysis, it is intuitively clear +from this discussion that the following holds in the conical case: as β varies in + +THE SCALAR CURVATURE IN WEDGE SPACES +13 +I/n−1 +Mg , the Fredholm index of (4.13) assumes its minimal value in the “innermost” +non-indicial interval corresponding to µ = 0, namely, In−1 := (2 − n, 0). Also, +we may compute this minimal Fredholm index by observing that, as explained +in [dL22a, Corollary 3.6], (4.13) is essentially self-adjoint (and hence Fredholm of +index 0) if β = βn−1; see Proposition 4.4. For the sake of comparison with the +corresponding argument below in the general wedge case, we briefly reproduce +this reasoning here. +We recall that the minimal domain of Cg,β as defined by (4.14) is +Dmin(Cg,β) = +� +u ∈ xβH0 +b(X); ∃{un} ⊂ C∞ +cpt(Xs); un +H0 +b +−→ u, ∆gun is H0 +b − Cauchy +� +, +whereas its maximal domain is +Dmax(Cg,β) = +� +u ∈ xβH0 +b(X); ∆gu ∈ xβH0 +b(X) +� +. +It is known that Dmin(Cg,β) ⊂ Dmax(Cg,β) and that (closed) sub-spaces of +Q(Cg,β) := Dmax(Cg,β)/Dmin(Cg,β) +correspond to domains of closed extensions of Cg,β. The key observation now is +the following fact whose proof may be found in [Les97, Section 1.3], but again with +a different weight numerics: +• Q(Mg,β) is formed by contributions coming from the finite set +In−1 +β +:= In−1 +Mg ∩ (β, 2 + β). +In particular, Q(Cg,β) = {0}, and hence Cg,β has a unique closed extension, +if In−1 +β += ∅. +We now remark that (βn−1, 2+βn−1) ⊂ In−1 and hence In−1 +βn−1 = ∅ if n > 4. Thus, +Cg,βn−1 has a unique closed extension which is self-adjoint (and hence Fredholm +of index 0) because it is symmetric by Proposition 4.4. The next result then follows +from the discussion above and Theorem 4.6. +Theorem 4.7. Let (X, g) be a conical space satisfying n > 4. Then the unbounded +symmetric map Cg,βn−1 in (4.14) is essentially self-adjoint. Moreover, the bounded map +(4.13) is Fredholm of index 0 as long as β ∈ In−1. +As explained in [dL22b, Section 2], at least if n > 4 this information suffices to +carry out the proof of Theorem 2.7 in this conical case (for more details of the ar- +gument together with a checking on how the missing case n = 4 may be recovered +by an alternative method, we refer to the proof of Theorem 2.7 below). +We now turn to the (non-conical) purely wedge case (1 ≤ f < n − 1). Here, +things quickly get complicated because the indicial decomposition R = If +Mg ⊔I/f +Mg +fails to sharply determine the mapping properties of (4.13), as no analogue of The- +orem 4.6 is expected to hold. Naturally enough, now the singular stratum (Y, gY ) +also plays a role by contributing an extra term to the indicial operator (4.16), so +as to form the so-called normal operator NMg, which may be identified to −t2∆gc, +where gc = dt2 + t2gZ + δu, δu = |du|2, is the natural metric on (0, +∞)t × Z × Rb +u, + +14 +LEVI LOPES DE LIMA +viewed as the tangent cone arising from the one parameter family of dilations +Tρ(y0)(x, y, z) = (ρx, y0 + ρ(y − y0), z), y0 ∈ Y , as ρ → +∞. Thus, +NMg = −D2 +t − (f − 1) Dt − ∆gZ − t2∆δu. +Comparison with the indicial operator in (4.16) shows that the extra term in +NMg spoils the invariance under dilations in t, which reveals an essential de- +parture from the conical case, but notice that both invariances under dilations in +(t, u) and translations in u are retained. As in [Maz91] we explore this by first +Fourier transforming in the u-direction with ξ ∈ T ∗Rb +u as dual variable. We next +set ϑ = ξ/|ξ|δu ∈ S∗Rb +u, the spherical conormal bundle, and s = |ξ|δut so as to +obtain the equivalent “Bessel-type” normal operator +BMg(ϑ) = −D2 +s − (f − 1) Ds − ∆gZ + s2|ϑ|2 +δu. +Note that the explicit dependence on ϑ is illusory since |ϑ|δu = 1. Thus, BMg acts +on functions in CZ := [0, +∞)s × Z, the (infinite) cone over Z endowed with the +conical metric ds2 + s2gZ. +To proceed, consider the spaces +Hσ,β,l(CZ) = {u ∈ D′(CZ); φu ∈ sβHσ +b(CZ), (1 − φ)u ∈ s−lHσ(CZ)}, +where φ ∈ C∞ +cpt(CZ) with φ = 1 near s = 0. Thus, we may view BMg as an operator +(4.19) +BMg : Hσ,β,l(CZ) → Hσ−2,β,l(CZ), +whose mapping properties are closely tied to Fredholmness for (4.13). Indeed, the +next criterion follows from the general theory developed in [Maz91, Sch98, ES12]. +Theorem 4.8. The map (4.13) is Fredholm provided β ∈ I/f +Mg and (4.19) is invertible (for +such β). If (4.19) is only known to be injective then (4.13) is semi-Fredholm (that is, it has +a closed range and is essentially injective in the sense that its kernel is finite dimensional). +Thus, establishing Fredholmness of (4.13), which should be thought of as a pre- +liminary step in probing its mapping properties, gets reduced to finding a range +of weights for which invertibility of (4.19) holds true. +Note that BMg is ellip- +tic as in [Maz91, Definition 5.3] with the same indicial operator as Mg. Thus, +[Maz91, Lemma 5.5] implies that the normal operator in (4.19) is Fredholm pro- +vided β ∈ I/f +Mg. It is known that the index of BMg does not depend on the pair +(σ, l) and remains the same as long as β varies in a given connected component of +I/f +Mg. Similarly, by [Maz91, Corollary 5.7] the kernel of BMg does not depend on +(σ, l) as well, although it might change when β crosses If +Mg. Fortunately, we shall +see that this kernel turns out to be trivial in an appropriate range of weights. +Proposition 4.9. The normal operator BMg in (4.19) is injective if β > 1 − f, f > 1. +Proof. By separation of variables, we find that a solution w ∈ Hσ,β,l(CZ) of the +homogeneous equation BMgw = 0 decomposes as +w = +� +µ∈Spec(∆gZ ) +wµ, +wµ ∈ Hσ,β,l +µ +(CZ), + +THE SCALAR CURVATURE IN WEDGE SPACES +15 +where Hσ,β,l +µ +(CZ) ⊂ Hσ,β,l(CZ) selects the eigenspace of ∆gZ associated to µ. Thus, +it suffices to check that each wµ vanishes. Since BMgwµ = 0, wµ can be expressed +in terms of the modified Bessel functions Iν(s) and Kν(s), where +ν = ν± = ± +� +γ2 +f + µ. +Precisely, +wµ(s) = sγf (c1Iν(s) + c2Kν(s)) , +where ci is a constant. Since ν ̸= 0, the asymptotical behavior of these Bessel +functions are as in the table below. +s → 0 +s → +∞ +Iν(s) +∼ sν +∼ es/s +Kν(s) +∼ s−|ν| +∼ e−s/√s +The exponential growth of Iν at infinity forces c1 = 0, whereas the exponential +decay of Kν poses no restriction on c2. Since wµ(s) ∼ sγf −|ν| as s → 0 and +γf − |ν| ≤ γf − |γf| = 2γf = 1 − f, +the result follows. +□ +Although injectivity of the normal operator suffices for our purposes as it pro- +vides, via Theorem 4.8, a left generalized inverse for Mg,β to which the argument +in the proof of Theorem 4.11 may be applied, it is known that the normal operator +is surjective in the “innermost” non-indicial interval If, at least if f > 1. Since this +information is not used in the sequel, we omit its proof. +Theorem 4.10. The normal operator (4.19) is surjective if β ∈ If, f > 1. +In view of Theorem 4.8, it follows from the various results established above +that (4.13) is Fredholm provided β ∈ If, f > 1 (if we ignore Theorem 4.10, it is at +least semi-Fredholm and essentially injective). Unfortunately, this does not suffice +to proceed as in the conical case in order to detect self-adjoint extensions starting +from Q(Cg,β); see [MV12, Section 2.5] for a discussion of the issues involved. We +may, however, combine this information with the reasoning in [ARS22, Section 1], +which by its turn is based on arguments in [GM03, ALMP12, AGR16, ALMP18], to +check that in most cases the Laplacian Cg,βf in (4.14), which is symmetric by Propo- +sition 4.4, has good mapping properties. As applied to our context, the idea is that +the existence of a (left) generalized inverse for (4.13) with β ∈ (βf, 2 + βf) leads +to an extra regularity in the weighted Sobolev scale for elements of Dmax(Cg,βf ) +which forces them to actually belong to Dmin(Cg,βf ) by characterizations of this +latter space appearing in [GM03, Proposition 3.6] and [GKM13, Theorem 4.2]. +Theorem 4.11. Let (X, g) be a wedge space satisfying either 3 ≤ f ≤ n − 1 or f = 1 +and the cone angle is at most 2π. Then the Laplacian Cg,βf defined in (4.14) is essentially +self-adjoint. Moreover, letting Dg,βf be the domain of this self-adjoint extension, the map +(4.20) +Cg,βf + λ : Dg,βf → xβf H0 +b(X) +is Fredholm of index 0 for any λ ∈ R. + +16 +LEVI LOPES DE LIMA +Proof. Let us initially assume that f > 3 so that both βf and 2+βf lie in If. We have +seen that Mg,β defined in (4.13) is semi-Fredholm and essentially injective pro- +vided β ∈ If. In particular, there exists a left generalized inverse G : xβH0 +b(X) → +xβH2 +b(X) for Mg,β, which means that +G∆g = Id − Πβ : xβH2 +b(X) → xβH2 +b(X), +β ∈ If, +where G = −Gx2 and Πβ is the projection onto the kernel of Mg,β. Now take +u ∈ Dmax(Cg,βf ) so that u = G∆gu + Πβu. From the diagram +x2+βf H0 +b(X) +xβH0 +b(X) +xβH2 +b(X) +Dmax(Cg,βf ) +G +−x2∆g +−x2∆g +G∆g +where the inclusion in the upper row comes from the Sobolev embedding in Re- +mark 4.3, we see that G∆gu ∈ xβH2 +b(X). Also, since Mg,βΠβu = 0 one expects that +Πβu is much more regular than it appears. Indeed, it already follows from [Maz91, +Corollary 3.24] that Πβu ∈ xβf Hσ +b(X), σ ≥ 0. With some more work we may infer +that Πβu ∈ xβH2 +b(X) as well (this relies on the analysis in [Maz91, Section 7] and +uses that no indicial root lies in the interval (βf, β], where the normal operator is +known to be injective). Hence, by setting ε = 2 + βf − β we obtain the inclusion +(4.21) +Dmax(Cg,βf ) ⊂ +� +0<ε<2 +x2+βf −εH2 +b(X), +that is, elements of Dmax(Cg,βf ) enjoy an extra “weighted regularity”. We now +claim that this leads to +(4.22) +D := +� +0<ε<2 +x2+βf −εH2 +b(X) ⊂ Dmin(Cg,βf ). +These inclusions imply Dmax(Cg,βf ) ⊂ Dmin(Cg,βf ), that is, Cg,βf is essentially self- +adjoint. +To prove (4.22) we will make use of the well-known fact that, since Cg,βf = −∆g +is symmetric by Proposition 4.4, u ∈ Dmax(Cg,βf ) is in Dmin(Cg,βf ) if and only if +⟨∆gu, v⟩xβf H0 +b(X) = ⟨u, ∆gv⟩xβf H0 +b(X) , +for any v ∈ Dmax(Cg,βf ); compare with [ALMP12, Lemma 5.12]. If u ∈ D set +un := x1/nu, n ∈ N, so that un ∈ x2+βf H2 +b(X). Also, for any ε ∈ (0, 2), +un → u +in +x2+βf −εH2 +b(X), +which gives +(4.23) +xε∆gun → xε∆gu +in +xβf H0 +b(X). + +THE SCALAR CURVATURE IN WEDGE SPACES +17 +Moreover, since ε �→ 2 − ε is a bijection of (0, 2), besides (4.23) we also have +x−εDmax(Cg,βf ) ⊂ xβf H2 +b(X). Hence, if v ∈ Dmax(Cg,βf ) we get +⟨∆gun, v⟩xβf H0 +b(X) += +� +xε∆gun, x−εv +� +xβf H0 +b(X) +→ +� +xε∆gu, x−εv +� +xβf H0 +b(X) += +⟨∆gu, v⟩xβf H0 +b(X) . +Also, +⟨∆gun, v⟩xβf H0 +b(X) = ⟨un, ∆gv⟩xβf H0 +b(X) → ⟨u, ∆gv⟩xβf H0 +b(X) , +where in the first step we used that un ∈ Dmin(Cg,βf ) and in the last one that +un → u in xβf +εH2 +b(X) ⊂ xβf H0 +b(X). Thus, +⟨∆gu, v⟩xβf H0 +b(X) = ⟨u, ∆gv⟩xβf H0 +b(X) , +which means that u ∈ Dmin(Cg,βf ) and hence proves (4.22). The fact that (4.20) is +Fredholm now follows from standard arguments based on the compact inclusion +Dg,βf = Dmax(Cg,βf ) ⊂ xβf H0 +b(X), which follows from (4.21) and Remark 4.3. +Finally, using that I3 = (−2, 0) = (β3, 2 + β3), the limiting case f = 3 may be +treated by the same argument since (4.21) still holds true even if both βf and 2+βf +are indicial roots. +We now consider the cone-edge case (f = 1), in which the link is a circle of +length L (the cone angle). It follows from (4.18) that ζ± +0,1 = 0, so the “inner- +most” non-indicial interval disappears. +If we assume that L < 2π then both +β1 = −1 ∈ I− +1 := (−2π/L, 0) and 2 + β1 = 1 ∈ I+ +1 := (0, 2π/L) are non-indicial; +note that I± +1 now jointly play the role of “innermost” non-indicial intervals. Using +that K0(s) ∼ − log(s/2) as s → 0, Bessel asymptotics implies that the correspond- +ing normal operator is injective for β > −2π/L. Hence, we may invoke Theorem +4.8 to ensure that Mg,β is semi-Fredholm and essentially injective for any non- +indicial β > −2π/L, which in particular provides a left generalized inverse for +Mg,β, β ∈ (−1, 1)\{0}, to which the argument in the previous paragraph may be +applied. Thus, essential self-adjointness holds provided L ≤ 2π. +□ +Remark 4.12. If we further assume that f > 3, so that 2+βf is not an indicial root, +then [GKM13, Theorem 4.2] actually identifies Dg,βf , the domain of the unique +self-adjoint extension of Cg,βf , to x2+βf H2 +b(X); see also [HLV18, Theorem 1.1] for +another proof of this result which is functional analytical in nature and altogether +avoids the use of the (so pervasive!) micro-local artillery. But we insist that this +extra piece of information is not needed here, so we content ourselves with the +formulation of Theorem 4.11, which suffices for our purposes. +Remark 4.13. An obvious reason why the method above does not directly apply +to the case f = 2 is that β2 = −3/2 lies far away from the range β > −1 where the +normal operator is known to be injective, so that (4.21) is unavailable (note that +the “innermost” non-indicial interval here is (−1, 0)). A similar difficulty appears +in the cone-edge case with cone angle L > 2π. A possible way to circumvent this, +equally convenient for our purposes, consists in determining a range of weights +where Cg,β would be surjective, but we will not pursue this because, as pointed +out in Remarks 2.8 and 2.9, those cases are not really needed here. + +18 +LEVI LOPES DE LIMA +Remark 4.14. The compact inclusion Dg,βf ⊂ xβf H0 +b(X) implies not only Fred- +holmness of (4.20) but also that its spectrum is discrete with each eigenspace of +finite multiplicity. +Remark 4.15. Instead of Mg, we may work with the operator +� +Mg += +x−βf Mgxβf += +−D2 +x + 2Dx + βf(2 + βf) − ∆gZ − x2∆gY + o(1), +acting on the weighted Sobolev scale based on the wedge volume element dvolg, +so that the “innermost” indicial roots associated to µ = 0 are 2+βf and −βf. Since +the “symmetry weight” now occurs ate β = 0 and f ≥ 3 implies that (0, 2) has no +indicial root, in this case Theorem 4.11 follows from [ARS22, Proposition 1.5]. +Remark 4.16. It is instructive to compare the analysis above with the complete edge +case, in which instead of (2.2) we take +g|U(x, y, z) = x−2 � +dx2 + x2gZ(z) + gY (y) + o(x) +� +. +This includes the conformally compact, asymptotically hyperbolic case with [gY ] +as conformal boundary (when Z collapses into a point). Here we are interested in +establishing the mapping properties of the operator P = −∆g + λ, where now λ is +a smooth function on Xs converging to a fixed value λ∗ ∈ R as one approaches Y . +As instances of geometric problems where this kind of question arises we mention +the singular Yamabe problem [MS91] and the rigidity of hyperbolic space in the +context of the AdS/CFT correspondence [Qin03]. As usual, we work with the +appropriate weighted Sobolev scale xδHσ +e (X) based on the edge volume element +dvole := x−1dxdvolgY dvolgZ. To simplify matters, let us assume that b ≥ 1 and +b2/4 + λ∗ > 0. Since +P|U = −D2 +x + bDx − ∆gZ − x2∆gY + λ + o(1), +the indicial symbol is +IP(ζ) = −ζ2 + bζ − ∆gZ + λ∗, +so that the “innermost” non-indicial interval is �I = (δ−, δ+), where +δ± = b +2 ± +�� b +2 +�2 ++ λ∗. +Also, a computation shows that the normal operator BP may be identified to +−∆�g + λ∗, where �g = ghyp + gZ is the product metric on Hb+1 × Z. Now, if +u ∈ xδH0 +e(X), δ > δ−, satisfies BPu = 0 then u ∈ xb/2H0 +e(X) = L2(X, dvolg), +so the invertibility of BP in �I boils down to checking that the L2 spectrum of +(Hb+1 × Z, �g) is disjoint from (−∞, b2/4) [McK70]. It then follows from the micro- +local analysis in [MS91] that P is Fredholm of index 0 in this range of weights; see +also [MM91] for a potential theoretic approach to this same result and note that +[Lee06] also treats the conformally compact case by “low-tech” methods. But no- +tice that here the vanishing of the index is readily guessed given that the essential +self-adjointness of P at δ = b/2 ∈ �I is a classical accomplishment [Gaf51] (recall +that g is complete). In particular, the analogue of the delicate analysis leading to +Theorem 4.11 is not needed. + +THE SCALAR CURVATURE IN WEDGE SPACES +19 +5. AN EXISTENCE RESULT: THE PROOF OF THEOREM 2.7 +We present here the proof of Theorem 2.7. We start with a result confirming that +the prescription problem for the scalar curvature of wedge metrics is invariant +under diffeomorphisms of bounded distortion; compare with [KW75, Theorem +2.1] and [dL22b, Proposition 2.2]. +Proposition 5.1. Let φ, φ′ ∈ C0(Xs) ∩ L∞(Xs) ֒→ xβH0 +b(X), β ≤ 0. If min φ < +φ′ < max φ then for any ε > 0 there exists a diffeomorphism Ψ : Xs → Xs of bounded +distortion (in particular, preserving the quasi-isometry class of wedge metrics) such that +∥φ ◦ Ψ − φ′∥xβH0 +b(X) < ε. +Now let (X, g) be a wedge space as in Theorem 2.7. By Theorem 3.9 and Re- +marks 3.8 and 3.10, we may assume that f ̸= 2 and Rg = −1, where 1 is the +function identically equal to 1. If ǫ > 0 is small enough, consider the smooth map +A : Bǫ(1) ⊂ Dg,βf → xβf H0 +b(X, dvolb) +given by +A(u) := R +u +4 +n−2 g = −u−αn � +c−1 +n ∆gu + u +� +, +αn = n + 2 +n − 2. +Note that A(1) = −1. +Proposition 5.2. The linearization ˙A1 : Dg,βf → xβf H0 +b(X) is an isomorphism. +Proof. A short computation gives +(5.24) +˙A1 = c−1 +n (−∆g + λn) , +λn = +4cn +n − 2 = +1 +n − 1, +which is Fredholm of index 0 by Theorem 4.11 (after a possible rescaling of the +circle link if f = 1). Alternatively, we may use Theorem 4.7 if f = n − 1, n > 4. +Note that self-adjointness of Cg,βf guarantees that integration by parts does not +yield a contribution coming from the singular stratum. Thus, if ˙A1u = 0 we get +− ∥∇gu∥2 +xβf H0 +b(X) = λn ∥u∥2 +xβf H0 +b(X) , +a contradiction unless u = 0, which confirms injectivity of ˙A1. The result now +follows from Fredholm alternative. +□ +By the Inverse Function Theorem, there exists ε > 0 and a neighborhood U ⊂ +Cg,β of 1 such that A|U : U → Bε(−1) ⊂ xβf H0 +b(X) is a diffeomorphism. Also, if +F is the prescribed function then there exists K > 0 such that K min F < −1 < +K max F and Proposition 5.1 gives a diffeomorphism Ψ : Xs → Xs of bounded +distortion such that KF ◦ Ψ ∈ Bε(−1), which implies that KF ◦ Ψ may be realized +as the scalar curvature function of some wedge metric �g. Thus, K1/2(Ψ−1)∗�g is +the required wedge metric (whose scalar curvature is F). Finally, we note that this +approach turns out to be effective enough to allow us to prove Theorem 2.7 in the +remaining conical case n = 4. Indeed, it suffices to take β = β3 and use that, again +by Theorem 4.11, Mg,β3 as in (4.14) is essentially self-adjoint. +Remark 5.3. Again, the regularity theory in [ACM14, Section 3] ensures that the +conformal metric produced in Theorem 2.7 lies in the same quasi-isometry class as +the original wedge metric and hence is wedge as well; compare with Remark 3.10. + +20 +LEVI LOPES DE LIMA +Remark 5.4. Although surjectivity of (5.24) would suffice to produce the metric �g +above via the Implicit Function Theorem, the finer isomorphism property shows +that the space of conformal factors yielding wedge metrics whose scalar curvature +functions are close to −1 may be locally parameterized by an open ball centered +at 1 ∈ Dg,βf . In particular, local uniqueness for the associated non-linear problem +holds. As another example of this sort of phenomenon, we note that with no re- +striction at all on the geometry of the link if f ≥ 3, the perturbative method above +may be used to check that for any v close enough to 1 there is a (unique) u close to +1 solving the semi-linear elliptic equation +∆gu + u = vuα, +α > 1. +Remark 5.5. A routine procedure that we omit here allows us to recast both Theo- +rem 4.11 and the proof of Theorem 2.7 in the language of weighted H¨older spaces, +which is more convenient to directly handle the regularity of solutions that we +actually refrained from spelling out in the Sobolev analysis above. +6. A TOPOLOGICAL OBSTRUCTION: THE PROOF OF THEOREM 2.10 +We now look at topological obstructions to the existence of wedge metrics with +(strictly) positive scalar curvature in a given wedge space. Our aim is to present +the proof of Theorem 2.10 and for this we will use a version of Index Theory as +developed in [AGR16], so we assume from now on that Xs is spin. Thus, in the +presence of a wedge metric g, we may consider the corresponding Dirac operator +acting on the associated spin bundle +ð : Γ(SX) → Γ(SX). +As before, we may define the weighted Sobolev scale xβHσ +b(SX) formed by ap- +propriate distributional spinors, so the question remains of determining for which +values of β the map +(6.25) +ð : xβHσ +b(SX) → xβHσ−1 +b +(SX) +is at least semi-Fredholm and essentially injective, as this is the first step in trying +to establish good mapping properties. The argument below, leading to Theorem +6.1, adapts to the Dirac setting the approach adopted in Section 4 for Laplace-type +operators and should be thought of as a variation of the reasoning in [AGR16]. +It follows from [AGR16, Lemma 2.2] that, in the wedge region U, +(6.26) +xð = c(∂x) +� +Dx + f +2 + ðZ + xðY +� ++ o(x), +where c is Clifford product and ðZ (respectively, ðY ) is the Dirac operator of (Z, gZ) +(respectively, (Y, gY )). Replacing Dx by ζ in the right-hand side above and sending +x → 0, we see that the corresponding indicial symbol is +Ið(ζ) = ζ + f +2 + ðZ. +Since we aim at obstructing positive scalar curvature metrics, we may assume that +Rg|U ≥ 0. It follows from (3.10) that RgZ ≥ f(f − 1) > 0 if f > 1. A well-known + +THE SCALAR CURVATURE IN WEDGE SPACES +21 +estimate [Fri00, Section 5.1] then gives the spectral gap +(6.27) +Spec(ðZ) ∩ +� +−f +2 , f +2 +� += ∅, +known as the “geometric Witt condition”. As explained in [dL22b, dL22a], at least +in the conical case (f = n − 1), this analysis suffices to guarantee that (6.25) is +Fredholm of index 0 whenever β ∈ Jn−1 := (1 − n, 0) since (6.27) prevents the +appearance of indicial roots in this interval. In the general wedge case, however, +one should take into account the effects coming from the corresponding normal +operator +Bð(ϑ) = c(∂s) +� +Ds + f +2 + ðZ +� ++ sic(ϑ), +ϑ ∈ S∗Rb +u. +Luckily, the general mapping theory of wedge elliptic operators [Maz91, Sch98] +tells us that the pattern to follow here is quite similar to the case of the Laplacian +studied in Section 4, which will allow us to easily transplant the previous analy- +sis to the present context. Firstly, we check that when acting on the appropriate +Sobolev scale on CZ, Bð is injective for β > −f and surjective for β ∈ Jf := (−f, 0), +the “innermost” non-indicial interval in this setting; see the proof of [AGR16, +Lemma 3.10], but be aware that the indicial numerics there is different from ours +essentially because their weighted Sobolev spaces are based on the volume ele- +ment dvolg instead of dvolb; another approach to this issue appears in [HLV18, +Section 6]. We insist, however, that only the injectivity of Bð on Jf is really needed +in the sequel, and this is an easy consequence of elementary Bessel asymptotics (as +in the proof of Proposition 4.9). Thus, (6.25) is semi-Fredholm and essentially in- +jective for β ∈ Jf by the appropriate analogue of Theorem 4.8. Secondly, we com- +bine this information (translated into the existence of a left generalized inverse for +(6.25) in this range of weights) with the argument in the proof of [AGR16, Theorem +3.11], which is the Dirac version of [ARS22, Section 1], to establish the following +result. +Theorem 6.1. Let (X, g) be a spin wedge space with Rg|U ≥ 0. If f = 1 assume also +that the cone angle is at most 2π. Then the unbounded map +(6.28) +ð : Γcpt(Xs) ⊂ xβf H0 +b(SX) → xβf H0 +b(SX) +is essentially self-adjoint and the corresponding self-adjoint extension is Fredholm. +Proof. As already advertised, this is just a matter of transplanting the analysis for +the Laplacian in Section 4 to this Dirac setting. In particular, have Remark 4.5 in +mind and note that 2 + βf should be replaced by 1 + βf, as dictated by the left- +hand side of (6.26). If f > 1 then both βf and 1 + βf lie in Jf and we may proceed +exactly as in the proof of Theorem 4.11 (where the corresponding assumption is +f > 3). Notice that f > 1 is also required here to justify the spectral estimate +leading to (6.27). Nonetheless, (6.27) holds true in the cone-edge case f = 1 if the +cone angle is at most 2π and we choose the bounding spin structure on the link +circle; see [Cho89, AGR16] for discussions on this subtlety. In any case, with this +extra information at hand, our reasoning above may also be transplanted to cover +the limiting case f = 1 since J1 = (−1, 0) = (β1, 1 + β1) (compare again with the +proof of Theorem 4.11, where the “limiting” case is f = 3). +□ + +22 +LEVI LOPES DE LIMA +Remark 6.2. As in Remark 4.12, it follows from [GKM13, HLV18] that the domain +of the self-adjoint extension of ð above is x1+βf H1 +b(SX), at least if f > 1. If f = 1 +and the cone angle is strictly less that 2π then 0 = 1 + β1 is not an indicial root and +the corresponding domain is H1 +b(SX). +We now make use of the theory above to find topological obstructions to the +existence of wedge metrics of positive scalar curvature. Let us still denote by ð the +self-adjoint realization of (6.28). If n = 2k is even, it is well -known that ð splits as +ð = +� +0 +ð− +ð+ +0 +� +where ð± are the realizations of the chiral Dirac operators +ð± : Γcpt(S± +X) → Γcpt(S∓ +X) +corresponding to the chiral decomposition SX = S+ +X ⊕ S− +X. It then follows from +Theorem 6.1 that these chiral Dirac operators are Fredholm and adjoint to each +other, so it makes sense to consider the associated index +ind ð+ = dim ker ð+ − dim ker ð−. +More generally, if E is a Hermitian vector bundle over Xs endowed with a com- +patible connection, we may consider the twisted Dirac operator +ð ⊗ E = +� +0 +ð−⊗ E +ð+⊗ E +0 +� +In general, twisting with E spoils self-adjointness since ðgZ ⊗ E|Z should some- +how appear in the expressions of the corresponding indicial symbol and normal +operator, thus compromising the analysis above. However, if E is admissible in the +sense that E|U is trivial (and endowed with a flat connection) then ð⊗E|U is a sum +of r := rankE copies of ð|U and, as explained in the proof of [dL22b, Proposition +4.1], Theorem 6.1 and Remark 6.2 hold verbatim for ð ⊗ E, so we can define the +corresponding index +ind ð+⊗ E = dim ker ð+⊗ E − dim ker ð−⊗ E. +It turns out that this fundamental invariant can be explicitly computed in terms +of the underlying geometry by means of heat asymptotics [Cho85, Les97, AGR16]. +Indeed, if Θr is the trivial bundle of rank r and �E = E −Θr is the associated virtual +bundle then [AGR16, Main Theorem] tells us that +(6.29) +ind ð+⊗ �E = r ind ð+ + +ˆ +Xs +�A(T Xs) ∧ ch �E, +where +ind ð+ = +ˆ +Xs +�A(T Xs) + +ˆ +Y +�A(T Y ) +� +−1 +2ηðZ + +ˆ +Z +T �A(T Xs) +� +, +�A is the �A-class, T means transgression, ηðZ is the eta invariant of ðZ and ch �E is +the Chern character of �E. + +THE SCALAR CURVATURE IN WEDGE SPACES +23 +According to [AGR16, Theorem 1.3], if g has (strictly) positive scalar curvature +everywhere then ind ð+ = 0, so that (6.29) reduces to +(6.30) +ind ð+⊗ �E = +ˆ +Xs +�A(T Xs) ∧ ch �E, +for any admissible bundle E. Notice that �A(T Xs) only contributes to the right- +hand side with “lower order” terms, which aligns with the comments in Remark +2.12. We now introduce the class of wedge spaces to which Theorem 2.10 applies; +compare with [dL22b, Section 3], where the conical case is discussed, and also with +[Gro96], where this notion was originally conceived in the smooth category. +Definition 6.3. A wedge space (X, g) has infinite K-area if for any ǫ > 0 there exists +an admissible, ǫ-flat bundle over X which is homologically nontrivial in the sense +that at least one of its Chern numbers does not vanish. +A key point here is that having infinite K-area is a quasi-isometric property of +the wedge space (X, g). In any case, with (6.30) at hand and proceeding exactly as +in [dL22b, Section 5.1], the proof of Theorem 2.10 follows immediately. +REFERENCES +[ACM14] +Kazuo Akutagawa, Gilles Carron, and Rafe Mazzeo. The Yamabe problem on stratified +spaces. 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HUMBERTO MONTE, S/N, BLOCO 914, 60455-760, FORTALEZA, CE, BRAZIL. +Email address: levi@mat.ufc.br + diff --git a/x9E4T4oBgHgl3EQfYQwo/content/tmp_files/load_file.txt b/x9E4T4oBgHgl3EQfYQwo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f125f46010729f2caea47b3313cd72318a6446e6 --- /dev/null +++ b/x9E4T4oBgHgl3EQfYQwo/content/tmp_files/load_file.txt @@ -0,0 +1,920 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf,len=919 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='05047v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='DG] 12 Jan 2023 THE SCALAR CURVATURE IN WEDGE SPACES: EXISTENCE AND OBSTRUCTIONS LEVI LOPES DE LIMA ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We study the scalar curvature of incomplete wedge metrics in certain stratified spaces with a single singular stratum (wedge spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Building upon sev- eral well established technical tools for this category of spaces (the corresponding Yamabe, elliptic and index theories) we provide existence and obstruction results for such metrics under suitable positivity assumptions on the underlying geom- etry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' This is meant to be a follow-up to a previous paper of ours (AGAG, 2022), where the case of spaces with an isolated conical singularity was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' CONTENTS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Wedge spaces and statements of the main results 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The Yamabe problem in the wedge setting 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The wedge elliptic theory for the scalar Laplacian 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' An existence result: the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' A topological obstruction: the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10 20 References 23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' INTRODUCTION The general problem of prescribing the scalar curvature function in a given smooth closed manifold is a central theme in Riemannian Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Since in prin- ciple this metric invariant only affects the underlying geometry at a local level with no direct influence on its large scale behavior, it is expected that a huge amount of functions might be realized as the scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In a sense this has been con- firmed by Kazdan and Warner [KW75], who showed that any function on a closed manifold of dimension n ≥ 3 which is negative somewhere is the scalar curva- ture of some metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Versions of this result in the setting of compact manifolds with boundary have been established in [CV19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' These contributions should be contrasted with the well-known topological obstructions to the existence of met- rics with positive scalar curvature in the spin setting stemming from the works of Lichnerowicz [Lic63], Hitchin [Hit74] and Gromov-Lawson [GL80b, Gro96] and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' de Lima has been supported by CNPq 312485/2018-2 and FUNCAP/CNPq/PRONEX 00068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='00/15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' 1 2 LEVI LOPES DE LIMA relying upon the remarkable properties of the Dirac operator acting on spinors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' see also [BH20] for similar obstructions in the presence of a boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The analysis in [dL22b] has suitably extended the aforementioned results to the category of spaces with an isolated conical singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The purpose of this note is to investigate further extensions of these contributions to the more general category of (incomplete) edge spaces with a single singular stratum, also named wedge spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The main results here are Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10 dealing respectively with existence and obstructions of wedge metrics with prescribed scalar curvature on such spaces under certain positivity assumptions on the underlying geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In particular, we provide a unified treatment that retrieves the conical case studied in [dL22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Also, as explained in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='12, the obstruction in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10 in a sense complements those obtained in [AGR16, BPR21] by using similar tools (the wedge index theory in [AGR16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In order to carry out the arguments needed to establish Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7, certain as- pects of the solution of the Yamabe problem in this wedge category are reviewed in Section 3, following [ACM14] closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Another key ingredient here is the mapping theory of geometric elliptic operators on such spaces (specifically, the Laplacian acting on functions and the Dirac operator acting on spinors), which is described in Sections 4 and 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' see, among others, [Maz91, Mel93, Les97, Sch98, Gri01, ES12] for the general theory and also [dL22a] for an informal account of this topic in the conical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' This theory is used in Section 4 to establish good mapping proper- ties for the scalar Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' More precisely, we check here that in all cases of in- terest this operator is essentially self-adjoint when viewed as an unbounded and symmetric operator acting on a certain weighted Sobolev space, with the corre- sponding self-adjoint extension being Fredholm (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' This justifies the integration by parts needed to carry out the perturbative scheme leading to Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We remark that surjectivity would suffice in this part of the argument, but we prefer to establish the finer mapping properties for at least two reasons: they provide a local description of the space of solutions of an associated non-linear problem (see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4) and, more fundamentally, the corresponding analysis may be easily transplanted to the Dirac setting, where surjectivity does not suffice and both self-adjointness and Fredholmness are crucial to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The mapping properties mentioned above should routinely follow from results in the existing literature (as in the accounts for the Hodge Laplacian in [MV12, ARS22]) but since we were unable to locate a specific source carrying out the anal- ysis for the scalar Laplacian, we supply a fairly detailed guide to the quite involved proof, as this also will allow us to keep track of the rather subtle role (the dimen- sion of) the link manifold plays in achieving self-adjointness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' see Remarks 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Among the various routes available, we have chosen to adopt as a key input here an argument employed in [ALMP12, ALMP18, AGR16] for Dirac-type oper- ators and in [ARS22] for the Hodge Laplacian, both relying on certain regularity results in [GM03, GKM13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' This same reasoning is used in Section 6 to probe the mapping properties of the Dirac operator needed in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Besides its relevance to the applications mentioned above, we believe that the ap- proach we take here has an independent interest as it might be implemented in THE SCALAR CURVATURE IN WEDGE SPACES 3 other non-linear geometric problems where Laplace-type operators show up at an infinitesimal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We now briefly describe the strategy behind the proofs of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Regard- ing Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7, we adapt the argument laid down in [KW75] in the smooth set- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The first step is to find a wedge metric with constant negative (say −1) scalar curvature (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In the smooth category, this corresponds to the “easy” case of the solution of the classical Yamabe problem [LP87], hence always solv- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Unfortunately, finding such a metric in the wedge category with the avail- able technology (the singular Yamabe theory in [ACM14]) turns out to be consid- erably much harder and seems to require some kind of positivity condition on the geometry of the link manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The simplest possibility, which suffices for the applications we have in mind, is to assume that the link metric has constant posi- tive scalar curvature (the more general case treated in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7, which merely assumes that the link metric is Yamabe positive, may be reduced to this simpler case in view of the well-known solution of the Yamabe problem for closed mani- folds [Sch84, LP87];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In any case, with this wedge metric (say g) at hand, the next step consists in showing that any function “close” enough to −1 may be realized as the scalar function of some conformal wedge metric (if this is the case, it is straightforward to check that the natural action of the group of diffeo- morphisms of bounded distortion on the space of wedge metrics takes care of the general case, in which the prescribed scalar curvature function is negative some- where;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' This step involves a perturbative argument relying on the invertibility of the Laplace-type operator −∆g + λ, λ > 0, arising as the liner- ization at g of the associated non-linear problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Again, in the smooth category the invertibility of this operator in the standard Sobolev scale is well established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In our case, however, one is led to work in a Sobolev scale that takes into account the structure of the singular stratum (the wedge) and it is precisely here that the mapping theory mentioned above is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The outcome of applying this the- ory to our problem appears in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11, confirming that in most cases the Laplacian has good mapping properties, which leads to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Finally, we mention that this same mapping theory is also needed to establish self-adjointness and Fredholmness for the Dirac operator (Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1), which enables the use of the index theory from [AGR16] employed in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The author would like to thank S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Almaraz for helpful dis- cussions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' WEDGE SPACES AND STATEMENTS OF THE MAIN RESULTS We start by describing the class of spaces we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' For simplicity, all smooth manifolds appearing below are assumed to be oriented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' A stratified space with a single singular stratum is a topological space X satisfying: (1) X is a (compact and connected) metric space with distance function d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' (2) X = Y ⊔Xs, where Xs is a (open and dense) smooth manifold of dimension n ≥ 3 (the smooth stratum of X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' 4 LEVI LOPES DE LIMA (3) Y is a (connected and boundaryless) smooth manifold of dimension b ≥ 0 (the singular stratum);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' (4) there exists a (connected) neighborhood U of Y (the wedge region) such that: X\\U is a smooth manifold with boundary Y •;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' there exists a retraction πY : U → Y with πY |U\\Y being smooth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' (5) there exists a “radial function” x = xY : U → [0, 1) such that x−1(0) = Y and with x|U\\Y being smooth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' (6) πY is a locally trivial fibration whose typical fiber is the (truncated) cone CZ = [0, 1) × Z/ ∼ over a closed connected manifold Z (here, ∼ means that {0} × Z has been collapsed into a point), with atlas (φ, V), where each φ : π−1 Y (V) → V × CZ is a trivialization and the corresponding transition functions preserve x (so that, restricted to each fiber, x ◦ φ−1 corresponds to the natural projection CZ → [0, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In particular, for each x ∈ (0, 1) one has a submersion x−1 Y (x) → Y whose typical fiber is Z, so if we send x → 1 we obtain a submersion Y • → Y again with typical fiber Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Hence, the manifold X\\U has a “fibred boundary” Y • which may be viewed as the “resolution” of the singular locus Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We assume that x has been smoothly extended to Xs so that x|Xs\\U ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Notice that n = b + f + 1, where f is the dimension of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Due to our use of the singular Yamabe theory in [ACM14], in the following we will make the key assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1) b ≤ n − 2 ⇐⇒ f ≥ 1 We now introduce the appropriate geometric data in X (more precisely, in the smooth stratum Xs = X\\Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' For simplicity we assume that πY is trivial and we fix a trivializing chart providing an identification U ≃ Y × CZ ≃ Y × ([0, 1) × Z/∼) , with x corresponding to the projection onto the [0, 1)-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Locally around each cone fiber over a fixed fiber of the submersion Y • → X we may introduce coor- dinates (x, y, z), where (y, z) are (local) coordinates on Y × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' This allows us to consider a class of adapted metrics on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' An (incomplete) wedge metric in a stratified space X as above is a Riemannian metric g on its smooth stratum Xs such that: (1) there holds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='2) g|U(x, y, z) = dx2 + x2gZ(z) + gY (y) + o(x), x → 0, where gY and gZ are fixed metrics in Y and Z, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' (2) the distance induced by g coincides with d|Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Under suitable, but still quite restrictive, assumptions we may also treat the case in which the metric gZ varies with y ∈ Y in a smooth way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' For in- stance, we may assume more generally that the family of varying metrics gZ(y, ·), is isospectral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' For simplicity, however, we prefer to avoid this complication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' THE SCALAR CURVATURE IN WEDGE SPACES 5 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' A pair (X, g) as above is a wedge space and the closed Riemannian manifold (Z, gZ) is the link of the singular stratum Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In the cone-edge case f = 1, the length of the linking circle is the cone angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We denote by dvolg the volume element associated to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We may define Sobolev spaces Hk(X, dvolg), k ≥ 0 an integer, in the usual way (just take the closure of Lipschitz functions under the usual Sobolev norm induced by dvolg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' If f ≥ 3 we recall that the conformal Laplacian of (Z, gZ) is the elliptic operator Lf gZ = −∆gZ + f − 2 4(f − 1)RgZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Here and in the following, ∆ stands for the Laplacian and R for the scalar curva- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We say that (Z, gZ) is Yamabe positive if Lf gZ is positive definite (viewed as a self-adjoint operator acting on L2(Z, dvolgZ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' It is known that Yamabe positivity is a conformal property of gZ: it is equivalent to the conformal class [gZ] of gZ carrying a metric with positive scalar curvature [LP87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' If this is the case then the solution of the Yamabe prob- lem for closed manifolds [Sch84, LP87] allows us to replace gZ by a Yamabe metric g′ Z ∈ [gZ], the conformal class of gZ, without affecting the wedge character of the underlying metric (in fact, the corresponding wedge metrics are easily seen to be quasi-isometric to each other, with the quasi-isometry bounds depending only on bounds on the conformal factor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In particular, after a further rescaling of the link we may assume that Rg′ Z = f(f − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' This quasi-isometric replacement of wedge metrics, induced by conformal deformations on the link, is crucial here since only for g′ Z as a link metric we may use Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9, which relies on Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7, to find a wedge metric with constant negative scalar curvature to which the pertur- bative scheme leading to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 below may be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We may now state our first result, which provides an existence theorem for wedge metrics with prescribed scalar curvature under appropriate assumptions on the link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Let (X, g) be a wedge space whose link (Z, gZ) satisfies either f = 1 or f ≥ 3 and it is Yamabe positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Then any smooth and bounded function which is negative somewhere in Xs\\U is the scalar curvature of some wedge metric on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The Yamabe positivity of (Z, gZ) in case f ≥ 3 relates to the already mentioned difficulty in using the singular Yamabe theory in [ACM14] to find a background wedge metric with constant negative scalar curvature to which the perturbative scheme in Section 5 could be applied (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9 (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As men- tioned in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6, after possibly passing to a Yamabe metric g′ Z in the confor- mal class [gZ] of gZ satisfying Rg′ Z = f(f − 1), Yamabe positivity suffices to carry out the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In any case, the fact that the class of closed mani- folds of dimension f ≥ 3 carrying metrics with positive scalar curvature is stable under surgeries of co-dimension at least 3 [GL80a, SY79] provides many examples of wedge spaces to which Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' On the other hand, if f = 2 then Yamabe positivity morally corresponds to asking that the link surface Z is (topo- logically) a sphere, precisely the case covered by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9 (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' see also Remark 6 LEVI LOPES DE LIMA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' But notice that now (Z, g′ Z) is a round sphere, which means that the original wedge manifold is quasi-isometric (in the “conformal” sense of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6) to a smooth manifold, in which case the conclusion of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 already follows from [KW75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Of course, this justifies the omission of this case in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Finally, as it is apparent from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9 (1), if f = 1 the existence of a Yamabe wedge metric with negative scalar curvature can always be taken for granted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As far as essential self-adjointness of the Laplacian operator is con- cerned, the analytic machinery employed in Section 4 works fine universally (that is, with no restriction on the link) if f ≥ 3, but it does not seem to deliver this specific mapping property if f = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' see Remarks 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='13 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Note that mere surjectivity would suffice for our purpose of extending Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 to this case but this is not needed here anyway due to the fact that, as explained in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='8, the existence of a Yamabe wedge metric forces the surface link to be a sphere, in which case the result follows from [KW75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' On the other hand, the cone-edge case (f = 1) is treated here and, as expected, self-adjointness is fulfilled if the cone angle is at most 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Again, this may always be achieved by a quasi-isometry of the underlying wedge space induced by a rescaling the circle link (as in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6), which suffices for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' For other instances of the usage of this “at most 2π” condition in the cone-edge setting, with relevant applications to rigidity phenomena in Geometry, we refer to [HK98, MM11, Don12, LM19, CLT21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The existence results in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 should be compared with our next contri- bution, which provides topological obstructions to the existence of wedge metrics with (strictly) positive scalar curvature on certain spin wedge spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' For the no- tion of infinite K-area, see Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Let (Xn, g), n = 2k, be a spin wedge manifold with infinite K-area and whose link satisfies either 1 < f ≤ n − 1 or f = 1 and the cone angle is at most 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Then X carries no wedge metric with (strictly) positive scalar curvature (in the given quasi-isometry class of wedge metrics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Also, the same conclusion holds true for n odd if X × S1 has infinite K-area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The arguments presented in [dL22b, Sections 2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='2] may be easily adapted to provide versions of the theorems above in case the wedge space carries a boundary ∂X disjoint from the wedge region U: one gains minimality of ∂X in the analogue of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 and should require mean convexity of ∂X for the analogue of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The obstruction in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10 may be thought of as being “com- plementary” to [AGR16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3] and [BPR21, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3] in a sense that we now discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' There exist two fundamental lines of inquiry concerning obstruc- tions to the existence of metrics of positive scalar curvature in the spin category, which in a sense reflect the quite diverse topological contributions to the Atiyah- Singer index formula for Dirac operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In the untwisted case, the only contri- bution comes from the tangent bundle, which has been explored by Lichnerow- icz [Lic63] to check that the �A-genus obstructs such metrics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' this line of thought has been refined by Hitchin [Hit74], with the corresponding obstruction coming from KO-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' On the other hand, it has been shown by Gromov and Law- son [GL80a, GL80b, GL83] that twisting the Dirac operator with almost flat vector bundles leads to a “complementary” obstruction for enlargeable manifolds (for THE SCALAR CURVATURE IN WEDGE SPACES 7 example, this works for tori, whose �A-genus vanish).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Later on, Gromov [Gro96] was able to somehow quantify this latter proposal by means of the notion of “in- finite K-area” (compare with Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Hence, whereas the obstructions in [AGR16, BPR21] referred to above provide versions of the Lichnerowicz-Hitchin approach in the wedge category, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10 aligns with Gromov’s philosophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' THE YAMABE PROBLEM IN THE WEDGE SETTING A first step towards the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 involves constructing a wedge metric with constant negative scalar curvature on the given wedge space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Here we explain how this result (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9 below) follows from the singular Yamabe theory developed in [ACM14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Given a wedge space (X, g) as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4, we denote by [g]w the space of wedge metrics �g in X which are conformal to g (in the sense that there exists u : Xs → R smooth and positive such that �g = u 4 n−2 g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The corresponding Yamabe problem asks: there exists �g ∈ [g]w with the property that its scalar curvature R�g is constant?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As in the smooth case, this admits a variational formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We fix a back- ground wedge metric g in the given conformal class and consider the quadratic form Q : H1(X, dvolg) → R, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3) Q(u) = ˆ X � |du|2 + cnRgu2� dvolg, cn = n − 2 4(n − 1), and the constraint sphere Bn∗ = {u ∈ H1(X, dvolg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' ∥u∥n∗ = 1}, n∗ = 2n n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We note that C∞ cpt(Xs) ⊂ H1(X, dvolg) densely, which is a consequence of the di- mensional assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' see [ACM14, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' This justifies the integration by parts leading to the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Critical points of Q|Bn∗ precisely correspond to (weak) solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4) Lgu = µu n+2 n−2 , µ ∈ R, where Lg := −∆g + cnRg is the conformal Laplacian (here, ∆g is the Laplacian of the background metric g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In particular, if u is smooth and further satisfies 0 < c−1 ≤ u(x, y, z) ≤ c for some c > 0 and (x, y, z) ∈ U then �g := u 4 n−2 g is a solution of the corresponding Yamabe problem (in the conformal class [g]w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The preferred way to produce critical points for Q|Bn∗ is by minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Un- fortunately, the Direct Method in the Calculus of Variations can not be applied here due to the fact that the continuous embedding H1(X, dvolg) ⊂ Lp(X, dvolg) is only compact for p < n∗ [ACM14, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, a minimizer, if it exists, must be located by alternative methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Also, from past experience with 8 LEVI LOPES DE LIMA the smooth case fully discussed in [LP87], we expect that a minimizer should exist only in case the “total energy” of [g]w, as measured by the global Yamabe invariant (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='5) Yglo(X, [g]w) := inf u∈Bn∗ Q(u), which is a conformal invariant of (X, g), lies below a certain threshold value (this is just a manifestation of the ubiquitous bubbling off phenomenon characteristic of conformally invariant problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' A major contribution in [ACM14] is precisely to identify this critical threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' To explain this latter point we consider, for each V ⊂ X open, Y (V ) = inf �ˆ V � |du|2 + cnRgu2� dvolg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' u ∈ H1 0(V ∩ Xs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' ∥u∥n∗ = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In particular, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='5), Y (X) = Yglo(X, [g]w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Notice that in principle we might have Y (X) = −∞ (in other words, Q|Bn∗ might not be bounded from below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The local Yamabe invariant of (X, [g]w) is given by Yloc(X, [g]w) = inf x∈X lim r→0 Y (Br(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' There always holds Yloc(X, [g]w) ≤ Yn, where Yn is the Yamabe in- variant of the round metric in the unit sphere Sn, which follows from the fact that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6) lim r→0 Y (Br(x)) = Yn, x ∈ Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Although some progress has been made in this regard [Mon17, AM22], the ac- tual computation of the local Yamabe invariant is notoriously hard, the reason being that, amazingly enough, it depends globally on the link (Z, gZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Precisely, Yloc(X, [g]w) = Y(Rb × CZ, [dy2 + dx2 + x2gZ]) = Y(Hb+1 × Z, [ghyp + gZ]), where (Hb+1, ghyp) is hyperbolic space and Y denotes the standard Yamabe in- variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Nonetheless, some of its qualitative properties may be established as a consequence of certain integrability conditions on the scalar curvature of the back- ground wedge metric, which also imply that Q|Bn∗ is bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' [ACM14] Assume that either (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7) Rg ∈ Lq(Xs, dvolg), for some q > n/2, or (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='8) sup r>0 rq−n ˆ Br(x) |Rg|qdvolg ≤ C, for some q > 1 and all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Then Yloc(X, [g]w) > 0 and Yglo(X, [g]w) > −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The main result in [ACM14], as applied to wedge spaces, yields the following criterion for the existence of minimizers for the Yamabe functional in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' [ACM14] Assume that either (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7) or (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='8) holds and that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9) Yglo(X, [g]w) < Yloc(X, [g]w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Then there exists a minimizer for Q|Bn∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' THE SCALAR CURVATURE IN WEDGE SPACES 9 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We stress that the theory in [ACM14] applies to a much larger class of singular spaces than that considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Also, in the smooth case, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6) that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='5 reproduces the classical Aubin’s criterion for the existence of minimizers [LP87, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In order to use Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='5 to solve the Yamabe problem for a given wedge conformal class, we must first come to grips with the integrability conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7)- (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='8) on the scalar curvature of the background metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As already observed in [ACM14, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4], it follows from the asymptotic expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10) Rg ∼ (RgZ − f(f − 1)) x−2 + O(x−1), x → 0, that either (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7) or (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='8) hold true if the link (Z, gZ) satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11) RgZ = f(f − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As a consequence, we obtain the following criterion for the existence of Yamabe wedge metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' [ACM14] Let (X, g) be a wedge space whose link (Z, gZ) satisfies one of the following conditions: (1) f = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' (2) f = 2 and (Z, gZ) is the round unit sphere;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' (3) RgZ = f(f − 1) if 3 ≤ f ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Then there exists a minimizer for Q|Bn∗ as long as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Under the conditions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 (2), the wedge metric g actually extends smoothly to Y and the result reduces to the classical criterion due to Aubin [LP87, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Hence, under the assumptions on the link displayed in the previous theorem, ensuring that the scalar curvature meets the appropriate integrability condition, the general solution of the Yamabe problem gets reduced to checking the validity of the Aubin-type condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9), a task that, as already observed, lies beyond the current technology due to the fact that the local Yamabe invariant is mostly inac- cessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Nonetheless, since here we are interested in merely constructing a wedge metric with constant negative scalar curvature, we may take advantage of the free- dom to suitably modify the background metric to which Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 should be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Let (X, g) be a wedge space whose link (Z, gZ) satisfies one of the following conditions: (1) f = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' (2) f = 2 and Z is (topologically) a sphere;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' (3) (Z, gZ) is Yamabe positive if 3 ≤ f ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Then Xs carries a wedge metric with constant negative scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We first observe that, as explained in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6, if item (3) above is satis- fied then we can replace gZ by a Yamabe metric whose constant scalar curvature is f(f − 1), so as to ensure that item (3) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Also, if f = 2 we may also reduce to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 (2) by replacing the link by the unit sphere (but recall 10 LEVI LOPES DE LIMA that here the new wedge metric turns out to be smooth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' After these preliminaries, we proceed by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3) with u a suitable constant to get Yglo(X, [g]w) ≤ C ˆ X Rgdvolg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As in [Bes07, Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='32] we can inject a sufficiently large amount of neg- ative scalar curvature around some point in Xs\\U (without further altering the wedge region) so as to make the new background metric, still denoted by g, to satisfy ´ X Rgdvolg < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, Yglo(X, [g]w) < 0 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 applies since Yloc(X, [g]w) > 0 and hence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Leading the minimizer to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4) we immediately see that the (constant) scalar curvature of the corresponding Yamabe wedge metric is negative, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The conformal factor, say u, obtained in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9 is strictly posi- tive and remains uniformly bounded from above as x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, the Yamabe met- ric so obtained is quasi-isometric to the original background wedge metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In fact, the regularity theory in [ACM14, Section 3] guarantees that u(x, y, z) = u0(y)+o(1) as x → 0, where u0 : Y → R is smooth and strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Hence, after possibly implementing a conformal deformation on gY and a (y-dependent) change in the radial coordinate, the new metric still satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='2), so that the corresponding link metric remains independent of y ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' It follows from the analysis in [ACM14, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3] that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 (3) holds true under the weaker assumption that the first eigenvalue λ0(Ln gZ) of the operator Ln gZ = −∆gZ + cnRgZ, cn = n − 2 4(n − 1), is exactly equal to cnf(f −1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' indeed, it suffices to assume λ0(Ln gZ) > 0 if f = n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Since in this case there exist positive constants A e B depending only on f and n such that Lf gZ = ALn gZ − B∆gZ, any of these assumptions on Ln gZ actually imply that (Z, gZ) is Yamabe positive, which is the requirement in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9 (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Hence, as far as the conclusion of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 is concerned, nothing is gained if we use this finer result based on λ0(Ln gZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' THE WEDGE ELLIPTIC THEORY FOR THE SCALAR LAPLACIAN We describe how the wedge elliptic theory applies to Laplace-type operators on a wedge space (X, g) as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As hinted at in the Introduction, this involves considering ∆g as acting on a suitable weighted Sobolev scale and then employ- ing the powerful micro-local methods in [Maz91, Les97, Sch98, ES12], with further developments in [GM03, GKM13, ALMP12, AGR16, ALMP18, ARS22], to check that ∆g, viewed as an unbounded, symmetric operator, has good mapping prop- erties (essential self-adjointness and Fredholmness) for a carefully chosen value of the weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We emphasize that surjectivity would suffice for the application we have in mind (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7), but we have chosen to establish finer mapping prop- erties not only because, as explained in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4, they provide a precise (local) description of the space of solutions of certain semi-linear elliptic equations on (X, g), but also because we intend to transplant the analysis to the Dirac operator THE SCALAR CURVATURE IN WEDGE SPACES 11 treated in Section 6, where self-adjointness and Fredholmness are crucial in appli- cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Also, as the statement of our ultimate goal, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11, makes it clear, in this section we pose no restriction on the geometry of the link (Z, gZ) if f ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We next consider the b-density dvolb = x−1dxdvolgY dvolgZ in the wedge region U and extend it to Xs in the obvious manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Given an integer k ≥ 0 and a cutoff function ϕ with ϕ ≡ 1 near Y and ϕ ≡ 0 outside U, we define Hk b(X) to be the space of all distributions u ∈ D′(Xs) such that: (1 − ϕ)u lies in the standard Sobolev space Hk(Xs, dvolg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' there holds (x∂x)j(x∂y)µ∂ν z (ϕu)(x, y, z) ∈ L2(X, dvolb), j + |µ| + |ν| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' (Weighted Sobolev scale) If β ∈ R we set xβHk b(X) = � v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' x−βv ∈ Hk b(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Using interpolation and duality we may define xβHσ b(X) for any σ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' This turns out to be a Sobolev scale of Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' For instance, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='12) ⟨u, v⟩xβH0 b(X) = ˆ X x−2βuv dvolb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In particular, one has the continuous inclusion xβ′Hσ′ b (X) ⊂ xβHσ b(X), β′ ≥ β, σ′ ≥ σ, which is compact if strict inequalities hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Also, if σ > n/2 then any u ∈ xβHσ b(X) is continuous in Xs and satisfies u(x) = O(xβ) as x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We are interested here not only in the bounded, weighted Laplacian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='13) Mg,β := −x2∆g : xβHσ b(X) → xβHσ−2 b (X), induced by the natural action of Mg := −x2∆g on the weighted Sobolev scale, but also in the unbounded Laplacian (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='14) Cg,β := −∆g : C∞ cpt(X) ⊂ xβH0 b(X) → xβH0 b(X), viewed as a densely defined operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Their analysis will eventually hinge on the pair of f-dependent constants βf = −f + 1 2 , γf = 1 − f 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The Laplacian Cg,β is symmetric if β = βf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Since in the wedge region dvolb relates to the Riemannian volume element dvolg of the underlying wedge metric g by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='15) dvolb = x2βf dvolg, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='12) that ⟨∆gu, v⟩xβH0 b(X) = ˆ X x−2(β−βf)v∆gu dvolg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' 12 LEVI LOPES DE LIMA The result follows if we take u, v ∈ C∞ cpt(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The assertion in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4 also holds true if we replace ∆g by any operator which is formally self-adjoint with respect to L2(X, dvolg) (the Dirac operator, for instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' At least in the conical case (f = n − 1) the mapping properties of Mg,β are completely determined by the spectral resolution of ∆gZ in a way that we now explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We first note that for a general wedge space, Mg,β|U = −D2 x − (f − 1) Dx − ∆gZ − x2∆gY + o(1), where Dx = x∂x and of course the term x2∆gY should be omitted in the conical case (Y collapses into a point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In any case, after discarding the last two terms with an explicit dependence on x we obtain the indicial operator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='16) IMg,β = −D2 x − (f − 1) Dx − ∆gZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Also, if we replace Dx by ζ ∈ C we get the indicial symbol of Mg,β, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='17) IMg,β(ζ) = −ζ2 − (f − 1)ζ − ∆gZ, a one-parameter family of elliptic operators on (Z, gZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Let us consider the spectrum of ∆gZ, Spec(∆gZ ) = {µ ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' ∃u ̸= 0 satisfying − ∆gZu = µu} ⊂ [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' For each µ ∈ Spec(∆gZ), let {ζ± µ,f} be the (real) roots of the indicial equation ζ2 + (f − 1)ζ − µ = 0, which is obtained by equating to zero the restriction of the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='17) to each eigenspace of ∆gZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Explicitly, the indicial roots are (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='18) ζ± µ,f = γf ± � γ2 f + µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The corresponding indicial set is If Mg = � µ∈ Spec(∆gZ ) {ζ± µ,f}, which is a discrete subset of R (because Z is closed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Finally, the non-indicial set is I/f Mg = R\\If Mg, a countable union of bounded, open intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In the conical case, we have the following well-known result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='13) is Fredholm if and only if β ∈ I/n−1 Mg , with the corresponding Fredholm index remaining the same as long as β varies in a given connected component of I/n−1 Mg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' This follows from [Maz91, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' An outline of its proof, along the lines of the theory developed in [Sch98, Les97], may be found in [dL22b, dL22a], albeit the weight numerics there is slightly different from ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Also, see [SS01] for the specific information regarding the dependence of the Fredholm index on β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Although its proof requires a somewhat delicate analysis, it is intuitively clear from this discussion that the following holds in the conical case: as β varies in THE SCALAR CURVATURE IN WEDGE SPACES 13 I/n−1 Mg , the Fredholm index of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='13) assumes its minimal value in the “innermost” non-indicial interval corresponding to µ = 0, namely, In−1 := (2 − n, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Also, we may compute this minimal Fredholm index by observing that, as explained in [dL22a, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='13) is essentially self-adjoint (and hence Fredholm of index 0) if β = βn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' For the sake of comparison with the corresponding argument below in the general wedge case, we briefly reproduce this reasoning here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We recall that the minimal domain of Cg,β as defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='14) is Dmin(Cg,β) = � u ∈ xβH0 b(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' ∃{un} ⊂ C∞ cpt(Xs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' un H0 b −→ u, ∆gun is H0 b − Cauchy � , whereas its maximal domain is Dmax(Cg,β) = � u ∈ xβH0 b(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' ∆gu ∈ xβH0 b(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' It is known that Dmin(Cg,β) ⊂ Dmax(Cg,β) and that (closed) sub-spaces of Q(Cg,β) := Dmax(Cg,β)/Dmin(Cg,β) correspond to domains of closed extensions of Cg,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The key observation now is the following fact whose proof may be found in [Les97, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3], but again with a different weight numerics: Q(Mg,β) is formed by contributions coming from the finite set In−1 β := In−1 Mg ∩ (β, 2 + β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In particular, Q(Cg,β) = {0}, and hence Cg,β has a unique closed extension, if In−1 β = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We now remark that (βn−1, 2+βn−1) ⊂ In−1 and hence In−1 βn−1 = ∅ if n > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, Cg,βn−1 has a unique closed extension which is self-adjoint (and hence Fredholm of index 0) because it is symmetric by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The next result then follows from the discussion above and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Let (X, g) be a conical space satisfying n > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Then the unbounded symmetric map Cg,βn−1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='14) is essentially self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Moreover, the bounded map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='13) is Fredholm of index 0 as long as β ∈ In−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As explained in [dL22b, Section 2], at least if n > 4 this information suffices to carry out the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 in this conical case (for more details of the ar- gument together with a checking on how the missing case n = 4 may be recovered by an alternative method, we refer to the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We now turn to the (non-conical) purely wedge case (1 ≤ f < n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Here, things quickly get complicated because the indicial decomposition R = If Mg ⊔I/f Mg fails to sharply determine the mapping properties of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='13), as no analogue of The- orem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6 is expected to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Naturally enough, now the singular stratum (Y, gY ) also plays a role by contributing an extra term to the indicial operator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='16), so as to form the so-called normal operator NMg, which may be identified to −t2∆gc, where gc = dt2 + t2gZ + δu, δu = |du|2, is the natural metric on (0, +∞)t × Z × Rb u, 14 LEVI LOPES DE LIMA viewed as the tangent cone arising from the one parameter family of dilations Tρ(y0)(x, y, z) = (ρx, y0 + ρ(y − y0), z), y0 ∈ Y , as ρ → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, NMg = −D2 t − (f − 1) Dt − ∆gZ − t2∆δu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Comparison with the indicial operator in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='16) shows that the extra term in NMg spoils the invariance under dilations in t, which reveals an essential de- parture from the conical case, but notice that both invariances under dilations in (t, u) and translations in u are retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As in [Maz91] we explore this by first Fourier transforming in the u-direction with ξ ∈ T ∗Rb u as dual variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We next set ϑ = ξ/|ξ|δu ∈ S∗Rb u, the spherical conormal bundle, and s = |ξ|δut so as to obtain the equivalent “Bessel-type” normal operator BMg(ϑ) = −D2 s − (f − 1) Ds − ∆gZ + s2|ϑ|2 δu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Note that the explicit dependence on ϑ is illusory since |ϑ|δu = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, BMg acts on functions in CZ := [0, +∞)s × Z, the (infinite) cone over Z endowed with the conical metric ds2 + s2gZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' To proceed, consider the spaces Hσ,β,l(CZ) = {u ∈ D′(CZ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' φu ∈ sβHσ b(CZ), (1 − φ)u ∈ s−lHσ(CZ)}, where φ ∈ C∞ cpt(CZ) with φ = 1 near s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, we may view BMg as an operator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='19) BMg : Hσ,β,l(CZ) → Hσ−2,β,l(CZ), whose mapping properties are closely tied to Fredholmness for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Indeed, the next criterion follows from the general theory developed in [Maz91, Sch98, ES12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='13) is Fredholm provided β ∈ I/f Mg and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='19) is invertible (for such β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' If (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='19) is only known to be injective then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='13) is semi-Fredholm (that is, it has a closed range and is essentially injective in the sense that its kernel is finite dimensional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, establishing Fredholmness of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='13), which should be thought of as a pre- liminary step in probing its mapping properties, gets reduced to finding a range of weights for which invertibility of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='19) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Note that BMg is ellip- tic as in [Maz91, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3] with the same indicial operator as Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, [Maz91, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='5] implies that the normal operator in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='19) is Fredholm pro- vided β ∈ I/f Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' It is known that the index of BMg does not depend on the pair (σ, l) and remains the same as long as β varies in a given connected component of I/f Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Similarly, by [Maz91, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7] the kernel of BMg does not depend on (σ, l) as well, although it might change when β crosses If Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Fortunately, we shall see that this kernel turns out to be trivial in an appropriate range of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The normal operator BMg in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='19) is injective if β > 1 − f, f > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' By separation of variables, we find that a solution w ∈ Hσ,β,l(CZ) of the homogeneous equation BMgw = 0 decomposes as w = � µ∈Spec(∆gZ ) wµ, wµ ∈ Hσ,β,l µ (CZ), THE SCALAR CURVATURE IN WEDGE SPACES 15 where Hσ,β,l µ (CZ) ⊂ Hσ,β,l(CZ) selects the eigenspace of ∆gZ associated to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, it suffices to check that each wµ vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Since BMgwµ = 0, wµ can be expressed in terms of the modified Bessel functions Iν(s) and Kν(s), where ν = ν± = ± � γ2 f + µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Precisely, wµ(s) = sγf (c1Iν(s) + c2Kν(s)) , where ci is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Since ν ̸= 0, the asymptotical behavior of these Bessel functions are as in the table below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' s → 0 s → +∞ Iν(s) ∼ sν ∼ es/s Kν(s) ∼ s−|ν| ∼ e−s/√s The exponential growth of Iν at infinity forces c1 = 0, whereas the exponential decay of Kν poses no restriction on c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Since wµ(s) ∼ sγf −|ν| as s → 0 and γf − |ν| ≤ γf − |γf| = 2γf = 1 − f, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' □ Although injectivity of the normal operator suffices for our purposes as it pro- vides, via Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='8, a left generalized inverse for Mg,β to which the argument in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11 may be applied, it is known that the normal operator is surjective in the “innermost” non-indicial interval If, at least if f > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Since this information is not used in the sequel, we omit its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The normal operator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='19) is surjective if β ∈ If, f > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In view of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='8, it follows from the various results established above that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='13) is Fredholm provided β ∈ If, f > 1 (if we ignore Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10, it is at least semi-Fredholm and essentially injective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Unfortunately, this does not suffice to proceed as in the conical case in order to detect self-adjoint extensions starting from Q(Cg,β);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' see [MV12, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='5] for a discussion of the issues involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We may, however, combine this information with the reasoning in [ARS22, Section 1], which by its turn is based on arguments in [GM03, ALMP12, AGR16, ALMP18], to check that in most cases the Laplacian Cg,βf in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='14), which is symmetric by Propo- sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4, has good mapping properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As applied to our context, the idea is that the existence of a (left) generalized inverse for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='13) with β ∈ (βf, 2 + βf) leads to an extra regularity in the weighted Sobolev scale for elements of Dmax(Cg,βf ) which forces them to actually belong to Dmin(Cg,βf ) by characterizations of this latter space appearing in [GM03, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='6] and [GKM13, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Let (X, g) be a wedge space satisfying either 3 ≤ f ≤ n − 1 or f = 1 and the cone angle is at most 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Then the Laplacian Cg,βf defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='14) is essentially self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Moreover, letting Dg,βf be the domain of this self-adjoint extension, the map (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='20) Cg,βf + λ : Dg,βf → xβf H0 b(X) is Fredholm of index 0 for any λ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' 16 LEVI LOPES DE LIMA Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Let us initially assume that f > 3 so that both βf and 2+βf lie in If.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We have seen that Mg,β defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='13) is semi-Fredholm and essentially injective pro- vided β ∈ If.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In particular, there exists a left generalized inverse G : xβH0 b(X) → xβH2 b(X) for Mg,β, which means that G∆g = Id − Πβ : xβH2 b(X) → xβH2 b(X), β ∈ If, where G = −Gx2 and Πβ is the projection onto the kernel of Mg,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Now take u ∈ Dmax(Cg,βf ) so that u = G∆gu + Πβu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' From the diagram x2+βf H0 b(X) xβH0 b(X) xβH2 b(X) Dmax(Cg,βf ) G −x2∆g −x2∆g G∆g where the inclusion in the upper row comes from the Sobolev embedding in Re- mark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3, we see that G∆gu ∈ xβH2 b(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Also, since Mg,βΠβu = 0 one expects that Πβu is much more regular than it appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Indeed, it already follows from [Maz91, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='24] that Πβu ∈ xβf Hσ b(X), σ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' With some more work we may infer that Πβu ∈ xβH2 b(X) as well (this relies on the analysis in [Maz91, Section 7] and uses that no indicial root lies in the interval (βf, β], where the normal operator is known to be injective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Hence, by setting ε = 2 + βf − β we obtain the inclusion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='21) Dmax(Cg,βf ) ⊂ � 0<ε<2 x2+βf −εH2 b(X), that is, elements of Dmax(Cg,βf ) enjoy an extra “weighted regularity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We now claim that this leads to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='22) D := � 0<ε<2 x2+βf −εH2 b(X) ⊂ Dmin(Cg,βf ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' These inclusions imply Dmax(Cg,βf ) ⊂ Dmin(Cg,βf ), that is, Cg,βf is essentially self- adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' To prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='22) we will make use of the well-known fact that, since Cg,βf = −∆g is symmetric by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4, u ∈ Dmax(Cg,βf ) is in Dmin(Cg,βf ) if and only if ⟨∆gu, v⟩xβf H0 b(X) = ⟨u, ∆gv⟩xβf H0 b(X) , for any v ∈ Dmax(Cg,βf );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' compare with [ALMP12, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' If u ∈ D set un := x1/nu, n ∈ N, so that un ∈ x2+βf H2 b(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Also, for any ε ∈ (0, 2), un → u in x2+βf −εH2 b(X), which gives (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='23) xε∆gun → xε∆gu in xβf H0 b(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' THE SCALAR CURVATURE IN WEDGE SPACES 17 Moreover, since ε �→ 2 − ε is a bijection of (0, 2), besides (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='23) we also have x−εDmax(Cg,βf ) ⊂ xβf H2 b(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Hence, if v ∈ Dmax(Cg,βf ) we get ⟨∆gun, v⟩xβf H0 b(X) = � xε∆gun, x−εv � xβf H0 b(X) → � xε∆gu, x−εv � xβf H0 b(X) = ⟨∆gu, v⟩xβf H0 b(X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Also, ⟨∆gun, v⟩xβf H0 b(X) = ⟨un, ∆gv⟩xβf H0 b(X) → ⟨u, ∆gv⟩xβf H0 b(X) , where in the first step we used that un ∈ Dmin(Cg,βf ) and in the last one that un → u in xβf +εH2 b(X) ⊂ xβf H0 b(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, ⟨∆gu, v⟩xβf H0 b(X) = ⟨u, ∆gv⟩xβf H0 b(X) , which means that u ∈ Dmin(Cg,βf ) and hence proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The fact that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='20) is Fredholm now follows from standard arguments based on the compact inclusion Dg,βf = Dmax(Cg,βf ) ⊂ xβf H0 b(X), which follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='21) and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Finally, using that I3 = (−2, 0) = (β3, 2 + β3), the limiting case f = 3 may be treated by the same argument since (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='21) still holds true even if both βf and 2+βf are indicial roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We now consider the cone-edge case (f = 1), in which the link is a circle of length L (the cone angle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='18) that ζ± 0,1 = 0, so the “inner- most” non-indicial interval disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' If we assume that L < 2π then both β1 = −1 ∈ I− 1 := (−2π/L, 0) and 2 + β1 = 1 ∈ I+ 1 := (0, 2π/L) are non-indicial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' note that I± 1 now jointly play the role of “innermost” non-indicial intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Using that K0(s) ∼ − log(s/2) as s → 0, Bessel asymptotics implies that the correspond- ing normal operator is injective for β > −2π/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Hence, we may invoke Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='8 to ensure that Mg,β is semi-Fredholm and essentially injective for any non- indicial β > −2π/L, which in particular provides a left generalized inverse for Mg,β, β ∈ (−1, 1)\\{0}, to which the argument in the previous paragraph may be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, essential self-adjointness holds provided L ≤ 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' If we further assume that f > 3, so that 2+βf is not an indicial root, then [GKM13, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='2] actually identifies Dg,βf , the domain of the unique self-adjoint extension of Cg,βf , to x2+βf H2 b(X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' see also [HLV18, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1] for another proof of this result which is functional analytical in nature and altogether avoids the use of the (so pervasive!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=') micro-local artillery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' But we insist that this extra piece of information is not needed here, so we content ourselves with the formulation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11, which suffices for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' An obvious reason why the method above does not directly apply to the case f = 2 is that β2 = −3/2 lies far away from the range β > −1 where the normal operator is known to be injective, so that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='21) is unavailable (note that the “innermost” non-indicial interval here is (−1, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' A similar difficulty appears in the cone-edge case with cone angle L > 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' A possible way to circumvent this, equally convenient for our purposes, consists in determining a range of weights where Cg,β would be surjective, but we will not pursue this because, as pointed out in Remarks 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9, those cases are not really needed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' 18 LEVI LOPES DE LIMA Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The compact inclusion Dg,βf ⊂ xβf H0 b(X) implies not only Fred- holmness of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='20) but also that its spectrum is discrete with each eigenspace of finite multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Instead of Mg, we may work with the operator � Mg = x−βf Mgxβf = −D2 x + 2Dx + βf(2 + βf) − ∆gZ − x2∆gY + o(1), acting on the weighted Sobolev scale based on the wedge volume element dvolg, so that the “innermost” indicial roots associated to µ = 0 are 2+βf and −βf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Since the “symmetry weight” now occurs ate β = 0 and f ≥ 3 implies that (0, 2) has no indicial root, in this case Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11 follows from [ARS22, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' It is instructive to compare the analysis above with the complete edge case, in which instead of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='2) we take g|U(x, y, z) = x−2 � dx2 + x2gZ(z) + gY (y) + o(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' This includes the conformally compact, asymptotically hyperbolic case with [gY ] as conformal boundary (when Z collapses into a point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Here we are interested in establishing the mapping properties of the operator P = −∆g + λ, where now λ is a smooth function on Xs converging to a fixed value λ∗ ∈ R as one approaches Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As instances of geometric problems where this kind of question arises we mention the singular Yamabe problem [MS91] and the rigidity of hyperbolic space in the context of the AdS/CFT correspondence [Qin03].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As usual, we work with the appropriate weighted Sobolev scale xδHσ e (X) based on the edge volume element dvole := x−1dxdvolgY dvolgZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' To simplify matters, let us assume that b ≥ 1 and b2/4 + λ∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Since P|U = −D2 x + bDx − ∆gZ − x2∆gY + λ + o(1), the indicial symbol is IP(ζ) = −ζ2 + bζ − ∆gZ + λ∗, so that the “innermost” non-indicial interval is �I = (δ−, δ+), where δ± = b 2 ± �� b 2 �2 + λ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Also, a computation shows that the normal operator BP may be identified to −∆�g + λ∗, where �g = ghyp + gZ is the product metric on Hb+1 × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Now, if u ∈ xδH0 e(X), δ > δ−, satisfies BPu = 0 then u ∈ xb/2H0 e(X) = L2(X, dvolg), so the invertibility of BP in �I boils down to checking that the L2 spectrum of (Hb+1 × Z, �g) is disjoint from (−∞, b2/4) [McK70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' It then follows from the micro- local analysis in [MS91] that P is Fredholm of index 0 in this range of weights;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' see also [MM91] for a potential theoretic approach to this same result and note that [Lee06] also treats the conformally compact case by “low-tech” methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' But no- tice that here the vanishing of the index is readily guessed given that the essential self-adjointness of P at δ = b/2 ∈ �I is a classical accomplishment [Gaf51] (recall that g is complete).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In particular, the analogue of the delicate analysis leading to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11 is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' THE SCALAR CURVATURE IN WEDGE SPACES 19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' AN EXISTENCE RESULT: THE PROOF OF THEOREM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 We present here the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We start with a result confirming that the prescription problem for the scalar curvature of wedge metrics is invariant under diffeomorphisms of bounded distortion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' compare with [KW75, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1] and [dL22b, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Let φ, φ′ ∈ C0(Xs) ∩ L∞(Xs) ֒→ xβH0 b(X), β ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' If min φ < φ′ < max φ then for any ε > 0 there exists a diffeomorphism Ψ : Xs → Xs of bounded distortion (in particular, preserving the quasi-isometry class of wedge metrics) such that ∥φ ◦ Ψ − φ′∥xβH0 b(X) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Now let (X, g) be a wedge space as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9 and Re- marks 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10, we may assume that f ̸= 2 and Rg = −1, where 1 is the function identically equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' If ǫ > 0 is small enough, consider the smooth map A : Bǫ(1) ⊂ Dg,βf → xβf H0 b(X, dvolb) given by A(u) := R u 4 n−2 g = −u−αn � c−1 n ∆gu + u � , αn = n + 2 n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Note that A(1) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The linearization ˙A1 : Dg,βf → xβf H0 b(X) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' A short computation gives (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='24) ˙A1 = c−1 n (−∆g + λn) , λn = 4cn n − 2 = 1 n − 1, which is Fredholm of index 0 by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11 (after a possible rescaling of the circle link if f = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Alternatively, we may use Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 if f = n − 1, n > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Note that self-adjointness of Cg,βf guarantees that integration by parts does not yield a contribution coming from the singular stratum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, if ˙A1u = 0 we get − ∥∇gu∥2 xβf H0 b(X) = λn ∥u∥2 xβf H0 b(X) , a contradiction unless u = 0, which confirms injectivity of ˙A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The result now follows from Fredholm alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' □ By the Inverse Function Theorem, there exists ε > 0 and a neighborhood U ⊂ Cg,β of 1 such that A|U : U → Bε(−1) ⊂ xβf H0 b(X) is a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Also, if F is the prescribed function then there exists K > 0 such that K min F < −1 < K max F and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1 gives a diffeomorphism Ψ : Xs → Xs of bounded distortion such that KF ◦ Ψ ∈ Bε(−1), which implies that KF ◦ Ψ may be realized as the scalar curvature function of some wedge metric �g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, K1/2(Ψ−1)∗�g is the required wedge metric (whose scalar curvature is F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Finally, we note that this approach turns out to be effective enough to allow us to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 in the remaining conical case n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Indeed, it suffices to take β = β3 and use that, again by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11, Mg,β3 as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='14) is essentially self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Again, the regularity theory in [ACM14, Section 3] ensures that the conformal metric produced in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 lies in the same quasi-isometry class as the original wedge metric and hence is wedge as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' compare with Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' 20 LEVI LOPES DE LIMA Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Although surjectivity of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='24) would suffice to produce the metric �g above via the Implicit Function Theorem, the finer isomorphism property shows that the space of conformal factors yielding wedge metrics whose scalar curvature functions are close to −1 may be locally parameterized by an open ball centered at 1 ∈ Dg,βf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In particular, local uniqueness for the associated non-linear problem holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As another example of this sort of phenomenon, we note that with no re- striction at all on the geometry of the link if f ≥ 3, the perturbative method above may be used to check that for any v close enough to 1 there is a (unique) u close to 1 solving the semi-linear elliptic equation ∆gu + u = vuα, α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' A routine procedure that we omit here allows us to recast both Theo- rem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11 and the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='7 in the language of weighted H¨older spaces, which is more convenient to directly handle the regularity of solutions that we actually refrained from spelling out in the Sobolev analysis above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' A TOPOLOGICAL OBSTRUCTION: THE PROOF OF THEOREM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10 We now look at topological obstructions to the existence of wedge metrics with (strictly) positive scalar curvature in a given wedge space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Our aim is to present the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10 and for this we will use a version of Index Theory as developed in [AGR16], so we assume from now on that Xs is spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, in the presence of a wedge metric g, we may consider the corresponding Dirac operator acting on the associated spin bundle ð : Γ(SX) → Γ(SX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As before, we may define the weighted Sobolev scale xβHσ b(SX) formed by ap- propriate distributional spinors, so the question remains of determining for which values of β the map (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='25) ð : xβHσ b(SX) → xβHσ−1 b (SX) is at least semi-Fredholm and essentially injective, as this is the first step in trying to establish good mapping properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' The argument below, leading to Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1, adapts to the Dirac setting the approach adopted in Section 4 for Laplace-type operators and should be thought of as a variation of the reasoning in [AGR16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' It follows from [AGR16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='2] that, in the wedge region U, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='26) xð = c(∂x) � Dx + f 2 + ðZ + xðY � + o(x), where c is Clifford product and ðZ (respectively, ðY ) is the Dirac operator of (Z, gZ) (respectively, (Y, gY )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Replacing Dx by ζ in the right-hand side above and sending x → 0, we see that the corresponding indicial symbol is Ið(ζ) = ζ + f 2 + ðZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Since we aim at obstructing positive scalar curvature metrics, we may assume that Rg|U ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10) that RgZ ≥ f(f − 1) > 0 if f > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' A well-known THE SCALAR CURVATURE IN WEDGE SPACES 21 estimate [Fri00, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1] then gives the spectral gap (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='27) Spec(ðZ) ∩ � −f 2 , f 2 � = ∅, known as the “geometric Witt condition”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As explained in [dL22b, dL22a], at least in the conical case (f = n − 1), this analysis suffices to guarantee that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='25) is Fredholm of index 0 whenever β ∈ Jn−1 := (1 − n, 0) since (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='27) prevents the appearance of indicial roots in this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In the general wedge case, however, one should take into account the effects coming from the corresponding normal operator Bð(ϑ) = c(∂s) � Ds + f 2 + ðZ � + sic(ϑ), ϑ ∈ S∗Rb u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Luckily, the general mapping theory of wedge elliptic operators [Maz91, Sch98] tells us that the pattern to follow here is quite similar to the case of the Laplacian studied in Section 4, which will allow us to easily transplant the previous analy- sis to the present context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Firstly, we check that when acting on the appropriate Sobolev scale on CZ, Bð is injective for β > −f and surjective for β ∈ Jf := (−f, 0), the “innermost” non-indicial interval in this setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' see the proof of [AGR16, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10], but be aware that the indicial numerics there is different from ours essentially because their weighted Sobolev spaces are based on the volume ele- ment dvolg instead of dvolb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' another approach to this issue appears in [HLV18, Section 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We insist, however, that only the injectivity of Bð on Jf is really needed in the sequel, and this is an easy consequence of elementary Bessel asymptotics (as in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Thus, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='25) is semi-Fredholm and essentially in- jective for β ∈ Jf by the appropriate analogue of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Secondly, we com- bine this information (translated into the existence of a left generalized inverse for (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='25) in this range of weights) with the argument in the proof of [AGR16, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11], which is the Dirac version of [ARS22, Section 1], to establish the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Let (X, g) be a spin wedge space with Rg|U ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' If f = 1 assume also that the cone angle is at most 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Then the unbounded map (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='28) ð : Γcpt(Xs) ⊂ xβf H0 b(SX) → xβf H0 b(SX) is essentially self-adjoint and the corresponding self-adjoint extension is Fredholm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As already advertised, this is just a matter of transplanting the analysis for the Laplacian in Section 4 to this Dirac setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In particular, have Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='5 in mind and note that 2 + βf should be replaced by 1 + βf, as dictated by the left- hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' If f > 1 then both βf and 1 + βf lie in Jf and we may proceed exactly as in the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11 (where the corresponding assumption is f > 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Notice that f > 1 is also required here to justify the spectral estimate leading to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Nonetheless, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='27) holds true in the cone-edge case f = 1 if the cone angle is at most 2π and we choose the bounding spin structure on the link circle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' see [Cho89, AGR16] for discussions on this subtlety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In any case, with this extra information at hand, our reasoning above may also be transplanted to cover the limiting case f = 1 since J1 = (−1, 0) = (β1, 1 + β1) (compare again with the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='11, where the “limiting” case is f = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' □ 22 LEVI LOPES DE LIMA Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' As in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='12, it follows from [GKM13, HLV18] that the domain of the self-adjoint extension of ð above is x1+βf H1 b(SX), at least if f > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' If f = 1 and the cone angle is strictly less that 2π then 0 = 1 + β1 is not an indicial root and the corresponding domain is H1 b(SX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We now make use of the theory above to find topological obstructions to the existence of wedge metrics of positive scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Let us still denote by ð the self-adjoint realization of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' If n = 2k is even, it is well -known that ð splits as ð = � 0 ð− ð+ 0 � where ð± are the realizations of the chiral Dirac operators ð± : Γcpt(S± X) → Γcpt(S∓ X) corresponding to the chiral decomposition SX = S+ X ⊕ S− X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' It then follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1 that these chiral Dirac operators are Fredholm and adjoint to each other, so it makes sense to consider the associated index ind ð+ = dim ker ð+ − dim ker ð−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' More generally, if E is a Hermitian vector bundle over Xs endowed with a com- patible connection, we may consider the twisted Dirac operator ð ⊗ E = � 0 ð−⊗ E ð+⊗ E 0 � In general, twisting with E spoils self-adjointness since ðgZ ⊗ E|Z should some- how appear in the expressions of the corresponding indicial symbol and normal operator, thus compromising the analysis above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' However, if E is admissible in the sense that E|U is trivial (and endowed with a flat connection) then ð⊗E|U is a sum of r := rankE copies of ð|U and, as explained in the proof of [dL22b, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1], Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1 and Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='2 hold verbatim for ð ⊗ E, so we can define the corresponding index ind ð+⊗ E = dim ker ð+⊗ E − dim ker ð−⊗ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' It turns out that this fundamental invariant can be explicitly computed in terms of the underlying geometry by means of heat asymptotics [Cho85, Les97, AGR16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Indeed, if Θr is the trivial bundle of rank r and �E = E −Θr is the associated virtual bundle then [AGR16, Main Theorem] tells us that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='29) ind ð+⊗ �E = r ind ð+ + ˆ Xs �A(T Xs) ∧ ch �E, where ind ð+ = ˆ Xs �A(T Xs) + ˆ Y �A(T Y ) � −1 2ηðZ + ˆ Z T �A(T Xs) � , �A is the �A-class, T means transgression, ηðZ is the eta invariant of ðZ and ch �E is the Chern character of �E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' THE SCALAR CURVATURE IN WEDGE SPACES 23 According to [AGR16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3], if g has (strictly) positive scalar curvature everywhere then ind ð+ = 0, so that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='29) reduces to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='30) ind ð+⊗ �E = ˆ Xs �A(T Xs) ∧ ch �E, for any admissible bundle E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Notice that �A(T Xs) only contributes to the right- hand side with “lower order” terms, which aligns with the comments in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' We now introduce the class of wedge spaces to which Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10 applies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' compare with [dL22b, Section 3], where the conical case is discussed, and also with [Gro96], where this notion was originally conceived in the smooth category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' A wedge space (X, g) has infinite K-area if for any ǫ > 0 there exists an admissible, ǫ-flat bundle over X which is homologically nontrivial in the sense that at least one of its Chern numbers does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' A key point here is that having infinite K-area is a quasi-isometric property of the wedge space (X, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' In any case, with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='30) at hand and proceeding exactly as in [dL22b, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='1], the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='10 follows immediately.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content=' Email address: levi@mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='ufc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} +page_content='br' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9E4T4oBgHgl3EQfYQwo/content/2301.05047v1.pdf'} diff --git a/xtFQT4oBgHgl3EQfAjU_/content/tmp_files/2301.13223v1.pdf.txt b/xtFQT4oBgHgl3EQfAjU_/content/tmp_files/2301.13223v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bdb0d52148c8f2c69cfa62307a49cd1e0b6f3862 --- /dev/null +++ b/xtFQT4oBgHgl3EQfAjU_/content/tmp_files/2301.13223v1.pdf.txt @@ -0,0 +1,1278 @@ +Astronomy & Astrophysics manuscript no. eso323_serafinelli +©ESO 2023 +February 1, 2023 +The NuSTAR view of the changing look AGN ESO 323-G77 +Roberto Serafinelli1, Valentina Braito2, 3, James N. Reeves3, 2, Paola Severgnini2, +Alessandra De Rosa1, Roberto Della Ceca2, and Tracey Jane Turner4 +1 INAF - Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133, Roma, Italy +e-mail: roberto.serafinelli@inaf.it +2 INAF - Osservatorio Astronomico di Brera, Via Brera 28, 20121, Milano, Italy & Via Bianchi 46, Merate (LC), Italy +3 Department of Physics, Institute for Astrophysics and Computational Sciences, The Catholic University of America, Washington, +DC, 20064, USA +4 Eureka Scientific, Inc, 2452 Delmer St. Suite 100, Oakland, CA 94602, USA +Received XXX; accepted YYY +ABSTRACT +The presence of an obscuring torus at parsec-scale distances from the central black hole is the main ingredient for the Unified Model +of Active Galactic Nuclei (AGN), as obscured sources are thought to be seen through this structure. However, the Unified Model +fails to describe a class of sources that undergo dramatic spectral changes, transitioning from obscured to unobscured and vice-versa +through time. The variability in such sources, so-called Changing Look AGN (CLAGN), is thought to be produced by a clumpy +medium at much smaller distances than the conventional obscuring torus. ESO 323-G77 is a CLAGN that was observed in various +states through the years with Chandra, Suzaku, Swift-XRT and XMM-Newton, from unobscured (NH < 3×1022 cm−2) to Compton-thin +(NH ∼ 1 − 6 × 1023 cm−2) and even Compton-thick (NH > 1 × 1024 cm−2), with timescales as short as one month. We present the +analysis of the first NuSTAR monitoring of ESO 323-G77, consisting of 5 observations taken at different timescales (1, 2, 4 and 8 +weeks from the first one) in 2016-2017, in which the AGN was caught in a persistent Compton-thin obscured state (NH ∼ 2 − 4 × 1023 +cm−2). We find that a Compton-thick reflector is present (NH,refl = 5 × 1024 cm−2), most likely associated with the presence of the +putative torus. Two ionized absorbers are unequivocally present, located within maximum radii of rmax,1 = 1.5 pc and rmax,2 = 0.01 pc. +In one of the observations, the inner ionized absorber is blueshifted, indicating the presence of a possible faster (vout = 0.2c) ionized +absorber, marginally detected at 3σ. Finally, we are able to constrain the coronal temperature and the optical depth of ESO 323-G77, +obtaining kTe = 38 keV or kTe = 36 keV, and τ = 1.4 or τ = 2.8, depending on the coronal geometry assumed. +Key words. X-rays: galaxies – galaxies: active – galaxies:individual:ESO 323-G77 +1. Introduction +Observations of active galactic nuclei (AGN) reveal the pres- +ence of two main classes of sources. Type 1 AGN are sources +for which the optical spectra show both narrow (FWHM≤ 1000 +km s−1) and broad (FWHM> 1000 km s−1) lines, while type 2 +AGN are objects whose spectra only manifest narrow lines. This +suggests that in type 1 AGN the broad line region (BLR) is visi- +ble, while type 2 AGN have the BLR covered by obscuring ma- +terial. The dichotomy between type 1 and type 2 objects led to +a unification scheme based on the orientation of the AGN (e.g., +Antonucci 1993; Urry & Padovani 1995), where the central en- +gine is surrounded by an axisymmetric absorber, called the torus, +and the amount of obscuration is entirely due to the line of sight +angle with respect to the AGN axis. +According to the unification model, the column density NH +measured in X-ray spectra should follow this simple physical +scheme. However, in many AGN the amount of obscuration +in the X-rays is variable on a wide range of timescales (e.g., +Risaliti et al. 2002; Markowitz et al. 2014; Laha et al. 2020), +suggesting that the unification model is too simplistic to prop- +erly describe the whole phenomenon in detail. In particular, in +some cases the X-ray absorbing medium is variable on very +short timescales (days/weeks), which implies that the obscur- +ing medium is clumpy and located at much smaller distances +than the torus, possibly consistent with the BLR (e.g., Risaliti +et al. 2007; Bianchi et al. 2009; Maiolino et al. 2010; Sanfrutos +et al. 2013; Marinucci et al. 2013; Walton et al. 2014). In other +cases, the X-ray absorption variability timescale is of the order +of months or years (e.g., Piconcelli et al. 2007; Rivers et al. 2011; +Coffey et al. 2014; Rivers et al. 2015; Ricci et al. 2016; Pizzetti +et al. 2022), suggesting an origin from the putative circumnu- +clear torus. However, these results strongly depend on the obser- +vation sampling time; frequently adopted monthly observational +monitoring may lose the variations at lower timescales. These +findings suggest that the X-ray obscurer is not a single homo- +geneous entity, but rather the observational product of multiple +layers of absorbing material from the BLR and the torus. +Moreover, there is mounting evidence for a clumpiness of +the circumnuclear torus (e.g., Tristram et al. 2007), which would +imply that the probability of observing the central engine is al- +ways non-zero (e.g., Elitzur 2008, 2012). The X-ray obscuration +can therefore occur due to individual clumps passing through the +line of sight, either in the BLR or in the circumnuclear torus. +ESO 323-G77 is a nearby Seyfert 1 galaxy at redshift z = +0.015, with a complex and highly variable absorber. A ∼ 20 +ks observation by XMM-Newton in 2006 unveiled complex ab- +sorpion and emission features that revealed the presence of out- +flowing material (Jiménez-Bailón et al. 2008). Subsequent ob- +servations with XMM-Newton (2013), Chandra (2011), Swift- +XRT (2006) and Suzaku (2011) revealed a wide range of spectral +shapes, mainly driven by variations of the column density of a +Article number, page 1 of 10 +arXiv:2301.13223v1 [astro-ph.GA] 30 Jan 2023 + +A&A proofs: manuscript no. eso323_serafinelli +Table 1. The NuSTAR observations considered in this work. The expo- +sures listed here are to be read as net exposures per single FPM module. +Epoch +OBSID +Date +Exposure (s) +1 +60202021002 +2016-12-14 +39360 +2 +60202021004 +2016-12-20 +42531 +3 +60202021006 +2017-01-04 +43403 +4 +60202021008 +2017-02-03 +43295 +5 +60202021010 +2017-03-31 +38231 +neutral absorber at several timescales (Miniutti et al. 2014; San- +frutos et al. 2016). +The spectral shape of the source ranges from an unobscured +state (NH < 1022 cm−2) in four Chandra observations taken in +2010, a moderately absorbed state (NH ∼ 3 × 1022 − 1023 cm−2) +for the 2006 XMM-Newton observation and two 2006 Swift-XRT +snapshots, a Compton-thin obscured state (NH ∼ 1 − 6 × 1023 +cm−2) observed by XMM-Newton in 2013 and in one Swift-XRT +pointing in 2006, and finally a Compton-thick obscured state +(NH > 1024 cm−2) in the Suzaku observation taken in 2011. +Miniutti et al. (2014) argued that low column density states +(NH ≲ 1023 cm−2) are due to the presence of a clumpy obscuring +torus, while the states with larger column densities are produced +by obscuration by clumps of a closer medium, likely co-spatial +with the BLR. This is reminiscent of other changing look sources +such as NGC 1365 (e.g., Risaliti et al. 2007). +Here we report the spectral analysis of the first Nuclear Spec- +troscopic Telescope Array (NuSTAR, Harrison et al. 2013) ob- +servations of ESO 323-G77. The paper is organized as follows. +In Sect. 2 we describe the data used for this work and the data +reduction pipeline. Sect. 3 describes the spectral analysis and all +the models tested for the data. In Sect. 4 we discuss the spectral +models adopted, and in Sect. 5 we summarize our results. +We adopt a standard flat cosmology with H0 = 70 km s−1 +Mpc−1, Ωm = 0.3 and ΩΛ = 0.7. +2. Data reduction +We analyze here a campaign of five NuSTAR observations per- +formed between December 2016 and March 2017 for a total of +∼ 200 ks. Each observation has an exposure of approximately 40 +ks, taken at 1,2,4 and 8 weeks from the first one (see Table 1 for +details). The observations were coordinated with ∼ 2 ks of Neil +Gehrels Swift Observatory (Gehrels et al. 2004) snapshots taken +with the X-ray Telescope (XRT). The NuSTAR spectra were re- +duced using the standard HEASOFT v6.28 command NUPIPELINE +from the NUSTARDAS software package, using the most recent +CALDB version. We filtered passages through the South Atlantic +Anomaly by setting the task NUCALSAA ’optimized’ mode. The +two Focal Plane Module (FPM) source spectra A and B were +extracted from a circular region with a radius of 40”, centered +on the source, while the background spectra were extracted from +two circular regions with a radius of 45” each, on the same chip. +The two FPMA and FPMB spectra were combined and the re- +sulting spectrum was binned to a minimum of 50 counts per bin. +The energy band considered for our fits is in the range E = 3−65 +keV. The spectra of the Swift-XRT observations were extracted +with the HEASOFT command XSELECT, selecting a circular region +with a 30” radius. Background spectra were also extracted with +the same procedure, but selecting a source-free circular region +of 70” radius. The XRTMKARF task was used to produce ancil- +lary files, and the response was provided by the CALDB reposi- +tory. Given the negligible variability, the XRT spectra were all +1 +10 +10−5 +10−4 +10−3 +keV (Photons cm−2 s−1 keV−1) +Energy (keV) +Fig. 1. Unfolded spectrum of the data analyzed here, adopting a simple +model with an absorbed continuum powerlaw with Γ = 2. Red, blue, +cyan, black and magenta spectra mark the NuSTAR spectra of Epochs 1 +to 5, respectively. The grey spectrum is the Swift-XRT one. +combined together, and grouped at a minimum of 10 counts per +energy bin. The energy range 0.5 − 10 keV was considered for +the spectral fits. The folded spectra, adopting a simple powerlaw +with photon index Γ = 2, are shown in Fig. 1. +3. Spectral analysis +All spectral fits are performed using the software XSPEC +v12.12.0 (Arnaud 1996). We adopt a constant Galactic absorp- +tion described by a column density of NH += 7.75 × 1020 +cm−2 (HI4PI Collaboration et al. 2016), which is modelled with +TBABS, in all our models. In all models, a cross-correlation con- +stant between XRT and NuSTAR (CXRT/NuS TAR) is adopted. In +every model the best-fit for this constant is CXRT/NuS TAR = +0.8 ± 0.1. All errors on the best-fit parameters are given with +a 90% confidence level, i.e. ∆χ2 = 2.71. +3.1. Slab-reflection model +We first tested a simple absorbed continuum powerlaw plus a +scattered powerlaw, with tied photon index. Fe Kα at E = 6.4 +keV, and Fe Kβ at E = 7.06 keV emission lines are also in- +cluded, with fixed centroid energies and width (σ = 0.03 keV). +The Fe Kβ line normalization is fixed at 13% of the Kα line (e.g., +Palmeri et al. 2003). However, this simple model does not prop- +erly describe the current data set. As a very flat photon index +(Γ ∼ 1.35) and an unacceptable statistic (χ2/dof = 1751/907) +are obtained, it is clear that this model does not properly fit the +data. Moreover, an equivalent width of EW> 250 eV is obtained +for the Fe Kα line, which is a signature of the presence of a re- +flection component in obscured sources (e.g., Krolik et al. 1994). +Therefore, we test a model that includes an absorbed power- +law and a neutral reflector. The slab-reflection model PEXRAV +(Magdziarz & Zdziarski 1995) is used with Fe Kα and Fe Kβ +emission lines modelled by two ZGAUSS components, plus an ab- +sorbed main powerlaw ZPHABS*CABS*ZPOW, and a soft-scattered +powerlaw ZPOW. The overall model is +(1) +TBabs ∗ (const1 ∗ cabs ∗ zphabs ∗ zpow1 ++ pexrav + zgauss1 + zgauss2 + const2 ∗ zpow2). +The five NuSTAR spectra are fitted together, keeping all param- +eters tied among different epochs to the ones of Epoch 4, which +Article number, page 2 of 10 + +Roberto Serafinelli et al.: The NuSTAR view of the changing look AGN ESO 323-G77 +is the brightest observation, with the exception of the column +density NH of the absorber and the normalizations of the main +powerlaw and the reflection component, to take their variability +into account. The Swift-XRT spectrum has all parameters tied to +Epoch 4, as in all models considered in this work. We assume +that the Fe K lines do not vary, as in most absorbed AGN (e.g., +Fukazawa et al. 2016), and we keep the Fe Kβ normalization +fixed at 13% of the value of the Fe Kα. All parameters of the +scattered powerlaw are kept tied to the ones of the main one, +whereas the two constants, CONST1 kept fixed at 1 at all epochs, +while CONST2 is fitted for Epoch 4 and not allowed to vary, in +order to take their ratio into account. +The +continuum +is +characterized +by +a +photon +index +Γ += +1.75 ± 0.03 and a normalization that varies from +npl = 9+5 +−4 × 10−4 to npl = (2.0 ± 0.3) × 10−3 photons cm−2 s−1 +keV−1, while the second constant is kept free and is ∼ 10−2. +PEXRAV models a pure reflection component from an infinite +slab, meaning that the reflection constant is fixed to R = −1. +The photon index of the reflection component is tied to the +one of the main continuum, while the cut-off energy is fixed +to Ecut = 500 keV. The line of sight absorption is given by +NH ∼ (7 − 11) × 1023 cm−2, depending on the observation, and +therefore the model cabs is included to take into account the +suppression of the continuum due to electron scattering, which +is non-negligible at column densities larger than 5 × 1023 cm−2 +(e.g., Yaqoob 2012), with column density fixed to the value of +ZPHABS. +The Fe Kα emission line centroid is found at E = 6.28+0.06 +−0.07 +keV, with width σ += +0.2 ± 0.1 keV and normalization +nFeKα += (1.3 ± 0.4) × 10−5 photons cm−2 s−1 keV−1. The +range of the equivalent width of the Fe Kα emission line is +EW∼ 0.2−0.4 keV. The cut-off energy in the reflection spectrum +is fixed at Ecut = 500 keV, while the photon index is tied to +that of the continuum component. The abundances are fixed to +solar ones, and the reflector normalizations vary in the range +nrefl ∼ (4 − 5) × 10−3 photons cm−2 s−1 keV−1. +The +model +has +an +overall +goodness +of +fit +of +χ2/dof = 978/909 = 1.07. +3.2. Toroidal model MYTORUS +The disk-reflection model provides an acceptable goodness of +fit. However, as pointed out by Yaqoob (2012), the model is in- +adequate to describe the reflector in detail. Indeed, the reflection +spectrum assumes an infinite line-of-sight column density and it +does not consider the finite nature of the reflector, as it was cre- +ated assuming a point source illuminating an infinite slab. +Therefore, in the following we adopt a detailed toroidal re- +flection model, MYTORUS (Murphy & Yaqoob 2009). This model +assumes a toroidal geometry characterized by a column density +NH and a fixed covering factor of 0.5, corresponding to a torus +opening angle of 60◦. Since the column density of ESO 323-G77 +is variable, we adopt the decoupled standard model, in which +the column density of the absorber NH,abs is different from the +column density of the reflector NH,refl (Yaqoob 2012). As a first +step we multiply the continuum power law by the zeroth-order +component of the model, namely the XSPEC table MYTZ1. This +table allows us to evaluate the line-of-sight column density NH +of the absorber. We consider the angle θ, which is the inclina- +1 All MYTORUS tables are available at http://mytorus.com/ +model-files-mytorus-downloads.html. The MYTZ model can be +downloaded with the table mytorus_Ezero_v00.fits +1 +10 +0.5 +1 +1.5 +Data/Model +Rest Energy (keV) +Fig. 2. Data-to-model ratio for the model in Eq. 2, where the reflector +is modelled with MYTS0, and only the neutral absorber is considered. +There are still significant ratios in the whole analyzed band, in particular +the curvature is not well modelled by a single neutral absorber. +tion angle between the polar axis of the absorber and the line +of sight. For the MYTZ, we fix θ = 90◦, which corresponds to +a line of sight direction for the absorber. We model the Comp- +ton hump continuum due to neutral reflection with the additive +table MyTS02. We fix θ = 0◦ to assume that this reflected com- +ponent does not come from the line of sight. The column den- +sity of this component is independent from the line of sight one +(decoupled model), and the normalization and photon index are +kept fixed to the ones of the continuum. The Fe Kα and Fe Kβ +emission lines of the line-of-sight reflection are included with +the additive table MyTL03, with fixed value θ = 0◦, normaliza- +tion and Γ tied to the absorbed continuum values. We multiply +MyTL0 by the convolution model GSMOOTH, to take into account +the broadening of the iron line. We fix the line width in the model +to σ = 0.03 keV, following the upper limit found by Sanfrutos +et al. (2016) with Chandra HETG. The fit has a global statistic +of χ2/dof = 1129/915 = 1.23. +We also allow for a forward scattering component on the +line of sight, namely another Compton-reflected continuum with +fixed θ = 90◦ (hereafter MyTS90). We assume that the column +density (NH,0) of this component coincides with the line-of-sight +NH. This additional reflection component is also accompanied by +a table with iron lines MyTL90, where the column density is tied +to NH,0. MyTL90 is also multiplied by a GSMOOTH model with +fixed σ = 0.03 keV. The normalizations and photon indices of +MyTS90 and MyTL90 are also tied to the one of the main power- +law. The goodness of fit is given by χ2/dof = 1162/915 = 1.25. +This means that the reflection due to the absorbing material on +the line of sight is not required in our model. In all models from +here on, we will only consider the reflection component out of +the line of sight. +The model is therefore +(2) +TBabs ∗ ((const1 ∗ MyTZ ∗ zpow1 ++ MyTS0 + gsmooth ∗ MyTL0) + const2 ∗ zpow2) +We find a line-of-sight NH,abs ranging from (3.2±0.3)×1023 cm−2 +in Epoch 4 up to (5.5±0.5)×1023 cm−2 at Epoch 2. The out of line +of sight column density of the reflector is NH,refl = 4.0+0.3 +−0.7 × 1024 +2 File name mytorus_scatteredH200_v00.fits +3 File name mytl_V000010nEp000H200_v00.fits +Article number, page 3 of 10 + +A&A proofs: manuscript no. eso323_serafinelli +−2 0 2 +σ +−2 0 2 +σ +−2 0 2 +σ +−2 0 2 +σ +10 +5 +−2 0 2 +σ +Rest Energy (keV) +Fig. 3. Residuals when the model with only one ionized absorber is +fitted. The observations are ordered top to bottom from the first to the +last taken. An absorption complex at ∼ 7 keV is observed in Epochs 1, +3 and 4. At Epoch 5, the absorption complex is observed at ∼ 8.5 keV, +suggesting a possible outflowing velocity of v ∼ 0.2c. +0.1 +0.15 +0.2 +0.25 +0.3 +1.7 +1.8 +1.9 +Γ +NH,abs (1024 cm−2) +X +Fig. 4. Contour plot of the spectral slope Γ and the line of sight column +density NH obtained for Epoch 4, as obtained from the zero-order MY- +TORUS model. The red, green and blue line represent 68% (1σ), 95% +(2σ) and 99.7% (3σ) contours. +cm−2. We also obtain a flatter photon index Γ = 1.61 ± 0.3, with +respect to the one obtained with the slab-reflection model. +3.3. Ionized absorbers +Fig. 2 shows the residuals of the model in Eq. 2. There are sig- +nificant residuals in the E ∼ 5 − 10 keV energy range and above +E ∼ 20 keV, showing that it does not properly fit the curvature of +the spectrum, which means that additional components might be +needed. Since past observations of this source reported the pres- +ence of ionized absorbers (Jiménez-Bailón et al. 2008; Miniutti +et al. 2014; Sanfrutos et al. 2016), we consider the addition of +one of such features. We denote this absorber as Zone 1. We +adopt a grid of photoionized absorbers, produced with the XS- +TAR (Kallman & Bautista 2001) photoionization code. The grid +spans a relatively wide ionization (log(ξ/erg cm s−1) ∼ 2 − 6) +and column density (NH ∼ 5 × 1022 − 5 × 1024 cm−2) range. The +turbulent velocity adopted to generate the grid is vturb = 3000 +km s−1. We first allow the ionization and the column density to +10−6 +10−5 +10−4 +normalized counts s−1 keV−1 cm−2 +1 +10 +0.5 +1 +1.5 +ratio +Rest Energy (keV) +Fig. 5. Normalized spectra and data-to-model ratio of ESO 323-G77. +The MYTORUS model is shown here. The same color code used in +Fig. 3 is adopted, with the addition of the Swift-XRT spectrum, shown +in grey. +1 +10 +10−4 +10−3 +2×10−5 +5×10−5 +2×10−4 +5×10−4 +keV (Photons cm−2 s−1 keV−1) +Rest Energy (keV) +Fig. 6. Unfolded spectra of ESO 323-G77, determined from the Swift- +XRT and NuSTAR data, based on the MYTORUS model. +vary among different observations, but we do not find significant +changes in either parameters, therefore we fix both parameters to +the ones of Epoch 4. The addition of this component improves +the fit by ∆χ2/∆dof = 159/2, with the overall goodness of fit be- +ing χ2/dof = 970/913. The photon index is 1.74±0.06. The col- +umn density of this absorber is given by NH,z1 = (5.8±0.5)×1023 +cm−2 and the ionization is log ξz1/(erg cm s−1) = 2.6 ± 0.1. +The addition of the Zone 1 absorber significantly reduces the +curvature residuals in Fig. 2. The residuals with the new model +in the E = 4 − 13 keV band are shown in Fig. 3, where the +data still show significant residuals in the Fe Kα spectral region +(E = 6 − 10 keV) in almost all observations. Most observa- +tions show an absorbing structure around 6.5 − 7 keV, which +may be due to absorbing material. This is particularly notice- +able near 7 keV in epochs 3 and 4 (see Fig. 3, panels 3 and +4). Moreover, a second more ionized absorber was reported in +Jiménez-Bailón et al. (2008), Miniutti et al. (2014) and Sanfru- +tos et al. (2016), which could be responsible for this absorb- +ing feature. We thus add a second absorber, which we label as +Zone 2, using the same XSTAR grid used for the first one. We +initally assumed that also this more ionized absorber did not +Article number, page 4 of 10 + +Roberto Serafinelli et al.: The NuSTAR view of the changing look AGN ESO 323-G77 +Table 2. Best-fit parameters of the final MYTORUS model shown in Eq. 3. The goodness of fit is χ2/dof = 927/906. The COMPTT parameters are +taken from the best-fit model shown in Eq. 5, assuming a comptonized continuum produced by a corona with a slab geometry. +Parameter +Epoch 1 +Epoch 2 +Epoch 3 +Epoch 4 +Epoch 5 +Central source (zpow) +Γ +− +− +− +1.79+0.04 +−0.06 +− +norm (10−3 photons cm−2 s−1 keV−1) +4.2 ± 0.7 +3.4 ± 0.6 +3.9 ± 0.7 +4.0 ± 0.7 +3.5 ± 0.6 +Fscat/Fnucl +− +− +− +2.3+0.4 +−0.3 × 10−2 +− +Central source (compTT, slab corona) +kT (keV) +− +− +− +38 ± 2 +− +τ +− +− +− +1.4 ± 0.1 +− +Neutral absorber (MYTORUS) +MyTZ +NH,abs (1023 cm−2) +2.6+0.4 +−0.5 +3.7+0.6 +−0.5 +2.5 ± 0.5 +1.9 ± 0.4 +3.4+0.6 +−0.3 +Reflection (MYTORUS) +MyTS0 +NH,refl (1024 cm−2) +− +− +− +5.0+2.8 +−1.3 +− +Ionized absorbers (xstar) +Zone 1 (external) +NH,z1 (1023 cm−2) +− +− +− +3.5+0.6 +−0.7 +− +log ξ (erg cm s−1) +− +− +− +2.4 ± 0.1 +− +Zone 2 (internal) +NH,z2 (1023 cm−2) +2+7 +−1 +< 14 +6+16 +−4 +2+4 +−1 +2+7 +−1 +log ξ (erg cm s−1) +− +− +− +4.0+0.5 +−0.2 +− +v/c +0 +0 +0 +0 +0.21+0.02 +−0.03 +CXRT/NuS TAR +0 +0 +0 +0.8 ± 0.1 +0 +Observed fluxes +Fobs +2−10 keV (erg cm−2 s−1) +2.7 × 10−12 +1.8 × 10−12 +2.5 × 10−12 +3.1 × 10−12 +2.0 × 10−12 +Unabsorbed fluxes +Funabs +2−10 keV (erg cm−2 s−1) +4.6 × 10−12 +3.0 × 10−12 +4.4 × 10−12 +5.5 × 10−12 +4.7 × 10−12 +Reflection flux +Frefl +3−65 keV (erg cm−2 s−1) +− +− +− +9.5 × 10−12 +− +vary between the 5 epochs. The addition of this absorber im- +proves the statistic by ∆χ2/∆dof = 26/2 to χ2/dof = 944/911. +We obtain a photon index of Γ = 1.81+0.06 +−0.07, a column density +of NH,z2 = 1.6+14.5 +−0.7 × 1023 cm−2 and a ionization parameter of +log ξz2/(erg cm s−1) = 4.0+1.1 +−0.2. Given its higher ionization, we +assume that Zone 2 is closer to the black hole with respect to +Zone 1. Since NH and log ξ are notoriously degenerate, we keep +the ionization at all epochs fixed to the one of Epoch 4, while all +column densities are allowed to vary independently. The good- +ness of fit slightly improves to χ2/dof = 938/907. +Finally, as shown in Fig. 3, the absorber in Epoch 5 appears +as a blueshifted absorption line at Erest ∼ 8.5 keV, which is a +clear signature of a non-zero velocity. Hence, we free the ve- +locity of the Zone 2 absorber in Epoch 5, in order to take this +blueshift into account. We obtain zobs = −0.18 ± 0.02, which +corresponds4 to a velocity v = (0.21+0.02 +−0.03)c. The goodness of +fit further improves by ∆χ2/∆dof = 11/1 to a final value of +χ2/dof = 927/906 = 1.02. The column density of the absorber +in Zone 2 is constrained in four out of five observations, ranging +from NH,z2 = 2+7 +−1 × 1023 cm−2 (Epoch 1) to NH,z2 = 6+16 +−4 × 1023 +4 The observed shift zobs is related to the rest-frame blueshift zabs +by the relation zabs = (1 + zobs)/(1 + zc) − 1, where zc = 0.015 is +the cosmological redshift. The velocity of the absorber is given by +v/c = (z2 +abs + 2zabs)/(z2 +abs + 2zabs + 2). +Article number, page 5 of 10 + +A&A proofs: manuscript no. eso323_serafinelli +cm−2 (Epoch 3). We note that in Epoch 2 we can place only +an upper limit on the column density. Indeed, Epoch 2 does not +show a clear absorption signature in Fig. 3 (blue curve). The ion- +ization parameter is log ξz2/erg cm s−1 = 4.0+0.5 +−0.2. +The final model is therefore +(3) +TBabs ∗ ((const1 ∗ xstar1 ∗ MyTZ ∗ xstar2 ∗ zpow1 ++ MyTS0 + gsmooth ∗ MyTL0) + const2 ∗ zpow2) +where XSTAR1 and XSTAR2 are the ionized absorbers in Zone +1 and Zone 2, respectively. The photon index of the spectrum, af- +ter the addition of these two ionized absorbers, is Γ = 1.79+0.04 +−0.06. +Fig. 4 shows the contour plot of Γ with the MyTZ column den- +sity NH for the brightest observation of the campaign, i.e. Epoch +4. The contour plot shows that both the photon index Γ and the +absorbing column density NH,abs are well constrained at 3σ con- +fidence level. The best-fit parameters obtained with this model +are summarized in Table 2. The normalized spectrum with data- +to-model ratios and the unfolded spectrum are shown in Figs. 5 +and 6. +We also tested an alternative approach in which the absorber +column density NH,abs is kept tied among the observations, while +the photon index Γ is allowed to vary. Unsurprisingly, the col- +umn density is NH = (2.6 ± 0.5) × 1023 cm−2, which is the mean +value of the NH found independently when the parameter is al- +lowed to vary between observations. We find various values of +the photon index, ranging from Γ = 1.51 ± 0.06 for Epoch 3 to +Γ = 1.84+0.05 +−0.04 for Epoch 4. However, we obtain a worse fit statis- +tic of χ2/dof = 983/906, which means that an absorber variation +is favored. Notably, the smaller photon indices are also the ones +with greater absorption and viceversa, resulting in an apparent +steeper when brighter effect. This effect is driven by the absorp- +tion variability, as the source has historically experienced in the +past, and should not be confused with the continuum softer when +brighter effect, driven by intrinsic Γ variations (e.g., Sobolewska +& Papadakis 2009; Serafinelli et al. 2017). +3.4. Alternative model for the reflector: BORUS +We also test for a spherical reprocessor, using the model BORUS +(Balokovi´c et al. 2018). We consider a continuum described by +a cut-off power law, ZCUTOFFPL, with a line of sight absorp- +tion modelled by ZPHABS, and reflector described by the table +BORUS025. We also include the two ionized absorbers located in +Zone 1 and Zone 2. Also in this model we allow the column den- +sity of the Zone 2 high-ionization absorber NH to vary between +observations, while we assume the ionization parameter to re- +main constant between the observations of the campaign. The +model used is +(4) +TBabs ∗ ((const1 ∗ xstar1 ∗ zphabs ∗ xstar2 ∗ zcutoffpl1 ++ borus02) + const2 ∗ zcutoffpl2). +We obtain a photon index Γ = 1.79+0.04 +−0.06, consistent with the +value obtained with the MYTORUS model. As the cut-off energy +Ecut is unconstrained, we fix it to a fiducial Ecut = 500 keV. The +neutral column density varies from NH,abs = (2.7 ± 0.3) × 1023 +cm−2 (Epochs 1 and 2) to (3.4 ± 0.3) × 1023 cm−2 (Epoch +5), roughly consistent with the ones found with the MY- +TORUS model. The column density of the reprocessor is +NH = 2.7+0.4 +−0.8 × 1024 cm−2, which is consistent to the value found +in the MYTORUS model. The covering factor of the reprocessor +5 All BORUS tables can be downloaded from the website https:// +sites.astro.caltech.edu/~mislavb/download +is given by C f = 0.90+0.02 +−0.03. Finally, we obtain consistent values +for the column density and the ionization parameter of the +ionized absorber in Zone 1. The ionization parameter of the +absorber in Zone 2 is also consistent with the one obtained with +the MYTORUS model. The column density of the absorber in +Zone 2 is also consistent, althought with large uncertainties. The +goodness of fit of this model is given by χ2/dof = 923/901. +3.5. Comptonizing plasma continuum +It is also interesting to investigate the coronal parameters of this +source, as these are often elusive for obscured sources. There- +fore, we investigate physical Comptonization models for the +continuum with both the MYTORUS and BORUS models. Starting +from the MYTORUS model in Eq. 3, We adopted the same config- +uration and free parameters, but we replaced the power law con- +tinuum with COMPTT (Titarchuk 1994). We also adopted the ap- +propriate MYTORUS table, namely we adopt the tables MYTSTT +0 +6 +and MyTLTT +0 +7, and we use them the same way we used MYTS0 +and MYTL0 in Sect. 3.2. The model is then +(5) +TBabs ∗ ((const1 ∗ xstar1 ∗ MyTZ ∗ xstar2 ∗ compTT1 ++ MyTSTT +0 + gsmooth ∗ MyTLTT +0 ) + const2 ∗ compTT2). +We first explore the slab coronal geometry by fixing the +value of the parameter approx to 0.5. We do not find significant +differences in any other parameter obtained in the previous +section. The coronal temperature with this fit is kT = 38 ± 2 +keV, while the optical depth is τ = 1.4 ± 0.1. The goodness +of fit of this model is given by χ2/dof = 920/906. Typically, +assuming a spherical geometry in COMPTT, the best-fit coronal +parameters would be a similar temperature, but a larger optical +depth (e.g., Tortosa et al. 2018). However, the MYTSTT +θ +tables +do not include larger values of τ, and therefore it is not possible +to explore the parameters of a spherical geometry. +However, the spherical geometry might be explored within +the BORUS model shown in Eq. 4. BORUS12 is produced with the +thermal comptonization continuum model NTHCOMP (Magdziarz +& Zdziarski 1995), which assumes a spherical geometry for the +corona. Hence, we also use such model for the continuum, and +the model is therefore: +TBabs ∗ ((const1 ∗ xstar1 ∗ zphabs ∗ xstar2 ∗ nthcomp1 ++ borus12) + const2 ∗ nthcomp2). +(6) +with a goodness of fit of χ2/dof = 915/901. We obtain Γ = +1.73+0.01 +−0.05 and a coronal temperature of kT = 36+13 +−8 keV. Remark- +ably, this value is consistent with the COMPTT temperature ob- +tained assuming a slab geometry in the MYTORUS model, even +adopting a different continuum model. +3.6. Relativistic reflection +The presence of a possible relativistic iron line in the X-ray spec- +tra of this AGN was inferred by Jiménez-Bailón et al. (2008) +during an unabsorbed state. Therefore, we test the possibility +that such component could also be detected in an absorbed state, +and we add the relativistic reflection component RELXILL (García +et al. 2014; Dauser et al. 2014) to the model in Eq. 3. The global +6 File name mytorus_scatteredkT034_v00.fits +7 File name mytl_V000010nEp000kT034_v00.fits +Article number, page 6 of 10 + +Roberto Serafinelli et al.: The NuSTAR view of the changing look AGN ESO 323-G77 +Table 3. Best-fit parameters of the final BORUS model shown in Eq. 4. The goodness of fit is χ2/dof = 923/901. The NTHCOMP parameters are +taken from the best-fit model shown in Eq. 6, assuming a Comptonized continuum produced by a spherical corona. +Parameter +Epoch 1 +Epoch 2 +Epoch 3 +Epoch 4 +Epoch 5 +Central source (zcutoffpl) +Γ +− +− +− +1.79+0.04 +−0.06 +− +norm (10−3 photons cm−2 s−1 keV−1) +3.0+0.3 +−0.5 +2.4+0.3 +−0.5 +2.8+0.3 +−0.5 +3.0+0.3 +−0.4 +2.5+0.3 +−0.4 +Fscat/Fnucl +− +− +− +3.1+0.5 +−0.4 × 10−2 +− +Central source (nthcomp) +Γ +− +− +− +1.73+0.01 +−0.05 +− +kT (keV) +− +− +− +36+13 +−8 +− +Neutral absorber +Absorption (zphabs) +NH (1023 cm−2) +2.7 ± 0.2 +2.7 ± 0.3 +2.8 ± 0.2 +3.0 ± 0.2 +3.4 ± 0.3 +Reflection (borus02) +NH (1024 cm−2) +− +− +− +2.7+0.4 +−0.8 +− +Covering factor (C f ) +− +− +− +0.90+0.02 +−0.03 +− +Ionized absorbers (xstar) +Zone 1 (external) +NH (1023 cm−2) +− +− +− +2.6+0.9 +−0.8 +− +log ξ (erg cm s−1) +− +− +− +2.37+0.05 +−0.25 +− +Zone 2 (internal) +NH (1023 cm−2) +< 2.2 +< 1.4 +4 ± 2 +1.6+0.1 +−0.8 +2+2 +−1 +log ξ (erg cm s−1) +− +− +− +4.0+0.2 +−0.1 +− +v/c +0 +0 +0 +0 +0.21 ± 0.02 +fit improves by ∆χ2/∆dof = 38/7. All parameters with the ex- +ception of the normalization are kept tied between observations. +We assume a frozen cut-off energy Ecut = 500 keV, a disk exter- +nal radius of Rout = 400Rg, where Rg = GM/c2 is the gravita- +tional radius, a 45◦ inclination (Schmid et al. 2003), a solar iron +abundance and an emissivity index of −3. The spin of the black +hole is unconstrained, for which therefore we freeze a = 0, and +we obtain a disk internal radius Rin < 12Rg, consistent with the +findings of Miniutti et al. (2014). The disk ionization parameter +is log(ξ/erg cm s−1) > 3. A steeper photon index Γ = 1.87+0.04 +−0.09 is +found, although consistent with the one found with the model in +Eq. 3 at 90% confidence level. The normalization of the relativis- +tic component is unconstrained in Epoch 4, normrelx,4 < 8×10−6 +photons cm−2 s−1 keV−1, while in Epochs 2, 3 and 5 it is roughly +constant (normrelx,2,3,5 = 6+7 +−4 × 10−6 photons cm−2 s−1 keV−1), +and in Epoch 1 it is normrelx,1 = 10+6 +−5 × 10−6 photons cm−2 s−1 +keV−1. We do not find significant differences in the absorbing +column density from Table 2. However, the two reflectors are +degenerate, and therefore we find a lower limit for the neutral +reflector column density, NH,refl > 4 × 1024 cm−2, even though it +is consistent with the value of Table 2. +We also tested RELXILL as an additional reflection compo- +nent in the model where we assume a comptonizing continuum +COMPTT (Eq. 5), to test the possible influence on the measure of +kT and τ. The temperature of the corona is kT = 26 ± 9 keV, and +τ = 1.5+0.3 +−0.1, consistent within the 3σ contour plot of these two +parameters for the model without a disk reflection component +(see Fig. 7). Very similar results are obtained by testing RELXILL +on the two models that use BORUS for the neutral reflection. +We stress that the RELXILL component contributes to ≲ 10% +of the 2 − 10 keV observed flux, and the main changes in this +model are in the spectral region between 3 and 5 keV, where +NuSTAR is less sensitive. Also, many parameters of the relativis- +tic reflection model are unconstrained due to the complex model +and numerous degeneracies with the neutral reflector. We point +that, in order to accurately measure the parameters of the ionized +relativistic reflection within the framework of such a complex +spectral model, a broad band spectrum and an improved energy +resolution would be needed. For instance, a simultaneous XMM- +Newton and NuSTAR observation would be ideal to observe the +Fe Kα spectral region in detail. +Article number, page 7 of 10 + +A&A proofs: manuscript no. eso323_serafinelli +4. Discussion +4.1. Comparison between MYTORUS and BORUS models +The MYTORUS model has been built assuming a toroidal shape, +asymmetric on the azimuthal axis. The covering factor in such +model is kept fixed by assuming that the torus opening angle is +θOA = 60◦, which means that its value is C f = cos(θOA) = 0.5 +(Murphy & Yaqoob 2009). Conversely, BORUS has a spherical +geometry for the reprocessor, with polar cutouts corresponding +to a variable opening angle θOA, and therefore is able to fit a value +for the C f , ranging from C f = 0.1 to C f = 1 (Balokovi´c et al. +2018). The two best-fit values of the average column density of +the reflector are slightly different, NH,MYT = 5.0+2.8 +−1.3 × 1024 cm−2 +and NH,borus = 2.7+0.4 +−0.8 ×1024 cm−2. Moreover, the covering factor +found with the BORUS model is not consistent with the value of +C f = 0.5 assumed in the MYTORUS one, and this might explain +the difference in the column density estimate. +In order to properly compare the two models, we construct a +BORUS version of the MYTORUS decoupled model. We consider +an out of line of sight reflector by setting the torus inclination to +cos θ = 0.95, which is the maximum value allowed by the BORUS +model. This corresponds to an inclination angle of θ = 18◦, dif- +ferently from the MYTORUS value θ = 0. The covering factor is +fixed to the MYTORUS value C f = 0.5. As expected, the inclina- +tion discrepancy is not crucial (see also Marchesi et al. 2019) and +we obtain NH,borus = 5+3 +−1 × 1024 cm−2 ≃ NH,MYT. We stress that +this model has been built with the sole purpose of comparing the +column density of the torus for the MYTORUS to the one obtained +with BORUS, since the goodness of fit is χ2/dof = 944/902, +marginally worse than the model presented in Eq. 4. +However, this configuration is more realistic than the one +with a covering factor of Cv ∼ 0.9, since the latter would im- +ply that ∼ 90% of the sightline intercepts a Compton-thick col- +umn density. As a consequence, a Compton-thick state would +be observed far more frequently. In fact, while this source has +been observed several times, it has been caught in a Compton- +thick state only once in 2011 by Suzaku. This would be possible +if we were looking at this Seyfert galaxy with an exceptional, +extremely polar, line of sight, whereas Schmid et al. (2003) esti- +mated a 45◦ angle for the inclination. Therefore a lower covering +factor is likely a more realistic scenario for this source. +4.2. Compton-thin absorber and Compton-thick reflector +Both models indicate that the absorbing material is Compton- +thin, with column density ranging from NH,abs ∼ 2 × 1023 cm−2 +up to NH,abs ∼ 4 × 1023 cm−2. This AGN was already caught +in this state by one Swift-XRT snapshot in 2006 and by XMM- +Newton in 2013. However, as shown by Miniutti et al. (2014), +the source is able to change from a relatively unobscured state +(NH,abs ∼ 2 − 4 × 1022 cm−2) up to a Compton-thick state. +Previous analyses of ESO 323-G77 have hinted that low ob- +scuration states (NH ≲ 1023 cm−2) might be caused by the pres- +ence of the obscuring torus, while higher obscuration states are +likely due to absorption by cold intra-clump material located in +the broad line region (Miniutti et al. 2014; Sanfrutos et al. 2016). +However, given that we do not observe a change of state during +the campaign analyzed in this work, but only moderate changes +in the absorber column density NH,abs we are not able to argue in +favour or against this hypothesis. +The unprecedented effective area of NuSTAR in the E > 10 +keV band allows us to properly study the reflection compo- +nent of the X-ray spectrum of this source. In particular, both +the MYTORUS and BORUS models clearly point to the presence +of a Compton-thick reflector with NH = 5.0+2.8 +−1.3 × 1024 cm−2 or +NH = 2.7+0.4 +−0.8 × 1024, depending on the model. If the absorption +is indeed produced by BLR clumps or intra-clump cold material, +this result indicates that the constant Compton-thick reflector is +located farther away from the central X-ray source, and it should +be associated with the classic torus. +4.3. Ionized absorbers +Similar to the results presented in Jiménez-Bailón et al. (2008), +Miniutti et al. (2014) and Sanfrutos et al. (2016), our data shows +the presence of two ionized absorbers. +We can estimate the location of these ionized absorbers using +standard arguments. For instance, the maximum distance from +the black hole can be estimated by considering that the size of +the absorbing clump Rclump cannot be larger than the distance, +i.e. Rclump = NH/n < rmax, where n is the density of the clump +(e.g., Crenshaw & Kraemer 2012; Serafinelli et al. 2021). From +the ionization parameter definition the maximum distance from +the black hole can be written as +rmax = Lion +NHξ. +(7) +The first one, located in what we denote as Zone 1, is char- +acterized by an ionization parameter of ξ ∼ 250 erg cm s−1. +The ionizing luminosity in the E = 13.6 eV− 13.6 keV energy +band is Lion ≃ (2.6 ± 0.2) × 1044 erg s−1 and the column den- +sity is NH ≃ 3 × 1023 cm−2. Therefore, using Eq. 7, we obtain +rmax,1 = 1.4+0.4 +−0.9 pc. +The second ionized absorber, located in Zone 2 is char- +acterized by a larger ionization parameter, ξ ≃ 104 erg cm +s−1. The average column density is given by NH ≃ 6 × 1023 +cm−2. Therefore, using Eq. 7 we obtain a maximum distance of +rmax,2 = 1.0+0.9 +−0.8 × 10−2 pc. +We condider an Eddington ratio of log λEdd = −0.56 and a +black hole mass of MBH = 2.5 × 107 M⊙ (Wang & Zhang 2007), +from which we can compute log Lbol ≃ 44.93.This means, as- +suming that Lbol/L5100Å ∼ 10 (e.g., Collin et al. 2002), that the +optical luminosity is log L5100Å ≃ 43.93. We consider the rela- +tion between the broad line region size and the optical luminosity +introduced by Bentz et al. (2009) +log RBLR(light days) = −21.3 + 0.519 log L5100Å +and we obtain a broad line region radius of RBLR ≃ 0.02 pc. We +therefore obtain that the moderately ionized absorber in Zone 1 +could be located outside the broad line region at r1 ≲ 1.5 pc, +while the more ionized absorber in Zone 2 is likely co-spatial or +within the BLR. +In the scenario in which the cold absorber either co-spatial +with one of the two ionized absorbers or sandwiched between +them (Sanfrutos et al. 2016), the cold absorber would be located +between the outer BLR, consistently with the model proposed by +Miniutti et al. (2014), and pc-scale distances. In the latter case, +a possible scenario would be the presence of an inner thick re- +flecting ring surrounded by a thinner absorbing layer at pc-scale +(e.g., Buchner et al. 2019). Recent mid-infrared results (Leftley +et al. 2021) found evidence of the presence of polar warm dust +at a distance r ≳ 1.5 pc, which is consistent with this scenario. +The outer layer would also be clumpy, allowing for the observed +long-term variability, a similar scenario to the one proposed for +NGC 7479 by Pizzetti et al. (2022). +Article number, page 8 of 10 + +Roberto Serafinelli et al.: The NuSTAR view of the changing look AGN ESO 323-G77 +30 +35 +40 +45 +1 +1.2 +1.4 +1.6 +1.8 +2 +τ +kT (keV) +X +Fig. 7. Contour plot of the optical depth τ versus the coronal tempera- +ture kTe, assuming a slab coronal geometry with the COMPTT Comp- +tonization model. The red, green and blue lines represent 68% (1σ), +95% (2σ) and 99.7% (3σ) confidence levels, respectively. +4.4. Ultra-fast outflow +The velocity of the absorber in Zone 2 is v ≲ 9000 km s−1, +consistent with the values measured by Jiménez-Bailón et al. +(2008) and Sanfrutos et al. (2016) of v ≃ 2000 km s−1, in +Epochs 1-4. However, in Epoch 5, we notice a moderately +relativistic velocity v ∼ 0.21c, with a level of ∆χ2/dof = 11/1. +This is a tentative indication that we are observing an absorber +outflowing at high velocity, a phenomenon that is commonly +known as ultra-fast outflows (UFOs) and are fairly common +(∼ 40%) in Seyfert galaxies and quasars (e.g., Pounds et al. +2003; Braito et al. 2007; Tombesi et al. 2010; Gofford et al. +2013; Nardini et al. 2015; Tombesi et al. 2015; Serafinelli +et al. 2019). Moreover, UFOs are known to be extremely +variable (e.g., Reeves et al. 2014; Matzeu et al. 2017; Braito +et al. 2018, 2022), therefore it is not surprising that the UFO +appears within a relatively short timescale in an AGN that never +showed signs of its presence before. However, given its modest +(∼ 3σ) detection here, further observations would be required to +confirm the detection of the UFO feature or its variability. +4.5. Coronal parameters +The X-ray continuum is well known to be produced by inverse +Compton on UV seed photons gaining energy by a very hot elec- +tron corona (e.g., Haardt & Maraschi 1991, 1993). The elec- +tron temperature therefore plays a crucial role in regulating the +Comptonization of UV seed photons. Indeed, the main contin- +uum breaks at the so-called cut-off energy Ecut, which is tied to +the temperature by the relation Ecut = 2 − 3 kTe, depending on +the geometry of the corona (e.g., Petrucci et al. 2001). +When the COMPTT model is adopted to model the continuum, +assuming a slab geometry for the corona, in the MYTORUS model +shown in Eq. 5, we find that the temperature of the corona is +kTe = 38 ± 2 keV, with an optical depth τ = 1.4 ± 0.1. The τ − Γ +contour plot is shown in Fig. 7. Since the grids do not allow +much larger values of τ, the only way to study the spherical ge- +ometry is to use the NTHCOMP Comptonization continuum with +the BORUS model (Eq. 6), and we find a consistent temperature, +although with larger errors, kTe = 36+13 +−8 keV. We can estimate +the optical depth using the following equation, valid for a spher- +ical optically thick (τ > 1) corona (Zdziarski et al. 1996): +Γ = +� +9 +4 + +511 keV +kT τ(1 + τ +3) − 1 +2. +Using the best-fit values of the BORUS model, summarized in +Table 3, we obtain τ ≃ 2.8. +These are fairly standard values, as the coronal temperature +is known to span from kT ∼ 3 keV up to kT ∼ 450 keV (e.g., +Matt et al. 2015; Tortosa et al. 2018, 2022, Serafinelli et al., in +prep.). However, even though some authors have recently un- +veiled coronal temperatures in isolated obscured sources (e.g., +Middei et al. 2021) and samples of Seyfert 2 galaxies (e.g., +Balokovi´c et al. 2020), they are not easily constrained, since they +are often degenerate with the reflection spectrum cut-off. +5. Summary and conclusions +We have presented the spectral analysis of a campaign of 5 NuS- +TAR observations of the Seyfert 1.2 galaxy ESO 323-G77. We +summarize our results in the following +– The source has been observed in a persistent obscured, but +Compton-thin state, due to the presence of neutral obscuring +material on the line of sight, with column density in the range +NH ∼ 2 − 4 × 1023 cm−2. +– We find a Compton-thick reflector both modelling it with MY- +TORUS and BORUS. The two NH,refl values are not consistent, +but this result is dependent on the covering factor of the re- +flector, which is assumed as C f = 0.5 in MYTORUS and fit- +ted (C f = 0.90+0.02 +−0.03) in BORUS. By fixing a more realistic +C f = 0.5 in BORUS, the two results are consistent. +– Two ionized absorbers are needed in our models, consistent +with Jiménez-Bailón et al. (2008), Miniutti et al. (2014) and +Sanfrutos et al. (2016). The ionized absorber identified with +Zone 1 is located at a distance of about r1 ∼ 1.5 pc from the +black hole, most likely outside the broad line region, whose +size is estimated as RBLR ≃ 0.02 pc. The ionized absorber in +Zone 2 instead is located at r2 ≃ 10−2 pc, either co-spatial or +within the BLR. +– Assuming that the cold absorber is either at the same distance +of one of the two ionized absorbers, or at an intermediate +one, its location can be placed between the outer BLR and at +pc-scale distances. In the first case, this would be consistent +with the model proposed by Miniutti et al. (2014), consisting +of cold absorbing intra-clump material in the BLR. In the +second case, the most likely scenario is pc-scale Compton- +thin absorbing material surrounding a Compton-thick reflec- +tor (Buchner et al. 2019), which is supported by recent mid- +infrared detection of polar dust at r ≳ 1.5 pc (Leftley et al. +2021). +– The ionized absorber in Zone 2 is blueshifted at Epoch 5, to +the value zobs ≃ −0.18, which suggests an outflowing veloc- +ity of vout ≃ 0.2c. +– The coronal temperature is constrained in both models, find- +ing kTe ≃ 37 keV, both assuming a slab and a spherical +corona. The optical depth is τ ≃ 1.4 when the slab coronal +geometry is assumed, and τ ≃ 2.8 for a spherical corona. +– We find hints of the possible presence of a relativistic reflec- +tion component from the accretion disk. However, this com- +ponent contributes to ≲ 10% of the observed 2−10 keV flux, +and it mostly affects the 3 − 5 keV energy band. Hence, the +parameters of the disk reflection component are very difficult +to constrain, and higher energy resolution data are needed to +further study this feature. +Article number, page 9 of 10 + +A&A proofs: manuscript no. eso323_serafinelli +The campaign was not able to observe any significant change of +state (e.g., obscured to unobscured), as the source has undergone +several times in the past (Miniutti et al. 2014). However, longer +campaigns should be able to observe the source passing from +obscured to unobscured or vice-versa, setting an upper limit to +the obscurer location. Future high-resolution instruments such +as the microcalorimeter Resolve on board XRISM (XRISM Sci- +ence Team 2020) will be able to measure the properties of the ab- +sorbers with much more detail, particularly on their location and +outflowing velocity. Moreover, future hard X-ray (E = 2 − 200 +keV) instruments such as the High Energy X-ray Probe (HEX- +P, Madsen et al. 2018) will allow us to measure the reflection +parameters with unprecedented accuracy. +Acknowledgements. The authors thank the referee for useful comments that +improved the quality of this paper. RS, VB, PS, ADR, and RDC acknowl- +edge financial contribution from the agreements ASI-INAF n.2017-14-H.0 and +n.I/037/12/0. This research has made use of data and software provided by +the High Energy Astrophysics Science Archive Research Center (HEASARC), +which is a service of the Astrophysics Science Division at NASA/GSFC and the +High Energy Astrophysics Division of the Smithsonian Astrophysical Observa- +tory. This research has made use of the NuSTAR Data Analysis Software (NUS- +TARDAS) jointly developed by the ASI Space Science Data Center (SSDC, +Italy) and the California Institute of Technology (Caltech, USA). 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N., & Magdziarz, P. 1996, Monthly Notices of the +Royal Astronomical Society, 283, 193 +Article number, page 10 of 10 + diff --git a/xtFQT4oBgHgl3EQfAjU_/content/tmp_files/load_file.txt b/xtFQT4oBgHgl3EQfAjU_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d5d57ffbc63f46e8edb017009bf0e78f46ae7292 --- /dev/null +++ b/xtFQT4oBgHgl3EQfAjU_/content/tmp_files/load_file.txt @@ -0,0 +1,1203 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf,len=1202 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' eso323_serafinelli ©ESO 2023 February 1, 2023 The NuSTAR view of the changing look AGN ESO 323-G77 Roberto Serafinelli1, Valentina Braito2, 3, James N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Reeves3, 2, Paola Severgnini2, Alessandra De Rosa1, Roberto Della Ceca2, and Tracey Jane Turner4 1 INAF - Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133, Roma, Italy e-mail: roberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='serafinelli@inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='it 2 INAF - Osservatorio Astronomico di Brera, Via Brera 28, 20121, Milano, Italy & Via Bianchi 46, Merate (LC), Italy 3 Department of Physics, Institute for Astrophysics and Computational Sciences, The Catholic University of America, Washington, DC, 20064, USA 4 Eureka Scientific, Inc, 2452 Delmer St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Suite 100, Oakland, CA 94602, USA Received XXX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' accepted YYY ABSTRACT The presence of an obscuring torus at parsec-scale distances from the central black hole is the main ingredient for the Unified Model of Active Galactic Nuclei (AGN), as obscured sources are thought to be seen through this structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, the Unified Model fails to describe a class of sources that undergo dramatic spectral changes, transitioning from obscured to unobscured and vice-versa through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The variability in such sources, so-called Changing Look AGN (CLAGN), is thought to be produced by a clumpy medium at much smaller distances than the conventional obscuring torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' ESO 323-G77 is a CLAGN that was observed in various states through the years with Chandra, Suzaku, Swift-XRT and XMM-Newton, from unobscured (NH < 3×1022 cm−2) to Compton-thin (NH ∼ 1 − 6 × 1023 cm−2) and even Compton-thick (NH > 1 × 1024 cm−2), with timescales as short as one month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We present the analysis of the first NuSTAR monitoring of ESO 323-G77, consisting of 5 observations taken at different timescales (1, 2, 4 and 8 weeks from the first one) in 2016-2017, in which the AGN was caught in a persistent Compton-thin obscured state (NH ∼ 2 − 4 × 1023 cm−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We find that a Compton-thick reflector is present (NH,refl = 5 × 1024 cm−2), most likely associated with the presence of the putative torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Two ionized absorbers are unequivocally present, located within maximum radii of rmax,1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 pc and rmax,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='01 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' In one of the observations, the inner ionized absorber is blueshifted, indicating the presence of a possible faster (vout = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2c) ionized absorber, marginally detected at 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Finally, we are able to constrain the coronal temperature and the optical depth of ESO 323-G77, obtaining kTe = 38 keV or kTe = 36 keV, and τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 or τ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8, depending on the coronal geometry assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' X-rays: galaxies – galaxies: active – galaxies:individual:ESO 323-G77 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Introduction Observations of active galactic nuclei (AGN) reveal the pres- ence of two main classes of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Type 1 AGN are sources for which the optical spectra show both narrow (FWHM≤ 1000 km s−1) and broad (FWHM> 1000 km s−1) lines, while type 2 AGN are objects whose spectra only manifest narrow lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' This suggests that in type 1 AGN the broad line region (BLR) is visi- ble, while type 2 AGN have the BLR covered by obscuring ma- terial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The dichotomy between type 1 and type 2 objects led to a unification scheme based on the orientation of the AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Antonucci 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Urry & Padovani 1995), where the central en- gine is surrounded by an axisymmetric absorber, called the torus, and the amount of obscuration is entirely due to the line of sight angle with respect to the AGN axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' According to the unification model, the column density NH measured in X-ray spectra should follow this simple physical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, in many AGN the amount of obscuration in the X-rays is variable on a wide range of timescales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Risaliti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Markowitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Laha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2020), suggesting that the unification model is too simplistic to prop- erly describe the whole phenomenon in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' In particular, in some cases the X-ray absorbing medium is variable on very short timescales (days/weeks), which implies that the obscur- ing medium is clumpy and located at much smaller distances than the torus, possibly consistent with the BLR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Risaliti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Maiolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Sanfrutos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Marinucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Walton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' In other cases, the X-ray absorption variability timescale is of the order of months or years (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Piconcelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Rivers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Coffey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Rivers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Ricci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Pizzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2022), suggesting an origin from the putative circumnu- clear torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, these results strongly depend on the obser- vation sampling time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' frequently adopted monthly observational monitoring may lose the variations at lower timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' These findings suggest that the X-ray obscurer is not a single homo- geneous entity, but rather the observational product of multiple layers of absorbing material from the BLR and the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Moreover, there is mounting evidence for a clumpiness of the circumnuclear torus (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Tristram et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2007), which would imply that the probability of observing the central engine is al- ways non-zero (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Elitzur 2008, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The X-ray obscuration can therefore occur due to individual clumps passing through the line of sight, either in the BLR or in the circumnuclear torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' ESO 323-G77 is a nearby Seyfert 1 galaxy at redshift z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='015, with a complex and highly variable absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' A ∼ 20 ks observation by XMM-Newton in 2006 unveiled complex ab- sorpion and emission features that revealed the presence of out- flowing material (Jiménez-Bailón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Subsequent ob- servations with XMM-Newton (2013), Chandra (2011), Swift- XRT (2006) and Suzaku (2011) revealed a wide range of spectral shapes, mainly driven by variations of the column density of a Article number, page 1 of 10 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='13223v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='GA] 30 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' eso323_serafinelli Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The NuSTAR observations considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The expo- sures listed here are to be read as net exposures per single FPM module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Epoch OBSID Date Exposure (s) 1 60202021002 2016-12-14 39360 2 60202021004 2016-12-20 42531 3 60202021006 2017-01-04 43403 4 60202021008 2017-02-03 43295 5 60202021010 2017-03-31 38231 neutral absorber at several timescales (Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' San- frutos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The spectral shape of the source ranges from an unobscured state (NH < 1022 cm−2) in four Chandra observations taken in 2010, a moderately absorbed state (NH ∼ 3 × 1022 − 1023 cm−2) for the 2006 XMM-Newton observation and two 2006 Swift-XRT snapshots, a Compton-thin obscured state (NH ∼ 1 − 6 × 1023 cm−2) observed by XMM-Newton in 2013 and in one Swift-XRT pointing in 2006, and finally a Compton-thick obscured state (NH > 1024 cm−2) in the Suzaku observation taken in 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2014) argued that low column density states (NH ≲ 1023 cm−2) are due to the presence of a clumpy obscuring torus, while the states with larger column densities are produced by obscuration by clumps of a closer medium, likely co-spatial with the BLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' This is reminiscent of other changing look sources such as NGC 1365 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Risaliti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Here we report the spectral analysis of the first Nuclear Spec- troscopic Telescope Array (NuSTAR, Harrison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2013) ob- servations of ESO 323-G77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2 we describe the data used for this work and the data reduction pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3 describes the spectral analysis and all the models tested for the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 4 we discuss the spectral models adopted, and in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 5 we summarize our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We adopt a standard flat cosmology with H0 = 70 km s−1 Mpc−1, Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 and ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Data reduction We analyze here a campaign of five NuSTAR observations per- formed between December 2016 and March 2017 for a total of ∼ 200 ks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Each observation has an exposure of approximately 40 ks, taken at 1,2,4 and 8 weeks from the first one (see Table 1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The observations were coordinated with ∼ 2 ks of Neil Gehrels Swift Observatory (Gehrels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2004) snapshots taken with the X-ray Telescope (XRT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The NuSTAR spectra were re- duced using the standard HEASOFT v6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='28 command NUPIPELINE from the NUSTARDAS software package, using the most recent CALDB version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We filtered passages through the South Atlantic Anomaly by setting the task NUCALSAA ’optimized’ mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The two Focal Plane Module (FPM) source spectra A and B were extracted from a circular region with a radius of 40”, centered on the source, while the background spectra were extracted from two circular regions with a radius of 45” each, on the same chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The two FPMA and FPMB spectra were combined and the re- sulting spectrum was binned to a minimum of 50 counts per bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The energy band considered for our fits is in the range E = 3−65 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The spectra of the Swift-XRT observations were extracted with the HEASOFT command XSELECT, selecting a circular region with a 30” radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Background spectra were also extracted with the same procedure, but selecting a source-free circular region of 70” radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The XRTMKARF task was used to produce ancil- lary files, and the response was provided by the CALDB reposi- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Given the negligible variability, the XRT spectra were all 1 10 10−5 10−4 10−3 keV (Photons cm−2 s−1 keV−1) Energy (keV) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Unfolded spectrum of the data analyzed here, adopting a simple model with an absorbed continuum powerlaw with Γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Red, blue, cyan, black and magenta spectra mark the NuSTAR spectra of Epochs 1 to 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The grey spectrum is the Swift-XRT one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' combined together, and grouped at a minimum of 10 counts per energy bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The energy range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 − 10 keV was considered for the spectral fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The folded spectra, adopting a simple powerlaw with photon index Γ = 2, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Spectral analysis All spectral fits are performed using the software XSPEC v12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0 (Arnaud 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We adopt a constant Galactic absorp- tion described by a column density of NH = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='75 × 1020 cm−2 (HI4PI Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2016), which is modelled with TBABS, in all our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' In all models, a cross-correlation con- stant between XRT and NuSTAR (CXRT/NuS TAR) is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' In every model the best-fit for this constant is CXRT/NuS TAR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' All errors on the best-fit parameters are given with a 90% confidence level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' ∆χ2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Slab-reflection model We first tested a simple absorbed continuum powerlaw plus a scattered powerlaw, with tied photon index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Fe Kα at E = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 keV, and Fe Kβ at E = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='06 keV emission lines are also in- cluded, with fixed centroid energies and width (σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='03 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The Fe Kβ line normalization is fixed at 13% of the Kα line (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Palmeri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, this simple model does not prop- erly describe the current data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' As a very flat photon index (Γ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='35) and an unacceptable statistic (χ2/dof = 1751/907) are obtained, it is clear that this model does not properly fit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Moreover, an equivalent width of EW> 250 eV is obtained for the Fe Kα line, which is a signature of the presence of a re- flection component in obscured sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Krolik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Therefore, we test a model that includes an absorbed power- law and a neutral reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The slab-reflection model PEXRAV (Magdziarz & Zdziarski 1995) is used with Fe Kα and Fe Kβ emission lines modelled by two ZGAUSS components, plus an ab- sorbed main powerlaw ZPHABS*CABS*ZPOW, and a soft-scattered powerlaw ZPOW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The overall model is (1) TBabs ∗ (const1 ∗ cabs ∗ zphabs ∗ zpow1 + pexrav + zgauss1 + zgauss2 + const2 ∗ zpow2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The five NuSTAR spectra are fitted together, keeping all param- eters tied among different epochs to the ones of Epoch 4, which Article number, page 2 of 10 Roberto Serafinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' : The NuSTAR view of the changing look AGN ESO 323-G77 is the brightest observation, with the exception of the column density NH of the absorber and the normalizations of the main powerlaw and the reflection component, to take their variability into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The Swift-XRT spectrum has all parameters tied to Epoch 4, as in all models considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We assume that the Fe K lines do not vary, as in most absorbed AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Fukazawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2016), and we keep the Fe Kβ normalization fixed at 13% of the value of the Fe Kα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' All parameters of the scattered powerlaw are kept tied to the ones of the main one, whereas the two constants, CONST1 kept fixed at 1 at all epochs, while CONST2 is fitted for Epoch 4 and not allowed to vary, in order to take their ratio into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The continuum is characterized by a photon index Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='03 and a normalization that varies from npl = 9+5 −4 × 10−4 to npl = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3) × 10−3 photons cm−2 s−1 keV−1, while the second constant is kept free and is ∼ 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' PEXRAV models a pure reflection component from an infinite slab, meaning that the reflection constant is fixed to R = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The photon index of the reflection component is tied to the one of the main continuum, while the cut-off energy is fixed to Ecut = 500 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The line of sight absorption is given by NH ∼ (7 − 11) × 1023 cm−2, depending on the observation, and therefore the model cabs is included to take into account the suppression of the continuum due to electron scattering, which is non-negligible at column densities larger than 5 × 1023 cm−2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Yaqoob 2012), with column density fixed to the value of ZPHABS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The Fe Kα emission line centroid is found at E = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='28+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='07 keV, with width σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1 keV and normalization nFeKα = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4) × 10−5 photons cm−2 s−1 keV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The range of the equivalent width of the Fe Kα emission line is EW∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The cut-off energy in the reflection spectrum is fixed at Ecut = 500 keV, while the photon index is tied to that of the continuum component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The abundances are fixed to solar ones, and the reflector normalizations vary in the range nrefl ∼ (4 − 5) × 10−3 photons cm−2 s−1 keV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The model has an overall goodness of fit of χ2/dof = 978/909 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Toroidal model MYTORUS The disk-reflection model provides an acceptable goodness of fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, as pointed out by Yaqoob (2012), the model is in- adequate to describe the reflector in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Indeed, the reflection spectrum assumes an infinite line-of-sight column density and it does not consider the finite nature of the reflector, as it was cre- ated assuming a point source illuminating an infinite slab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Therefore, in the following we adopt a detailed toroidal re- flection model, MYTORUS (Murphy & Yaqoob 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' This model assumes a toroidal geometry characterized by a column density NH and a fixed covering factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5, corresponding to a torus opening angle of 60◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Since the column density of ESO 323-G77 is variable, we adopt the decoupled standard model, in which the column density of the absorber NH,abs is different from the column density of the reflector NH,refl (Yaqoob 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' As a first step we multiply the continuum power law by the zeroth-order component of the model, namely the XSPEC table MYTZ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' This table allows us to evaluate the line-of-sight column density NH of the absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We consider the angle θ, which is the inclina- 1 All MYTORUS tables are available at http://mytorus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='com/ model-files-mytorus-downloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The MYTZ model can be downloaded with the table mytorus_Ezero_v00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='fits 1 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 Data/Model Rest Energy (keV) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Data-to-model ratio for the model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2, where the reflector is modelled with MYTS0, and only the neutral absorber is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' There are still significant ratios in the whole analyzed band, in particular the curvature is not well modelled by a single neutral absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' tion angle between the polar axis of the absorber and the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' For the MYTZ, we fix θ = 90◦, which corresponds to a line of sight direction for the absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We model the Comp- ton hump continuum due to neutral reflection with the additive table MyTS02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We fix θ = 0◦ to assume that this reflected com- ponent does not come from the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The column den- sity of this component is independent from the line of sight one (decoupled model), and the normalization and photon index are kept fixed to the ones of the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The Fe Kα and Fe Kβ emission lines of the line-of-sight reflection are included with the additive table MyTL03, with fixed value θ = 0◦, normaliza- tion and Γ tied to the absorbed continuum values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We multiply MyTL0 by the convolution model GSMOOTH, to take into account the broadening of the iron line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We fix the line width in the model to σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='03 keV, following the upper limit found by Sanfrutos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2016) with Chandra HETG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The fit has a global statistic of χ2/dof = 1129/915 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We also allow for a forward scattering component on the line of sight, namely another Compton-reflected continuum with fixed θ = 90◦ (hereafter MyTS90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We assume that the column density (NH,0) of this component coincides with the line-of-sight NH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' This additional reflection component is also accompanied by a table with iron lines MyTL90, where the column density is tied to NH,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' MyTL90 is also multiplied by a GSMOOTH model with fixed σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='03 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The normalizations and photon indices of MyTS90 and MyTL90 are also tied to the one of the main power- law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The goodness of fit is given by χ2/dof = 1162/915 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' This means that the reflection due to the absorbing material on the line of sight is not required in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' In all models from here on, we will only consider the reflection component out of the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The model is therefore (2) TBabs ∗ ((const1 ∗ MyTZ ∗ zpow1 + MyTS0 + gsmooth ∗ MyTL0) + const2 ∗ zpow2) We find a line-of-sight NH,abs ranging from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3)×1023 cm−2 in Epoch 4 up to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5)×1023 cm−2 at Epoch 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The out of line of sight column density of the reflector is NH,refl = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7 × 1024 2 File name mytorus_scatteredH200_v00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='fits 3 File name mytl_V000010nEp000H200_v00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='fits Article number, page 3 of 10 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' eso323_serafinelli −2 0 2 σ −2 0 2 σ −2 0 2 σ −2 0 2 σ 10 5 −2 0 2 σ Rest Energy (keV) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Residuals when the model with only one ionized absorber is fitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The observations are ordered top to bottom from the first to the last taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' An absorption complex at ∼ 7 keV is observed in Epochs 1, 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' At Epoch 5, the absorption complex is observed at ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 keV, suggesting a possible outflowing velocity of v ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='9 Γ NH,abs (1024 cm−2) X Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Contour plot of the spectral slope Γ and the line of sight column density NH obtained for Epoch 4, as obtained from the zero-order MY- TORUS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The red, green and blue line represent 68% (1σ), 95% (2σ) and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7% (3σ) contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We also obtain a flatter photon index Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3, with respect to the one obtained with the slab-reflection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Ionized absorbers Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2 shows the residuals of the model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' There are sig- nificant residuals in the E ∼ 5 − 10 keV energy range and above E ∼ 20 keV, showing that it does not properly fit the curvature of the spectrum, which means that additional components might be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Since past observations of this source reported the pres- ence of ionized absorbers (Jiménez-Bailón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Sanfrutos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2016), we consider the addition of one of such features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We denote this absorber as Zone 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We adopt a grid of photoionized absorbers, produced with the XS- TAR (Kallman & Bautista 2001) photoionization code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The grid spans a relatively wide ionization (log(ξ/erg cm s−1) ∼ 2 − 6) and column density (NH ∼ 5 × 1022 − 5 × 1024 cm−2) range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The turbulent velocity adopted to generate the grid is vturb = 3000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We first allow the ionization and the column density to 10−6 10−5 10−4 normalized counts s−1 keV−1 cm−2 1 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 ratio Rest Energy (keV) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Normalized spectra and data-to-model ratio of ESO 323-G77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The MYTORUS model is shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The same color code used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3 is adopted, with the addition of the Swift-XRT spectrum, shown in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 1 10 10−4 10−3 2×10−5 5×10−5 2×10−4 5×10−4 keV (Photons cm−2 s−1 keV−1) Rest Energy (keV) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Unfolded spectra of ESO 323-G77, determined from the Swift- XRT and NuSTAR data, based on the MYTORUS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' vary among different observations, but we do not find significant changes in either parameters, therefore we fix both parameters to the ones of Epoch 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The addition of this component improves the fit by ∆χ2/∆dof = 159/2, with the overall goodness of fit be- ing χ2/dof = 970/913.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The photon index is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='74±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The col- umn density of this absorber is given by NH,z1 = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5)×1023 cm−2 and the ionization is log ξz1/(erg cm s−1) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The addition of the Zone 1 absorber significantly reduces the curvature residuals in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The residuals with the new model in the E = 4 − 13 keV band are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3, where the data still show significant residuals in the Fe Kα spectral region (E = 6 − 10 keV) in almost all observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Most observa- tions show an absorbing structure around 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 − 7 keV, which may be due to absorbing material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' This is particularly notice- able near 7 keV in epochs 3 and 4 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3, panels 3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Moreover, a second more ionized absorber was reported in Jiménez-Bailón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2008), Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2014) and Sanfru- tos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2016), which could be responsible for this absorb- ing feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We thus add a second absorber, which we label as Zone 2, using the same XSTAR grid used for the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We initally assumed that also this more ionized absorber did not Article number, page 4 of 10 Roberto Serafinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' : The NuSTAR view of the changing look AGN ESO 323-G77 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Best-fit parameters of the final MYTORUS model shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The goodness of fit is χ2/dof = 927/906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The COMPTT parameters are taken from the best-fit model shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 5, assuming a comptonized continuum produced by a corona with a slab geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Parameter Epoch 1 Epoch 2 Epoch 3 Epoch 4 Epoch 5 Central source (zpow) Γ − − − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='06 − norm (10−3 photons cm−2 s−1 keV−1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6 Fscat/Fnucl − − − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 × 10−2 − Central source (compTT, slab corona) kT (keV) − − − 38 ± 2 − τ − − − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1 − Neutral absorber (MYTORUS) MyTZ NH,abs (1023 cm−2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 Reflection (MYTORUS) MyTS0 NH,refl (1024 cm−2) − − − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 − Ionized absorbers (xstar) Zone 1 (external) NH,z1 (1023 cm−2) − − − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7 − log ξ (erg cm s−1) − − − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1 − Zone 2 (internal) NH,z2 (1023 cm−2) 2+7 −1 < 14 6+16 −4 2+4 −1 2+7 −1 log ξ (erg cm s−1) − − − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2 − v/c 0 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='21+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='03 CXRT/NuS TAR 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1 0 Observed fluxes Fobs 2−10 keV (erg cm−2 s−1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7 × 10−12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 × 10−12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 × 10−12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1 × 10−12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0 × 10−12 Unabsorbed fluxes Funabs 2−10 keV (erg cm−2 s−1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6 × 10−12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0 × 10−12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 × 10−12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 × 10−12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7 × 10−12 Reflection flux Frefl 3−65 keV (erg cm−2 s−1) − − − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 × 10−12 − vary between the 5 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The addition of this absorber im- proves the statistic by ∆χ2/∆dof = 26/2 to χ2/dof = 944/911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We obtain a photon index of Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='81+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='07, a column density of NH,z2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6+14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7 × 1023 cm−2 and a ionization parameter of log ξz2/(erg cm s−1) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Given its higher ionization, we assume that Zone 2 is closer to the black hole with respect to Zone 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Since NH and log ξ are notoriously degenerate, we keep the ionization at all epochs fixed to the one of Epoch 4, while all column densities are allowed to vary independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The good- ness of fit slightly improves to χ2/dof = 938/907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Finally, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3, the absorber in Epoch 5 appears as a blueshifted absorption line at Erest ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 keV, which is a clear signature of a non-zero velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Hence, we free the ve- locity of the Zone 2 absorber in Epoch 5, in order to take this blueshift into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We obtain zobs = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='02, which corresponds4 to a velocity v = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='21+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='03)c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The goodness of fit further improves by ∆χ2/∆dof = 11/1 to a final value of χ2/dof = 927/906 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The column density of the absorber in Zone 2 is constrained in four out of five observations, ranging from NH,z2 = 2+7 −1 × 1023 cm−2 (Epoch 1) to NH,z2 = 6+16 −4 × 1023 4 The observed shift zobs is related to the rest-frame blueshift zabs by the relation zabs = (1 + zobs)/(1 + zc) − 1, where zc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='015 is the cosmological redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The velocity of the absorber is given by v/c = (z2 abs + 2zabs)/(z2 abs + 2zabs + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Article number, page 5 of 10 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' eso323_serafinelli cm−2 (Epoch 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We note that in Epoch 2 we can place only an upper limit on the column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Indeed, Epoch 2 does not show a clear absorption signature in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3 (blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The ion- ization parameter is log ξz2/erg cm s−1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The final model is therefore (3) TBabs ∗ ((const1 ∗ xstar1 ∗ MyTZ ∗ xstar2 ∗ zpow1 + MyTS0 + gsmooth ∗ MyTL0) + const2 ∗ zpow2) where XSTAR1 and XSTAR2 are the ionized absorbers in Zone 1 and Zone 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The photon index of the spectrum, af- ter the addition of these two ionized absorbers, is Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 4 shows the contour plot of Γ with the MyTZ column den- sity NH for the brightest observation of the campaign, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Epoch 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The contour plot shows that both the photon index Γ and the absorbing column density NH,abs are well constrained at 3σ con- fidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The best-fit parameters obtained with this model are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The normalized spectrum with data- to-model ratios and the unfolded spectrum are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We also tested an alternative approach in which the absorber column density NH,abs is kept tied among the observations, while the photon index Γ is allowed to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Unsurprisingly, the col- umn density is NH = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5) × 1023 cm−2, which is the mean value of the NH found independently when the parameter is al- lowed to vary between observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We find various values of the photon index, ranging from Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='06 for Epoch 3 to Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='84+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='04 for Epoch 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, we obtain a worse fit statis- tic of χ2/dof = 983/906, which means that an absorber variation is favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Notably, the smaller photon indices are also the ones with greater absorption and viceversa, resulting in an apparent steeper when brighter effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' This effect is driven by the absorp- tion variability, as the source has historically experienced in the past, and should not be confused with the continuum softer when brighter effect, driven by intrinsic Γ variations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Sobolewska & Papadakis 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Serafinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Alternative model for the reflector: BORUS We also test for a spherical reprocessor, using the model BORUS (Balokovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We consider a continuum described by a cut-off power law, ZCUTOFFPL, with a line of sight absorp- tion modelled by ZPHABS, and reflector described by the table BORUS025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We also include the two ionized absorbers located in Zone 1 and Zone 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Also in this model we allow the column den- sity of the Zone 2 high-ionization absorber NH to vary between observations, while we assume the ionization parameter to re- main constant between the observations of the campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The model used is (4) TBabs ∗ ((const1 ∗ xstar1 ∗ zphabs ∗ xstar2 ∗ zcutoffpl1 + borus02) + const2 ∗ zcutoffpl2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We obtain a photon index Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='06, consistent with the value obtained with the MYTORUS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' As the cut-off energy Ecut is unconstrained, we fix it to a fiducial Ecut = 500 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The neutral column density varies from NH,abs = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3) × 1023 cm−2 (Epochs 1 and 2) to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3) × 1023 cm−2 (Epoch 5), roughly consistent with the ones found with the MY- TORUS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The column density of the reprocessor is NH = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 × 1024 cm−2, which is consistent to the value found in the MYTORUS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The covering factor of the reprocessor 5 All BORUS tables can be downloaded from the website https:// sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='edu/~mislavb/download is given by C f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='90+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Finally, we obtain consistent values for the column density and the ionization parameter of the ionized absorber in Zone 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The ionization parameter of the absorber in Zone 2 is also consistent with the one obtained with the MYTORUS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The column density of the absorber in Zone 2 is also consistent, althought with large uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The goodness of fit of this model is given by χ2/dof = 923/901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Comptonizing plasma continuum It is also interesting to investigate the coronal parameters of this source, as these are often elusive for obscured sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' There- fore, we investigate physical Comptonization models for the continuum with both the MYTORUS and BORUS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Starting from the MYTORUS model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3, We adopted the same config- uration and free parameters, but we replaced the power law con- tinuum with COMPTT (Titarchuk 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We also adopted the ap- propriate MYTORUS table, namely we adopt the tables MYTSTT 0 6 and MyTLTT 0 7, and we use them the same way we used MYTS0 and MYTL0 in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The model is then (5) TBabs ∗ ((const1 ∗ xstar1 ∗ MyTZ ∗ xstar2 ∗ compTT1 + MyTSTT 0 + gsmooth ∗ MyTLTT 0 ) + const2 ∗ compTT2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We first explore the slab coronal geometry by fixing the value of the parameter approx to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We do not find significant differences in any other parameter obtained in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The coronal temperature with this fit is kT = 38 ± 2 keV, while the optical depth is τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The goodness of fit of this model is given by χ2/dof = 920/906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Typically, assuming a spherical geometry in COMPTT, the best-fit coronal parameters would be a similar temperature, but a larger optical depth (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Tortosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, the MYTSTT θ tables do not include larger values of τ, and therefore it is not possible to explore the parameters of a spherical geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, the spherical geometry might be explored within the BORUS model shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' BORUS12 is produced with the thermal comptonization continuum model NTHCOMP (Magdziarz & Zdziarski 1995), which assumes a spherical geometry for the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Hence, we also use such model for the continuum, and the model is therefore: TBabs ∗ ((const1 ∗ xstar1 ∗ zphabs ∗ xstar2 ∗ nthcomp1 + borus12) + const2 ∗ nthcomp2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (6) with a goodness of fit of χ2/dof = 915/901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We obtain Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='73+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='05 and a coronal temperature of kT = 36+13 −8 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Remark- ably, this value is consistent with the COMPTT temperature ob- tained assuming a slab geometry in the MYTORUS model, even adopting a different continuum model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Relativistic reflection The presence of a possible relativistic iron line in the X-ray spec- tra of this AGN was inferred by Jiménez-Bailón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2008) during an unabsorbed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Therefore, we test the possibility that such component could also be detected in an absorbed state, and we add the relativistic reflection component RELXILL (García et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Dauser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2014) to the model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The global 6 File name mytorus_scatteredkT034_v00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='fits 7 File name mytl_V000010nEp000kT034_v00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='fits Article number, page 6 of 10 Roberto Serafinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' : The NuSTAR view of the changing look AGN ESO 323-G77 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Best-fit parameters of the final BORUS model shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The goodness of fit is χ2/dof = 923/901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The NTHCOMP parameters are taken from the best-fit model shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 6, assuming a Comptonized continuum produced by a spherical corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Parameter Epoch 1 Epoch 2 Epoch 3 Epoch 4 Epoch 5 Central source (zcutoffpl) Γ − − − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='06 − norm (10−3 photons cm−2 s−1 keV−1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 Fscat/Fnucl − − − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 × 10−2 − Central source (nthcomp) Γ − − − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='73+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='05 − kT (keV) − − − 36+13 −8 − Neutral absorber Absorption (zphabs) NH (1023 cm−2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 Reflection (borus02) NH (1024 cm−2) − − − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 − Covering factor (C f ) − − − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='90+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='03 − Ionized absorbers (xstar) Zone 1 (external) NH (1023 cm−2) − − − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 − log ξ (erg cm s−1) − − − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='37+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='25 − Zone 2 (internal) NH (1023 cm−2) < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 4 ± 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 2+2 −1 log ξ (erg cm s−1) − − − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1 − v/c 0 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='02 fit improves by ∆χ2/∆dof = 38/7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' All parameters with the ex- ception of the normalization are kept tied between observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We assume a frozen cut-off energy Ecut = 500 keV, a disk exter- nal radius of Rout = 400Rg, where Rg = GM/c2 is the gravita- tional radius, a 45◦ inclination (Schmid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2003), a solar iron abundance and an emissivity index of −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The spin of the black hole is unconstrained, for which therefore we freeze a = 0, and we obtain a disk internal radius Rin < 12Rg, consistent with the findings of Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The disk ionization parameter is log(ξ/erg cm s−1) > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' A steeper photon index Γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='87+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='09 is found, although consistent with the one found with the model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 3 at 90% confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The normalization of the relativis- tic component is unconstrained in Epoch 4, normrelx,4 < 8×10−6 photons cm−2 s−1 keV−1, while in Epochs 2, 3 and 5 it is roughly constant (normrelx,2,3,5 = 6+7 −4 × 10−6 photons cm−2 s−1 keV−1), and in Epoch 1 it is normrelx,1 = 10+6 −5 × 10−6 photons cm−2 s−1 keV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We do not find significant differences in the absorbing column density from Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, the two reflectors are degenerate, and therefore we find a lower limit for the neutral reflector column density, NH,refl > 4 × 1024 cm−2, even though it is consistent with the value of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We also tested RELXILL as an additional reflection compo- nent in the model where we assume a comptonizing continuum COMPTT (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 5), to test the possible influence on the measure of kT and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The temperature of the corona is kT = 26 ± 9 keV, and τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1, consistent within the 3σ contour plot of these two parameters for the model without a disk reflection component (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Very similar results are obtained by testing RELXILL on the two models that use BORUS for the neutral reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We stress that the RELXILL component contributes to ≲ 10% of the 2 − 10 keV observed flux, and the main changes in this model are in the spectral region between 3 and 5 keV, where NuSTAR is less sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Also, many parameters of the relativis- tic reflection model are unconstrained due to the complex model and numerous degeneracies with the neutral reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We point that, in order to accurately measure the parameters of the ionized relativistic reflection within the framework of such a complex spectral model, a broad band spectrum and an improved energy resolution would be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' For instance, a simultaneous XMM- Newton and NuSTAR observation would be ideal to observe the Fe Kα spectral region in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Article number, page 7 of 10 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' eso323_serafinelli 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Comparison between MYTORUS and BORUS models The MYTORUS model has been built assuming a toroidal shape, asymmetric on the azimuthal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The covering factor in such model is kept fixed by assuming that the torus opening angle is θOA = 60◦, which means that its value is C f = cos(θOA) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 (Murphy & Yaqoob 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Conversely, BORUS has a spherical geometry for the reprocessor, with polar cutouts corresponding to a variable opening angle θOA, and therefore is able to fit a value for the C f , ranging from C f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1 to C f = 1 (Balokovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The two best-fit values of the average column density of the reflector are slightly different, NH,MYT = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 × 1024 cm−2 and NH,borus = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 ×1024 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Moreover, the covering factor found with the BORUS model is not consistent with the value of C f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 assumed in the MYTORUS one, and this might explain the difference in the column density estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' In order to properly compare the two models, we construct a BORUS version of the MYTORUS decoupled model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We consider an out of line of sight reflector by setting the torus inclination to cos θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='95, which is the maximum value allowed by the BORUS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' This corresponds to an inclination angle of θ = 18◦, dif- ferently from the MYTORUS value θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The covering factor is fixed to the MYTORUS value C f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' As expected, the inclina- tion discrepancy is not crucial (see also Marchesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2019) and we obtain NH,borus = 5+3 −1 × 1024 cm−2 ≃ NH,MYT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We stress that this model has been built with the sole purpose of comparing the column density of the torus for the MYTORUS to the one obtained with BORUS, since the goodness of fit is χ2/dof = 944/902, marginally worse than the model presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, this configuration is more realistic than the one with a covering factor of Cv ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='9, since the latter would im- ply that ∼ 90% of the sightline intercepts a Compton-thick col- umn density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' As a consequence, a Compton-thick state would be observed far more frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' In fact, while this source has been observed several times, it has been caught in a Compton- thick state only once in 2011 by Suzaku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' This would be possible if we were looking at this Seyfert galaxy with an exceptional, extremely polar, line of sight, whereas Schmid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2003) esti- mated a 45◦ angle for the inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Therefore a lower covering factor is likely a more realistic scenario for this source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Compton-thin absorber and Compton-thick reflector Both models indicate that the absorbing material is Compton- thin, with column density ranging from NH,abs ∼ 2 × 1023 cm−2 up to NH,abs ∼ 4 × 1023 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' This AGN was already caught in this state by one Swift-XRT snapshot in 2006 and by XMM- Newton in 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, as shown by Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2014), the source is able to change from a relatively unobscured state (NH,abs ∼ 2 − 4 × 1022 cm−2) up to a Compton-thick state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Previous analyses of ESO 323-G77 have hinted that low ob- scuration states (NH ≲ 1023 cm−2) might be caused by the pres- ence of the obscuring torus, while higher obscuration states are likely due to absorption by cold intra-clump material located in the broad line region (Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Sanfrutos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, given that we do not observe a change of state during the campaign analyzed in this work, but only moderate changes in the absorber column density NH,abs we are not able to argue in favour or against this hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The unprecedented effective area of NuSTAR in the E > 10 keV band allows us to properly study the reflection compo- nent of the X-ray spectrum of this source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' In particular, both the MYTORUS and BORUS models clearly point to the presence of a Compton-thick reflector with NH = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 × 1024 cm−2 or NH = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 × 1024, depending on the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' If the absorption is indeed produced by BLR clumps or intra-clump cold material, this result indicates that the constant Compton-thick reflector is located farther away from the central X-ray source, and it should be associated with the classic torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Ionized absorbers Similar to the results presented in Jiménez-Bailón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2008), Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2014) and Sanfrutos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2016), our data shows the presence of two ionized absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We can estimate the location of these ionized absorbers using standard arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' For instance, the maximum distance from the black hole can be estimated by considering that the size of the absorbing clump Rclump cannot be larger than the distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Rclump = NH/n < rmax, where n is the density of the clump (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Crenshaw & Kraemer 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Serafinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' From the ionization parameter definition the maximum distance from the black hole can be written as rmax = Lion NHξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (7) The first one, located in what we denote as Zone 1, is char- acterized by an ionization parameter of ξ ∼ 250 erg cm s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The ionizing luminosity in the E = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6 eV− 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6 keV energy band is Lion ≃ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2) × 1044 erg s−1 and the column den- sity is NH ≃ 3 × 1023 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Therefore, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 7, we obtain rmax,1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='9 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The second ionized absorber, located in Zone 2 is char- acterized by a larger ionization parameter, ξ ≃ 104 erg cm s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The average column density is given by NH ≃ 6 × 1023 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Therefore, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 7 we obtain a maximum distance of rmax,2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 × 10−2 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We condider an Eddington ratio of log λEdd = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='56 and a black hole mass of MBH = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 × 107 M⊙ (Wang & Zhang 2007), from which we can compute log Lbol ≃ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='This means, as- suming that Lbol/L5100Å ∼ 10 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Collin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2002), that the optical luminosity is log L5100Å ≃ 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We consider the rela- tion between the broad line region size and the optical luminosity introduced by Bentz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2009) log RBLR(light days) = −21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='519 log L5100Å and we obtain a broad line region radius of RBLR ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='02 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We therefore obtain that the moderately ionized absorber in Zone 1 could be located outside the broad line region at r1 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 pc, while the more ionized absorber in Zone 2 is likely co-spatial or within the BLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' In the scenario in which the cold absorber either co-spatial with one of the two ionized absorbers or sandwiched between them (Sanfrutos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2016), the cold absorber would be located between the outer BLR, consistently with the model proposed by Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2014), and pc-scale distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' In the latter case, a possible scenario would be the presence of an inner thick re- flecting ring surrounded by a thinner absorbing layer at pc-scale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Buchner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Recent mid-infrared results (Leftley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2021) found evidence of the presence of polar warm dust at a distance r ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 pc, which is consistent with this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The outer layer would also be clumpy, allowing for the observed long-term variability, a similar scenario to the one proposed for NGC 7479 by Pizzetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Article number, page 8 of 10 Roberto Serafinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' : The NuSTAR view of the changing look AGN ESO 323-G77 30 35 40 45 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 2 τ kT (keV) X Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Contour plot of the optical depth τ versus the coronal tempera- ture kTe, assuming a slab coronal geometry with the COMPTT Comp- tonization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The red, green and blue lines represent 68% (1σ), 95% (2σ) and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='7% (3σ) confidence levels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Ultra-fast outflow The velocity of the absorber in Zone 2 is v ≲ 9000 km s−1, consistent with the values measured by Jiménez-Bailón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2008) and Sanfrutos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2016) of v ≃ 2000 km s−1, in Epochs 1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, in Epoch 5, we notice a moderately relativistic velocity v ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='21c, with a level of ∆χ2/dof = 11/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' This is a tentative indication that we are observing an absorber outflowing at high velocity, a phenomenon that is commonly known as ultra-fast outflows (UFOs) and are fairly common (∼ 40%) in Seyfert galaxies and quasars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Pounds et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Braito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Tombesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Gofford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Nardini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Tombesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Serafinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Moreover, UFOs are known to be extremely variable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Reeves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Matzeu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Braito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2018, 2022), therefore it is not surprising that the UFO appears within a relatively short timescale in an AGN that never showed signs of its presence before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, given its modest (∼ 3σ) detection here, further observations would be required to confirm the detection of the UFO feature or its variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Coronal parameters The X-ray continuum is well known to be produced by inverse Compton on UV seed photons gaining energy by a very hot elec- tron corona (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Haardt & Maraschi 1991, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The elec- tron temperature therefore plays a crucial role in regulating the Comptonization of UV seed photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Indeed, the main contin- uum breaks at the so-called cut-off energy Ecut, which is tied to the temperature by the relation Ecut = 2 − 3 kTe, depending on the geometry of the corona (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Petrucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' When the COMPTT model is adopted to model the continuum, assuming a slab geometry for the corona, in the MYTORUS model shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 5, we find that the temperature of the corona is kTe = 38 ± 2 keV, with an optical depth τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The τ − Γ contour plot is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Since the grids do not allow much larger values of τ, the only way to study the spherical ge- ometry is to use the NTHCOMP Comptonization continuum with the BORUS model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 6), and we find a consistent temperature, although with larger errors, kTe = 36+13 −8 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We can estimate the optical depth using the following equation, valid for a spher- ical optically thick (τ > 1) corona (Zdziarski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 1996): Γ = � 9 4 + 511 keV kT τ(1 + τ 3) − 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Using the best-fit values of the BORUS model, summarized in Table 3, we obtain τ ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' These are fairly standard values, as the coronal temperature is known to span from kT ∼ 3 keV up to kT ∼ 450 keV (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Matt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Tortosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2018, 2022, Serafinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, even though some authors have recently un- veiled coronal temperatures in isolated obscured sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Middei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2021) and samples of Seyfert 2 galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Balokovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2020), they are not easily constrained, since they are often degenerate with the reflection spectrum cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Summary and conclusions We have presented the spectral analysis of a campaign of 5 NuS- TAR observations of the Seyfert 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2 galaxy ESO 323-G77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We summarize our results in the following – The source has been observed in a persistent obscured, but Compton-thin state, due to the presence of neutral obscuring material on the line of sight, with column density in the range NH ∼ 2 − 4 × 1023 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' – We find a Compton-thick reflector both modelling it with MY- TORUS and BORUS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The two NH,refl values are not consistent, but this result is dependent on the covering factor of the re- flector, which is assumed as C f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 in MYTORUS and fit- ted (C f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='90+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='03) in BORUS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' By fixing a more realistic C f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 in BORUS, the two results are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' – Two ionized absorbers are needed in our models, consistent with Jiménez-Bailón et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2008), Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2014) and Sanfrutos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The ionized absorber identified with Zone 1 is located at a distance of about r1 ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 pc from the black hole, most likely outside the broad line region, whose size is estimated as RBLR ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='02 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The ionized absorber in Zone 2 instead is located at r2 ≃ 10−2 pc, either co-spatial or within the BLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' – Assuming that the cold absorber is either at the same distance of one of the two ionized absorbers, or at an intermediate one, its location can be placed between the outer BLR and at pc-scale distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' In the first case, this would be consistent with the model proposed by Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' (2014), consisting of cold absorbing intra-clump material in the BLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' In the second case, the most likely scenario is pc-scale Compton- thin absorbing material surrounding a Compton-thick reflec- tor (Buchner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2019), which is supported by recent mid- infrared detection of polar dust at r ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='5 pc (Leftley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' – The ionized absorber in Zone 2 is blueshifted at Epoch 5, to the value zobs ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='18, which suggests an outflowing veloc- ity of vout ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' – The coronal temperature is constrained in both models, find- ing kTe ≃ 37 keV, both assuming a slab and a spherical corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The optical depth is τ ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='4 when the slab coronal geometry is assumed, and τ ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='8 for a spherical corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' – We find hints of the possible presence of a relativistic reflec- tion component from the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, this com- ponent contributes to ≲ 10% of the observed 2−10 keV flux, and it mostly affects the 3 − 5 keV energy band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Hence, the parameters of the disk reflection component are very difficult to constrain, and higher energy resolution data are needed to further study this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Article number, page 9 of 10 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' eso323_serafinelli The campaign was not able to observe any significant change of state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', obscured to unobscured), as the source has undergone several times in the past (Miniutti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' However, longer campaigns should be able to observe the source passing from obscured to unobscured or vice-versa, setting an upper limit to the obscurer location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Future high-resolution instruments such as the microcalorimeter Resolve on board XRISM (XRISM Sci- ence Team 2020) will be able to measure the properties of the ab- sorbers with much more detail, particularly on their location and outflowing velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Moreover, future hard X-ray (E = 2 − 200 keV) instruments such as the High Energy X-ray Probe (HEX- P, Madsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2018) will allow us to measure the reflection parameters with unprecedented accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' The authors thank the referee for useful comments that improved the quality of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' RS, VB, PS, ADR, and RDC acknowl- edge financial contribution from the agreements ASI-INAF n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='2017-14-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='0 and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content='I/037/12/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' This research has made use of data and software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical Observa- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' This research has made use of the NuSTAR Data Analysis Software (NUS- TARDAS) jointly developed by the ASI Space Science Data Center (SSDC, Italy) and the California Institute of Technology (Caltech, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' We acknowl- edge the use of public data from the Swift data archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' References Antonucci, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 1993, ARA&A, 31, 473 Arnaud, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' A.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Balokovi´c, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Marinucci, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2015, MNRAS, 447, 3029 Matzeu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Reeves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2021, A&A, 647, A102 Miniutti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Sanfrutos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Beuchert, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2014, MNRAS, 437, 1776 Murphy, K.' metadata={'source': 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al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2015, Science, 347, 860 Palmeri, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Mendoza, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Kallman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Bautista, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' A.' metadata={'source': 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+page_content=' 2001, ApJ, 556, 716 Piconcelli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Bianchi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Guainazzi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Fiore, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', & Chiaberge, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2007, A&A, 466, 855 Pizzetti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Torres-Alba, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Marchesi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2022, ApJ, 936, 149 Pounds, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Reeves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', King, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2003, MNRAS, 345, 705 Reeves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Braito, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Gofford, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2014, ApJ, 780, 45 Ricci, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Bauer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Arevalo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2016, ApJ, 820, 5 Risaliti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Elvis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Fabbiano, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2007, ApJ, 659, L111 Risaliti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Elvis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', & Nicastro, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2002, ApJ, 571, 234 Rivers, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Balokovi´c, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Arévalo, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2015, ApJ, 815, 55 Rivers, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Markowitz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', & Rothschild, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2011, ApJ, 742, L29 Sanfrutos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Miniutti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Agís-González, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2013, MNRAS, 436, 1588 Sanfrutos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Miniutti, G.' metadata={'source': 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+page_content=' 2017, A&A, 600, A101 Sobolewska, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' & Papadakis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 2009, MNRAS, 399, 1597 Titarchuk, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=' 1994, ApJ, 434, 570 Tombesi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} +page_content=', Cappi, M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xtFQT4oBgHgl3EQfAjU_/content/2301.13223v1.pdf'} diff --git a/ytAzT4oBgHgl3EQfQvv4/content/tmp_files/2301.01206v1.pdf.txt b/ytAzT4oBgHgl3EQfQvv4/content/tmp_files/2301.01206v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fc99c07f25710705ad753374e9ad660795efcff2 --- /dev/null +++ b/ytAzT4oBgHgl3EQfQvv4/content/tmp_files/2301.01206v1.pdf.txt @@ -0,0 +1,589 @@ +Speed up the inference of diffusion models via +shortcut MCMC sampling +Gang Chen +Department of Computer Science +SUNY at Buffalo +Buffalo, NY 14260 +newhorizontal@gmail.com +Abstract +Diffusion probabilistic models have generated high quality image synthesis re- +cently. However, one pain point is the notorious inference to gradually obtain +clear images with thousands of steps, which is time consuming compared to other +generative models. In this paper, we present a shortcut MCMC sampling algo- +rithm, which balances training and inference, while keeping the generated data’s +quality. In particular, we add the global fidelity constraint with shortcut MCMC +sampling to combat the local fitting from diffusion models. We do some initial +experiments and show very promising results. Our implementation is available at +https://github.com//vividitytech/diffusion-mcmc. +1 +Introduction +Leveraging deep generative models to generate high quality images has becoming the dominant +approach in machine learning community. For example, generative adversarial networks (GANs) [1], +PixelCNN [2] and variational autoencoders [3] have shown impressive image and speech synthesis +results. Diffusion probabilistic models [4] have recently gained popularity over a variety of applica- +tions on computer vision and machine learning domain. And it also obtains state-of-the-art Inception +score and FID score [5; 6; 7] on image generation, as well as best results on density estimation +benchmarks [8]. Diffusion models are well defined with Markov chain assumption and are efficient +to train. But it is time consuming to generate high quality images, which may take thousands of steps +to the best of our knowledge. This paper presents an approach to speed up the inference of diffusion +models. Instead of thousands of steps to produce samples, we constrain the number of inference +steps, which can be randomly sampled from these thousand steps (we call shortcut MCMC) and +then generate images to match the data. Both denoising diffusion probabilistic models (DDPMs) and +variational diffusion models (VDMs) train a similar denoising deep nets, which focus on local model +characteristics and thus long sampling steps needed to produce high quality images. +Compared to VDMs, we introduce the shortcut MCMC sampling and add the fidelity term in the loss +function so that the final synthesized image match the original data. This new fidelity term is more +like a global constraint and quality control while generating images in a shortcut manner. Thus, our +method can balance the training and inference stages, and mitigates the inference burden significantly. +We do some initial analysis and show promising results on synthesis dataset. +Preprint. +arXiv:2301.01206v1 [cs.CV] 18 Dec 2022 + +Figure 1: The noised data with increasing noise level until random Gaussian distribution. +2 +Background +The diffusion models [4; 5] are composed of forward process and reverse (backward) process. Given +the data x0 ∼ q(x0), the forward (diffusion) process follows a Markov chain +q(xt|x0) = N(xt, αtx0 + σtI), +q(x1:T |x0) = +T +� +t=1 +q(xt|xt−1) +(1) +where αt = +� +1 − σ2 +t , and (αt, σt) is the signal and noise pair at time step t. the Markov chain +q(xt|xt−1) is Gaussian +q(xt|xt−1) = N(αt|t−1, σ2 +t|t−1I) +(2) +where αt|t−1 = αt/αt−1 and σ2 +t|t−1 = σ2 +t − α2 +t|t−1σ2 +t−1 according to VDMs [8]. The reverse (or +backward) process is to learn p(x0) = +� +p(x0:T )dx1:T ), where p(xT ) is Gaussian N(xT ; 0, I): +p(xt−1|xt) = N(xt−1; µθ(xt, t), σθ(xt, t)), +p(x0:T ) = p(xT ) +T +� +t=1 +p(xt−1|xt) +(3) +Fig 1 shows the examples while increasing noise signal over the original data. By optimizing the +variational lower bound, VDMs [8] chooses the conditional model distributions below +p(xt−1|xt) = q(xt−1|xt, x0) +(4) +which can be induced according to the KL divergence. In the inference stage, we can replace x0 with +its prediction ˆx0(xt; t) using denoising diffusion models. +3 +Model +In this section, we will introduce our approach based on the variational lower bound and the shortcut +MCMC sampling to skip multiple steps to speed up inference. We consider the finite time steps and +it can be easily extended to continuous scenario. +3.1 +Objective lower bound +In the case of finite T steps, we maximize the variational lower bound of marginal likelihood below +L(x0; θ) = Eq(z|x)[log p(x0|z)] − DKL(q(xT |x0)||p(xT )) − +T +� +t=2 +DKL(q(xt−1|xt, x0)|| log p(xt−1|xt)) +(5) +where z = (x1, x2, ..., xT ), and for detail induction, please refer Appendix A. Compared to VDMs, +we have an additional fidelity term Eq log p(x0|z), which maps the latent (prior) Gaussian noise to +data distribution. This is similar to GANs model, which can generate data from latent distribution. +However, for diffusion model, it depends on the hyperparameter T that will take thousands of steps +(e.g. T = 1000) to produce synthesized data. In other words, it is 3 orders of magnitude slower than +GANs when both use the similar deep neural nets architecture in the inference stage. +As for the diffusion loss, it leverages KL-divergence to match p(xt−1|xt) with the forward process +posterior q(xt−1|xt, x0). Since both the forward posterior and p(xt−1|xt) are Gaussians, with same +variance assumption, then the KL loss can be minimized using the deep denoise model +DKL(q(xs|xt, x0)|| log p(xs|xt)) = 1 +2(α2 +s +σ2s +− α2 +t +σ2 +t +)||ϵ − ˆϵθ(xt, t)||2 +(6) +2 + +t=0 +t=1 +t=T +forward +backward +k=K +k=K-1 +k=1 +k=0 +Figure 2: The forward process over T steps and the reverse process with shortcut MCMC sampling +(red line). +where 0 < s < t ≤ T, and (αs, σs) and (αt, σt) are signal and noise pairs respectively at time step s +and t. +In the following part, we will focus on the fidelity term log p(x0|z), and we want the data generated +from the latent space match the original data distribution. +3.2 +Shortcut MCMC sampling +The fidelity term Eq log p(x0|z) is hard to optimize, because its complexity is determined by the +depth of the generative model and its neural nets architecture. In the training stage, we always set a +large T, such as T = 1000. We use the forward posterior to match p(xt−1|xt). In other words, we +have N(xt−1; µθ(xt, t), σθ(xt, t)) and needs to recover the data step by step. +For any time step s and t ∈ [1, T] and s < t, we have q(xs|xt, x0) = N(xs; µθ(xt; s, t), σ2 +θ(s, t)I), +with mean and variance as below +µθ(xt; s, t) = αt|sσ2 +s +σ2 +t +xt + +αsσ2 +t|s +σ2 +t +x0, +σ2 +θ(s, t) = σ2 +t|sσ2 +s/σ2 +t +(7) +Using KL divergence, p(xs|xt) = q(xs|xt, x0), and we need to replace x0 with ˆx0(xt, t) in the +inference. After do some mathematical operations in Appendix B, we have the following formula +p(xs) = αsx0 + σsϵ +(8) +Thus, we can sample xs at any time step s. In the best scenario, the marginal distribution p(xt) from +the reverse process matches the forward one q(xt). Since we have p(xt) ∼ q(xt), we approximate +p(xt) with the same formula in Eq. 1 and we can sample xs from the constructed ˆx0. Since the +latent variable z = (x1, ..., xT ), it will be time-consuming. To speed up the inference, we can skip +steps to produce data while using MCMC sampling. Specifically, we random sample K time steps +{t1, .., tK} from [1, T]. Then we use the prediction ˆxtk to get the next sample ˆxtk−1 according to the +equation above. Thus we have the fidelity loss +Eq log p(x0|z) = ||x0 − ˆx0||2 +(9) +where ˆx0 is predicted from the shortcut MCMC sampling. By minimizing this loss, we add the global +constraint to the deep denoise models, and further improve the data approximation quality. +3.3 +Algorithm +We summarize our approach in Algorithm. 1. Compared to DDPMs and VDMs, we add the fidelity +term which imposes a global constraint to our generated samples and use shortcut MCMC sampling +to speed up the inference. +3 + +Algorithm 1 Training +Initialize denoise neural networks and its parameters +for epoch = 1 to N do +x0 ∼ q(x0) +sample t ∼ Uniform(1,..., T) +take step to minimize ||ϵ − ˆϵθ(xt, t)||2, where xt = αtx0 + σtϵ +random sample K steps (not need to be equal distance), t0, t1, ..., tK ∼ Uniform(1,..., T) +for k = K to 1 do +predict ˆx0 = (xtk − σtkˆϵθ(xtk, t))/αtk +update xtk−1 ∼ αtk−1 ˆx0 + σtk−1ϵ +end for +take gradient step to minimize ||x0 − ˆx0||2 +end for +Return model and parameters. +In the inference stage, we just sample ϵ ∼ N(0, I), then we sample K time steps from [1, T] and +sample xtk ∼ αtk ˆx0 + σtkϵ, where ˆx0 is predicted from the denoise neural network in the previous +tk−1. Thus, our method has the potential to speed up inference at least an order of magnitude fast. +4 +Experimental results +We did initial experiments on synthetic dataset. In this experiment, we create the swirl dataset with +1024 points, shown in Fig 1. As for the model architecture, we use 3 layer MLP, with Fourier feature +expansion as the inputs. We set K = 10 for all the training in all the experiments below. +In the first experiment in Fig 3, we train the model with the shortcut MCMC sample. In the inference +stage, we set T = 200 and sample K = 10 time steps, then we generate our results with only 10 steps +inference. The result in Fig 3 shows that our approach not only converge fast, but also reconstruct +better results. +In the second experiments, we train with K = 10, and in the inference we set K the same value as T, +K = T = 200 for step by step comparison. It indicates that with the same time steps, our approach +converge fast and yield better results in Fig 4. For example, our approach recover the data well at +K = 100. +5 +Conclusion +In this paper, we propose a fast approach for diffusion models in the inference stage. To this end, we +add a fidelity term as the global constraint over the diffusion models, and present a shortcut MCMC +sampling method to speed up the inference. The experiments show promising results on both data +quality and fast inference time. +6 +Appendix +A +The maximum likelihood x0 is +log p(x0) = log +� +z +p(x0, z) = log +� +z +p(x0, z)q(z|x) +q(z|x) = log +� +z +q(z|x)p(x0, z) +q(z|x) +≥ +� +q(z|x) log p(x, z) +q(z|x) = Eq(z|x)[log p(x, z) +q(z|x) ] = Eq(z|x)[log p(x|z)p(z) +q(z|x) +] += Eq(z|x)[log p(x|z)] − Eq(z|x)[log p(z) +q(z|x)] +(10) +4 + +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +k=0 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +k=2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +k=4 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +k=7 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +k = 10 +(a) VDMs +(b) ours with MCMC sampling +the time step +Figure 3: The left column is from VDMs[8], the right column is from our approach. We use T = 200 +in the inference stage, and K=10 to sample 10 time steps. Then we compare the corresponding 5 +generated images between VDMs and our method. +5 + +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +k =20 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +k=60 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +k = 100 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +k =160 +2 +0 +2 +2 +0 +2 +2 +0 +2 +2 +0 +2 +k=200 +(a) VDMs +(b) ours with MCMC sampling +the time step +Figure 4: We use T = 200 in the inference stage, and K=200 for the full time steps comparison. We +can see our method can generate very good samples and converge fast then VDMs. +6 + +where we assume the latent z = (x1, x2, ..., xT ). Overall, we want to maximize the variational lower +bound. The first term is reconstruction loss, which is our fidelity term in the paper. The second term +is the KL divergence between p(z) and q(z|x), which we want to minimize. +As for the second term we can do some decomposition to get KL divergence between p(xs|xt) and +q(xs|xt, x0) in the following analysis: +Ex0:T ∼q(x0:T )[log +p(x1:T ) +q(x1:T |x0)] +=Ex0:T ∼q(x0:T )[− log q(x1:T |x0) + log p(x1:T )] +=Ex0:T ∼q(x0:T ) +� +− log[q(xT |x0) +T +� +t=2 +q(xt−1|xt, x0)] + log[p(xT ) +T +� +t=2 +p(xt−1|xt)] +� += − DKL(q(xT |x0)||p(xT )) − +T +� +t=2 +DKL(q(xt−1|xt, x0)|| log p(xt−1|xt)) +(11) +B +p(xs|xt) = q(xs|xt, x = ˆxθ(zt; t)) +(12) +Since the reverse process is also Gaussian, we then have +p(xs|xt) = N(xs; µθ(xt; s, t), σ2 +Q(s, t)I) +(13) +µθ(xt; s, t) = αt|sσ2 +s +σ2 +t +xt + +αsσ2 +t|s +σ2 +t +ˆxθ(xt; t) += +1 +αt|s +xt − +σ2 +t|s +αt|sσt +ˆϵθ(xt; t) += +1 +αt|s +(αtx + σtϵ) − +σ2 +t|s +αt|sσt +ˆϵθ(xt; t) += αsx + +1 +αt|s +(σtϵ − +σ2 +t|s +σt +ˆϵθ(xt; t)) += αsx + +1 +αt|sσt +(σ2 +t ϵ − σ2 +t|sˆϵθ(xt; t)) +(14) +Since p(xs|xt) = +µθ(xt; s, t) = αt|sσ2 +s +σ2 +t +xt + +αsσ2 +t|s +σ2 +t +x0 += αt|sσ2 +s +σ2 +t +(αtx0 + σtϵt) + +αsσ2 +t|s +σ2 +t +x0 += αtσ2 +s +σ2 +t +x0 + αt|sσ2 +s +σt +ϵt + +αsσ2 +t|s +σ2 +t +x0 += αsx0 + αt|sσ2 +s +σt +ϵt +(15) +7 + +We know that the variance at time s, σ2 +θ(s, t) = σ2 +t|sσ2 +s/σ2 +t , then we can get by sampling p(xs|xt) = +N(xs; µθ(xt; s, t), σ2 +θ(s, t)I) +xs = µθ(xt; s, t) + σθ(s, t)ϵs += αsx0 + αt|sσ2 +s +σt +ϵt + σθ(s, t)ϵs += αsx0 + αt|sσ2 +s +σt +ϵt + σt|sσs +σt +ϵs +(16) +since ϵt and ϵs from the same Gaussian noise, when we reduce the steps we can merge these two +independent Gaussian distributions, the new variance can be formulated as: +(αt|sσ2 +s +σt +)2 + (σt|sσs +σt +)2 += +α2 +t|sσ4 +s +σ2 +t ++ +σ2 +t|sσ2 +s +σ2 +t +=σ2 +s +σ2 +t +(α2 +t|sσ2 +s + σ2 +t|s) +=σ2 +s +(17) +we can see that xs ∼ αsx0 + σsϵ +So the most important step is to estimate accurate x in the inference stage. we borrow the idea from +signal decomposition. The forward process of diffusion model is to add noise to the original signal +until it approximate random Gaussian distribution, while the backward process is to denoise the +merged the signal to recover the original data. While the data is noising, the recovered ˆx, but it will +be better with more denoising steps. +References +[1] Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil +Ozair, Aaron Courville, and Yoshua Bengio. Generative Adversarial Networks. In NIPS, 2014. +[2] Aäron van den Oord, Nal Kalchbrenner, Oriol Vinyals, Lasse Espeholt, Alex Graves, and Koray +Kavukcuoglu. Conditional image generation with pixelcnn decoders. In NIPS, 2016. +[3] Diederik P. Kingma and Max Welling. Auto-Encoding Variational Bayes. In 2nd International +Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, +Conference Track Proceedings, 2014. +[4] Jascha Sohl-Dickstein, Eric A. Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsu- +pervised learning using nonequilibrium thermodynamics. In Francis R. Bach and David M. Blei, +editors, Proceedings of the 32nd International Conference on Machine Learning, ICML 2015, +Lille, France, 6-11 July 2015, volume 37 of JMLR Workshop and Conference Proceedings, pages +2256–2265. JMLR.org, 2015. +[5] Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In +H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, Advances in Neural +Information Processing Systems, volume 33, pages 6840–6851. Curran Associates, Inc., 2020. +[6] Alexander Quinn Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic +models. In Pat Langley, editor, Proceedings of the 17th International Conference on Machine +Learning (ICML 2021), pages 8162–8171. PMLR, 2021. +[7] Yang Song, Jascha Sohl-Dickstein, Diederik P. Kingma, Abhishek Kumar, Stefano Ermon, and +Ben Poole. Score-based generative modeling through stochastic differential equations. In 9th +International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May +3-7, 2021. OpenReview.net, 2021. +8 + +[8] Diederik P Kingma, Tim Salimans, Ben Poole, and Jonathan Ho. Variational diffusion models. +In NIPS, 2021. +9 + diff --git a/ytAzT4oBgHgl3EQfQvv4/content/tmp_files/load_file.txt b/ytAzT4oBgHgl3EQfQvv4/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d91b59abb0978d987088dccf4b39f7a796e3affb --- /dev/null +++ b/ytAzT4oBgHgl3EQfQvv4/content/tmp_files/load_file.txt @@ -0,0 +1,187 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf,len=186 +page_content='Speed up the inference of diffusion models via shortcut MCMC sampling Gang Chen Department of Computer Science SUNY at Buffalo Buffalo, NY 14260 newhorizontal@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content='com Abstract Diffusion probabilistic models have generated high quality image synthesis re- cently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' However, one pain point is the notorious inference to gradually obtain clear images with thousands of steps, which is time consuming compared to other generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' In this paper, we present a shortcut MCMC sampling algo- rithm, which balances training and inference, while keeping the generated data’s quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' In particular, we add the global fidelity constraint with shortcut MCMC sampling to combat the local fitting from diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' We do some initial experiments and show very promising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Our implementation is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content='com//vividitytech/diffusion-mcmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 1 Introduction Leveraging deep generative models to generate high quality images has becoming the dominant approach in machine learning community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' For example, generative adversarial networks (GANs) [1], PixelCNN [2] and variational autoencoders [3] have shown impressive image and speech synthesis results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Diffusion probabilistic models [4] have recently gained popularity over a variety of applica- tions on computer vision and machine learning domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' And it also obtains state-of-the-art Inception score and FID score [5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 7] on image generation, as well as best results on density estimation benchmarks [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Diffusion models are well defined with Markov chain assumption and are efficient to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' But it is time consuming to generate high quality images, which may take thousands of steps to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' This paper presents an approach to speed up the inference of diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Instead of thousands of steps to produce samples, we constrain the number of inference steps, which can be randomly sampled from these thousand steps (we call shortcut MCMC) and then generate images to match the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Both denoising diffusion probabilistic models (DDPMs) and variational diffusion models (VDMs) train a similar denoising deep nets, which focus on local model characteristics and thus long sampling steps needed to produce high quality images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Compared to VDMs, we introduce the shortcut MCMC sampling and add the fidelity term in the loss function so that the final synthesized image match the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' This new fidelity term is more like a global constraint and quality control while generating images in a shortcut manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Thus, our method can balance the training and inference stages, and mitigates the inference burden significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' We do some initial analysis and show promising results on synthesis dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content='01206v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content='CV] 18 Dec 2022 Figure 1: The noised data with increasing noise level until random Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 2 Background The diffusion models [4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 5] are composed of forward process and reverse (backward) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Given the data x0 ∼ q(x0), the forward (diffusion) process follows a Markov chain q(xt|x0) = N(xt, αtx0 + σtI), q(x1:T |x0) = T � t=1 q(xt|xt−1) (1) where αt = � 1 − σ2 t , and (αt, σt) is the signal and noise pair at time step t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' the Markov chain q(xt|xt−1) is Gaussian q(xt|xt−1) = N(αt|t−1, σ2 t|t−1I) (2) where αt|t−1 = αt/αt−1 and σ2 t|t−1 = σ2 t − α2 t|t−1σ2 t−1 according to VDMs [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' The reverse (or backward) process is to learn p(x0) = � p(x0:T )dx1:T ), where p(xT ) is Gaussian N(xT ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 0, I): p(xt−1|xt) = N(xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' µθ(xt, t), σθ(xt, t)), p(x0:T ) = p(xT ) T � t=1 p(xt−1|xt) (3) Fig 1 shows the examples while increasing noise signal over the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' By optimizing the variational lower bound, VDMs [8] chooses the conditional model distributions below p(xt−1|xt) = q(xt−1|xt, x0) (4) which can be induced according to the KL divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' In the inference stage, we can replace x0 with its prediction ˆx0(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' t) using denoising diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 3 Model In this section, we will introduce our approach based on the variational lower bound and the shortcut MCMC sampling to skip multiple steps to speed up inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' We consider the finite time steps and it can be easily extended to continuous scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content='1 Objective lower bound In the case of finite T steps, we maximize the variational lower bound of marginal likelihood below L(x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' θ) = Eq(z|x)[log p(x0|z)] − DKL(q(xT |x0)||p(xT )) − T � t=2 DKL(q(xt−1|xt, x0)|| log p(xt−1|xt)) (5) where z = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=', xT ), and for detail induction, please refer Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Compared to VDMs, we have an additional fidelity term Eq log p(x0|z), which maps the latent (prior) Gaussian noise to data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' This is similar to GANs model, which can generate data from latent distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' However, for diffusion model, it depends on the hyperparameter T that will take thousands of steps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' T = 1000) to produce synthesized data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' In other words, it is 3 orders of magnitude slower than GANs when both use the similar deep neural nets architecture in the inference stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' As for the diffusion loss, it leverages KL-divergence to match p(xt−1|xt) with the forward process posterior q(xt−1|xt, x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Since both the forward posterior and p(xt−1|xt) are Gaussians, with same variance assumption, then the KL loss can be minimized using the deep denoise model DKL(q(xs|xt, x0)|| log p(xs|xt)) = 1 2(α2 s σ2s − α2 t σ2 t )||ϵ − ˆϵθ(xt, t)||2 (6) 2 t=0 t=1 t=T forward backward k=K k=K-1 k=1 k=0 Figure 2: The forward process over T steps and the reverse process with shortcut MCMC sampling (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' where 0 < s < t ≤ T, and (αs, σs) and (αt, σt) are signal and noise pairs respectively at time step s and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' In the following part, we will focus on the fidelity term log p(x0|z), and we want the data generated from the latent space match the original data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content='2 Shortcut MCMC sampling The fidelity term Eq log p(x0|z) is hard to optimize, because its complexity is determined by the depth of the generative model and its neural nets architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' In the training stage, we always set a large T, such as T = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' We use the forward posterior to match p(xt−1|xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' In other words, we have N(xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' µθ(xt, t), σθ(xt, t)) and needs to recover the data step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' For any time step s and t ∈ [1, T] and s < t, we have q(xs|xt, x0) = N(xs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' µθ(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' s, t), σ2 θ(s, t)I), with mean and variance as below µθ(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' s, t) = αt|sσ2 s σ2 t xt + αsσ2 t|s σ2 t x0, σ2 θ(s, t) = σ2 t|sσ2 s/σ2 t (7) Using KL divergence, p(xs|xt) = q(xs|xt, x0), and we need to replace x0 with ˆx0(xt, t) in the inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' After do some mathematical operations in Appendix B, we have the following formula p(xs) = αsx0 + σsϵ (8) Thus, we can sample xs at any time step s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' In the best scenario, the marginal distribution p(xt) from the reverse process matches the forward one q(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Since we have p(xt) ∼ q(xt), we approximate p(xt) with the same formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 1 and we can sample xs from the constructed ˆx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Since the latent variable z = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=', xT ), it will be time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' To speed up the inference, we can skip steps to produce data while using MCMC sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Specifically, we random sample K time steps {t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content='., tK} from [1, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Then we use the prediction ˆxtk to get the next sample ˆxtk−1 according to the equation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Thus we have the fidelity loss Eq log p(x0|z) = ||x0 − ˆx0||2 (9) where ˆx0 is predicted from the shortcut MCMC sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' By minimizing this loss, we add the global constraint to the deep denoise models, and further improve the data approximation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content='3 Algorithm We summarize our approach in Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Compared to DDPMs and VDMs, we add the fidelity term which imposes a global constraint to our generated samples and use shortcut MCMC sampling to speed up the inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 3 Algorithm 1 Training Initialize denoise neural networks and its parameters for epoch = 1 to N do x0 ∼ q(x0) sample t ∼ Uniform(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=', T) take step to minimize ||ϵ − ˆϵθ(xt, t)||2, where xt = αtx0 + σtϵ random sample K steps (not need to be equal distance), t0, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=', tK ∼ Uniform(1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=', T) for k = K to 1 do predict ˆx0 = (xtk − σtkˆϵθ(xtk, t))/αtk update xtk−1 ∼ αtk−1 ˆx0 + σtk−1ϵ end for take gradient step to minimize ||x0 − ˆx0||2 end for Return model and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' In the inference stage, we just sample ϵ ∼ N(0, I), then we sample K time steps from [1, T] and sample xtk ∼ αtk ˆx0 + σtkϵ, where ˆx0 is predicted from the denoise neural network in the previous tk−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Thus, our method has the potential to speed up inference at least an order of magnitude fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 4 Experimental results We did initial experiments on synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' In this experiment, we create the swirl dataset with 1024 points, shown in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' As for the model architecture, we use 3 layer MLP, with Fourier feature expansion as the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' We set K = 10 for all the training in all the experiments below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' In the first experiment in Fig 3, we train the model with the shortcut MCMC sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' In the inference stage, we set T = 200 and sample K = 10 time steps, then we generate our results with only 10 steps inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' The result in Fig 3 shows that our approach not only converge fast, but also reconstruct better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' In the second experiments, we train with K = 10, and in the inference we set K the same value as T, K = T = 200 for step by step comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' It indicates that with the same time steps, our approach converge fast and yield better results in Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' For example, our approach recover the data well at K = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 5 Conclusion In this paper, we propose a fast approach for diffusion models in the inference stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' To this end, we add a fidelity term as the global constraint over the diffusion models, and present a shortcut MCMC sampling method to speed up the inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' The experiments show promising results on both data quality and fast inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 6 Appendix A The maximum likelihood x0 is log p(x0) = log � z p(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' z) = log � z p(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' z)q(z|x) q(z|x) = log � z q(z|x)p(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' z) q(z|x) ≥ � q(z|x) log p(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' z) q(z|x) = Eq(z|x)[log p(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' z) q(z|x) ] = Eq(z|x)[log p(x|z)p(z) q(z|x) ] = Eq(z|x)[log p(x|z)] − Eq(z|x)[log p(z) q(z|x)] (10) 4 2 0 2 2 0 2 2 0 2 2 0 2 k=0 2 0 2 2 0 2 2 0 2 2 0 2 k=2 2 0 2 2 0 2 2 0 2 2 0 2 k=4 2 0 2 2 0 2 2 0 2 2 0 2 k=7 2 0 2 2 0 2 2 0 2 2 0 2 k = 10 (a) VDMs (b) ours with MCMC sampling the time step Figure 3: The left column is from VDMs[8],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' the right column is from our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' We use T = 200 in the inference stage, and K=10 to sample 10 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Then we compare the corresponding 5 generated images between VDMs and our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 5 2 0 2 2 0 2 2 0 2 2 0 2 k =20 2 0 2 2 0 2 2 0 2 2 0 2 k=60 2 0 2 2 0 2 2 0 2 2 0 2 k = 100 2 0 2 2 0 2 2 0 2 2 0 2 k =160 2 0 2 2 0 2 2 0 2 2 0 2 k=200 (a) VDMs (b) ours with MCMC sampling the time step Figure 4: We use T = 200 in the inference stage, and K=200 for the full time steps comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' We can see our method can generate very good samples and converge fast then VDMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 6 where we assume the latent z = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=', xT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Overall, we want to maximize the variational lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' The first term is reconstruction loss, which is our fidelity term in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' The second term is the KL divergence between p(z) and q(z|x), which we want to minimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' As for the second term we can do some decomposition to get KL divergence between p(xs|xt) and q(xs|xt, x0) in the following analysis: Ex0:T ∼q(x0:T )[log p(x1:T ) q(x1:T |x0)] =Ex0:T ∼q(x0:T )[− log q(x1:T |x0) + log p(x1:T )] =Ex0:T ∼q(x0:T ) � − log[q(xT |x0) T � t=2 q(xt−1|xt, x0)] + log[p(xT ) T � t=2 p(xt−1|xt)] � = − DKL(q(xT |x0)||p(xT )) − T � t=2 DKL(q(xt−1|xt, x0)|| log p(xt−1|xt)) (11) B p(xs|xt) = q(xs|xt, x = ˆxθ(zt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' t)) (12) Since the reverse process is also Gaussian, we then have p(xs|xt) = N(xs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' µθ(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' s, t), σ2 Q(s, t)I) (13) µθ(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' s, t) = αt|sσ2 s σ2 t xt + αsσ2 t|s σ2 t ˆxθ(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' t) = 1 αt|s xt − σ2 t|s αt|sσt ˆϵθ(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' t) = 1 αt|s (αtx + σtϵ) − σ2 t|s αt|sσt ˆϵθ(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' t) = αsx + 1 αt|s (σtϵ − σ2 t|s σt ˆϵθ(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' t)) = αsx + 1 αt|sσt (σ2 t ϵ − σ2 t|sˆϵθ(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' t)) (14) Since p(xs|xt) = µθ(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' s, t) = αt|sσ2 s σ2 t xt + αsσ2 t|s σ2 t x0 = αt|sσ2 s σ2 t (αtx0 + σtϵt) + αsσ2 t|s σ2 t x0 = αtσ2 s σ2 t x0 + αt|sσ2 s σt ϵt + αsσ2 t|s σ2 t x0 = αsx0 + αt|sσ2 s σt ϵt (15) 7 We know that the variance at time s, σ2 θ(s, t) = σ2 t|sσ2 s/σ2 t , then we can get by sampling p(xs|xt) = N(xs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' µθ(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' s, t), σ2 θ(s, t)I) xs = µθ(xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' t) + σθ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' t)ϵs = αsx0 + αt|sσ2 s σt ϵt + σθ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' t)ϵs = αsx0 + αt|sσ2 s σt ϵt + σt|sσs σt ϵs (16) since ϵt and ϵs from the same Gaussian noise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' when we reduce the steps we can merge these two independent Gaussian distributions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' the new variance can be formulated as: (αt|sσ2 s σt )2 + (σt|sσs σt )2 = α2 t|sσ4 s σ2 t + σ2 t|sσ2 s σ2 t =σ2 s σ2 t (α2 t|sσ2 s + σ2 t|s) =σ2 s (17) we can see that xs ∼ αsx0 + σsϵ So the most important step is to estimate accurate x in the inference stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' we borrow the idea from signal decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' The forward process of diffusion model is to add noise to the original signal until it approximate random Gaussian distribution, while the backward process is to denoise the merged the signal to recover the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' While the data is noising, the recovered ˆx, but it will be better with more denoising steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' References [1] Ian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Generative Adversarial Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' In NIPS, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' [2] Aäron van den Oord, Nal Kalchbrenner, Oriol Vinyals, Lasse Espeholt, Alex Graves, and Koray Kavukcuoglu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Conditional image generation with pixelcnn decoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' In NIPS, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' [3] Diederik P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' Kingma and Max Welling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAzT4oBgHgl3EQfQvv4/content/2301.01206v1.pdf'} +page_content=' 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